Chapter 18. Electrolyte Solutions

Water, water, every where, Nor any drop to drink.

S.T. Coleridge, “The Rime of the Ancient Mariner”

Electrolyte solutions are as common as seawater. They exist in every biological organism, in underground reservoirs, and in numerous industrial processes. Thermodynamic models of electrolyte solutions can range from extremely simple to extremely complex. At the simple end of the spectrum are solubility product constants, familiar from introductory chemistry courses. At the complex end of the spectrum, we recognize that concentrated solutions can lead to ion-ion interactions that influence the activity coefficients, altering the simultaneous reaction and phase equilibria that pervade the entire subject. Our goal in this chapter is to introduce the vocabulary and the manner of setting up typical problems involving electrolytes.

Chapter Objectives: You Should Be Able to...

1. Compute the equilibrium concentrations of ionic species in “ideal” electrolyte solutions.

2. Compute the effect of salts on boiling point, freezing point, osmotic pressure, and acid/base dissociation.

3. Understand and articulate the reason for charges on biological molecules and the isoelectric point.

4. Quantitatively describe the “salting in” and “salting out” effects and the common ion effect for solubility.

5. Estimate the true and apparent mole fractions of gaseous solutes in electrolyte solutions.

6. Apply the extended Debye-Hückel model for activity coefficients in dilute solutions.

18.1. Introduction to Electrolyte Solutions

Briefly, an electrolyte is a substance that dissociates into charged species in a liquid phase. The behavior can occur in solution, or in the case of ionic liquids used as nonvolatile solvents, occurs in the pure state. Electrolytes exist in biological and industrial systems and are thus important to our everyday life.

Sodium chloride almost totally dissociates into sodium and chloride ions in dilute aqueous solutions near room temperature. Due to this strong dissociation, sodium chloride is described in these circumstances with a strong electrolyte model by assuming that the dissociation is complete. On the other hand, acetic acid, the dominant ingredient in household vinegar, partially dissociates into hydrogen ions and acetate ions in dilute aqueous solution and is modeled as a weak electrolyte. A key caution is that the literature tends to describe compounds as strong or weak electrolytes. However, the extent of dissociation depends on the environment. For example, Fig. 18.1 shows the distribution of species in sulfuric acid as a function of concentration measured experimentally1 and modeled.2 Both hydrogens dissociate only in very dilute solution (which is difficult to discern from the figure). The second dissociation is suppressed at moderate and high concentrations. The first dissociation disappears at high concentrations and sulfuric acid exhibits limited dissociation when pure. Experimental information on the degree of dissociation is important as well as thermodynamic models to represent the behavior to characterize the strong or weak dissociation under the conditions of interest.

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Figure 18.1. Speciation of sulfuric acid in aqueous solutions as measured by experiments1 and modeled by OLI Systems, Inc.2 Note the square root scale to emphasize the dilute region.

The extremely polar nature of water and the capability of water to adjust its partial charges by adjusting the H-O bond distance make it capable of hydrating or solvating the ions. The water surrounding the ion is known as the water of hydration or solvation and the ions are described as solvated. Fundamental studies show that three to five water molecules hydrate ions in dilute to moderately concentrated solutions. The number is inexact due to the difficulty of the experiments and the fact that the hydration shell is not forming stoichiometric bonds, so the whole hydrated ion is constantly undergoing exchange resulting in various packing effects and various sizes. The key point is to understand that ions must be solvated in solution, and the degree of electrolyte dissociation depends on the ability of a solvent to dissolve ions and the competing driving force for the electrolyte to stay undissociated. Solvents must have a high dielectric constant to be capable of solvating ions. Sodium chloride has an infinitesimal solubility in hexane because the ions are not solvated effectively.

Electrolytes can also have important effects in inhomogeneous fluids like surfaces and colloids. For example, the ions may align next to a surface of opposite charge causing a phenomenon known as an electric double layer. Ions can also strongly affect surfactants in micelles, emulsions, and microemulsions. We restrict our attention here to bulk, homogeneous systems, which are affected in relatively straightforward ways by reaction and dilution.

18.2. Colligative Properties

Boiling points, freezing points and osmotic pressure are sometimes termed colligative properties. The adjective “colligative” describes phenomena that are dependent on molar concentration and ignore the solution nonidealities. This of course is an approximation. We understand from previous chapters discussing nonelectrolytes that solution nonidealities can be important. However, in the case of dissolved solids such as sodium chloride, which have negligible vapor pressure under common conditions, the partial pressure of water/solvent determines the boiling point because the salt does not contribute measurably to the vapor composition. Likewise, the osmotic pressure and freezing points are dominated by concentration effects. We know that sodium chloride is commonly used to melt snow and ice on sidewalks and roads in cold climates. The salt works because the freezing temperature of the solution is lower than that of pure water. It is more effective on a molar basis than a compound that does not dissociate because it decreases the mole fraction of water to a greater extent, thus decreasing the chemical potential of water. François-Marie Raoult wrote about electrolytes and their melting point depression compared to molecular solutes: “These facts show that, contrary to what I thought until now, the general law of freezing does not apply to the salts dissolved in water. They tend to show that it applies to radicals (ions) constituting the salts, almost as if these radicals (ions) were simply mixtures in dissolutions.”3 To be fair, however, the degree of melting depression is not always the most critical factor when selecting a system for melting depression or boiling point elevation. Automobile radiators commonly use ethylene glycol as an antifreeze. On a molar basis, the ethylene glycol is less effective than a dissociating salt in lowering the freezing point and increasing the boiling point. Nevertheless, the liquid additive is used instead of salt for practical reasons like avoiding solid precipitation or corrosion (especially for NaCl).


Raoult’s study of electrolyte dissociation led to further developments by Jacobius van’t Hoff and Wilhelm Ostwald.



Example 18.1. Freezing point depression

Compare NaCl (used on icy roads), ethylene glycol (used in car radiators), and glucose (used by hibernating frogs) as alternatives for freezing point depression. Consider 5 g of each for 0.1 L (5.55 mol) of water and then compare 0.1 mol of each in 0.1 L of water. For the molar basis, compare the masses used and the effectiveness. Assume NaCl totally dissociates, and use an ideal solution approximation.

Solution

The melting point is calculated with Eqn. 14.24. To calculate mole fractions, the molecular weights are NaCl 58.44, ethylene glycol (EG) 62.07, and glucose 180.16. For 5 g of each, the molar amounts are 0.0855 mol, 0.0805 mol, and 0.027 mol, respectively. The mole fractions of water in the solutions (recall that NaCl forms two moles of ions!) are xH2O = 5.55/(5.55 + 2·0.0855) = 0.970, xH2O = 5.55/(5.55 + 0.0805) = 0.986, xH2O = 0.995, respectively, and

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The freezing points for 5 g of each are 270.0 K, 271.7 K, 272.7 K for depressions of 3.2°C, 1.5°C, and 0.5°C respectively. NaCl is more effective than an equivalent mass of EG. Frogs must generate a very concentrated solution of glucose to keep from freezing while hibernating (though concentrated glucose also forms a metastable subcooled liquid easily). For 0.1 mol of each, the mol fractions are xH2O = 0.965, xH2O = 0.982, xH2O = 0.982, with freezing points of 269.6 K, 271.3 K, 271.3 K for depressions of 3.6°C, 1.9°C, and 1.9°C, respectively. There is no difference between the last two solutes because they do not dissociate. The masses needed for 0.1 mol are 5.8 g, 6.2 g, 18.0 g. On a mass basis, NaCl is more effective than glucose even though only one-third as much is used. For EG and glucose, 0.1 mol of each gives the same melting depression, but the mass of glycol is about one-third because the molecular weight is smaller.


Osmotic pressure was discussed in nonelectrolytes in Section 11.13 on page 449. For electrolyte systems a primary difference is that the dissociation of strong electrolytes creates a larger effect on osmotic pressure at the same molar concentration. Look back at Eqn. 11.71. The osmotic pressure is related to the logarithm of the activity of water. When a monovalent electrolyte dissociates, it doubles the effect on the activity relative to an undissociated molecular species (with the ideal solution approximation). Sodium chloride (MW = 58.4 g/mol) and propanol (MW = 60.1 g/mol) have similar molecular weights, but when dissolved in water, to achieve a given osmotic pressure, only about half as much mass of salt is required at low concentrations.


Example 18.2. Example of osmotic pressure

Consider the solutes from Example 18.1 assuming complete dissociation of NaCl and ideal solutions. (a) Compare the osmotic pressure for 0.1 mol of each in 0.1 L of water at 298.15 K. (b) What concentration of NaCl (wt%) is isotonic with human blood?

Solution

a. The mole fractions have been calculated in Example 18.1 as 0.965, 0.982, and 0.982. The osmotic pressure is given by Eqn. 11.71. The osmotic pressure for an ideal solution is

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Inserting the mole fractions of each, the osmotic pressures are 4.89 MPa, 2.5 MPa, 2.5 MPa. In a reverse osmosis system, a solution of NaCl requires much more pressure to purify than a solution of a nonelectrolyte with the same apparent concentration.

b. Isotonicity with human blood is defined in Section 11.13 on page 449 as having a concentration that is 0.308 mol/L of solute. Since two ions are obtained for each NaCl that dissociates, this corresponds to 0.154 mol/L of NaCl, or 8.99 g/L. Assuming the concentration is sufficiently low, a dilute aqueous solution corresponds to a density of 1000 g/L. Therefore, the weight fraction is 9/1000 = 0.009 or 0.9wt%. This is commonly known as “physiological saline” or just “saline.”


Vapor-liquid equilibria is also affected by electrolytes. In many cases the electrolyte can be considered to be nonvolatile, such as with sodium chloride. Below we consider the equilibrium condition where salt is only in the liquid phase. In actual application, some salt may be entrained in aerosol droplets as is well known in ocean-side communities where corrosion from salty aerosols is common, but this is not an equilibrium phenomenon. On the other hand, many electrolytes are volatile, such as HCl and acetic acid, so the following analysis will not apply in exactly the same manner.


Example 18.3. Example of boiling point elevation

Consider the solutes from Example 18.1. Compare the bubble points for 0.1 mol of each in 0.1 L of water at 1.013 bar. Consider complete dissociation of NaCl and ideal solutions. Ignore volatility of EG.

Solution

This will be a bubble-temperature calculation. Because the solutes are nonvolatile (ignoring volatility of EG), yH2O = 1. The bubble-pressure condition is

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Using the Antoine equation for water and the mole fractions from Example 18.1, the bubble temperatures are found by using an iterative solver to be 101°C, 100.5°C, and 100.5°C, respectively. Again, the salt has a larger effect due to its dissociation.


Typically, the analysis of electrolyte dissociation is treated in the reaction equilibrium framework termed speciation, modeling the dissociation into ionic species as a chemical reaction. When multiple phases exist, simultaneous reaction and phase equilibria must be solved. The general term electrolyte refers to a species that dissociates into ions in solution.

18.3. Speciation and the Dissociation Constant

The term speciation refers to a cataloging of the species that exist in solution. The species are characterized by writing dissociation reactions that identify the species and material balance constraints that exist in solution. To introduce the concepts of speciation, consider the dissociation of water:

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Thus, in pure water, the species in solution are H2O, H+, and OH. From introductory chemistry courses, we are familiar with this reaction and the equilibrium constant at 25°C:

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Another important concept in speciation is that the solution must satisfy a charge balance, and the net charge in solution must be zero. While the charge balance is trivial for this example, it becomes important in setting up the mathematical solutions to the material balances.

Eqn. 18.5 contains some implicit assumptions that are rarely explained in introductory chemistry books. From our discussion in Chapter 17, we know that a rigorous calculation should use activities for a chemical reaction:

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Comparing the last two equations, we recognize that Eqn. 18.6 can be simplified to Eqn. 18.5 if the activities of ions are replaced with concentrations, and if the activity of undissociated water is unity. Certainly the approximations of Eqn. 18.5 are valid under common conditions or the introductory textbooks would have been in error. But what is meant by the activity of ions, and why does the activity of water not appear in Eqn. 18.5? To understand the simplifications, we must understand the various concentration conventions. The subtleties are important because the values of the equilibrium constants that characterize the reactions are coupled to the concentration and activity scales.

To partially clarify the relation between Eqns. 18.5 and 18.6, it is necessary to recall that activity is dimensionless but molarity has dimensions of mol/L. This issue is subtly handled by using molal concentrations and defining the standard state for electrolytes to be an ideal solution at m° = 1 molal. Therefore, Eqn. 18.6 can be successively converted using Image where the m° states are equal to 1 molal, the molal activity coefficients are ignored, and the molar concentration is used as an approximation to the molality. Water disappears from the relation because the water standard state is taken as purity and a neutral solution is virtually pure water.4 Supporting discussion is provided in Sections 18.1318.15 and summarized in Section 18.24.

Measuring Speciation

Quantitative speciation is important in development of a proper thermodynamic model. Various techniques are used, including absorption, NMR and Raman spectroscopies, conductance, emf, solubilities, and rates of reaction. Most techniques require estimation of the activity coefficients to extrapolate to infinite dilution where the Ka is calculated from ideal solution approximations. Because modeling extrapolation is always necessary to “measure” the constants, considerable scatter among literature values is common.

18.4. Concentration Scales and Standard States

To discuss the concentration of an electrolyte, some terminology conventions are important for clarity. For example, when 0.01 mole of sodium chloride is added to water and diluted to 1 liter at 25°C, the solution results in 0.02 mol/L of ions because it acts as a strong electrolyte. However, we need a method to communicate the solubility of an electrolyte as an entity or on a superficial basis or apparent basis or nominal basis. These terms are used interchangeably in the literature; they refer to the totality of electrolytes in solution as if they were molecular, without dissociation. Formal concentrations (mol/L) are used in chemistry to refer to the molar concentration based on the chemical formula of the substance. The terms “undissociated” and “un-ionized” are different from “apparent.” For species that partially dissociate, the sum of the undissociated and dissociated species comprises the apparent composition. For example, acetic acid does not completely dissociate in aqueous solution, and to communicate most clearly, the terms “nominal,” “superficial,” “as an entity,” and “apparent” are used to refer to the total amount in solution.5 VLE of a weak electrolyte like acetic acid occurs between the undissociated species in the vapor and the undissociated species in the liquid (as we will show later), but a “apparent” perspective would simply consider that there is acetic acid in the vapor and liquid and the apparent activity coefficient of acetic acid is different from what it would be without dissociation. The true distribution of species should be determined by reaction equilibria and Le Châtelier’s principle to characterize the NH4OH that forms and the NH4+ and OH- speciation. Although the apparent perspective may seem oversimplified, the apparent concentration is very important in engineering calculations because it is often the most accessible measure of concentration when multiple species are present. In this text, we strive to consistently refer to “apparent concentrations,” but we may use the equivalent term “superficial concentrations.”


Image The apparent basis refers to the hypothetical situation as a molecular species.


The NaCl solution of the preceding paragraph would be described as a 0.01 M apparent concentration of NaCl. On the other hand, the 0.01 M apparent solution of NaCl would have true concentrations of 0.01 M Na+ and 0.01 M Cl. The convention of terminology is to describe modeled concentrations as “true” concentrations even though the modeled true concentrations often vary for different models for weak electrolytes. [Apparently, some “true” concentrations are more true than others ;-).] In a good model, true concentrations from experiments are represented accurately. Note how the apparent concentration is the only quantity on which everybody can agree.

Concentration Scales For Electrolytes

Throughout introductory chemistry texts, the convention is to express the concentrations in Eqn. 18.5 using molarity. For introductory courses, frequently all the calculations are at room temperature (25 °C), and thus temperature effects are disregarded. For biological systems, the electrolytes are almost always very near room temperature and the density of the solutions is almost always constant, so the convention is to use molarity. However, this choice of concentration as a composition scale has a disadvantage when the temperature changes in industrial processes because temperature affects density, which affects molar concentrations. Thus, an increase in temperature decreases molar concentrations of all species, even when the solution composition does not change. Therefore, for fundamental calculations as a function of temperature, alternative concentration scales are required.

In the case of electrolytes for industrial reactions, it is often easiest to perform calculations based on the number of solute (i.e., electrolyte) molecules relative to the number of solvent (e.g., water) molecules. This can be done with mole fraction or molality. Molality has a disadvantage that the quantity diverges to infinity at high concentration. Nevertheless, it is the dominant convention in the older electrolyte literature for nonideal solution behavior and dissociation constants. Molality approaches molarity at low concentrations near room temperature and thus is a natural extension from the use of molarity. Also, molality has a convenient magnitude at common concentrations. Molality is the moles of species existing in 1 kg of solvent (e.g., water) molecules. If water is the solvent, the number of moles per kg of solvent is 55.509. We use the notation “m” for molality and “M” for molarity. For example, a one molal aqueous solution of sodium chloride is prepared by adding 1 mole of NaCl (58.44 g) to 1 kg of water. The molarity of that solution is slightly less than 1 M since the total volume after addition is greater than 1 liter. Molality and molarity are subtly but distinctly different. The ratio of solute to solvent molecules in all 1 molal solutions will be the same for all solutes in a given solvent. To clarify, a comparison of molality, molarity, and mole fractions for NaCl solutions is provided in Table 18.1. The molality and molarity approach each other at low aqueous concentration near room temperature. For dilute solutions, the molarity and molality can be interchanged as a good approximation. The mole fraction scale, typically with a Henry’s law standard state, is preferred when working with concentrated solutions.6 Molality has a disadvantage that it goes to infinity when the water concentration goes to zero. Formulas for interconversions between concentration scales are summarized in Section 18.24

Table 18.1. Apparent Mole Fraction, Molality, Density and Molarity for Aqueous Sodium Chloride Solutionsa

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a. Densities are from Washburn, E.W., ed., 1926–1930. International Critical Tables, National Research Council, vol. 3, p. 79.


Image We use the notation “m” for molality and “M” for molarity.


Standard State for Electrolyte Systems

When molality or molarity is used for concentration, the standard state is selected such that the activity coefficients of the electrolyte species, including the undissociated species, go to unity at infinite dilution. This is similar to the Henry’s law standard state discussed in Section 11.12.7 An unsymmetric convention (cf. Section 11.12) is used in that a Lewis-Randall standard state is applied for the solvent (typically water) and the activity coefficient of the solvent goes to unity when the solvent is pure.

To understand how the conventions are useful, look back at Table 18.1. Note that the mole fraction of water is approximately xw = 0.9 when the solution is 3 m in NaCl (remember NaCl dissociates!). So the use of an activity coefficient of unity for water is reasonable for approximate calculations. The use of an activity coefficient of unity for the electrolytes is exact only at infinite dilution, but can be used as a rough approximation for introductory calculations. We take that approach initially, using ideal solution calculations and the infinite dilution standard state for electrolytes and the Lewis-Randall standard state for the solvent, and later introduce the more rigorous calculation using activity coefficients.

A caution is that various conventions of molarity, molality, and mole fractions exist in the literature.8 Therefore, when using equilibrium constants, the reader must be careful to understand the conventions used to tabulate the values, and must carefully convert the constants if a different scale/convention is to be used for calculations. The convention can sometimes be inferred if the constant is given with units. Interconversion of constants is discussed in Section 18.25.

Solubility and degree of dissociation depend on the solvent. The majority of published equilibrium constants are for aqueous systems. However, many solvents, including amines, pyridines, ammonia, alcohols, esters, and carboxylic acids, are also capable of solvating ions to varying extents. Because the environment of the ion is critical in determination of the extent of solvation, care should be used when modeling dissociation in nonaqueous solvents.

18.5. The Definition of pH

The pH of a solution is defined to be

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where the activity is expressed on the molal scale. Methods for calculating the activity are covered later. Commonly as an approximation, the concentration (mol/L) is substituted, and the “p” notation stands for the negative of the common logarithm:

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We use this common approximation for introductory examples. The primary use of pH is to characterize a solution as acidic or basic.

Relation of H and OH in Water

Recall at room temperature9

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The “p” notation is extended to other ions and equilibrium constants. When working with dilute solutions, this means that –log(aH+ aOH) = – log(aH+) = –log(aOH) = –logKa,w, or

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At room temperature, pKa,w = 14. Thus, when the solution is acidic, the concentration of [OH] < 10–7 mol/L, and when the solution is basic, the concentration of [H+] < 10–7 mol/L. Often one of these concentrations can be neglected when taking sums or differences when it is small relative to other terms.

Importance of Solvent

In water, the pH typically varies in the range 0–14. However, because of the definition, strong acids at high concentrations can have negative pH. In ammonia, however, the pH varies from 0 to 32. Therefore, the environment is important in determining the range of pH.

18.6. Thermodynamic Network for Electrolyte Equilibria

Equilibrium Constants

Suppose an electrolyte has a chemical formula C2A where C+ is a monovalent cation and A2– is a bivalent anion. Succinic acid, H2Succ, is an example of an electrolyte with this formula. An equilibrium network can be created as shown in Fig. 18.2. An electrolyte with an arbitrary C2A composition is shown, with the expectation that readers can properly generalize the network for a specific electrolyte. If completely general notation was used, the figure notation would be unwieldy. While consideration of simultaneous equilibrium between solid, liquid, and vapor phases is rarely necessary, certain subsets of the general network are very common. For example, when an aqueous solution of succinic acid is saturated with solid H2Succ, the aqueous solution contains H2Succ, HSucc2–, and H+ and the prevalence of a particular species depends on the pH, but the H2Succ in the vapor is probably negligible.

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Figure 18.2. Reaction network for an example electrolyte where C is a monovalent cation and A is a divalent cation. The equilibrium constants are related as explained in the text.

Fig. 18.2 emphasizes that the undissociated electrolyte species is the key connection to vapor-liquid equilibria, and also can be used for solid-liquid equilibria. It is proven in Section 18.23 that the apparent chemical potential of C2A in the liquid phase Image, is equal to the chemical potential of the dissolved undissociated species μi(aq):

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This may seem odd in the case of a strong electrolyte where the amount of undissociated species is infinitesimal, but it is in fact rigorous. From an engineering perspective this apparent chemical potential is important because it is used to relate the apparent chemical potential to the component chemical potentials in the phases where the electrolyte is undissociated.

Note that the reaction for the solubility product constant

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can be obtained by summing the reactions for melting along with the two dissociation reactions. Since the Gibbs energies are related to the logarithms of the equilibrium constants, we may write

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Thus, the solubility of electrolytes is frequently written using the solubility product constant, Ksp instead of representing all the intermediate dissociation behavior. While this is rigorous thermodynamically using activities, an approximate solution is often used with concentrations rather than activities.

Whether a given electrolyte acts as a strong or weak electrolyte is sometimes unknown at the time a model is required for process design. In other cases, the electrolytes are assumed to act as strong electrolytes for convenience. A confusing aspect of the literature is that often the reader must infer whether a strong electrolyte model is applied depending on whether the dissociation constants are used. When a weak electrolyte is modeled using a strong electrolyte model, the adjustable activity model parameters can be forced to fit, but often the parameters are unusual. For example, when activity coefficients are determined for a weak electrolyte assuming that it is strong, the activity coefficients are very small. This is because the assumed “true” concentration of ions is much larger than the actual concentration, so the activity coefficients must be smaller than expected to match the experimental activities.

18.7. Perspectives on Speciation

An important consideration about speciation is the dissociation reaction stoichiometry of reactions that form H+. The concept of hydration was discussed in Section 18.1. Positive ions usually require water of hydration, and do not float freely in solution as implied by Eqn. 18.4. For example, the species H9O4+ is spectroscopically identifiable even at the normal boiling point of water.10 However, the generally accepted method of writing the reaction is given by Eqn. 18.4 with the understanding that it is actually hydrated. Nevertheless, omitting the water of hydration in the reactions can lead to unexpected results from calculations in mixed solvents. When water is the dominant solvent, there is sufficient water to hydrate the ions. However, at lower concentrations of water, Eqn. 18.4 has no requirement for water of hydration to be present. A more realistic method of writing reactions that form H+ is to write them including at least one water, for example,

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where H3O+(aq) is known as the hydronium ion. For acetic acid, the ionization forming hydronium can be written

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In this manner, at least some water of hydration must be present for ionization to occur. When the dissociation reactions are written this way, it has no effect on the equilibrium constants from the literature which are measured when the activity of water is essentially one. On the other hand, when Gibbs energies are used, the Gibbs energy and enthalpy of formation of the hydronium includes the corresponding energy and enthalpy of formation for water, resulting in the same Gibbs energies and enthalpies of reaction for Eqns. 18.14 and 18.4, and the same equilibrium behavior when water is the dominant component. When the speciation is calculated in a mixture with small concentrations of water where the activity of water deviates from unity, Eqns. 18.4 and 18.14 lead to different results. For example, the dissociation of H2SO4 requires water, as seen in Fig. 18.1, but a dissociation in terms of H+ would result a dissociation in pure sulfuric acid. The requirement for a solvent in the dissociation reaction is more realistic in aqueous systems.11 Despite the importance of the hydration water in calculations, the convention in the literature is to write the reactions using only the H+ ion, and we follow that convention here where we work with dilute aqueous solutions.

The nature of hydration changes with concentration. As the ion concentrations increases, the ions are often paired with counter-ions and are called ion pairs. Ion pairs often contain water between them. Under other circumstances water is excluded from within the pair and the pair is hydrated. The ion pair phenomenon is used in ion pair chromatography (IPC) to influence the retention time using ions that are bulky or interact strongly with the solid stationary phase.

Charge Balance

When ions are dissolved in solution, an extra constraint of charge balance is needed, which is often called the condition of electroneutrality. The net charge on a solution must be zero. Electroneutrality can be expressed using moles, molarity, or molality. This condition provides an important constraint that is used in all calculations, supplementing the component balances and equilibrium constraints. Note that the charge balance always uses concentrations, not activities.

Approximate Calculations

We begin quantitative discussion by using concentrations instead of activities. We expect that students can make the transition to activities as their skill level develops. Further, as we show, the calculations using concentrations are frequently the first step to a more rigorous solution including activities. To avoid clumsy notation, we write many equilibrium constant relations using concentrations instead of activities. When a solution is dilute, we also use concentration in place of molality without further description. Near the end of the chapter we provide a complete example using activities.

18.8. Acids and Bases

The terms strong and weak are used when referring to acids and bases in a manner similar to salts. The terms do not imply anything directly about pH. Rather, like other electrolytes, they refer to a compound’s degree of dissociation. When the dissociation constant is extremely large the acid/base is considered strong; when dissociation is incomplete the acid/base is considered weak.

Strong Acids/Bases and the Leveling Effect

The magnitude of the dissociation constant (or the associated Gibbs energy of dissociation) determines whether an acid/base is strong or weak. An acid is a proton source and reacts with the solvent (usually water) to create an increased activity of [H+] or [H3O+]; for example, AcOH in Eqn. 18.15 is a proton donor and an acid. HCl is a strong acid and dissociates totally at common concentrations to form H3O+ (denoted as H+) and Cl. A base acts as a proton sink, reacting with water to withdraw a proton and increase the hydroxide activity, as shown by the reaction of the weak base ammonia with water,

Image

Hydroxides are also common strong bases, such as NaOH or Ca(OH)2. These strong bases are a direct source of hydroxide because when they dissociate the [OH] must be high to balance the positive cation charges in solution.

The strength of an acid or base is determined by the solvent, which governs the degree of dissociation/reaction. Table 18.2 illustrates acid/base strength. Strong acids are to the upper left. Strong bases are to the lower right. Weak acids are in the left column below H3O+. Weak bases are in the right column above OH. The reaction of H3O+ with water may initially look like little has happened: Image. All acids in the left column above H3O+ are equally strong when they are dilute in water because the protons released from the acid immediately react with H2O to give hydronium ions. Thus, the strongest acid in water is H3O+ and any stronger acids are leveled to be the same strength. In an analogous way, OH is the strongest base that can exist in water. Any other proton sinks added to solution from stronger bases react immediately with water to give OH. (Often these reactions are violent and superbases must be handled with care near water.) Strong bases are leveled by water and do not produce a stronger base solution than an equivalent concentration of NaOH. Sometimes organic chemists use strong acids or strong bases in non-aqueous solvents to overcome the leveling effect.

Table 18.2. Reference Table for Relative Acid and Base Strengths at 25°C Based on pKa,water = –13.995. pKb = 13.995 – pKa

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Each acid in the left column in the table has a conjuate base in the right column. Similarly, each base in the right column has a conjugate acid in the left column. Strong acids have weak conjugate bases, and strong bases have weak conjugate acids.

Strong Acids

Consider the behavior of HCl in solution at an apparent molarity of CA. The behavior is characterized by the dissociation of HCl (essentially complete) and the dissociation of water. Further, a charge balance must exist in solution, yielding the following conditions

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Image
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The next step is important and we use it throughout our calculations. Though trivial here, later we find it important to combine the material balance and charge balance to arrive at a balance known as the proton condition. The principle is to insert the known constants from the problem statement, which often cancel to leave the intermediate and smaller concentrations. In this case no terms drop out, but the proton condition becomes

Image

Think about the size of the [H+] and [OH] and how they are coupled to Ka,w. Suppose that CA = 10–2 mol/L. Then, by the coupling of the equilibrium and charge balance, because HCl is an acid, we expect [OH] < 10–7 mol/L by Eqn. 18.10, thus [OH] << [Cl] = 10–2 mol/L. The charge balance involves terms that are very different in size. Now, when CA > 10-6.5, it is a good approximation to ignore [OH] and we can calculate pH = pCA. At low acid concentrations below about 10-6.5, we must insert the dissociation constant for water into the charge balance, to obtain [H+] = CA + Ka,w/[H+] and solve the resultant quadratic equation for [H+]. The behavior of strong acids is plotted in Fig. 18.3. Note that a good approximation for the strong acid curve on this log-log plot is a diagonal line with slope –1 running from (0,0) through (7, –7). It is an excellent approximation from (0,0) through (6.5,–6.5).

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Figure 18.3. Flood diagram showing the behavior of strong monoprotic acids, weak monoprotic acids, salts of weak monoprotic bases, and strong monovalent bases.

Strong Bases

Consider a strong monovalent base, such as [OH]. In a solution of apparent molarity CB of NaOH the constraints are

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Image
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Combining the material balance and charge balance results in the proton condition

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When CB > 10-6.5, it will be a good approximation to ignore [H+] and we can calculate pH = 14 – pCB. At low concentrations, the quadratic equation becomes [H+] + CB = Ka,w/[H+]. As evident in Fig. 18.3, a good approximation for the strong base curve on this log-log plot is a diagonal line with slope 1 running from (7,–7) through (14, 0).

Weak Monoprotic Acids

Consider a weak monoprotic acid such as acetic acid, pKa,A = 4.7. We denote as the undissociated acid as HA. The weak acid requires an additional dissociation reaction because it is incomplete. The material balance is written including the acid HA and conjugate base A, because the acid is only partially dissociated. The acid equilibrium constant is required to quantify the dissociation.

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The rigorous solution to pH at a given concentration can be obtained from Eqn. 18.26 by eliminating [HA] and [A] in terms of [H+] and [OH] and then eliminating [OH] using [OH] = Kw/ [H+], resulting in a cubic equation,

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Solutions of the cubic for various pKa,A values and concentrations are shown in the solid lines on the left side of Fig. 18.3. The graph can serve as a useful guide for monoprotic weak acids, but is only applicable for the charge balance of Eqn. 18.28 when no other ions are present. Rather than solving the cubic, or becoming too reliant on the graph, real problems usually involve other ions, and thus the Flood diagram and cubic have limited utility. Instead, we utilize two equations that relate [A] and [HA] to CA and [H+] and are always applicable, even when other ions are present. Combining the material balance with Eqn. 18.26 to eliminate [A] results in Ka,A = [H+](CA – [HA])/[HA] which becomes

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Combining the material balance with Eqn. 18.26 to eliminate [HA] results in Ka,A = [H+][A]/(CA – [A]) which becomes

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Note that the denominators of the last two equations are the same and we need to know the pH to solve for the concentrations. Example 18.4 shows how these equations are useful for problem solving.

Weak Monoprotic Bases

Consider a weak monovalent base such as acetate ion, denoted as A, pKa,B = 9.3, which might be added to solution as sodium acetate. The sodium acetate dissociates completely, making it a strong electrolyte, but the acetate equilibrates with water to form undissociated acid (HA) and hydroxide, Image, so sodium acetate is called a weak base. The material balance is written for the cation in this case as well as the two forms of A, and the governing equations are:

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In an analogous way to treating weak acids, the material balance and charge balance are combined to eliminate [HA] and [A] in terms of [H+] and [OH]. Then the dissociation constant of water is used to eliminate [OH], resulting in

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The same equation results from a weak neutral base with capacity for one ion, such as ammonia. Note, however, for ammonia that the base NH3 is neutral and the conjugate acid NH4+ is charged, but when the charge balance is modified and the same method is used, Eqn. 18.36 results. As with a weak acid, the pH values at various base concentrations have been plotted in Fig. 18.3. As with weak acids, the charge balance that leads to the cubic equation does not hold when other ions are present. However, we can solve problems using Eqns. 18.30 and 18.31. Thus, we develop a single method of solution regardless of whether acid or base is added to water. Note that

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As an example, consider the fluconazole shown in Fig. 18.4. This is a drug used for treating fungal infections. Fluconazole is a base that is protonated in water, depending on pH:

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Figure 18.4. Structure of Fluconazole.

By lowering the pH, the [OH] is lowered, driving the reaction to the right. At high pH values, the reaction shifts to the left, as we show in Example 18.4. The equilibrium shift affects solubility as we consider in Example 18.8 on page 726.


Example 18.4. Dissociation of fluconazole

Fluconazole is a drug used for treating fungal infections. Behavior of drugs at various pH conditions is important because the stomach system is at low pH, but the intestinal system has a higher pH. Thus, models for the dissociation and solubility are desirable. Fluconazole equilibrium written as Eqn 18.38 can be modeled with the expression

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where fluconazole and its ion and the hydroxyl are on the molality scale and water is on the Lewis-Randall scale. Determine the percentage of fluconazole dissociated at pH 7 and pH 1.5 when the apparent amount of fluconazole in aqueous solution is 1.5E-3m. The molecular weight of fluconazole is 306.27. Assume ideal solutions.

Solution

First consider the chemical reaction to identify the acid and base. In this case, the fluconazole is the base and the fluconazole+ is the acid. Therefore Eqn. 18.39 represents Ka,B. As with this example, it is common in the literature that acids and bases are not explicitly identified, and recognition is an important step in the solution. We can rewrite the reaction in the acid form as,

Fluconazole+ + H2O Image Fluconazole + H3O+

(We adopt this approach of writing reactions in the acid form as a standard method, as further implemented in Section 18.9 below.) For Eqn. 18.39, the equilibrium constant at 298.15 K is Ka,B = 6.181E-13, or Ka,A = 10–14/6.181E-13 = 0.01617, or pKa,A = 1.79. Therefore, the fluconazole is predominately protonated below pH = 1.8 and largely neutral above. The calculations should bear out this rule of thumb.

At pH = 7 we have from Eqn. 18.30,

[Fluc+] = Cfluc[H+] / ([H+] + Ka,A) = 1.5E-3m(10–7)/(10–7 + 0.01617) = 9.27E-9m. Therefore, [Fluc] = 1.5E-3m, and the fraction protonated is 9.27E-9/(1.5E-3)·100% = 0% (trace).

At pH = 1.5, the system is near the pKa,A and both terms are important in the denominator, [Fluc+] = Cfluc[H+] / ([H+] + Ka,A) = 1.5E-3m(10–1.5)/(10–1.5 + 0.01617) = 9.9E-4m. Thus, [Fluc] = 1.5E-3m – 9.9E-4m = 5.1E-4m. The concentration of protonated species is higher, as expected. The fraction protonated is 9.9E-4/(1.5E-3)·100% = 66%.


Perspectives on Calculations

Calculations for electrolyte systems can be challenging to converge because the concentrations of important species vary by several orders of magnitude. Each pH unit is an order of magnitude; thus, at pH 2 compared to pH 7 the [H+] is five orders of magnitude larger. Calculations using Excel Solver are insensitive to the latter condition if the Solver tolerance is set to 1E-3! Also, issues may occur if iterating on values near zero because the concentration of 1E-3 can easily jump to a negative value on the next iteration if not constrained. Iterations can also converge slowly for some situations.12 Because charges are involved, the net charge in a solution must be zero; a charge balance is required when iterating on concentrations. However, a charge balance often involves adding terms that are different by orders of magnitude. There are several general recommendations.

1. Develop a good initial guess using techniques that we discuss below. The pKa,A represents the pH condition where the reaction coordinate will be 50% between acid and conjugate base and thus their concentrations will be equal.

2. Constrain concentrations to be positive or use logarithms of concentrations for iterations when using automated equation solvers.


Image The pKa,A represents the pH condition where the reaction coordinate will be 50% between acid and conjugate base, and thus their concentrations will be equal.


3. Set the convergence criteria in Solver or optimization routine to an extremely small number (1E-30) and the number of iterations to a high number.

4. Check results to ensure convergence, especially if the specified number of iterations is reached.

How does one generate a good first guess when the concentrations differ by orders of magnitude? The best way to generate a good first guess is to use a Sillèn diagram. Sillèn diagrams, originally developed by Swedish chemist Lars Gunnar Sillèn in the 1950s, are quick to sketch. The next section discusses how to construct and use a Sillèn diagram. Often the results from the Sillèn are sufficiently accurate for routine practical applications or as first estimates for more detailed calculations using activity coefficients.

Other important aspects of the initial examples include the standard states, concentration units, and composition independence of K. The typical convention used for standard states of charged species is similar to Henry’s law, but subtly different. Without belaboring the details until later in the chapter, the activity coefficients for the charged species are unity at infinite dilution, and we will disregard the activity coefficients for the introductory examples. The corrections are typically small when the concentrations of ions (measured by the ionic strength) is low, less than approximately 10–2 m. We will use this approximation at even higher concentrations to develop problem-solving strategies. The reader should be cautioned that the activity coefficients for charged species can become large rapidly and can be very large, but typically above 5m. A good understanding of the standard state for water and uncharged solute species is also important. The standard states for water and uncharged solutes are different from that used for charged species. Uncharged solutes, such as molecular acids and bases, like acetic acid or ammonia, are treated with Henry’s law. This shares a similarity to the treatment of charged species, because the activity coefficients are unity. Water, on the other hand, is treated relative to the Lewis-Randall standard state of pure water. Because the water concentration on a mole fraction basis is nearly one, it will be sufficient to approximate the activity coefficient of water as one and the activity of water will be approximately one. In summary, we extend the concepts of using unsymmetric standard states introduced in Chapter 11.


Image Ions are treated with a molal standard state. Aqueous molecular solutes are treated with Henry’s law. Water is treated with the Lewis-Randall rule.


Another important approximation is that we use molar concentration rather than molality to work the examples early in the chapter. This follows the conventions used in introductory chemistry, and, as shown in Table 18.1, can be a good approximation at low concentrations. Technically, the units should be molality for the electrolyte species, and certainly the examples can be reworked with those units, but use of molality requires more unit conversions with little pedagogical advantage. The later examples in the chapter use molality to demonstrate the more rigorous approach.

Temperature, Pressure, and Composition Effects on K

Initially, the discussions and examples provide values for equilibrium constants. Commonly in introductory chemistry texts values are provided for 298.15 K, though the designation is often omitted in those texts. The values of Ka change with temperature as with any reactive system.

In Chapter 17, the equilibrium constant did not depend on pressure. This is not the case for electrolytes when the typical electrolyte standard state is used for ions, though it is common to neglect it as a first approximation. When an ion is dissolved in water at infinite dilution, the hydrogen bonding is disrupted resulting in a pressure-dependence for the infinite-dilution standard state. We do not develop the details further in this text, but readers should consult advanced texts or handbooks when working at high pressures.13

There are two conventions to correct for solution nonidealities. Extending the concepts of Chapter 17, the most rigorous method of including solution nonidealities is to use activity coefficients as we show later. However, another method used in the literature is to determine the dependence of K = Ka / Kγ on ionic strength of the solution, and then proceed with calculations using K rather than Ka. This is possible because to the first approximation the activity coefficients depend on charge and ionic strength and are independent of species (as we show in Section 18.15). Therefore, when working with values from the literature, some care is necessary to discern if the authors’ values for K are at the standard states or if they are corrected for ionic strength. When the dependence of ionic strength is included and K is substituted for Ka, the methods developed below can be used directly by replacing Ka with K in the calculations.

18.9. Sillèn Diagram Solution Method

Monoprotic Acids and Bases

Seven main steps are necessary to solve electrolyte problems using a Sillèn diagram (cf. Fig. 18.5), which is similar to a Flood diagram. We summarize the steps and then work an example for sodium acetate. Skim the procedure initially, and then follow closely with Example 18.5.

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Figure 18.5. Sillèn plot for Example 18.5, acetate with an overall concentration 0.01 M.

1. Create a coordinate system like the Flood diagram. (A template is available on the textbook web site.) Draw straight lines for the strong acid and strong base lines. The detail of the taper at pH = 7 should be ignored, and cross the lines. Note that the sum of the two lines is always –14 on the log scale and represents the ion product for water. Label these lines [H+] (left) and [OH] (right).

Some rules of thumb are helpful for plotting on common logarithmic coordinates. Note that when [B] = 2[A], the ordinate of [B] on a log10 scale is 0.3 units higher than [A]. Likewise, when [B] = 0.5[A], then the [B] ordinate will be 0.3 units lower. A factor of 5 is 0.7 units. And of course a factor of 10 is one unit. For convenience the pairs of (linear factor, log10 translation) are (2, 0.3), (3, 0.47), (4, 0.6), (5, 0.7), (6, 0.78), (7, 0.85), (8, 0.9), (9, 0.95), (10, 1).

2. Write the material balance for the dissociating species to relate the apparent species to the species in solution; for example, Eqn. 18.32.

3. Write the equilibria relations using the dissociation constants for weak acids or bases. If the acid/base is strong it will completely dissociate, and thus the relation is not needed. Always write the reactions in the acid form (even if bases are involved); for example, Eqn. 18.26 for acetate or acetic acid. Write the dissociation reaction for water. Using the acid form provides a consistent solution strategy, but is not theoretically required.

4. Write the electroneutrality constraint.

5. Sketch Eqns. 18.30 and 18.31 without calculations on the diagram using these steps. (See the example.) The steps are: (a) create a system point at CA (or CB) and pKa,A; the procedure always uses pKa,A, even for bases; (b) create an acid/base intersection point at (pKa,A, logCA – 0.3). (The value of 0.3 represents a decrease of 50% in the concentration, which is where the acid and base concentrations will match.); (c) sketch diagonal lines with slope +1 and –1 (parallel to the H+ and OH– lines) below CA that project through the system point but extend downwards starting about logCA – 1; (d) draw horizontal lines on either side of the system point leaving a gap of approximately 1 pH unit on either side of the system point and label the line on the left (low pH) as the acid and the line to the right (high pH) as the base; (e) connect the sloping lines with the horizontal lines with smooth curves that pass through the acid/base intersection point.

6. Decide which concentrations are largest and which are least significant. Let Ci be the apparent concentration. The goal is to simplify the balances and provide a good guess for true concentrations. This is almost always done by converting the charge balance to a proton condition by inserting the mass balance to eliminate terms that are largest and leave smaller terms that are more similar in magnitude. Use the diagram as a guide to decide which concentrations are insignificant in the pH range expected. The goal is to use the proton condition to identify the intersection of the positive and negative charges of the proton condition. Unless some of the diagonal curves are very close to each other this will be easy. There can be various proton conditions that are equally valid when many ions are present at similar concentrations. Hints: Remember that each unit on the log scale is an order of magnitude. Acids by themselves result in pH < 7; bases alone result in pH > 7; salts of a strong acid and weak base (e.g., NH4+Cl) are acidic; salts of a weak acid and strong base (e.g., NaOAc) are basic.

7. Check the result. The results can be checked by iterating on charge balance pH by inserting Eqns. 18.30 and 18.31 or the analogs.


Example 18.5. Sillèn diagram for HOAc and NaOAc

Sodium acetate, NaOAc, is dissolved in water at an apparent concentration of CB = 10-2 mol/L. Construct a Sillèn diagram and estimate the pH. For acetic acid, pKa,A = 4.76 at room temperature in dilute solutions.

Solution

Here we replace the generic A with OAc to denote acetate. The Sillèn diagram is presented in Fig. 18.5. The approximate solution (thick lines) is shown below superimposed on the exact equations (thin lines).

Step 1: The lines for [H+] and [OH] have been drawn and labeled in the figure.

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Step 5: See the diagram labels denoting steps 5(a) and 5(b). Referring to the procedure above indicates the system point (x) should be at CB and pKa,A. On the diagram below, use a straight edge to verify that the lines for step 5(c) extrapolate through the x from step 5(a). Note that the curves for step 5(e) are not shaded for clarity, but it is obvious that the curves could be easily drawn through o. Recall that step 5 plots Eqns. 18.30 and 18.31 without calculations.

Step 6: Develop the proton condition. This step is very important and can be the most confusing. It is best understood by using equations together with the diagram. Since we have dissolved the salt of a weak acid and strong base, we expect the pH to be above 7. Looking at the diagram in this range, [OAc] >> [HOAc] and we will be unable to reliably calculate [HOAc] = CB – [OAc] with the material balance because the last two terms are nearly equal. Let us use the material balance to eliminate the large terms [Na+] and [OAc] from the charge balance. Note that we can replace [Na+] with CB and [OAc] = CB – [HOAc]. This causes all the largest terms to drop from the charge balance, giving

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Eqns. 18.4018.43 are now all condensed to using Eqn. 18.44 with the graph, looking for where the proton condition is satisfied. Looking at the lines on the graph where pH > 7, it is obvious that [HOAc] is almost three orders of magnitude larger than [H+] above pH = 5. Thus, the left side of the proton condition becomes [HOAc] + [H+] = [HOAc] + ... where ... denotes a very small number. The proton condition becomes [HOAc] + ... = [OH], and the solution is given for practical purposes by the intersection of the [HOAc] curve with the [OH] curve as shown in the diagram. The approximate concentrations are

pH = 8.4, pOH = 14 – 8.4 = 5.6, [HOAc] = 10–5.6, [Na+] =[OAc] = 10–2

Step 7: The proton condition is in terms of Eqns. 18.27 and 18.30, and avoiding taking differences, [OH] = 10-14/[H+] = [H+] + [HOAc] = [H+] + CA[H+] / ([H+] + Ka.A)

Rearranging for successive substitution on [H+], and inserting the initial guess of pH = 8.4, iterate on the highest power of [H+],

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pH = 8.38. Plugging this back in results in no further changes. Recall that if successive substitution results in divergence rather than convergence, that the equation needs to be rearranged. See Appendix A, Section A.4.


This example has demonstrated that a relatively complex problem can be solved rapidly with a quick sketch. A key simplification used in this introductory example was that [H+] << [HOAc]. Do not generalize this approximation. If the same problem is repeated with CB = 10–4 mol/L, this translates the lines for [HOAc] and [OAc] downwards two orders of magnitude and the lines for [H+] and [HOAc] become very close. The approximation [H+] << [HOAc] is not valid then. Instead, the left side of the proton condition, [H+] + [HOAc], can be calculated at a selected point from the two lines. The [H+] + [HOAc] sum becomes a line through this point with the same slope as the individual lines. The answer is where this “summed” line crosses [OH].

A remarkable feature of this solution technique is that the solution to the four simultaneous equations did not require sophisticated algebra or a cubic equation. As an exercise, consider a solution of acetic acid, CA = 10–2 mol/L. The diagram is the same as used above. However, the material balance and proton condition are different. Use the diagram to show that the pH = 3.35 approximately.

Polyprotic Acids and Bases

The phosphoric system (H3PO4, H2PO4, HPO42–, PO43–) and the CO2 (CO2, HCO3, CO32–) systems are important for both biology and environmental applications. Succinic acid, a dicarboxylic acid produced by fermentation, is expected to become more widely produced via fermentation in future years, typically as a salt. Amino acids, the building blocks for proteins, combine a basic amine and a carboxylic acid on the same molecule. Let us begin by considering the nonvolatile phosphate system.

Phosphate System

The equilibria can be written (using all acid equilibrium constants, but without the A subscript for convenience),

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The material balance on phosphorous is

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Defining a variable αi to denote the fraction of each species relative to the total phosphate concentration where the subscript denotes the number of protons,

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Dividing the material balance by [H3PO4], we find the reciprocal of α3,

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Inverting and simplifying,

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Then α2 as the fraction of acid with three protons is

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Recognizing the recurring relation between the fractions, the arguments for one and no protons are

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The Sillèn diagram for the phosphate system is slightly more complicated than a monoprotic system, but can still be quickly drawn by hand. The concentration of each species i can rigorously be calculated at each pH by αiC where αi is calculated from Eqns. 18.5118.54. The exact concentrations are shown in Fig. 18.6 for C = 10–2 mol/L. The upper part of the Sillèn diagram is analogous to the monoprotic diagram. The difference is that the slopes change to +2 or –2 when the species concentration crosses the pKa,A for a neighboring dissociation. The curve passes through a y coordinate approximately 0.3 “log10 units” below the point where the upper extrapolated line crosses the neighboring pKa,A. Though not apparent from this diagram, at very low concentrations when the species concentration crosses additional pKa,A values, the slope increments (decrements) again by one.

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Figure 18.6. Phosphoric system at C = 10-2 mol/L discussed in Example 18.6.


Example 18.6. Phosphate salt and strong acid

A solution of NaH2PO4 and HCl is prepared such that the total phosphorous concentration is 1E-2 M and the total Cl concentration is 5E-3M. Calculate the pH and concentrations of species present.

Solution

Begin a problem with multiple ions with the material balance. The material balances on the sodium, chloride, and phosphate are:

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Note the coefficients on the ions in the charge balance. The Sillèn diagram for the phosphate system is shown below. It may seem daunting that so many species are present, but when you look at the Sillèn diagram, notice that only two phosphate species at a time are important at any pH range. This occurs because the pKa,A values are well separated. The curves are drawn with the exact relations, but can be quickly sketched. Practice the sketch using the rules given above and compare with Fig. 18.6.

Steps 1-5 of the procedure have already been executed.

Step 6. The proton condition is developed by eliminating [Na+] and [Cl] using the material balances since they are both known constants. The material balance for phosphate is also inserted, resulting in

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which becomes

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Understanding where to find the solution requires some thought and reasoning rather than a direct numerical manipulation. Both terms on the left side of the proton condition are almost equal at 2.5 < pH < 7. The values are added on the short dashed line marked “1” (since they are virtually equal in most of the range, the sum is double, or about 0.3 units higher on the log10 scale). Note that [H2PO4] does not appear in the proton condition. On the right-hand side, the term 5E-3 dominates at pH < 6. Solutions at high pH are impossible because the decreasing right-hand side is too small to balance the value of 5E-3 plus increasing concentrations of the negative phosphate and hydroxide ions in the proton condition. Therefore, the solution must be a low pH where the concentration of negative phosphate and hydroxide ions in the proton condition are small. The solution occurs where [H+] + [H3PO4] = 5E-3 (the line marked “2”), and pH = 2.6. The approximate concentrations from the diagram are [H+] = [H3PO4] = 2.5E-3. Eqn. 18.45 simplifies to [H2PO4] = Ka1 = 10–2.15, [HPO42–] = 10–6.5, [Na+] = 1E-2, [Cl] = 5E-3.

Step 7. The detailed calculations are often tedious. Inserting Eqn. 18.51 into the proton condition, where the first three terms on the right side are negligible,

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Inserting the initial guess,

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Repeating the iteration results in [H+] = 2.44E-3, pH = 2.613. Note how close we were with the graphical value of pH = 2.6.


Amino Acids

Amino acids are the fundamental building blocks from which all proteins are built. DNA encodes the formulas used to assemble 22 standard amino acids into the multitudes of proteins. Proteins with specific catalytic functions are called enzymes. Amino acids include at least one carboxylic acid group and one amine group. The 20 amino acids summarized in Fig. 18.7 are encoded in the universal genetic code. Together with selenocysteine and pyrrolysine which are encoded in special situations, the 22 amino acids link together to provide the functionalities required for biological life by use of various side chains. When biological machinery assembles amino acids into proteins, a carboxylic acid from one amino acid is covalently bonded to the amine on the next amino acid. One end of any protein backbone is an amine and the other end is a carboxylic acid. Note that some side chains in Fig. 18.7 include acidic and basic side chains. These acidic and basic side chains lead to charges on proteins, which change as a function of pH. Since biological systems usually have buffered pH near 7, which is above the pKa,A for the carboxylic acids, those groups are in the conjugate base form, leading to negative charges on the side chains. Similarly, basic groups below the pKa,A are protonated. For example, serum albumin, a globular (round) blood protein, has negative charges on the surface at physiological pH. At physiological pH, it has an intrinsic charge of -17 and binds 6 monovalent anions giving a total net charge of –23. At low pH, the intrinsic charge becomes positive and the protein binds more anions. Hence the charge and binding change with pH.14 Serum albumin is critical for controlling the intravascular hydrostatic pressure by regulating the osmotic pressure. About half of the osmotic pressure is controlled by Donnan equilibria, as described on page 725. Other proteins have a larger number of basic side groups than acidic groups. The basic side chains are protonated at neutral pH, contributing a positive charge.

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Figure 18.7. Summary of 20 amino acids encoded by the universal genetic code. The amine and carboxylic groups on the bottom of each molecule are where the amino acid is linked into the biomolecule. The acidic and basic side chains are shown uncharged.

Glycine

Glycine is the simplest amino acid. The side chain is simply a hydrogen; that is, there is no side chain. For glycine, pKa,A1 = 2.35 for the carboxylic acid and pKa,A2 = 9.78 for the amine. Thus, the amino acids combine the concepts of acids and bases with the concepts of a polyprotic system. The species at various pH values and nomenclature are shown in Fig. 18.8. Note that the species at neutral pH has a net charge of zero, but includes a positive and negative charge. Such a molecule with both positive and negative charges is called a zwitterion (German for “hybrid ion”). This combination of positive and negative charges results in large dipole moments for biological molecules at neutral pH. The relevant equilibria are (written using acid dissociation constants, but omitting the A subscript):

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Figure 18.8. Dominant species for glycine at various pH levels. pKa,A = 2.35 for the carboxylic acid and pKa,A = 9.78 for NH3+. The top line shows the nomenclature and the second line shows abbreviations.

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Example 18.7. Distribution of species in glycine solution

a. Calculate the pH of a 0.1 M solution of glycine.

b. What is the distribution of species for glycine at a physiological pH of 7.4?

Solution

A Sillèn diagram for glycine (Fig. 18.9) is sketched by the standard procedures.

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Figure 18.9. Sillèn diagram for 0.1 M glycine discussed in Example 18.7.

a. The relevant equilibria are given in Eqns. 18.64 and 18.65. The material balance is

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The charge balance is

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The pH is expected to be near neutral because the glycine added is neither an acid nor a base, though it has both functionalities. Look at the charge balance and the Sillèn plot of concentrations near neutral pH. On the left-hand side of the charge balance, the concentration of [H+] is about an order of magnitude smaller than [H2 Gly+] making the total positive charge concentration about 1.1[H2 Gly+]. On the right side, [Gly] is over three orders of magnitude larger than [OH]. Thus, the charge balance is effectively

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The answer is found at pH = 6.1. [OH] = 10–(14–6.1) = 10–7.9 M. The glycine species concentrations are quickly read from the graph, [HGly] = 10–2 M, [Gly] = 10–4.65 M, [H2 Gly+] = 10–4.75 M. Note the glycine is almost totally in the zwitterion form and the charged forms are about 3.5 orders of magnitude smaller. The final verification of the concentrations is left as a homework problem. Note that if the concentration of glycine was lower, [H+] would become more important in the charge balance.

b. First, [OH] = 10–(14–7.4) = 10–6.6 M. The glycine species concentrations are quickly read from the graph, [HGly] = 10–2 M, [Gly] = 10–3.4 M, [H2 Gly+] = 10–6.05 M.


Summary for Use of Sillèn Plots

Throughout this section, we have demonstrated the use of Sillèn plots for weak acids and bases, and polyprotic systems including amino acids. We have demonstrated that the equilibrium relations plotted on the diagram are not directly dependent on the other ions present in the system. The material balance is often combined with the charge balance to yield a proton condition. The proton condition focuses on the intermediate concentrations, by canceling out the overwhelmingly large concentrations, enhancing precision. The proton condition is used along with the plot to determine an approximate solution. In cases where the curves for the dominant species with the same charge sign in the proton condition are close to each other, the curve values are added together and replotted to find the solution graphically (cf. Example 18.6).

For polyprotic systems we have demonstrated that the distribution can easily be calculated by the Sillén method (cf. Example 18.6). For the polyprotic systems in the examples, the pKa values were well separated and each species has a pH where it is the dominant species and the species concentration is virtually equal to the overall concentration, C. For example, in the case of glycine, the intermediate species HGly is virtually equal to C in the range of 4 < pH < 8 (cf. Fig. 18.9). In some polyprotic systems, the pKa.A values are close together—for example, in succinic acid, where the species are H2Succ, HSucc, and Succ2–. In these cases, the plot maxima in the intermediate species concentration (HSucc in the case of succinic acid) will not extend all the way up to C. When the diagram is prepared, the regions with sloping lines should be drawn first. Then, if the maxima are not obvious, the full equations for the distribution can be developed as illustrated in Example 18.6 and then plotted for the intermediate species.

18.10. Applications

To introduce the concepts of electrolytes, we started in Section 18.2 with examples of freezing point depression, osmotic pressure, and boiling point elevation. Here we consider other applications where some of the subtler effects of charges are important.

Buffers

When salts and acids that share a common ion are present in solution, the solution is buffered. A buffered solution is resistant to changes in pH, and such behavior is critical in biology. For example, blood is buffered to be at pH 7.4 with carbonates, and slight deviations can cause severe illness. The buffering capacity is dependent on the concentrations of the acid and salt. For a given overall concentration, the buffering capacity is best understood relative to a titration curve. The buffering capacity (for a change in either pH direction) is greatest when the acid is “half” titrated. The fundamental explanation for the buffering phenomenon is because the titration curve is steepest when the acid is half titrated. For a monoprotic acid, this occurs when the acid and salt concentrations are equal, and the buffered pH = pKa,A unless the buffer is very dilute such that the acid/base lines are close to the [H+] or [OH] lines on a Sillèn diagram. Consider a weak monoprotic acid HA (species A) and the sodium salt NaA, (species B). A specific case would be a mixture of acetic acid and sodium acetate in water. The equations are an extension of Eqns. 18.2518.28:

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The sodium material balance is inserted into the charge balance, and solving for [A],

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If this is substituted for [A] in the acid balance,

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Substituting into the equilibrium Eqn. 18.70,

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When the [H+] and [OH] are much smaller than CA and CB the right-hand side simplifies, resulting in the Henderson-Hasselbalch equation:

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The Sillèn graph method can certainly be used, but the Henderson-Hasselbalch equation is convenient under proper conditions. When CA and CB are near [H+] or [OH], the equation can lead to absurd results. A better approximation is

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The equations can be used iteratively from an initial assumed value of [H+] or [OH].

Isoelectric Point and Ionic Strength

Proteins and biomolecules frequently have charged surfaces at neutral pH due to the carboxylic acid and amine side chains. Basic amines are protonated at neutral pH values and acidic carboxylic acids are deprotonated. When the biomolecules have a net charge, they repel each other, and are thus more soluble, enabling them to provide important biological functions by remaining soluble. Two important phenomena exhibited by charged molecules are the change in solubility as the pH is changed, and the dependence of solubility on ionic strength (salt concentration).

First of all, as the pH is varied, the charges on the side chains change. The pH value at which the biomolecule has no net charge is called the isoelectric point. Solubility is usually smallest at the isoelectric point because the lack of net charges permits the large macromolecules to approach each other and the large cooperative physical forces cause them to precipitate. Therefore, solubility typically increases rapidly on either side of the isoelectric point. The isoelectric point is often characterized by the pI which is the isoelectric point pH. Solubility of a milk protein β-lactoglobulin is shown in Fig. 18.10.

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Figure 18.10. Salting in and illustration of minimum solubility at the isoelectric point for milk protein β-lactoglobulin as a function of pH and ionic strength. Gronwall, A., 1942. C.R. Trav. Lab. Carlsberg, Ser. Chim. 24:185-200.

The solubility of biomolecules increases at low ionic strength when the ionic strength (ion concentration) increases. This effect is called salting in and occurs because the ions in solution screen the surface charges. This reduces attractions between positive and negative charges, even near the isoelectric point. There are usually positive and negative charges when the net charge is zero, (cf. amino acid and zwitterion section above) which lead to net attraction. However, at high ionic strength (high salt concentrations), the opposite effect is seen and increasing salt concentrations result in decreasing biomolecule solubility, known as salting out. Salting out occurs because the ionic strength is so high that it screens the repulsive forces that would normally prevent precipitation. Thus, the solubility increases with salt concentration at low salt loading, but decreases with salt concentration at high salt loading, causing a maximum in solubility at intermediate salt concentrations. Observe in Fig. 18.10 the salting in behavior and the minimum solubility near the isoelectric point at all ionic strengths.

Donnan Equilibria

Membranes can have interesting effects when they are impermeable to certain ions or charged species. DNA is a polyanion and requires cations to balance its negative charge. For example, consider a membrane impermeable to DNA shown in Fig. 18.11 where an arbitrary DNA of charge –z is shown in the presence of KCl. The chemical potential of the apparent KCl must be the same on both sides of the membrane, resulting in

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Figure 18.11. Illustration of Donnan Equilibria for DNA. DNA cannot cross the membrane. A larger concentration of ions will exist on the β side, creating higher pressure on the β side due to osmotic pressure.

Electroneutrality requires

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Using concentrations to approximate activities and combining the charge and equilibrium relations results in

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It is clear that the concentration of potassium on the β side is larger due to the minus sign in the parentheses. Similarly, manipulation for chlorine shows that the concentration on the α side is larger, due to the plus sign in the parentheses:

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What is the relative magnitude of the terms in parentheses in Eqns. 18.80 and 18.81? Substitute Eqns. 18.80 and 18.81 into the left side of Eqn. 18.78, and it becomes obvious that the product of the two terms in parentheses is 1. What may not be immediately clear is that mathematically there are more dissolved species on the β side, creating a higher osmotic pressure on the β side. Hemoglobin in red blood cells contributes to a Donnan effect because it is confined to the cells. Another interesting effect is that counter anions such as chloride, bicarbonate, and hydroxyl can pass through the membrane and impact the pH of blood because they are bases.

Solubility and Ksp

Dissociation of species can have a dramatic effect on solubility in water. Consider the behavior of fluconazole in Example 18.4. How might the dissociation affect the solubility? Since the solubility is dependent on the activity of the un-ionized species, the solubility goes up appreciably as the equilibrium shifts to the protonated form below the pKa,A.


Example 18.8. Dissociation and solubility of fluconazole

In Example 18.4 the dissociation of fluconazole (fluc) was considered. The solubility can be modeled using (on the molality scale)

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Determine the solubility of fluconazole at pH 7 and pH 1.5 and the distribution of species in solution at 298.15K. Assume ideal solutions.

Solution

This involves two simultaneous equilibria, dissociation and solubility. Note that the acid form of the reaction equilibrium constant is

Ka,A = [fluc][H+]/([fluc+]aH2O)

Since the pH is specified, the solution is so dilute (thus aH2O = 1), and the Ka,A is constant, the ratio [fluc]/[fluc+] is constant at a given pH. From the earlier example, at pH 7, the reciprocal is [fluc+]/[fluc] = 0, and at pH 1.5, [fluc+]/[fluc] = 9.9/5.1 = 1.94.

At 298.15 K, KSLE = 0.018. Using the ideal solution approximation, [fluc] = 0.018 m. This is independent of pH. At pH 7, virtually no [fluc+] is present and thus the solubility is 0.018 mol/L, or using the molecular weight, 0.018 mol/L(306.27 g/mol) = 5.5 g/L.

At pH 1.5, the [fluc] = 0.018 m and [fluc+] = 1.94[fluc], thus the total solubility is 2.94(0.018) = 0.0529 mol/L, or 0.0529(306.27) = 16.2 g/L. The pH makes a large difference in the solubility!


Common Ion Effect

When compounds in solution share a common ion and one of the compounds is near or at the solubility limit, addition of the other species can induce precipitation of the first. For example, consider a solution of water saturated with KCl. If a small amount of NaCl is added to the solution, additional KCl precipitates because the equilibrium is disrupted.

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Because the activity of Cl is increased when NaCl is added, the activity of K must be decreased to match Ksp, thus additional KCl precipitates. Likewise, KCl affects the solubility of NaCl near its solubility limit.

18.11. Redox Reactions

Chemical reactions involving electron transfers are fundamental steps in diverse applications ranging from biological systems to corrosion, batteries, and fuel cells. Electron losses are called oxidations and reactions involving gains in electrons are called reductions. A simple acronym to remember the conventions is OILRIG; oxidation is loss, reduction is gain. To balance electrons, an oxidation reaction is always coupled with a reduction reaction, and the combined reactions are termed redox reactions.

Redox reactions can be conducted in any phase. Even combustion of methane is a redox reaction where carbon is oxidized into CO2 and O2 is reduced into water. Though “oxidation” sounds like it is limited to reaction with oxygen, the process is much more general, relating to the loss of electrons. The gas phase combustion of methane does not produce useful electrons. However, biological oxidation carries out the oxidation of glucose in a series of smaller steps, capturing the electrons and by coupling the favorable oxidation to otherwise unfavorable reactions. In a battery, the redox reaction is enabled by electron flow through an external circuit and a voltage is generated. By permitting the spontaneous redox reaction through an external circuit, we obtain electrical power.

Li-ion batteries power the majority of portable devices today. They are constructed of electrodes of CoO2 and graphite (represented here as C6 to stress the aromatic ring structure and the way each ring can host one cation). Li+ and e are shuttled back and forth during discharge and charge. During battery discharge, an oxidation of graphite is occurring at the anode, LiC6 Image C6 + Li+ + e (remember that oxidation is loss, so electrons are generated at the anode), and reduction of Co from Co+4 to Co+3 at the cathode, CoO2 + Li+ + e Image LiCoO2. The overall discharge reaction is LiC6 + CoO2 Image C6 + LiCoO2. The electrons are carried through the external circuit while the Li+ are not capable of flowing through the circuit, but travel through an electrolyte solution between the electrodes in the opposite direction as the electrons. The combination of the electrodes and electrolyte is called an electrochemical cell or battery. The Li-ion cell produces about 3.9V. To charge the cell, an opposing voltage > 3.9V must be applied. The life of the cell depends upon chemical reversibility, side reactions, and physical changes that may affect transport or the ability of the Li+ to be hosted by the graphite sheets or the CoO2.

Half-Reactions

Redox reactions can be decoupled into the corresponding reduction and oxidation processes as we have indicated with the battery example above. For example, look at the overall reaction for the battery above and note that the reaction does not show the electrons! From the overall battery reaction, how do we determine the two half-reactions, how many electrons are transferred, the voltage, and the direction of electron flow?

As with other thermodynamic tables a reference is used to establish a relative scale. Because reduction reactions are always coupled to oxidation reactions, measurement of absolute potentials is not possible. The reference for redox reactions is to measure/tabulate reduction reactions relative to a standard H2 electrode. In the standard H2 electrode, an acid solution is used to establish aH+ = 1 (pH = 0) at 298.15K and H2(g) is bubbled through the solution at 1 bar partial pressure;15 the reaction 2H+ + 2e- Image H2(g) is conducted on a submersed platinum electrode. This reaction is a reduction of H+. The direction of the reaction depends on whether the reduction potential for the other electrode is greater than or less than that of hydrogen. The potential for hydrogen ion reduction is arbitrarily set to zero at the standard conditions cited above, permitting determination of other reduction potentials. The convention established by the International Union of Pure and Applied Chemistry, (IUPAC) is to always tabulate the potential for reduction reactions.16 The oxidation potential is the negative of the reduction potential. Standard reduction potentials determined at 298.15 K and molal activity of unity (usually near 1 m) for a variety of reactions are tabulated in Appendix E.17 The half-reaction with the higher reduction potential undergoes reduction at standard conditions in the electrochemical reaction, regardless of the number of electrons in the half-reaction. The other reaction proceeds in the direction of oxidation at standard conditions. The relation between the overall cell voltage and the Gibbs energy of reaction at equilibrium is

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Michael Faraday (1791–1867) established the concept of an electromagnetic field, and also popularized the terms “anode,” “cathode,” “electrode,” “ion.”


where ne is the number of electrons transferred in the balanced pair of redox reactions, F is Faraday’s constant 96,485 J/V, and = redox is the difference in reduction potentials of both half-reactions (or you may wish to think of adding the reduction potential and the oxidation potential). Note that the potential is measured per electron, and thus the potentials can be combined without balancing. However, the number of balanced electrons is necessary to relate the potential to the equilibrium constant, Gibbs energy, or nonstandard conditions. As discussed in Section 18.6, Gibbs energies for steps are additive for an overall reaction; the principle can be applied to the half-reactions, ΔG° = – neFE°red + neFE°ox, where red and ox both represent reduction potentials. A redox reaction at equilibrium is not useful because there are no driving forces. Instead, the direction of the reaction at nonstandard conditions is dependent on ΔG. The principles of Chapter 17 also apply, even under nonstandard conditions, and the reaction goes forward for ΔG < 0, and backward for ΔG > 0. We write

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This is often presented as the Nernst equation. Also, since reactions frequently occur near 298.15 K, and log is more convenient than ln for quick calculations, the equation is frequently written with values inserted,

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Walther Nernst (1864–1941) is credited with development of the third law of thermodynamics, for which he was awarded the 1920 Nobel Prize in chemistry.


where the last equation is limited to 298.15 K. The voltage of a cell is determined by E, not directly by .

Balancing Redox Reactions

Suppose that you want to design a battery and evaluate the voltage that would be generated. The conventional presentation of thermochemical information is organized in tables of half-cell reactions. After determining the two half-cell reactions and finding them in tables, the two half-cell reactions are balanced and then combined to determine the overall cell voltage. The steps to identify the half-reactions and reaction direction are: 1) identify the oxidized and reduced species using oxidation states (the procedure to determine oxidation states is summarized in Table 18.3)18; 2) break the reactions into two half-reactions; 3) for each half-reaction, balance the number of atoms for the species oxidized or reduced; 4) balance the change in oxidation state by adding the correct number of electrons to one side of each half-reaction; 5) balance oxygen by adding H2O to one side of each half-reaction; 6) balance the hydrogen by adding H+ to the appropriate side (the total charge on both sides of the reactions should now be the same.); 7) look up the reduction potential for each reaction and write the reaction with the smaller reduction potential as the oxidation reaction; 8) multiply the reactions by the smallest integers such that when they are added the electrons cancel; and 9) determine the Gibbs energy and equilibrium condition from Eqn. 18.85 or 18.86 for nonstandard conditions. Note that if the reaction is under basic conditions, it may be more appropriate to work with the basic form. Because OH and H2O both involve H and O, it is easiest to balance with H+ and then use the water dissociation reaction to convert the reaction as shown in Example 18.9.

Table 18.3. Procedure to Determine Oxidation States

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Image Steps to determine standard cell potential and direction of reaction.


Including Nonredox Reactions

Frequently, it is necessary to combine redox reactions with reactions that do not involve oxidation and reduction, such as a dissociation or solubility, or use the dissociation reaction of water to convert the acid form (reactions using H+) to a basic form (reactions using OH). This is rigorously done using the Gibbs energy of formation. To combine the reactions: (1) write all the individual reactions balanced so that they add to give the overall balanced reaction as explained above and include the desired nonredox reaction; (2) determine the Gibbs energies for the constituent balanced reactions; (3) add the reactions and Gibbs energies together and then divide the overall Gibbs energy by F and ne using the Nernst equation to find E°. The concepts for combining reactions are shown in Example 18.9 where acidic half-cell reactions are transformed to basic reactions.


Example 18.9. Alkaline dry-cell battery

Consumer portable electronics are commonly powered by ‘alkaline’ dry-cell batteries. These cells use an alkaline paste instead of an aqueous solution. The moisture content is low to minimize leakage, and the alkaline solution is used instead of acid because the degradation of the electrodes is slower in alkali compared to acid. The relevant species are Zn(s), ZnO(s), γ-MnO2(s), and α-MnOOH(s). A new battery has Zn(s) and γ-MnO2(s) electrodes.

a. Determine the balanced reactions for H+ and then transform them to use OH. Then provide the balanced overall reaction. (b) Determine the voltage generated by the cell when [OH] = 1 m and [OH] = 1.1 m, and the Gibbs energy of reaction.

Solution

The oxidation states of Zn are 0 for Zn(s) and +2 for ZnO(s); of Mn are +4 for MnO2(s) and +3 for α-MnOOH(s). Since the initial electrode is Zn(s) and γ-MnO2(s), Zn is being oxidized (losing electrons) and Mn is being reduced during battery use.

a. For Mn, the half-cell reduction reaction is found to be γ-MnO2(s) + H+ + e Image α-MnOOH(s), through the following procedure. Start with the Mn species (MnO2 and MnOOH) on each side of the reaction (more reduced on the right). The reduction requires one electron to go from +4 to +3, so one electron is added to the left. At this point, the O is already balanced, and one H+ is added to the left to balance hydrogen. The total charge is 0 on each side of the reaction. To convert to the base form, we add H2O Image H+ + OH, giving γ-MnO2(s) + H2O + e Image α-MnOOH(s) + OH and the total charge is –1 on each side of the reaction.

For the other electrode, the half-cell reduction reaction is found to be ZnO(s) + 2H+ + 2e Image Zn(s) + H2O through the following procedure. After writing the Zn species on each side (more reduced on the right), we note that the reaction requires two electrons and add them to the left, water is added on the right side to balance oxygen, then 2H+ are added to the left side to balance H. The total charge is 0 on each side. To convert to the base form, we add 2H2O Image 2H+ + 2OH, giving ZnO(s) + H2O + 2e Image Zn(s) + 2OH.

For the overall reaction, to balance electrons, two Mn must be reduced for each Zn oxidized. Combining, Zn(s) + 2γ-MnO2(s) + H2O Image ZnO(s) + 2α-MnOOH(s).

b. The voltage is found by taking the difference in reduction potentials found in Appendix E. The standard potential is found by the differences in reduction potentials, E° = 0.3 – (–1.26) = 1.56 V. The potential under operating conditions is given by

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Since all the species except for H2O are solids, they exist in the pure state as a first approximation. (In actual practice the MnOOH forms a solid solution with MnO2, but we ignore the effect here.) The activity of water is near 1 in the paste and [OH] does not appear, and thus it has no effect on the equilibrium voltage. Therefore, the battery should give a constant 1.56 V throughout its life.

Note that we are neglecting transport effects and the solid solution behavior. Thus, the actual voltage drops as the battery dies owing in part to these effects. The Gibbs energy of reaction is ΔG = –neFE = –2(96485)1.56 = –301 kJ/mol, a spontaneous reaction when the circuit is closed.


Fuel Cells

Fuel cells offer many potential advantages for energy usage. They are similar to a battery in that they involve oxidation at the anode and reduction at the cathode. Like the battery discussed above, they also involve transport of molecular cations between the electrodes. The primary difference is that the oxidizing and reducing species are considered to be “fuels” that either flow past the electrodes, or are fuels that can be replenished.

Fuel cell technology is in a state of rapid change. Typical issues revolve around the economical choice of fuels and the longevity of fuel cell devices. Nevertheless, the promise of converting chemical energy into electrical energy without the limitations of the Carnot cycle is a significant motivation. A biological fuel cell is considered in Example 18.11. The status of this topic is addressed in an online supplement with particular emphasis on the thermodynamic aspects of this technology.

18.12. Biological Reactions

Oxidation States and Degree of Reduction

Oxidation states, introduced in Section 18.11, provide an important balance condition for any chemical process, but particularly for biochemical reactions and fermentations. Recall that glucose oxidation to CO2 and H2O is an important energy-generating reaction in eurakyrotic cells to permit synthesis reactions. The oxidation of glucose or other foodstuffs provides electrons for reducing other species. For biological reaction networks, an electron balance can provide critical analysis of feasible products. CO2, H2O, N2, and O2 in any mixture cannot sustain biological life in the absence of other energy inputs. Therefore, such a mixture constitutes a useful reference point for a scale known as the degree of reduction.19 The degree of reduction provides a means to compare the overall electrons in a substance and the energy that can be gained by metabolically converting them to a mixture of CO2 and H2O. Combustion of a carbon-containing substance with the generic formula CfHaOb follows the balance,

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It is easy to show using stoichiometry that r = f + a/4 – b/2. The oxidation state of oxygen in O2 is 0, and in products is –2. Thus, four electrons are transferred to oxygen atoms for each mole of O2. The moles of electrons transferred to oxygen from the carbon compound are thus 4r = 4f + a – 2b, where the degree of reduction multipliers are +4 for C, +1 for H, and –2 for O. Nitrogen, sulphur, and phosphorous are often supplied to fermentations, and the reduction multipliers are selected such that reduction numbers are zero in the supply.20 When the N supply is ammonia, the multiplier for N is given a value of –3. For H2SO4 as the source, the multiplier for S is +6, and for phosphoric acid as a source, the multiplier for P is +5. To apply an electron balance, the reduction multipliers are used, not the oxidation states. Consider reaction of acetaldehyde. For acetaldyhde (C2H4O) the reduction calculation is 2(+4) + 4(+1) + 1(–2) = 10; for O2 the calculation is r·2·(–2) = (5/2)·2·(–2) = –10; for a net of 0 on the left-side. For CO2, 1(+4) + 2(–2) = 0, for H2O, 2(+2) + 1(–2) = 0, and the right-side is also 0. Though each side is not always zero, the two sides will balance.

For carbon-containing compounds the degree of reduction, γred, is often expressed per mole of carbon, (known as a basis of C-moles). For CfHaObNcSdPe,

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Thus for acetaldehyde above, γred = 10/2 = 5 per C-mole. Glucose (C6H12O6), has a degree of reduction of (4(6) + 12 – 2(6))/6 = 4 per C-mole. Hexane (C6H14) is a more highly reduced species, with a degree of reduction of (4(6) + 14)/6 = 6.33. For molecules containing multiple atoms of carbon, the degree of reduction can expressed in terms of moles or C-moles. For example, 180 g of glucose (Mw = 180), can be described as 1 mole of C6H12O6, or as six C-moles of CH2O, (Mw = 30). An average elemental formula for cell mass is CH1.8O0.5N0.2, with a degree of reduction of 4.2 per C-mole, slightly higher than glucose. For compounds not containing carbon, the degree of reduction is expressed per mole of that compound. For the compound HaObNcSdPe,

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A fermentation can be represented with a pseudo-reaction, balancing inputs and outputs. For example, on the basis of one C-mole of substrate CHaObNcSd,

CHaObNcSd + Yo(O2) + Yn(NH3) + Yaux(CHe OfNgSh) → Ybiomass(CHiOjNkSl) + Yproduct(CHm OnNpSq) + YCO2(CO2) + Yw(H2O) + YS(H2SO4)

where the Y values on the left are for the nutrients and on the right are for the products and by-products. The number of moles for each species is the value of the corresponding coefficient Y.

The number of electrons must balance for reactants and products using the degree of reduction relative scale. An electron balance is a useful method for performing mass balances on fermentation processes. The number of electrons in a feed or product is simply ∑(C-moles or Y)iγi. Thus, you can see that if you envision a biological process converting a mole of glucose to a mole of hexane, the fermentation needs an additional source of electrons to perform the reduction and the required C-mols of the substance supplying the electrons can be calculated. Some fermentations “fix” CO2, such as the succinic acid (C4H6O4) fermentation. CO2 and water have a degree of reduction of 0, and succinic acid has a degree of reduction of 3.5, so 3.5 electrons must be furnished from some other source for each C-mole of succinic acid produced (14 electrons per mole of succinic acid). Roels19 has presented a simple approximate correlation between the degree of reduction and availability and degree of reduction and heat of combustion. Applying the availability concepts from Section 4.12 (with different notation), Roels analyzes heat production and irreversibilities in aerobic and anaerobic fermentations. Grethlein, et al. used electron balances to determine CO and CO2 utilization in syngas (mixtures of H2, CO, CO2) fermentations to produce methanol.21 Shuler and Kargi show that the combination of the electron balance and elemental balances together can be used to determine the fraction of product, biomass by-product, and fraction lost to CO2.22

Binding Polynomials

To treat driving forces for reactions such as electron transfers and chemical equilibria, transformed Gibbs energies of formation are used along with their related apparent equilibrium constants. To perform the transformation, we use a binding polynomial when a species can exist in several bound states. Here we discuss binding polynomials that are helpful for relating the apparent molar concentration to the concentrations of individual species.

When we discussed H3PO4 in Section 18.9, we developed a recurring relation for the dissociation in Eqn. 18.50. In that section, we considered H3PO4 to be the “parent” molecule that lost successive hydrogens with each dissociation. However, an alternative perspective is to consider PO43– to be a binding receptor for H+ “ligands.” If we consider the addition of H+ to be successive binding reactions, the first binding constant is the reciprocal of the last dissociation constant, Image, and other binding/dissociations can be similarly related. From this perspective, the binding receptor, PO43– is the species of interest. A total balance on the species from this perspective replaces Eqn. 18.50 with the equivalent relation (left as a homework problem),

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Either of the arguments in parentheses is called a binding polynomial, Pbind. Many successive binding events can be represented by this recursion pattern with either the dissociation constants or the binding constants. The concept illustrated here for three protons as ligands can be generalized to other binding receptors and ligands. Each term in the binding polynomial is proportional to the concentration of a bound species, and the sum represents all possibilities. The fraction of the binding receptor in a given state is (in terms of the binding reaction constant),

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where we have generalized to an arbitrary ligand concentration [x], t is the maximum number of ligands, and we have generalized the bonding constant, (K0 = 1). For our phosphoric acid example, Image (recall for our example that the first binding is related to the third dissociation). In the analogy here, the H+ serves as the ligand. A key quantity in comparing models of binding to experiments is known as the average number of ligands bound per receptor as a function of ligand concentration. Using binding constants, the average ligands per receptor are calculated by the sum of the number of ligands multiplied by the fraction given by Eqn. 18.91,

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An equivalent expression can be obtained using Pbind in terms of the dissociation constants from Eqn. 18.90.23 If the binding constants (or dissociation constants, which are the reciprocal of each) are known, then the average binding number can be found as a function of ligand concentration. Note that the average binding number does not depend on the receptor concentration. The binding polynomials are used in transforming the individual Gibbs energies to the apparent Gibbs energy of formation for a family of receptors as we show later.

Energy Carriers in Biological Systems

When introducing biological reactions in Section 3.7, we mentioned the use of carbohydrates, fats, and proteins as food sources. We know that sugars are not “burned” using a single step in the human body. The human body could not survive the adiabatic temperatures of a single-step oxidation. However, biological systems oxidize sugars to CO2 and water. The reactions that disassemble these foods or other energy storage molecules are termed catabolic reactions. Reactions that build new structures are termed anabolic reactions. Biological systems have a clever way of carrying out the energy transformations. The body carries out the oxidation in small steps, using enzymes to pair endergonic steps with highly exergonic steps. Biological systems transfer and store energy by either 1) forming and breaking bonds; 2) performing redox reactions using electron carriers. A main carrier of energy captured by forming bonds is a molecule called adenosine triphosphate, or ATP as shown in Fig. 18.12. In the process of glycolysis, two ATP molecules are used to transfer two phosphates to glucose (a 6-carbon sugar), modifying it to facilitate subsequent isomerizations and production of two molecules of glyceraldehyde 3-phosphate. Then, the two aldehydes are oxidized to a phosphorylated carboxylic acid in coupled reactions, reducing nicotinamide adenine dinucleotide,24 NAD+ to NADH (see Fig. 18.12), storing electrons for other reactions. The two ATP molecules which earlier became ADP are regenerated. In the final step of glycolysis, pyruvate is produced, along with transforming two additional molecules of ADP to ATP, storing additional energy. CO2 is produced in subsequent reactions when the pyruvate is decarboxylated. Another biological electron carrier is flavin adenine dinucleotide, FAD (oxidized), which is reduced to FADH2 in other reactions. Similar to the NAD reduction that occurs on the nicotinamide, the riboflavin moiety is reduced, in this case with two hydrogens. The biological networks are quite complex, and this text is not intended to serve as an introduction to biological networks. The goal of the next few subsections is to explain the methods used for calculation of thermodynamic driving forces and equilibrium constants in biochemical reactions.

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Figure 18.12. (a) Structure of adenosine triphosphate in the form stable at high pH, ATP4–. Adenosine diphosphate (ADP) has two phosphates; adenosine monophosphate (AMP) has one; adenosine has none. (b)Nicotinamide adenine dinucleotide phosphate, NADP+. Nicotinamide also shown reduced, as in NADPH or NADH.

Adenosine triphosphate (ATP), diphosphate (ADP) and monophosphate (AMP) shown in Fig. 18.12 are primary carriers of energy in eukaryotic biological systems, and the distribution of phosphate species is important to represent. ATP can bind up to five protons at low pH (the four on the phosphates plus one on the NH2). As the pH is lowered, the average number of bindings will undergo a continuous increase until all “receptor” sites are filled at low pH. The individual species are denoted ATP4–, HATP3–, H2ATP2–, H3ATP, H4ATP, and H5ATP+. ATP can also bind other cations, one of principle importance being Mg2+, and the relevant species are MgATP2–, MgHATP, and Mg2ATP. Similar to the situation discussed above with phosphoric acid, each ATP dissociation has a known dissociation constant. The notation [ATP] will be used to represent the apparent concentration of ATP in all forms. The distribution of phosphate species is also important as discussed here because phosphates are transferred to/from molecules during many of the biological cycles. The pH and pMg are natural variables for determining the distribution of species using Pbind and Eqn. 18.91.

Biological Standard State and Apparent Equilibrium Constants

Biologists work on the molar concentration scale with standard state properties at 298.15 K, 1 bar and a standard concentration of 1 M, except water is kept on the Lewis-Randall scale, analogous to the molal treatment. The 1 M standard state is awkward in biological systems because the standard state of [H+] is a 1 M solution with a pH near 0. Gibbs energy changes based on such a standard state requires a large correction to physiological pH. However, it is convenient to transform the Gibbs energy such that pH and/or pMg may be held constant. Until this point in the text, we have utilized Gibbs energy for analyzing chemical and phase equilibria because it is minimized at constant T and P. If a system were at constant T and V, then the Helmholtz energy would be the correct property minimized, and if at constant S and V, then U would be minimized (note the relation between the natural variables and the minimized property). Biological systems are pH buffered, and when a biological reaction occurs, it occurs at a constant T, P, and pH, and often other ion concentrations are constant, such as Mg. The convention is to transform the Gibbs energy calculations to a pH (and ion concentration) of interest and use a potential that measures the driving forces at constant pH. The process of transformation is special for H+ “receptors” such as ATP4–, PO43–, and other species such as ADP and AMP which lead to a distribution of species. The collection of a given receptor populated with various numbers of ligands are known as a family of pseudoisomers. The transformed properties are denoted with ’. In a similar way, binding of Mg2+ is important for ATP, ADP, and AMP. An additional transformation can be made to provide Gibbs energies when pMg = –log[Mg2+] is held constant, denoted using the same ’.


Image Gibbs energies and redox potentials transformed to a specific pH and pMg are indicated using ’.


The transformed Gibbs energy has some interesting effects on the way that reactions are written and balanced. In this section, we present two main concepts: 1) balancing of reactions in the transformed framework; and 2) relationships between the apparent equilibrium constant and the equilibrium or nonequilibrium concentrations. In this section, we focus on applications where the apparent equilibrium constants are known or determined from apparent equilibrium concentrations. Details on the steps to calculate the apparent equilibrium constants from Gibbs energies of formation are provided in Section 18.17.

When the Gibbs energy is transformed for H, the pH of the solution is considered to be buffered and the surrounding solution is then a sink/reservoir for H+ ions. This means that when we write isolated chemical reactions, H is not conserved because the surrounding solution is a sink/reservoir. Therefore, a single reaction in this environment does not cause the pH to go up or down. When we write chemical reactions, we write them without balancing H or H+. Because we ignore a cation, H+, we also ignore the charge balance for chemical reactions. Analogous arguments apply if the transformation is done for Mg2+, and we ignore the balance on Mg.


Image For biological reactions using transformed Gibbs energies, the H, Mg balances, and charge balances are ignored.



Image Families of pseudoisomers are handled by using the apparent concentrations and apparent Gibbs energies.



Image Gibbs energies of formation and equilibrium constant for depend on pH, pMg, ionic strength, and T due to the conventions.



Image Transformed Gibbs energies in biology are rigorously related to concentrations.


Another convention of biological thermodynamics uses the apparent concentrations/Gibbs energies for families of pseudoisomers instead of tracking the individual species. This applies to species like phosphate and ATP. Looking at Eqn. 18.90, it is obvious that the distribution of phosphate species is completely determined by the buffered pH. Similar arguments apply to ATP, ADP, or other H+ receptors except that Mg2+ is simultaneously considered. The approach is to write equilibrium constants that use the apparent concentrations of H+ and Mg2+ receptors, and absorb the calculations of the distribution and electrolyte nonidealities into Gibbs energies of formation and the equilibrium constants. Details on the mathematics and thermodynamics are explained in Section 18.17, but the details are not important for applications. For applications, the important principle is to recognize that the Gibbs energies of formation and equilibrium constants change significantly with pH, pMg, ionic strength, and temperature. The quantities must be available or calculated at the specific conditions before the equilibrium calculations are performed. However, once they are available, they can be applied with easy hand calculations to determine driving forces or equilibrium conditions. The biological molar standard states and the transformations result in

Image

Consider the hydrolysis reaction of ATP to release a phosphate and produce ADP and phosphate,

Image

where the left-hand notation writes phosphate as phosphoric acid, and the right-hand notation writes phosphate as a generic Pi. Since ATP, ADP, and phosphoric acid are all distributions of receptor pseudoisomers, the right-side notation is more common in biological publications. The equilibrium constant, at a specified T, pH, and pMg would be written,

Image

Since the biological standard state uses molar concentrations, the transformed equilibrium constant does also. Note that water is included in the Gibbs energy calculation, but not in the equilibrium constant because the standard state for water is purity and the solution is nearly pure, even though it is transformed.


Example 18.10. ATP hydrolysis

a. Calculate the transformed standard state Gibbs energy of reaction and equilibrium constant Kc for hydrolysis of ATP at pHc = 7, pMg = 3, 298.15 K, ionic strength, I = 0.25 m, where the following data apply.

b. Show whether the reaction is endergonic or exergonic at the above conditions when the apparent concentrations area [ATP] = 0.00185 M, [ADP] = 0.0014 M, [Pi] = 0.001 M. If the reaction is exergonic, at what concentration of ADP does it reach equilibrium if the concentration of phosphate and ATP are constant?

a. In the human body, [ATP]/[ADP] ~ 10. Alberts, B.; Bray, D.; Hopkin, K.; Johnson, A.; Lewis, J.; Raff J.; Roberts, K.; Walter, P. Essential Cell Biology, 3rd ed., New York: NY, Garland Science, (2010), pg. 465.

Gibbs energies of formation at pHc = 7, pMg = 3, 298.15K, I = 0.25 mol/kg

Image

Solution

First, note that the Gibbs energy of water is different from the value in Appendix E because of the transformation. The transformed standard state Gibbs energy of reaction is –1426 – 1060 + 2298 + 156 = – 32 kJ/mol. The equilibrium constant will be Image.

b. The propensity for reaction at the given concentrations is Image

The reaction is even more strongly exergonic than the standard state. Equilibrium occurs when [ADP] = kc[ATP]/[Pi] = 5.3×108(0.00185)/0.001 = 9.8×108M. Of course, such a high concentration never happens, so the reaction is always favorable at reasonable concentrations. Instead of hydrolyzing ATP and “losing the energy,” the phosphate is transferred to glucose in a coupled reaction, the subject of a homework problem.



Example 18.11. Biological fuel cell

A biological fuel cell is a portable electrical source that can be refueled. Electrical current is generated by a biological redox couple. In an ideal fuel cell, the enzymes would be immobilized on the electrodes and maintain the same activity as if free. In the conceptualized fuel cell on the right, glucose is to be oxidized to gluconolactone in the right cell, catalyzed by immobilized glucose oxidase. Oxygen is excluded from the right cell to avoid loss of electrons by bulk oxidation. The left cell is saturated with air, and a reduction of O2 to H2O2 catalyzed by immobilized laccase is envisioned. Electrons are to flow through the external circuit and H+ is to flow through the membrane. Each side of the cell is buffered to pHc = 7, I = 0.25 M at T = 298.15 K. Suppose the concentrations on the right side are [glucose] = 0.1 M, [gluconolactone] = 0.05 M, and on the left side [H2O2] = 0.05 M. Determine the transformed standard state half-cell potentials and the voltage expected from the cell under stated concentrations. The standard state Gibbs energies of relevant species are shown below at the stated conditions.

Image
Image

Solution

Note that two hydrogen ions are generated by the oxidation and two are consumed by the reduction of O2. The two half-cell reactions are glucose Image gluconolactone + 2H+ + 2e, and O2(g) + 2H+ + 2e Image H2O2(aq). Note that we could use O2(aq) in the reaction, but that would require an extra calculation using Henry’s law. Since the solution is saturated, we may use the partial pressure in the gas phase where the standard state Gibbs energy is 0. The standard state reduction potential for the glucose reaction is ΔG′° = –427 + 496 = 69 kJ/mol, thus E′° = –ΔG′°/neF = –69000/2/96485 = –0.357 V.

For the oxygen reaction, the standard state reduction potential is ΔG′° = –52 = –52 kJ/mol (the Gibbs energy of formation for O2(g) is 0), thus E′° = –ΔG′°/neF = 52,000/2/96485 = 0.269 V. The potential expected from a standard state cell would be E′° = 0.269 + 0.357 = 0.626 V, which is favorable. Let us evaluate E under the proposed conditions. Using the Nernst equation,

Image

Thus, the cell is favorable. Note that other factors are important before the cell can be implemented, such as the rate that electrons can be produced, which requires preserving the activity (turnover number) of the enzymes. Immobilized enzymes have much slower kinetics compared with free enzymes. Many of the challenges have been summarized by Calabrese Barton, et. al.a


a. Calabrese Barton, S., Gallaway, J., Atanassov, P. 2004. Chem. Rev. 104:4867–4886.

18.13. Nonideal Electrolyte Solutions: Background

To this point in the chapter, we have considered solutions to be ideal or absorbed the nonidealities into the Gibbs energies or equilibrium constants. The representation of nonidealities is important for applications where the concentrations are above approximately 0.01 m. The literature through the 1970s has been largely developed by chemists and a variety of notations and models are used in the literature. More recently, chemical engineers have become actively involved in model development and applications. The remainder of this chapter limits the discussion to the extended Debye-Hückel model and its use as a starting point for more sophisticated models. There are common underlying themes in most of the literature. The use of a standard state in which the dilute activity coefficient goes to unity is common for electrolytes and often for molecular species, though the Lewis-Randall scale is used almost always for water. Due to the prevalent use of molality in literature, coverage of the chemical potential and activity coefficients on that scale is necessary.

As with nonelectrolytes, the chemical potential is the primary property that determines phase equilibria and is independent of the scale used to characterize the value. On the molality scale, the chemical potential of the electrolyte is written in a manner analogous to the Lewis-Randall rule on the mole fraction scale,

Image

where μio is replaced with Image and Image is the activity of the component.25 The details of Image are subtle as elaborated in Section 18.24. A key consideration is that molality has dimensions of mol/kg that are inconsistent with the (dimensionless) activity and activity coefficient. By defining the standard state at 1 molal, we are implicitly writing Image, but mo = 1 mol/kg. We defer the details because most problems are solved by incorporation of Image into the Gibbs energy of formation that is commonly tabulated. In other words, the infinite dilution chemical potential is subsumed into the Gibbs energy of formation and reflected in the computation of ΔGrxn that produces Ka. Then,

Image

As long as Image, ai = mi. This perspective highlights why the concept is unnecessary to solve problems using Ka at low concentrations as we have done early in the chapter. Because it is common to keep the Lewis-Randall standard state for water, reaction Eqn. 18.14 is written,

Image

Thus, the activity coefficients and Gibbs energies are related to the equilibrium constant that we have used earlier. Since a solution of pure H+ or H3O+ or OH cannot exist, we cannot easily use a pure state for either unless it is a hypothetical pure state. However, we can measure the behavior of ions in extremely dilute solutions by various means including spectroscopy and electrochemical cells. Even though measurements are taken at finite concentrations, the behavior can be extrapolated to infinite dilution. To illustrate, suppose the activity of H+ can be measured and plotted, as shown in Fig. 18.13. Then extrapolating the measurements to infinite dilution yields an ideal solution that is similar to Henry’s law. Similar to Henry’s law, the activity coefficient goes to unity at infinite dilution. At very dilute concentrations, the molality can usually be approximated by the molarity as we discussed above.

Image

Figure 18.13. Schematic representation of the activity of an ion at low concentration in a binary mixture. The standard states of ions for industrial calculations are typically 1 molal. For biological systems, they are usually 1 molar. The molality and molarity scales are used alternatively, not simultaneously. When using molal concentration, use molal activity. When using molar concentration, use the molar activity scale.

When first working with electrolytes, this standard state can be very confusing. Because the activity coefficient goes to unity at infinite dilution, a common misconception is that the standard state composition is infinitely dilute. However, the standard state composition is 1 molal with a slope taken such that the infinite dilution activity coefficient is unity. An ideal solution on the basis of the standard state follows the dashed line as shown in Fig. 18.13 and extrapolates to higher compositions. Any composition along the dashed line could be taken as the standard state and the activity coefficient would still be unity, but choosing 1 molal as the standard state permits us to (deceptively, but conveniently) drop mo numerically from many equations. Referring back to non-electrolyte systems, the Henry’s law standard state applies a similar concept, but extrapolates the ideal solution line to the fugacity of the hypothetical pure fluid. Like the discussion here, the slope of the Henry’s law ideal solution is selected such that the infinite dilution value also goes to unity. The difference is that the Henry’s law line is based on fugacity as the y-axis, whereas the electrolyte standard state is based on activity. Like Henry’s law and the Lewis-Randall rule, the activity coefficient quantifies the deviation from the ideal solution line. Readers should refer to Section 18.24 for further clarification of the relations between Henry’s law, the Lewis-Randall rule, and molality/molarity.

18.14. Overview of Model Development

The relation between chemical potentials of the ions and the molal activity coefficients is analogous to nonelectrolytes:

Image
Image

The activity coefficients of the ions are determined from the chemical potentials. The details of model development are beyond the intentions of the overview provided here. Briefly, the Gibbs energy of the solution is equal to the work performed in placing an ion in the solution of other ions. The work is determined from solving Poisson’s equation for electrostatics using various approximations for the charge density,

Image

where r is the radial position, Φ is the electric potential, ρ±(r) is the charge distribution as a function of radial distance, ε = εoD, where εo is the permittivity of a vacuum, and D is the dielectric constant. Different models result from different approximations to the charge distribution and the screening. The Extended Debye-Hückel model discussed next is one example of a solution.26 An excellent overview of the approximations and methods of solving the equation is available27 but is beyond the scope intended here. Briefly, once again, the Debye-Hückel approximation results from assuming that the charge distribution follows the low density radial distribution function form

g(r) ~ exp(–uCoul/kT)

The solvent is considered to be a continuum during the calculation represented by the dielectric constant. Various mathematical approximations are made to develop the solution, and different approximations lead to slightly different approximate formulations used in the literature. The Extended Debye-Hückel model is limited to dilute concentrations, generally below ionic strengths of 0.1 molal, and significant errors result from using the model outside this range. The excess Gibbs energy for the Extended Debye-Hückel theory is28

Image
Image
Image
Image

where e = 1.60218E–19 C, εo = 8.85419E–12 C2 N-1 m-2 is the permittivity of vacuum, εr is the dielectric constant or relative permittivity of the solvent, R is the gas constant in J/mol-K, T is the temperature in K, ρs is the density of the solvent in g/cm3, zi is the valence of the Coulombic charge on each ion type, and I is the ionic strength which characterizes the overall charges in the charge distribution. The parameter Aγ is not a Helmholtz energy. Dielectric constants for water as a function of temperature are provided in Appendix E. The parameter a represents the average distance of closest approach, which is larger than the ion size due to water hydration which is always present. The term 1/(BI½) is an approximate distance known as the screening, shielding, or Debye length. It represents the screening of the coulombic potential due to the presence of other ions. A common assumption is Ba = 1 (kg/mol)½, though in biological systems, Ba = 1.6 (kg/mol)½.

18.15. The Extended Debye-Hückel Activity Model

Activity Coefficients for Ions

The activity coefficients for ions are obtained by differentiating the excess Gibbs energy of Eqn. 18.102.29 The resultant formula is

Image

where the constants are defined in Eqns. 18.10318.105. The ionic strength is calculated based on the actual ion concentrations, which means that for weak electrolytes calculation of I must be repeated when the concentration of ions changes during iterations on concentration. The model predicts activity coefficients that are unity at infinite dilution of ions and decrease to a finite limit at high concentration. Experimentally, activity coefficients usually pass through a minimum at concentrations above 0.1 m, which is not captured by the model. Note that all species with the same charge will have the same activity coefficient values at a given ionic strength. More sophisticated models are available for higher concentrations as we discuss in Section 18.20.

Activity and Osmotic Coefficient for Water

The solvent activity coefficients from the Extended Debye-Hückel theory can be obtained by differentiation of the model for excess Gibbs energy. The result is

Image

Recall that the activity of the solvent is expressed on the Lewis-Randall standard state. The mole fractions are typically near unity, and thus many significant digits are required to characterize activity coefficients of solvent. Commonly the activity of the solvent is expressed in terms of the “practical” osmotic coefficient, Φ,

Image

In literature, activities of ions are often measured indirectly by measuring or controlling the partial pressure (isopiestic method) of water above the solution and then reporting the osmotic coefficient. The results are very sensitive to whether complete dissociation is assumed for activity calculation and in the summation in the denominator. Readers must pay careful attention to the assumptions applied in the experimental interpretation. The osmotic pressure can be converted to the ion activity using the Gibbs-Duhem equation to obtain the mean ionic activity coefficient described in Section 18.19. The osmotic coefficient approaches 1 at infinite dilution of ions. The osmotic coefficient is related to the osmotic pressure,

Image

Nonidealities for Nonelectrolyte Solutes

When nonelectrolytes exist in solution with electrolytes, such as with acetic acid, the undissociated acetic acid is typically treated with a molal standard state with the corresponding unity infinite dilution activity coefficient. To fit experimental data, the activity coefficient can be represented as Image where b is a constant fitted empirically. While this does not satisfy the Gibbs-Duhem equation, it is a common model.

18.16. Gibbs Energies for Electrolytes

In Chapter 17, we determined Ka from the Gibbs energy of formation. However, we also noted that occasionally results are summarized in terms of a temperature-dependent Ka. Treatment of the model using Ka requires less thermodynamic information, and it is quite common for electrolytes. Tabulating in terms of Gibbs energies requires consistency with a large database that is tedious to maintain. Such data bases are available for common ions only.

The Gibbs energy of reaction can be represented by

Image

Writing a completely general notation is difficult because different standard states are often used for different components. For example, we have already discussed using Image for cations and similar notation for anions. Writing a general sum is clumsy because of the different standard states used for components, so we leave the generic superscript ° and expect that readers apply the appropriate standard states.

Two key steps in understanding the tables for Gibbs energies are to consider the dissociation constant of water, and the selection of zero as the Gibbs energy of formation for H+. These steps and choices become clearer in upcoming descriptions. In Chapter 17, we introduced the use of Gibbs energy of the reaction as

Image

Consider again the dissociation constant for water, as shown in Eqn. 18.4. The equilibrium constant for this reaction at 298.15 K is well known as Ka = 10–14. The Gibbs energy of the reaction is thus:

Image

Now consider that the Gibbs energy of the reaction can be calculated by the Gibbs energies of formation at 298.15 K using the value from Eqn 18.112:

Image

Note that, like the chemical potentials, writing a general notation for Image is slightly imprecise and we use the default ° superscript, expecting readers to insert the ion standard state for ions. Note that the standard state Gibbs energy for pure water at 298.15 K has been inserted from the tables for nonelectrolytes. This detail makes an important connection with the standard tables for all other molecular components, and thus the values from the usual tables can be applied when compounds appear in reactions with electrolytes, as long as this convention is used.

Looking at Eqn. 18.113, two values are unknown, both Image and Image. The dilemma is resolved with the arbitrary choice at 298.15 K and 1 bar:

Image

This convention then determines the value

Image

With these values, the remainder of the tables can be developed. Other acid reactions involving H+ can then be characterized based on the degree of dissociation and the above standard selection of Image. For example, the value of Image can be determined by the dissociation behavior of HCl. Once the Gibbs energy of Cl can be determined the dissociation of NaCl will lead to the Gibbs energy of Na+. The remainder of the tables are developed using similar calculations.

Strategy for Using Gibbs Energies

There is important perspective regarding tabulation of standard state Gibbs energies. Have you considered how scientists created the tables for Image? Scientists used experimental equilibrium concentration measurements with models to calculate ai, and then inserted them into the equilibrium relation:

Image

Experiments were performed where Image (or Image) was known for all but one of the species. Then the value of Image for the species was determined from the experiment by difference. Calculations from multiple investigators using different reactions refined the values that we use from the tables today. When we solve an applied problem, we are using the equation in the opposite direction: looking up Image (and Image) and using models of ai to determine concentrations. Calculations are reliable as long as ai’s are calculated using methods consistent with the standard state Image and Image.

The steps to solving a problem usually involve reverting the procedures used to develop the tables. Tables are used to calculate Image (or Image) from standard state. Temperature and pressure corrections are applied to determine ΔGT, Image In K and then selecting methods to calculate ai consistent with the standard state. Concentrations are thus determined from the activity, often assuming ideal solutions. Frequently, the Gibbs energies of individual species are not calculated, or are not available. Rather, scientists report the values of Ka as used in Example 18.4. A reliable database of Gibbs energies is available as documented in footnote 13 of this chapter.

18.17. Transformed Biological Gibbs Energies and Apparent Equilibrium Constants

The transformed Gibbs energies in Section 18.12 are a convenient method to handle biological reactions but the details were not discussed earlier. The transformation of Gibbs energy to a field of buffered pH is analogous to the other Legendre transforms used previously. To obtain Gibbs energy from internal energy, starting from dU = TdSPdV, we introduced G = UTS + PV, resulting in a potential where T and P are the natural variables, dG = –SdT + VdP, and G is minimized when the natural variables are constrained. By introducing G’ = UTS + PV – NHμH+ we arrive at a potential that is a natural function of pH. The variable NH is the number of hydrogens in the molecule. To obtain a transform that is also a natural function of pMg = –log[Mg2+], we use G’ = UTS + PV – NHμH+ – NMgμMg2+. Standard-state transformed Gibbs energies of formation are developed for species, and they are used analogously to the untransformed Gibbs energies to calculate the Gibbs energy for reactions and the transformed equilibrium constant.

Also, to correct for solution nonidealities, the extended Debye-Hückel model is added, and a convention is to use Ba = 1.6 in the model. A further difference from previous models is that molar concentrations are used for the equilibrium constants, though it makes little difference numerically because solutions of biological molecules are typically dilute on a molar basis. The effect of nonidealities is typically calculated using the molal form of the Debye-Hückel model, using the overall solution molal ionic strength since the model does not differentiate between charged species.

Gibbs Energy Transformations for Species with a Single Form

For a species containing hydrogen or Mg,


Image Transformed Gibbs energy of formation at a specified pH and I. Workbook GprimeCalc.xlsx or MATLAB GprimeCalc.m are helpful.


Image

The Gibbs energy of formation appearing on the right side for j = i, H+, and Mg2+ is

Image

where Ba = 1.6 (kg/mol)½, pHc = –log[H+], pMg = –log[Mg2+], and I is measured in molality. By assuming that the heat of formation is independent of temperature in the small temperature range where biological reactions occur, the short-cut van’t Hoff equation can be applied before inserting the Gibbs energy of formation into

Image

The equations are presented in the reverse order compared to how they are used. Standard state values are inserted into Eqn. 18.119 for j = i, H+, and Mg2+. The results are inserted into Eqn. 18.118 for each, and then each of those results is inserted into Eqn. 18.117. Conversion of the Gibbs energy of formation is easy using the Excel workbook GprimeCalc.xlsx or MATLAB m-file GprimeCalc.m. Note that the solution nonidealities are a minor correction compared to the transformations on H+ and Mg2+. The ’ notation is used quite widely for this transformed Gibbs energy at any specified pH, though in some literature the transformation is restricted to pH 7. The context of applications must be studied to discern if Mg2+ is included.

Enthalpy Transformations for Species with a Single Form

The enthalpy is obtained via the Gibbs-Helmholtz relation, where the heat of formation at I = 0, pH = 0, and pMg = 0 is independent of temperature as a first approximation. The Debye-Hückel temperature dependence introduces a correction. The heat of formation is


Image Transformed enthalpy of formation at a specified pH and I. Workbook GprimeCalc.xlsx or MATLAB GprimeCalc.m are helpful.


Image

and for each enthalpy of formation on the right side, j = i, H+, and Mg2+:

Image

Gibbs Energy Transformations for Pseudoisomers

For families of receptors with different numbers of ligands, the transformed Gibbs energy of formation depends on the distribution of species, which changes with T, pHc, and pMg, but the dependence is easily represented by the binding polynomial. The Gibbs energy of formation for the apparent species takes a very simple modification relative to the Gibbs energy of formation of the completely bare receptor,

Image

where G′o f, T, i(1)(I, pHc, pMg) on the right is for the completely bare receptor as determined by Eqns. 18.11718.119. The binding polynomial can most easily be expressed in terms of the transformed Gibbs energies of formation of each pseudoisomer relative to the most bare receptor,

Image

where all Gibbs energies are for aqueous solutions using Eqns. 18.117 - 18.119 at a specified pHc, pMg, and ionic strength, but the designations have been omitted for brevity. Alternative forms of Eqn. 18.123 are sometimes written, but the given form is recommended to avoid exponentials of very large or very small numbers because the differences in Gibbs energies of formation are much smaller than the values.30,31 Note that the exponential terms in Eqn. 18.123 are equivalent to the products of the equilibrium constants as shown in Eqn. 18.90.

The pHc, pMg, and nonideal solution effects are already incorporated into the reactions, and thus the binding polynomial in terms of the transformed equilibrium constants for phosphate would give

Image

The fraction of receptor in each pseudoisomer form j in family i, can be calculated by two different methods. Pbind can be used directly, or we can use the transformed Gibbs energy of formation for the pseudoisomer with the transformed Gibbs energy of pseudoisomer form j,

Image

where all Gibbs energies are at the specified T, pH, and pMg. The argument of the first exp( ) is the same as used in 18.123, and the values of the first exp( ) are the same as the values for each term in Eqn. 18.124. The transformed enthalpy for the apparent species is most easily calculated by using the individual ri values where each pseudoisomer values is transformed by Eqn. 18.120,

Image

This section has been intended as an introduction to biological thermodynamics. Readers interested in more depth will find more details in the work of Alberty31,32,33 or Goldberg.34 While the transformation of the standard state potentials shifts the standard state values of Gibbs energies, such that, for a reaction, ΔGoT, i(I, pHc, pMg) ≠ ΔGoT, i(I, pHc, pMg), the actual driving force for a reaction at nonequilibrium conditions is the same,35 thus, ΔGT, i(I, pHc, pMg) = ΔGT, i(I, pHc, pMg) as in Eqn 18.93 and in the analogous untransformed relation. Also, this means that KcKc, but the reaction is at equilibrium at the same true concentrations regardless of the transformation. This relation again emphasizes that the true driving force for the reaction is not represented by the standard state values.


Example 18.12. Gibbs energy of formation for ATP

The Gibbs energies of formation of ATP species are available in Appendix E. Using the available Gibbs energies, calculate the apparent Gibbs energy of formation for ATP at T = 298.15 K, pHc = 7, pMgc = 3, I= 0.25 mol/kg, the pK’, and the percentage of each species present.

Solution

The Gibbs energies for the molal standard state are ATP4– = –2768.1, HATP3– = –2811.48, H2ATP2– = –2838.18, MgATP2– = –3258.68, MgHATP = –3287.5, and Mg2ATP = –3729.33 kJ/mol. The charges are –4, –3, –2, –2, –1, and 0, respectively. The number of H’s are 12, 13, 14, 12, 13, and 12, respectively. Inserting the values into GprimeCalc.xlsx, at the stated conditions, the transformed Gibbs energies of the species are ATP4– = –2291.9, HATP3– = –2288.8, H2ATP2– = –2270.7, MgATP2– = –2297.1, MgHATP = –2282.7, and Mg2ATP = –2288.8 kJ/mol. The rest of the problem must be solved with hand calculations.

The third dissociation reaction is HATP3– Image ATP4–,

Image

The product of the first and second is given by H2ATP2– Image ATP4–,

Image

The remaining terms for HnATP(–4+n) in the binding polynomial are not important at pH = 7 because they will be even smaller than the last term calculated since the pH is far above the pKa (review Example 18.6 on page 718 for an analogy with phosphoric acid). For the species involving Mg, defining KaMg for MgATP2– Image ATP4–,

Image

Defining KaMgH for MgHATP Image ATP4–,

Image

defining KaMg2 for Mg2ATP Image ATP4–,

Image

The binding polynomial is Pbind = 1 + 0.2863 + 2E–4 + 8.148 + 0.0244 + 0.2863 = 9.7452.

The fraction of each pseudoisomer is given by Eqn. 18.91:

Image
Image

The other species make up the remainder and are insignificant at pH = 7.

The Gibbs energy of formation for the apparent species is

Image

Note this is the value used in Example 18.10.


18.18. Coupled Multireaction and Phase Equilibria

Many texts are available to facilitate more advanced study. The following example uses thermochemical data from the OBIGT database documented in footnote 13 of this chapter. The example here is implement using the extended Debye-Hückel model and ignoring pressure corrections.

Chlorination of Water

Chlorination is one method of water treatment for drinking. When Cl2 dissolves in pure water, it undergoes reaction with water to simultaneously form the strong acid HCl and the weak acid hypochlorous acid (HClO). The reaction is sensitive to pH, and at low pH it is shifted toward molecular Cl2. At high pH, the reaction shifts to hypochlorous acid which is an oxidizer as well as a weak acid. Chlorine bleach is prepared by stabilizing the hypochlorous acid at high pH by reacting Cl2 with a solution of NaOH.

Consider the situation with pure water. There are three reactions to be taken into account:

Image
Image
Image

We can also write the liquid-vapor equilibria as “reactions,” noting the convention in the electrolyte literature is that the liquid phase is always the “product.” This is the convention used for Henry’s law constants (cf. Section 11.12) in Eqns. 18.130 and 18.131:

Image
Image

Example 18.13. Chlorine + water electrolyte solutions

Determine the concentration and species present when chlorine is in equilibrium with water at 298.15 K and 0.8 atm. Develop an approximate solution and then use extended Debye-Hückel with Ba = 1 (kg/mol)1/2. Thermodynamic properties from the OBIGT documented in footnote 13 of this chapter are tabulated in Table 18.4.

Table 18.4. Thermochemical Data for the Species

Image

Solution

We first work the problem assuming ideal solutions. This provides an approximate answer. Then we may use the activity coefficients to refine the answer. Using the Gibbs energies of formation, the equilibrium constants are: pKa1 = 3.339, pKa2 = 7.549, and KH(Cl2) = 0.0606, where KH is Henry’s constant for Cl2.

Since chlorine forms the strong acid HCl and the weak hypochlorous acid when dissolving in pure water, we expect pH < 7. Note that the weak hypochlorous acid should be almost totally protonated below pH = pK – 1 = 6.5. Since a strong acid HCl is being formed, this seems very likely. Let us proceed with that assumption. This enables us to disregard the dissociation of Eqn. 18.128 as a first approximation.

The three reaction equilibria are summarized in Eqns. 18.12718.129. The charge balance is

Image

where [ClO] is ignored because the dissociation of hypochlorous is small when the pH is small and [OH] is ignored when pH is small. Thus, the equilibria of Eqn. 18.127 can be approximated as

Image

Approximate Solution:

The partial pressure for water can be estimated by first assuming that the water is almost pure. This approximation can be refined later if we find significant concentrations of chlorine species. We also use molar concentrations to approximate molalities. Using Raoult’s law for water, Image. From the steam tables, Image bar = 0.0313 atm, and yH2O = 0.0313/0.8 = 0.039. Then yCl2 P = 0.8 – 0.0313 = 0.7687 atm, yCl2 = 1 – 0.039 = 0.961. Using Henry’s law coefficient (KH) for Cl2 at 298.15 K, the concentration of Cl2(aq) is (independent of pH):

Image

The concentration [HOCl(aq)] = 10–3.339(0.0484)/[H+]2 (Eqn. 18.133) at small values of pH is plotted in a Sillèn diagram. The weak acid dissociation of hypochlorous acid [OCl] is to be calculated from Eqn. 18.31 using the concentration of [HOCl] as a function of pH. As expected, the dissociation is small at low pH.

The weak acid curve in Fig. 18.14 is much different from curves in previous examples because, in this case, the overall concentration of weak acid is changing rapidly with pH. Now consider the material balance associated with Eqns. 18.127 and 18.128. Since Eqn. 18.128 does not occur to a significant extent, to a good approximation by the stoichiometry of Eqn. 18.127 [H+] = [Cl] = [HOCl]. This occurs at the intersection shown by the dotted lines. The approximate solution is pH = 1.55, [H+] = [Cl] = [HOCl] = 10–1.55 = 0.0282 mol/L. Note on the diagram that [OCl–] = 10–7.5 = 3.2E-8. Now, we can use these as initial guesses in a more rigorous answer.

Image

Figure 18.14. Determination of equilibria for the chlorine system.


Image CL2H2O.xlsx.


Calculation with Activity Coefficients:

Thermodynamic properties for the components are tabulated below and in the spreadsheet: CL2H2O.xlsx. Note that the data tabulated below include values for Cl2 and H2O in both the vapor and aqueous phases. The Gibbs energies are used to calculate the VLE distribution coefficients as a “reaction.”

To solve the nine equations (five equilibria, three atom balances, and one charge balance) simultaneously, we must identify nine unknowns. The nine unknowns selected here are the species listed in Table 18.4: the liquid moles of H2O, Cl2, HClO, H+, Cl, ClO, and OH, and the vapor moles of H2O(v) and Cl2(v). The basis is 1 liter of liquid water (niH2O = 55.51 moles) and niCl2 = 0.9 moles initially.

The detailed calculation are handled as a reactive flash. Three atom balances must be satisfied–H, O, Cl, along with the charge balance. The atom balances and charge balance are shown in Table 18.5 for the basis of 1 liter of liquid water and 0.9 moles of Cl2. The compositions for iteration are summarized in Table 18.6.

Table 18.5. Atom Balance and Electroneutrality Constraints

Image

Table 18.6. Electrolyte Component Mole Numbers and Activities at the Converged Composition

Image

The results from the approximate ideal solution calculation above are used as initial guesses. Excel Solver is used to adjust the moles of each species (in the second and sixth columns of Table 18.6) until all equations are simultaneously satisfied.

Note from Table 18.6 that the γ for all the ionic species is the same. This occurs because the Debye-Hückel model is too simple to make distinctions as long as all species have the same valence. The activity coefficients for Cl2 and HClO are assumed to be unity.

The calculations summarized in Table 18.6 show that the chlorine solubility is enhanced beyond what might be predicted from the Henry’s law constant alone due to formation of HClO and Cl in solution. The Cl in HClO and Cl together is about two-thirds of the Cl atoms in Cl2(aq) at P = 0.8 atm. Open the spreadsheet Solver to see how the constraints were implemented. Compare with the approximate answer to see that the approximate answer is pretty close. For example, [HClO] = 0.0312 versus 0.0282.


18.19. Mean Ionic Activity Coefficients

Mean ionic activity coefficients are often used for electrolytes modeling in the literature. The mean ionic activity coefficients provide an alternative method to express the activity of the apparent electrolyte species. This section provides the background to relate those activity coefficients to the ion activity coefficients. The chemical potential of the apparent electrolyte species is the same as the undissociated electrolyte species as shown in Supplement Section 18.23, however only the undissociated chemical potential is used here to keep the equations shorter.

Image

If we insert Eqns. 18.99 and 18.100,

Image

Looking at the last terms in parentheses, we can define,

Image

The molality and activity coefficients are averaged by taking the geometric mean. In the literature, these are commonly called the mean, and the clarification as the geometric mean is commonly omitted. The mean molality is

Image

where v ≡ (v+ + v). The mean ionic activity coefficient is found from the geometric mean of the ion activity coefficients:

Image

This results in the following relation for the chemical potential of the undissociated species

Image

Note that in the formula for chemical potential, the variable v appears before the ln term, unlike the similar expression for ions or nonelectrolyte species. Therefore, the mean ionic activity is not the same as the activity of the undissociated (or apparent) activity. There is no new information in these equations. They are simply an alternative method of expressing the activity coefficients, molalities, and chemical potentials. The mean molal activity coefficient is

Image

The final equation requires a proof that Image is related to Image. To see this, start with the charge balance, v+z+ + v_z_ = 0. Obtain one equation by multiplying by z+ and another by multiplying by z_. Add them and rearrange:

Image

The extended Debye-Hückel model is valid for 1-1 electrolytes up to about 0.1 molal, and to lower concentrations for species with multiple charges: The extended Debye-Hückel is compared with experimental mean ionic activity coefficients for NaCl in Fig. 18.15. Note that the experimental activity coefficients for NaCl are nearly 1 near 6 m. This happens to be about the solubility limit at room temperature, but the solution is quite nonideal at lower concentrations.

Image

Figure 18.15. Mean ionic activity coefficient for NaCl, KCl and those predicted by the extended Debye-Hückel (DH) model at 298K. Dashed line is the extended Debye-Hückel, Solid lines are the unsymmetric eNRTL using default parameters in ASPEN Plus ver. 7.1. Experimental activity coefficients are from Hamer, W.J.; Wu, Y.-C., 1972. J. Phys. Chem. Ref. Data, 1:1047.

The osmotic pressure can be manipulated using the Gibbs-Duhem equation to obtain the mean ionic activity coefficient. The derivation is beyond the scope intended here, but the equation is

Image

The use of the square root is a necessary mathematical manipulation. The integral may be done numerically. One difficulty is that the experimental data must extend to low concentrations.

18.20. Extending Activity Calculations to High Concentrations

This chapter has served as an introduction to electrolyte models, and the extended Debye-Hückel leaves much to be desired in its limitations to concentrations lower than 0.1 m. However, the model has been used as an introduction, and those who work with electrolytes can find more models in the literature. In the older literature, the model was primarily improved by making modifications to the Debye-Hückel approximations. For example, Bromley and Davies add to the activity coefficient a term CI, where C is a parameter and I is ionic strength. Fig. 18.15 illustrates that the parameter C must be system-dependent to represent the data. One suggestion is that the ionic strength modifies the dielectric constant of the medium. Others propose that the ions begin to interact with each other in a way that the Debye-Hückel model cannot capture. Molecular simulations are relatively complicated in the presence of long-range electrostatic interactions, delaying the conclusive resolution of such arguments. Since the mid-1980s significant success has been achieved by combining various versions of the Debye-Hückel model with activity models such as NRTL or UNIQUAC.36 The Debye-Hückel model is considered to represent the “long range” electrostatic interactions, and the conventional activity models are considered to represent the “short range” physical interactions. Often, the short-range model parameters are lumped to minimize the number of parameters to be adjusted. Plotted in Fig. 18.15 are the activity coefficients calculated with ASPEN Plus using the unsymmetric electrolyte-NRTL (eNRTL) model. The ASPEN electrolyte wizard was used to set up the dissociations and pull parameters from the database. Owing to the importance of electrolytes in industrial processes and corrosion management, and the complexities of correct modeling, companies such as OLI Systems, Inc., specialize in electrolyte modeling.

18.21. Summary

This chapter began with a review of acid-base behavior to stress the importance of pH on equilibrium. Compounds are in the acid form below the pKa,A and in the base form above. Techniques including Sillèn’s graphical method were provided to determine solution pH values and species distributions at various concentrations. We explained the origin of charges on biological molecules and why the charges change with pH, as well as the concept of zwitterions. Applications such as solubility, osmotic coefficients, and isoelectric point were developed.

Concepts of redox reactions were developed, relating the voltage to the Gibbs energy of reaction. Procedures were given to determine oxidation states, degree of reduction for molecules, and voltages in cells. The concept of oxidation and reduction in biological systems was introduced in the context of a biological fuel cell.

Binding polynomials were introduced as a method of representing simultaneous equilibria of families of pseudoisomers. The concepts of transformed Gibbs energies were introduced for biological systems buffered in pH or pMg.

Then solution nonidealities were introduced using the extended Debye-Hückel model. We finished the chapter by providing an example calculation of ATP distribution in nonideal solutions, and an example that couples phase equilibria with electrolyte equilibria. The later example demonstrates that for dilute solutions the graphical technique and simple arguments comparing the pH with the pKa,A values provides rapid estimates that are valuable for converging to more precise values. Because pH and species concentrations vary over many orders of magnitude, the approximate methods are important to use first, and in many cases they are adequate for approximate engineering work. Ultimately, activity coefficient modeling is important for accurate calculations. The chapter concludes with some supplemental sections that are extremely valuable for conversions of the units used in the literature.

Important Equations

The most important equations of this chapter are the material balance, reaction equilibria, and electroneutrality relations. Unfortunately, these are different for every electrolyte system, so there is not much point in listing them the way we usually do in the chapter summary. A key step in using Sillèn’s method is to use the acid form of dissociations for weak electrolytes,

Image
Image

A new equation is the electroneutrality constraint. It is not a surprising equation, but it can cause difficulty because some species may be present in very small quantities that make very big differences–pH, for instance. Solving for these small quantities often requires rearranging the equations into a proton condition to avoid the precision problems that come with adding small numbers to large numbers.

In Section 18.11, the Nernst equation is important to relate voltage to standard potentials and actual concentrations, and the number of electrons transferred:

Image

In Section 18.12 the Gibbs energy was transformed to use apparent concentrations. All the pH, pMg, and solution nonidealities were transferred to the Gibbs energies of reaction and the equilibrium constants:

Image

Besides the primary equations, the extended Debye-Hückel equation (Eqn. 18.107) is introduced in this chapter to account for nonideal behavior of ionic species. It is best limited to concentrations of 0.1 molal or less, but it conveys the concept that electrolyte solutions may deviate from ideality just as nonelectrolytes do. Going beyond Eqn. 18.107 would generally involve developing expertise beyond the introductory level.

Although the notation and reference states are obscure and frustrating, the implications are impressive. Salting in and salting out, protonation versus pH, osmosis, buffering, and leveling are just a few examples of implications that play significant roles in commonly encountered chemical systems, especially biological systems and corrosive environments. All the new jargon may seem overwhelming at first, but it can be assimilated if you only remember the three P’s: practice, practice, practice.

18.22. Supplement 1: Interconversion of Concentration Scales

Throughout this chapter, subscript s indicates solvent, and Mw,i represents molecular wt in (g/mol).

Image
Image
Image

relation of mole fraction to molality:

Image

The relation for molality leads to another commonly used substitution for xs,

Image

18.23. Supplement 2: Relation of Apparent Chemical Potential to Species Potentials

To understand the origin of the models for the mean activity coefficient, some discussion of the chemical potentials is necessary. Many of the concepts are extensions of the methods used for reaction engineering. For example, consider the general case of an electrolyte dissociating in solvent. When an electrolyte dissociates, the material balance gives the molality of positive and negative charges in solution,

Image
Image

where the subscript u represents the amount of electrolyte that is un-ionized. The change in Gibbs energy at fixed temperature and pressure is given by the changes in composition and the chemical potentials,

Image

which rearranges to

Image

The quantities μ+ and μ are analogous to the partial molar Gibbs energies of other components and the subscript w represents water or solvent. Since the positive and negative charges cannot change independently in a physical mixture as required by the rigorous definition of the partial molar quantity where all but one species is constrained when the derivative is evaluated, they are nonphysical, but they can be calculated by theory. Recall that the quantity dni relates to changes in the apparent composition. As a solution of fixed apparent concentration equilibrates (e.g., dni = 0), the un-ionized concentration changes until Gibbs energy is minimized and dG = 0. Thus, we conclude that at the equilibrium ionization the first term in parentheses is zero. Thus,

Image

This relationship between the “products” and “reactants” of the ionization is analogous to the relations developed for molecular reacting systems. This equality can be inserted into the second term, at equilibrium ionization resulting in

Image

On an apparent basis, we also must satisfy

Image

where μi is the apparent chemical potential. By comparison of the last two equations we conclude that the apparent chemical potential must equal the chemical potential of the un-ionized species, which also must equal the weighted sum of the chemical potentials of the ions:

Image

Thus, the approach for developing a model for the apparent chemical potential and apparent activity coefficient is based on developing models for the ions and then using the weighted sum.

18.24. Supplement 3: Standard States

An important principle of the following discussion is that the chemical potential should be a property of the state of the system. All models should result in an identical value for the chemical potential at the same state. The standard state provides a convenient reference condition, but is slightly different from a reference state because it is at the same temperature as the system.37

The typical convention for nonelectrolytes uses mole fractions and the Lewis-Randall standard state Image:

Image

An alternative convention is related to the Henry’s law standard state Image and the activity coefficient is known as the rational activity coefficient.38

Image

The activity coefficients on the two scales are related.

Image

Inserting the activity coefficient relation into Eqn. 18.158 results in

Image

Consider Eqn. 18.158. Inserting Eqn. 18.147 to replace the mole fraction,

Image

Because molality is not dimensionless, but activity is dimensionless, we must introduce some manipulations. We wish to introduce a molal activity coefficient, Image, to use with molal concentrations. The convention is to set the standard state as a hypothetical ideal solution, Image, at unit molality, mo = 1 mol/(kg solvent). Introducing the standard state concentration (twice):

Image

The unit value of mo is traditionally omitted and thus “transparent.” We define the molal activity coefficient:

Image

Note that Eqn. 18.147 can be reinserted into Eqn. 18.163 to eliminate xs, if desired. Inserting Eqn. 18.163 into Eqn. 18.162,

Image

We can see that the standard state reference potential must be given by the first two terms on the right-hand side of the equation:

Image

Substituting Eqn. 18.165 into Eqn. 18.164 results in the molal standard state and molal activity coefficient:

Image

The activities corresponding to the standard states are thus

Image
Image
Image

where the activities from the different scales are not equal at a given concentration because of the difference in standard states. Combining Eqns. 18.162 and 18.147,

Image

Finally, we note that the value of mo is dropped from all the final expressions in application, based on the assertion that its value is 1 molal by the definition of the standard state. This is the basis for the equations presented in Sections 18.4 and 18.13.

18.25. Supplement 4: Conversion of Equilibrium Constants

Equilibrium constants in electrolyte literature are often presented on the molal scale. For clarity in this section, we will use Ka,m to denote the molality equilibrium constant and Ka to denote the rational (Henry’s law scale) using mole fractions. Recall that the solvent (usually water) is on the Lewis-Randall scale. Using the molality scale for the electrolytes,

Image

On the rational (Henry’s law) mole fraction scale, we have

Image

To convert, consider the ln of each equation and take the difference. Inserting Eqn. 18.170,

Image

For a 1-1 electrolyte such as NaCl, Image, thus Image. If a Ka is desired on the Lewis-Randall scale, similar conversions can be done using infinite dilution activity coefficients.

18.26. Practice Problems

P18.1.

a. Compute the freezing point depression for an aqueous solutions that is 3 wt% NaCl.

b. Compute the boiling point elevation for an aqueous solutions that is 3 wt% NaCl.

c. Compute the osmotic pressure for an aqueous solutions that is 3 wt% NaCl.

18.27. Homework Problems

18.1. Calcium chloride is used occasionally as an alternative to sodium chloride for de-icing walkways. It is rumored to maintain puddles even a day or so after all evidence of sodium chloride has disappeared.

a. Compute the freezing point depression for aqueous solutions that are 5 wt% CaCl2 and NaCl.

b. Compute the boiling point elevation for aqueous solutions that are 5 wt% CaCl2 and NaCl.

c. Compute the osmotic pressure for aqueous solutions that are 5 wt% CaCl2 and NaCl.

18.2. Ammonia is a weak base, as indicated by the pKa,A and pKa,B values in Table 18.2. Determine the percentage of NH3 dissociated at pH 7 and pH 1.5 when the apparent amount of NH3 in aqueous solution is 0.15 m. Assume ideal solutions.

18.3. Sodium fluoride, NaF, is dissolved in water at an apparent concentration of CB = 10-3 mol/L. Construct a Sillèn diagram and estimate the pH. Refer to the pKa,A and pKa,B values in Table 18.2.

18.4. A solution of NaHCO3 and HCl is prepared such that the total carbon concentration is 1E-3 M and the total Cl concentration is 2E-3 M. Calculate the pH and concentrations of species present. Assume that the pressure is sufficiently that any evolved CO2 remains in solution. Estimate the partial pressure of the CO2 by

a. using Henry’s Law.

a. assuming the MAB model.

18.5. Plot the “apparent molality” of Cl2 in solution against the partial pressure of Cl2. The apparent molality is the sum of all Cl species in solution (Cl2 counts twice) divided by 2 (to put it on a Cl2 basis). Compare your plot to the experimental data of Whitney and Vivian (1941).39

18.6. Model a soft drink as a solution of water with CO2 dissolved at 298.15 K. In this way we ignore the sugar, flavor, and color. The Henry’s law constant for CO2 at 298.15 K is 0.035 (mol/kg-bar).

a. What pH and composition exist when the vapor phase is 3.5 bar absolute at room temperature ignoring O2 or N2 present? This approximates conditions in the unopened container.

b. After the soft drink is opened, and the liquid equilibrates with atmosphere, what pH and composition exist when the CO2 vapor mole fraction is yCO2 = 0.0003 (the normal ambient value) and the pressure is 1 bar?

18.7. Sodium bicarbonate, NaHCO3, commonly known as baking soda, is dissolved in water at 10–2 m at 298.15 K. Assume ideal solutions.

a. Determine the pH and the dominant species concentrations. For this part of the problem, ignore the potential loss of CO2 escaping from the solution as vapor.

b. Now evaluate whether CO2 may have a propensity to come out of solution at the conditions determined in (a) at 1 bar total pressure. yCO2 = 0.0003 is the normal ambient value.

18.8. Sodium carbonate is mixed into a solution of acetic acid and the container is rapidly closed before the container components react. The amount of sodium carbonate is such that the total sodium concentration is 1E-2 m and the total acetate concentration is also 1E-2 m. When the mixture equilibrates, the partial pressure of CO2 over the solution is measured to be 0.5 bar. Determine the pH and concentrations of the acetate species. The Henry’s law constant for CO2 at 298.15 K is 0.035 (mol/kg-bar).

18.9. Thermodynamic data for Gibbs energy of formation is shown below (kJ/mol for molal standard states) at 298.15 K. A saturated solution of NaCl is approximately an ideal solution.

Image

a. Use the Gibbs energy of formation to determine the solubility of NaCl in molality at 298.15 K. Treat the equilibrium between the solids and ions as a Ksp.

b. Determine the NaCl solubility (molal) when the concentration of KCl is 1 m.

c. Prove that the solution in part (b) is not saturated with KCl. (An ideal solution is actually a poor approximation for a saturated solution of KCl, but provide the proof based on an ideal solution.)

18.10. Suppose 0.1 mol of CO2 were mixed with 0.9 mol of Cl2 and 1 liter of water. What would be the concentrations of the aqueous species and the mole fractions in the vapor phase at 0.8 atm in that case?

18.11. Corrosion resistant alloys (such as nickel alloys and stainless steels) can be susceptible to crevice corrosion in solutions where no corrosion is observed in the bulk solution. For example, nickel base alloys are immune to corrosion in seawater; however, in areas where two pieces of this alloy are joined (typically by a flange and an o-ring) crevice corrosion may be observed. This phenomenon occurs as the result of two conditions, restricted mass transport and water hydrolysis which both act to make the solution inside a crevice more aggressive. Water hydrolysis occurs when metal cations react with water to form acid (H+):

Image

In a bulk solution diffusion, convection and migration transport the acid away from the surface and no damaging effects are observed. However, the restricted mass transport inside a crevice results in accumulation of metal ions under the crevice former and acidification. As a result, the alloy can be exposed to a very aggressive environment. The pH inside the crevice can be calculated from knowledge of empirically determined concentration quotients (Qxy) where:

Image

You will note that Qxy is similar to Ka in form; however, here we use concentrations and not activities. Qxy is another way to express K = Ka/Kγ as given in then end of Section 18.8. As a practical example of this phenomenon, consider austenitic stainless steels which are generally composed of Fe, Ni, and Cr. Corrosion results in the formation of metal cations in solution, the most aggressive cation for stainless steel being Cr+3.

a. Given that log(Q13) = –4.6 for Cr+3 at a temperature and ionic strength of interest, write the hydrolysis reaction (Eqn. 18.174). Then, solve the corresponding concentration quotient (Eqn. 18.175) to obtain a relation between pH and the concentration of [Cr+3].

b. Make a table of the crevice concentrations that result in pH = {6, 4, 2}.

c. Explain why the concentration of Image does not appear in Eqn. 18.175.

d. For Fe2+, log Q12 = –9.5, and for Ni2+, logQ12 = –10.5. Repeat (a) for each of these ions and compare the crevice concentrations with those in part (b).

18.12. Ruthenium (Ru) is a strong oxidation catalyst for organic compounds typically in the form RuO4(aq) represented as H2RuO5(aq), but it is a stoichiometric catalyst because it is reduced during the oxidation of the organic species. Ru species also undergo redox reactions with water and dissolved oxygen. Assume ideal solutions.

a. Show that RuO4(s) is not stable in contact with an air saturated solution of water and that it will revert to RuO2(s), independent of pH. (Hint: Water will not appear explicitly in the final reaction.)

b. Show that the solubility of H2RuO5(aq) is independent of pH in an aqueous solution saturated with air at 1 bar in contact with RuO2(s). Calculate the concentration of H2RuO5(aq) that would exist at equilibrium.

c. Consider the equilibrium of RuO4(aq) and H2RuO5(aq) in the presence of RuO2(s). Determine the coefficients for the equation Image.

d. Consider the equilibrium of RuO4(aq) and RuO42–(aq) in the presence of RuO2(s). Determine the coefficients for the equation Image.

18.13.

a. Rank the following molecules in order of increasing oxidation of carbon and give the oxidation state of C for each: CO2, -COH(aldehyde), -COOH(carboxylic acid), -CO(ketone), -COH(alcohol), -CH2-, -CH3, CH4.

b. Rank the following C5 molecules in order of decreasing degree of reduction, pentane, valeric acid, 2-pentanone, propyl-ethyl ether, 1-pentanol, 2-methyl butanol, and 1-pentanal.

18.14. The human body processes ethanol by oxidizing it to acetaldehyde via the NADox/NADred dehydrogenase redox reaction. The reaction is

NADox + ethanol Image NADred + acetaldehyde

The values for properties in the order the species appear in the reaction are

ΔG′of, 310.15(pHc =7.4, I=0.25) ={1163.9, 91.45, 1231.0, 43.2}kJ/mol
ΔHof, 310.15(pHc =7.4, I=0.25) ={–11.9, –291.2, –42.9, –214.1}kJ/mol

a. Determine the magnitude of the heat of reaction under the stated conditions.

b. Calculate the equilibrium constant. If we assume the ratio of the two forms of NAD are near unity, what is implied about the ratio of acetaldehyde:ethanol? What is the importance of sign of the Gibbs energy for the subsequent oxidation of acetaldehyde to acetic acid?40

18.15. The first step in biological glycolysis (the catabolic reaction for glucose consumption) involves addition of a phosphate to create glucose 6-phosphate2–. If the reaction were to occur in aqueous solution “chemically” (as compared to “biochemically’”), it would be written

Image

However, in a biological solution, ATP and ADP are carriers of phosphate. Another reaction in solution is the hydrolysis of ATP to ADP:

Image

The transformed values of the Gibbs energies of formation and enthalpy of formation (kJ/mol) at 298.15K, pHc 7.0, I 0.25m, pMg 3.0 are tabulated below along with the physiological concentrations.

Image

a. Evaluate the standard state Gibbs energy and enthalpy for Eqn. 18.176 and ΔG′ under the actual concentrations.

b. Repeat part (a) for Eqn. 18.177.

c. Biological glycolysis works by coupling the two reactions. Write the overall reaction and evaluate ΔG′ under physiological conditions.

18.16. When we discussed H3PO4 in Section 18.9, we developed a recurring relation for the dissociation in Eqn. 18.50. Later we gave with verbal argument a binding polynomial in Eqn. 18.90.

a. Write the series of binding reactions for PO43– and derive the Eqn. 18.50.

b. Create a plot of <i> vs. pH for PO43–.

18.17. Write a binding polynomial for CO2 in aqueous systems and determine the transformed standard state Gibbs energy of total CO2 at pHc = 7 and I = 0.25 m. Give the distribution of aqueous species at these conditions.

18.18.

a. Write the binding polynomial for ATP at 298.15K in terms of binding constants in the absence of Mg for application between 3 < pHc < 14. Assume ideal solutions. Hint: Use untransformed equilibrium constants calculated from the Gibbs energies of formation, and ignore the species that don’t have Gibbs energies tabulated in the appendix.

b. Convert the binding constants to dissociation constants and give the pKa,A for each dissociation constant.

c. Using the binding constants, calculate and plot the <i> vs. pH for ATP between 3 < pHc < 14. Mark the <i> at the pKa,A values determined in (b).

d. Give the fraction of each species at a pHc = 7.6.

18.19. Repeat problem 18.18, but use ADP.

18.20. Beginning with the untransformed Gibbs energies of formation, document the intermediate calculations for the value of apparent Gibbs energy of formation of ADP at the conditions of Example 18.10, using the extended Debye-Hückel activity coefficient model and transformed Gibbs energies. Also calculate the distribution of each species.

18.21. Repeat problem 18.20, but use H3PO4.

18.22. At pHc = 7, I = 0.25, beginning with the untransformed Gibbs energies of formation, document the intermediate calculations for the value of apparent Gibbs energy of formation of CO2 at the conditions using the extended Debye-Hückel activity coefficient model and transformed Gibbs energies. Also calculate the distribution of each species.

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