4. Stoichiometry

If you are thinking about someone while reading this book, you are most definitely in LOVE.

—Tim Newberger Undergraduate ChE Student W2013

Now that we have shown how the rate law can be expressed as a function of concentrations, we need only express concentration as a function of conversion in order to carry out calculations similar to those presented in Chapter 2 to size reactors. If the rate law depends on more than one species, we must relate the concentrations of the different species to each other. This relationship is most easily established with the aid of a stoichiometric table. This table presents the stoichiometric relationships between reacting molecules for a single reaction. That is, it tells us how many molecules of one species will be formed during a chemical reaction when a given number of molecules of another species disappears. These relationships will be developed for the general reaction

Recall that we have already used stoichiometry to relate the relative rates of reaction for Equation (2-1):

This stoichiometric relationship relating reaction rates will be used in Chapters 6 and 8.

In formulating our stoichiometric table, we shall take species A as our basis of calculation (i.e., the limiting reactant) and then divide through by the stoichiometric coefficient of A

in order to put everything on a basis of “per mole of A.”

Next, we develop the stoichiometric relationships for reacting species that give the change in the number of moles of each species (i.e., A, B, C, and D).

Figure 4-1 shows a starving artist’s rendition of a batch system in which we will carry out the reaction given by Equation (2-2). At time t = 0, we will open the reactor and place a number of moles of species A, B, C, and D, and inerts I(N_A0, N_B0, N_C0, N_D0, and N_I0, respectively) into the reactor.

Species A is our basis of calculation, and N_A0 is the number of moles of A initially present in the reactor. After a time t, N_A0X moles of A are consumed in the system as a result of the chemical reaction, leaving (N_A0 – N_A0X) moles of A in the system. That is, the number of moles of A remaining in the reactor after a conversion X has been achieved is

N_A = N_A0 – N_A0X = N_A0(1 – X)

We now will use conversion in a similar fashion to express the number of moles of B, C, and D in terms of conversion.

To determine the number of moles of each species remaining after N_A0X moles of A have reacted, we form the stoichiometric table (Table 4-1). This stoichiometric table presents the following information:

Column 1: the species in the reaction

Column 2: the number of moles of each species initially present

Column 3: the change in the number of moles brought about by reaction

Column 4: the number of moles remaining in the system at time t

Components of the stoichiometric table

To calculate the number of moles of species B remaining at time t, we recall that at time t the number of moles of A that have reacted is N_A0X. For every mole of A that reacts, b/a moles of B must react; therefore, the total number of moles of B that have reacted is

Because B is disappearing from the system, the sign of the “change” is negative. N_B0 is the number of moles of B initially in the system. Therefore, the number of moles of B remaining in the system, N_B, at a time t, is given in the last column of Table 4-1 as

The complete stoichiometric table delineated in Table 4-1 is for all species in the general reaction

Let’s take a look at the totals in the last column of Table 4-1. The stoichiometric coefficients in parentheses (d/a + c/a – b/a – 1) represent the change in the total number of moles per mole of A reacted. Because this term occurs so often in our calculations, it is given the symbol δ:

The parameter δ

Definition of δ

The total number of moles can now be calculated from the equation

N_T = N_T0 + δ(N_A0X)

We recall from Chapter 1 and Chapter 3 that the kinetic rate law (e.g., ) is a function solely of the intensive properties of the reacting system (e.g., temperature, pressure, concentration, and catalysts, if any). The reaction rate, –r_A, usually depends on the concentration of the reacting species raised to some power. Consequently, to determine the reaction rate as a function of conversion, X, we need to know the concentrations of the reacting species as a function of conversion, X.

We want C_j = h_j(X)

Batch concentration

After writing similar equations for B, C, and D, we use the stoichiometric table to express the concentration of each component in terms of the conversion X:

Because almost all batch reactors are solid vessels, the reactor volume is constant, so we can take V = V₀, then

We will soon see that Equation (4-6) also applies to liquid systems.

We further simplify these equations by defining the parameter Θ_j, which allows us to factor out N_A0 in each of the expressions for concentration

Feed
Equalmolar: Θ_B = 1
Stoichiometric:

for an equalmolar feed Θ_B = 1 and for a stoichiometric feed Θ_B = b/a.

Continuing for species C and D

Constant-volume batch concentrations

For constant-volume batch reactors, e.g., steel containers V = V₀, we now have concentration as a function of conversion. If we know the rate law, we can now obtain –r_A = f(X) to couple with the differential mole balance in terms of conversion in order to solve for the reaction time, t.

For liquid-phase reactions taking place in solution, the solvent usually dominates the situation. For example, most liquid-phase organic reactions do not change density during the course of the reaction and represent still another case for which the constant-volume simplifications apply. As a result, changes in the density of the solute do not affect the overall density of the solution significantly and therefore it is essentially a constant-volume reaction process, V = V₀ and υ = υ₀. Consequently, Equations (4-6) through (4-9) can be used for liquid-phase reactions as well. An important exception to this general rule exists for polymerization processes.

For liquids V = V₀ and υ = υ₀

To summarize for constant-volume batch systems and for liquid-phase reactions, we can use a rate law for reaction (2-2) such as –r_A = k_AC_AC_B to obtain –r_A = f(X) ; that is,

Substituting for the given parameters k, C_A0, and Θ_B, we can now use the techniques in Chapter 2 to size the CSTRs and PFRs for liquid-phase reactions.

Stoichiometric table for a flow system

where

and Θ_C, Θ_D, and Θ_I are defined similarly

Definition of concentration for a flow system

Units of υ are typically given in terms of liters per second, cubic decimeters per second, or cubic feet per minute. We now can write the concentrations of A, B, C, and D for the general reaction given by Equation (2-2) in terms of their respective entering molar flow rates (F_A0, F_B0, F_C0, F_D0), the conversion, X, and the volumetric flow rate, υ.

υ = υ₀

Then

and so forth for C_C and C_D.

Consequently, using any one of the rate laws in Chapter 3, we can now find –r_A = f(X) for liquid-phase reactions. However, for gas-phase reactions the volumetric flow rate most often changes during the course of the reaction because of a change in the total number of moles or a change in temperature or pressure. Hence, one cannot always use Equation (4-13) to express concentration as a function of conversion for gas-phase reactions.

A situation where one encounters a varying flow rate occurs quite frequently in gas-phase reactions that do not have an equal number of product and reactant moles. For example, in the synthesis of ammonia

4 mol of reactants gives 2 mol of product. In flow systems where this type of reaction occurs, the molar flow rate will be changing as the reaction progresses. Because equal numbers of moles occupy equal volumes in the gas phase at the same temperature and pressure, the volumetric flow rate will also change.

In the stoichiometric tables presented on the preceding pages, it was not necessary to make assumptions concerning a volume change in the first four columns of the table (i.e., the species, initial number of moles or molar feed rate, change within the reactor, and the remaining number of moles or the molar effluent rate). All of these columns of the stoichiometric table are independent of the volume or density, and they are identical for constant-volume (constant-density) and varying-volume (varying-density) situations. Only when concentration is expressed as a function of conversion does variable density enter the picture.

At the entrance to the reactor

Taking the ratio of Equation (4-14) to Equation (4-15) and assuming negligible changes in the compressibility factor, i.e., Z≅Z₀, during the course of the reaction we have upon rearrangement

Gas-phase reactions

We can now express the concentration of species j for a flow system in terms of its flow rate, F_j, the temperature, T, and total pressure, P.

The total molar flow rate is just the sum of the molar flow rates of each of the species in the system and is

We can also write equation (4-17) in terms of the mole fraction of species j, y_j, and the pressure ratio, p, with respect to the initial or entering conditions, i.e., sub “0”

The molar flow rates, F_j, are found by solving the mole balance equations. The concentration given by Equation (4-17) will be used for measures other than conversion when we discuss membrane reactors (Chapter 6) and multiple gas-phase reactions (Chapter 8).

Now, let’s express the concentration in terms of conversion for gas flow systems. From Table 4-2, the total molar flow rate can be written in terms of conversion and is

F_T = F_T0 + F_A0δX

We divide this equation through by F_T0

Then

where y_A0 is the mole fraction of A at the inlet (i.e., (F_A0/F_T0)), and where δ is given by Equation (4-1) and ε is given by

Relationship between δ and ε

Equation (4-21) holds for both batch and flow systems. To interpret ε, let’s rearrange Equation (4-20) at complete conversion (i.e., X = 1 and F_T = F_Tf)

Interpretation of ε

Substituting for (F_T/F_T0) in Equation (4-16) for the volumetric flow rate, υ, we have

Gas-phase volumetric flow rate

The concentration of species j in a flow system is

The molar flow rate of species j is

F_j = F_j0 + υ_j(F_A0X) = F_A0(Θ_j + υ_jX)

where υ_i is the stoichiometric coefficient, which is negative for reactants and positive for products. For example, for the reaction

Substituting for υ using Equation (4-23) and for F_j, we have

Rearranging

Gas-phase concentration as a function of conversion

Recall that y_A0 = F_A0/F_T0, C_A0 = y_A0C_T0, and ε is given by Equation (4-21) (i.e., ε = y_A0δ).

The stoichiometric table for the gas-phase reaction (2-2) is given in Table 4-3.

One of the major objectives of this chapter is to learn how to express any given rate law –r_A as a function of conversion. The schematic diagram in Figure 4-3 helps to summarize our discussion on this point. The concentration of the reactant species B in the generic reaction is expressed as a function of conversion in both flow and batch systems for various conditions of temperature, pressure, and volume.

As previously mentioned many, if not most, rate laws for catalyst reactions are given in terms of partial pressures. Luckily, partial pressures are easily related to conversion with the aid of the ideal gas law and Equation (4-17).

The following equation is used when the mole balance is written in terms of molar flow rates

However, when the mole balance is written in terms of conversion, we use Equation (4-25)

For example, the rate law for the hydrodemethylation of toluene (T) to form methane (M) and benzene (B) given by Equation (10-80) on page 440 can now be written in terms of conversion.

T + H₂ → M + B

If you haven’t decided which computer to buy or borrow, or don’t have your integral tables, you could resort to the graphical techniques in Chapter 2 and use a Levenspiel plot, () versus X, to achieve a specified conversion of toluene.

Need to first calculate X_e

2. In the case of ideal gases, Equation (S4-3) relates volumetric flow rate to conversion.

For the general reaction given by (S4-1), we have

Definitions of δ and ε

and

3. For incompressible liquids or for batch gas phase reactions taking place in a constant volume, V = V₀, the concentrations of species A and C in the reaction given by Equation (S4-1) can be written as

Equations (S4-7) and (S4-8) also hold for gas-phase reactions carried out in constant-volume batch reactors.

4. For gas-phase reactions, we use the definition of concentration (C_A = F_A/υ) along with the stoichiometric table and Equation (S4-3) to write the concentration of A and C in terms of conversion.

with

5. In terms of gas-phase molar flow rates, the concentration of species i is

Equation (S4-11) must be used for membrane reactors (Chapter 6) and for multiple reactions (Chapter 8).

6. Many catalytic rate laws are given in terms of partial pressure, e.g.,

The partial pressure is related to conversion through the stoichiometric table. For any species “i” in the reaction

For species A

Substituting in the rate law

1. Web P4–1_B Puzzle Problem “What Four Things Are Wrong with This Solution?”

• Learning Resources

1. Summary Notes for Chapter 4

2. Interactive Computer Games

Quiz Show II

3. Solved Problems

CDP4-B_B Microelectronics Industry and the Stoichiometric Table

• Living Example Problems

1. Example 4-5 Calculating the Equilibrium Conversion

• FAQ (Frequently Asked Questions)—In Updates/FAW icon section

• Professional Reference Shelf

(a) List the important concepts that you learned from this chapter. Make a list of concepts that you are not clear about and ask your instructor or colleague about them.

(b) Explain the strategy to evaluate reactor design equations and how this chapter expands on Chapters 2 and 3.

(a) Example 4-1. Would the example be correct if water were considered an inert? Explain.

(b) Example 4-2. How would the answer change if the initial concentration of glyceryl stearate were 3 mol/dm³? Rework Example 4-2 correctly using the information given in the problem statement.

(c) Example 4-3. Under what conditions will the concentration of the inert nitrogen be constant? Plot Equation (E4-5.2) in terms of (1/–r_A) as a function of X up to value of X = 0.99. What did you find?

(e) Example 4-4. Why is the equilibrium conversion lower for the batch system than the flow system? Will this always be the case for constant-volume batch systems? For the case in which the total concentration C_T0 is to remain constant as the inerts are varied, plot the equilibrium conversion as a function of the mole fraction of inerts for both a PFR and a constant-volume batch reactor. The pressure and temperature are constant at 2 atm and 340 K. Only N₂O₄ and inert I are to be fed. Go to the Living Example Problems and load Wolfram. (1) What values of K_C, C_A0, and ε cause X_ef to be the farthest away from X_eb?(2) At what value C_A0 will X_eb and X_ef be closest together?

(d) Example 4-5. (a) Using the molar flow rate of A of 3 mol/minute and Figure E4-5.1, calculate the PBR volume necessary for 40% conversion. (b) Next consider the entering flow rate of SO₂ is 1,000 mol/h. Plot () as a function of X to determine the PBR catalyst weight to achieve (1) 30% conversion, (2) 40% conversion, and (3)99% of the equilibrium conversion, i.e., X = 0.99X_e.

P4-2_A Load the Interactive Computer Games (ICG) Kinetic Challenge from the CRE Web site. Play the game and then record your performance number for the module that indicates your mastering of the material. Your professor has the key to decode your performance number. ICG Kinetics Challenge Performance # ____________.

P4-3_A The elementary reversible reaction

is carried out in a flow reactor where pure A is fed at a concentration of 4.0 mol/dm³. If the equilibrium conversion is found to be 60%,

(a) What is the equilibrium constant, K_C if the reaction is a gas phase reaction? (Ans.: Gas: K_C = 0.328 dm³/mol)

(b) What is the K_C if the reaction is a liquid-phase reaction? ( Ans.: Liquid: K_C = 0.469 dm³/mol)

(c) Write –r_A solely as a function of conversion (i.e., evaluating all symbols) when the reaction is an elementary, reversible, gas-phase, isothermal reaction with no pressure drop with k_A = 2 dm⁶/mol⋅s and K_C = 0.5 all in proper units.

(d) Repeat (c) for a constant-volume batch reactor.

P4-4_B Stoichiometry. The elementary gas reaction

2A + B → C

is carried out isothermally in a PFR with no pressure drop. The feed is equal molar in A and B, and the entering concentration of A is 0.1 mol/dm³. Set up a stoichiometric table and then determine the following.

(a) What is the entering concentration (mol/dm³) of B?

(b) What are the concentrations of A and C (mol/dm³) at 25% conversion of A?

(c) What is the concentration of B(mol/dm³) at 25% conversion of A? (Ans.: C_B = 0.1 mol dm³)

(d) What is the concentration of B(mol/dm³) at 100% conversion of A?

(e) If at a particular conversion the rate of formation of C is 2 mol/min/dm³, what is the rate of formation of A at the same conversion?

(f) Write –r_A solely as a function of conversion (i.e., evaluating all symbols) when the reaction is an elementary, irreversible, gas-phase, isothermal reaction with no pressure drop with an equal molar feed and with C_A0 = 2.0 mol/dm³ at, k_A = 2dm⁶/mol·s.

(g) What is the rate of reaction at X = 0.5?

P4-5_A Set up a stoichiometric table for each of the following reactions and express the concentration of each species in the reaction as a function of conversion, evaluating all constants (e.g., ε, Θ). Next, assume the reaction follows an elementary rate law, and write the reaction rate solely as a function of conversion, i.e., –r_A = f(X).

(a) For the liquid-phase reaction

the entering concentrations of ethylene oxide and water, after mixing the inlet streams, are 16.13 mol/dm³ and 55.5 mol/dm³, respectively. The specific reaction rate is k = 0.1dm³/mol·s at 300 K with E = 12,500cal/mol.

(1) After finding –r_A = f(X), calculate the CSTR space-time, τ, for 90% conversion at 300 K and also at 350 K.

(2) If the volumetric flow rate is 200 liters per second, what are the corresponding reactor volume? (Ans.: At 300 K: V = 439 dm³ and at 350 K: V = 22 dm³)

(b) For the isothermal, isobaric gas-phase pyrolysis

pure ethane enters a flow reactor at 6 atm and 1100 K. Set up a stoichiometric table and then write –r_A = f(X). How would your equation for the concentration and reaction rate, i.e., –r_A = f(X), change if the reaction were to be carried out in a constant-volume batch reactor?

(c) For the isothermal, isobaric, catalytic gas-phase oxidation

the feed enters a PBR at 6 atm and 260°C, and is a stoichiometric mixture of only oxygen and ethylene. Set up a stoichiometric table and then write as a function of partial pressures. Express the partial pressures and as a function of conversion for (1) a fluidized batch reactor and (2) a PBR. Finally, write solely as a function of the rate constant and conversion.

(d) Set up a stoichiometric table for the isothermal, isobaric, catalytic gas-phase reaction carried out in a fluidized CSTR.

The feed is stoichiometric and enters at 6 atm and 170°C. What catalyst weight is required to reach 80% conversion in a fluidized CSTR at 170°C and at 270°C? The rate constant is defined with respect to benzene and υ₀ = 50 dm³/min.

Fluidized CSTR

First write the rate law in terms of partial pressures and then express the rate law as a function of conversion.

P4-6_A Orthonitroanaline (an important intermediate in dyes—called fast orange) is formed from the reaction of orthonitrochlorobenzene (ONCB) and aqueous ammonia (see explosion in Figure E13-2.1 in Example 13-2).

The liquid-phase reaction is first order in both ONCB and ammonia with k = 0.0017 m³/kmol · min at 188°C with E = 11,273 cal/mol. The initial entering concentrations of ONCB and ammonia are 1.8 kmol/m³ and 6.6 kmol/m³, respectively (more on this reaction in Chapter 13).

(a) Set up a stoichiometric table for this reaction for a flow system.

(b) Write the rate law for the rate of disappearance of ONCB in terms of concentration.

(c) Explain how parts (a) and (b) would be different for a batch system.

P4-7_B Consider the following elementary gas-phase reversible reaction to be carried out isothermally with no pressure drop and for an equal molar feed of A and B with C_A0 = 2.0 mol/dm³.

(a) What is the concentration of B initially? C_B0 = _______ (mol/dm³)

(b) What is the limiting reactant? _______

(c) What is the exit concentration of B when the conversion of A is 25%? C_B = ______ (mol/dm³)

(d) Write –r_A solely as a function of conversion (i.e., evaluating all symbols) when the reaction is an elementary, reversible, gas-phase, isothermal reaction with no pressure drop with an equal molar feed and with C_A0 = 2.0 mol/dm³, k_A = 2dm⁶/mol²·s, and K_C = 0.5 all in proper units –r_A = _______.

(e) What is the equilibrium conversion?

(f) What is the rate when the conversion is

(1) 0%?

(2) 50%?

(3) 0.99 X_e?

P4-8_B The gas-phase reaction

is to be carried out isothermally first in a flow reactor. The molar feed is 50% H₂ and 50% N₂, at a pressure of 16.4 atm and at a temperature of 227°C.

(a) Construct a complete stoichiometric table.

(b) Express the concentrations in mol/dm³ of each for the reacting species as a function of conversion. Evaluate C_A0, δ, and ε, and then calculate the concentrations of ammonia and hydrogen when the conversion of H₂ is 60%. (Ans: C_H2 = 0.1 mol/dm³)

(c) Suppose by chance the reaction is elementary with k_N2 = 40dm³/mol/s. Write the rate of reaction solely as a function of conversion for (1) a flow reactor and for (2) a constant-volume batch reactor.

P4-9_B Calculate the equilibrium conversion and concentrations for each of the following reactions:

(a) The reversible reaction

is carried out in a flow reactor where pure A is fed at a concentration of 4.0 mol/dm³. If the equilibrium conversion is found to be 60%.

(1) What is the equilibrium constant, K_C, if the reaction is a gas phase reaction?

(2) What is the K_C if the reaction is a liquid-phase reaction?

(b) The gas-phase reaction

is carried out in a flow reactor with no pressure drop. Pure A enters at a temperature of 400 K and a pressure of 10 atm. At this temperature, K_C = 0.25(mol/dm³)².

(3) The gas-phase reaction is carried out in a constant-volume batch reactor.

(4) The gas-phase reaction is carried out in a constant-pressure batch reactor.

P4-10_C Consider a cylindrical batch reactor that has one end fitted with a frictionless piston attached to a spring (Figure P4-10_c). The reaction

with the rate law

is taking place in this type of reactor.

(a) Write the rate law solely as a function of conversion, numerically evaluating all possible symbols. (Ans.:–r_A = 5.03×10^–9[(1–X)³/(1 + 3X)^3/2] lb mol/ft³·s.)

(b) What is the conversion and rate of reaction when V = 0.2 ft³? (Ans.: X = 0.259, –r_A = 8.63× 10^–10 lb mol/ft³·s.)

Additional information:

Equal moles of A and B are present at t = 0

Initial volume: 0.15 ft³

Value of k₁: 1.0 (ft³/lb mol)² · s^–1

The spring constant is such that the relationship between the volume of the reactor and pressure within the reactor is

V = (0.1)(P)     (V in ft³, P in atm)

Temperature of system (considered constant): 140°F

Gas constant: 0.73 ft³·atm/lb mol·°R

KEILLOR, GARRISON and TIM RUSSELL, Dusty and Lefty: The Lives of the Cowboys (Audio CD), St. Paul, MN: Highbridge Audio, 2006.

FELDER, R. M., and R. W. ROUSSEAU, Elementary Principles of Chemical Processes, 4th ed. New York: Wiley, 2015, Chapter 4.

HIMMELBLAU, D. M., and J. D. RIGGS, Basic Principles and Calculations in Chemical Engineering, 7th ed. Upper Saddle River, NJ: Prentice Hall, 2004, Chapters 2 and 6.