1. Lewis, G.N., Randall, M. 1923. Thermodynamics and the Free Energy of Chemical Substances, New York: McGraw-Hill.
3. Denbigh, K., 1971. The Principles of Chemical Equilibrium, London: Cambridge University Press, p. 9.
4. The dipole is a model of the charge distribution on the molecule, and it is thus an approximate description of the charge distribution.
5. The polarizability is the linear proportionality constant in a model of how easily a dipole is “induced” when the molecule is placed in an electric field.
6. We qualify this criterion for the purposes of chemical engineering that there is no driving force for “meaningful” change because most of our systems are technically metastable (in a state of local equilibrium). For example, considering air expansion in a piston/cylinder expansion, we neglect the potential corrosion of the piston/cylinder by air oxidation when we state the system has reached mechanical equilibrium.
7. For a reactive system, C is replaced with the number of distinct species minus the number of independent reactions.
9. See an introductory physics text for further discussion of time-averaged force.
10. This is a pressure [=] force/area where motion is in 2-D and forces are in only two dimensions. In an alternative perspective molecules would only exist in a 2D plane. Then the divisor should be 2L2 and we multiply by area L2, and P2D [=] MPa-m, P2DL2 = nRT.
12. Calculation of these properties requires mastery of several fundamental concepts as well as application of calculus and will be deferred. We calculate energies for ideal gas in Chapter 2 and for real fluids in Chapter 8.
13. There is also a third law of thermodynamics, as discussed by Denbigh, K., 1981. The Principles of Chemical Equilibrium, London: Cambridge University Press, p. 416. The third law is of less direct interest in this introductory text, however.
1. Some texts refer to expansion/contraction work as PV work. This leads to confusion since Section 2.3 shows that work associated with flow is PV, and the types of work are distinctly different. We have chosen to use the term “expansion/contraction” for work involved in moving boundaries to help avoid this ambiguity.
2. Two other possibilities exist: 1) The piston may hit a stop before it has finished moving upward, a case that will be considered below, or; 2) The piston may fly out of the cylinder if the cylinder is too short, and there is no stop.
3. However, this is not a useful perpetual motion machine because the net effect on the surroundings and the piston is zero at the end of each cycle. If we tried to utilize the motion, we would damp it out.
4. Other possibilities include electric or magnetic fields, or mechanical springs, etc., which we do not address in this text.
5. The degrees of freedom discussed here are different from those discussed for the Gibbs phase rule.
6. This theory also requires experimental spectroscopic measurements, but those are quite different from the calorimetric measurement of enthalpy changes with respect to temperature.
8. Perry, R.M., Green, D.W., 2008. Chemical Engineer’s Handbook, 8th ed., New York: McGraw-Hill.
9. Poling, B.E., Prausnitz, J.M., O’Connell, J.P., 2001. The Properties of Gases and Liquids, 5th ed., New York: McGraw-Hill.
10. Pitzer, K.S., Lippmann, D.Z., Curl Jr., R.F., Huggins, C.M., Petersen, D.E. 1955. J. Am. Chem. Soc., 77:3433.
11. In the most detailed calculations, absolute zero is used as a reference state to create some thermodynamic tables. This is based on a principle known as the third law of thermodynamics, that states that entropy goes to zero for a perfect crystal at absolute zero. The difficulties in the rigorous calculations are mentioned above, and although the principles are straightforward, the actual calculations are beyond the scope of this book.
12. For calculation of ideal gas U and H, only a reference temperature is required; however, for the entropy introduced in the next chapter, a reference pressure is needed, so we establish the P requirement now.
13. Turbine design is a specialized topic. Introductions to the actual operation are most readily available in mechanical engineering thermodynamics textbooks, such as Jones, J.B., Dugan, R.E. 1996. Engineering Thermodynamics. Upper Saddle River, NJ: Prentice-Hall, pp. 734–745.
14. This may seem like common sense, but sometimes when calculations are performed, it is surprisingly easy to overlook the fact that a valid mathematical result might be physically impossible to obtain.
16. This problem is reconsidered as an adiabatic process in problem P3.14.
1. The reaction coordinate is in some texts called the extent of reaction. This is misleading because depending on conditions, it can be less than one at complete conversion, or it can be negative.
2. Another common measure of reaction progress is conversion. In reaction engineering, it is common to follow the conversion of a particular reactant species, say, species A. If XA is the conversion of A, then nA = nAin(1 – XA), and XA = a ξ/nAin, where a is the stoichiometric coefficient for A as written in the reaction.
3. cf. RealClimate.org (... “simple model”) and aip.org/history/climate/simple.htm, 8/2011
1. Denbigh, K. 1971. The Principles of Chemical Equilibrium. 3rd ed. New York: Cambridge University Press, p.33.
2. A simple system is not acted on by external force fields or inertial forces.
3. The term “configurational” is occasionally used in different contexts. We apply the term in the context of Denbigh, K. 1981. The Principles of Chemical Equilibrium, 4th ed. Cambridge University Press, pp. 54–55. Technically, the configurational entropy includes both the combinatorial contribution discussed here for ideal gases, and the entropy departure function discussed in Unit II. Note that configurational energy is equivalent to the energy departure function of Chapter 7 because the change in energy of spatially rearranging ideal gas particles is zero.
4. The distinctions between these types of entropy are discussed in more detail by Denbigh, K. 1981. The Principles of Chemical Equilibrium, 4th ed. Cambridge University Press, pg. 353.
5. Ideal gases are non-interacting. Non-interacting particles are oblivious to the presence of other particles and the energy is independent of the interparticle separations. In other words, potential energies are ignored.
6. Note that the number of particles and the energy are constant throughout the discussion presented here and the volume is specified at each stage. The constant energy for non-interacting particles means that the temperature will be constant; only the pressure will be reduced at larger volumes because it takes the molecules longer to get around the box and collide with a particular wall. We can think of this as an N, V, U perspective, and we will demonstrate that entropy is maximized at equilibrium within this perspective, but some other quantity might characterize equilibrium if we held other quantities constant.
7. In an isolated system at constant (U, V), entropy will be generated as equilibrium is approached; S will increase and will be maximized at equilibrium. If the system is closed but not isolated, the property which is minimized is determined by the property which is a natural function of the controlled variables: H is minimized for constant (S,P); A for constant (T,V); G for constant (T,P). A and G will be introduced in future chapters.
8. In statistics, this is called the number of permutations.
9. The formula for the particles in boxes is an example of a binomial coefficient, fundamental in the study of probability and statistics. Detailed development of the binomial distribution and the issue of indistinguishability can be found in any textbook or handbook on the subject.
10. In statistics this is called the number of combinations. It is also known as the multinomial coefficient.
11. Moore, Schroeder, 1997. Am. J. Phys. 65:26–36.
12. Callen, H.B. 1985. Thermodynamics and an Introduction to Thermostatistics, 2ed, Indianapolis IN: Wiley, p.333.
13. The Debye model is described by McQuarrie, D.A. 1976. Statistical Mechanics. Harper and Row.
15. Note, however, that Q is constrained to be reversible in the macroscopic definition, so it is not entirely arbitrary.
16. This relation is known as Fourier’s law and is studied in heat-transfer courses.
17. In multistage units, the stages may be considered individually.
18. Bejan, A. 2006. Advanced Engineering Thermodynamics, 3ed, New York: John Wiley & Sons.
19. Kondepudi, D. 2008. Introduction to Modern Thermodynamics. New York: Wiley, p. 386.
20. Though the structure building decreases entropy, the reactions are proceeding at a finite rate which generates entropy.
1. Cavitation occurs when vapor bubbles form in the inlet line of a liquid pump, and the bubbles prevent the pump from drawing the liquid into the pump cavity.
2. This criterion can be evaluated by looking at the dependence of temperature on pressure at constant enthalpy on a thermodynamic chart or table. In Chapter 6, we introduce principles for calculating this derivative from P, V, T properties.
1. Details are given by Tester, J.W., Modell, M. 1996. Thermodynamics and Its Applications. Upper Saddle River, NJ: Prentice-Hall.
2. Leithold, L., 1976. The Calculus with Analytical Geometry. 3rd ed. New York, NY: Harper & Row, p. 929.
3. Other descriptions include “condition for exact differential.” It was called by Rudolf Clausius the “condition of immediate integrability.”
1. Naturally, some compounds decompose before their critical point is reached, or like carbon or tungsten they have such a high melting temperature that such a measurement is impossible even at the present time.
2. The significance of the vapor pressure curve in determining the thermodynamic properties can be readily appreciated if you consider the difference between a vapor enthalpy and a liquid enthalpy. The detailed consideration of vapor pressure behavior is treated in Chapter 9.
3. Pitzer, K.S., Lippmann, D.Z., Curl Jr., R.F., Huggins, C.M., Petersen, D.E. 1955. J. Am. Chem. Soc. 77:3427–3433.
4. This is the Lee-Kesler equation. Lee, B. I., Kesler, M.G. 1975. AIChE J. 21:510.
5. Smith, J.M., Van Ness, H.C. 1975. Introduction to Chemical Engineering Thermodynamics, 3rd ed, New York: McGraw-Hill, p. 87.
6. Lee, B.I., Kesler, M.G. 1975. AIChE J. 21:510.
8. Vargaftik, N.B. 1975. Handbook of Physical Properties of Liquids and Gases. New York: Hemisphere Publishers.
10. To be more precise, however, it is impossible to pack spheres such that the space is completely filled. One example of highly efficient packing would be the body-centered-cubic (bcc) unit cell (Fig. 7.10). We can determine ηP for the bcc unit cell by noting that there are two atoms in the unit cell. The obvious atom is the one in the center. The second atom is actually the combination of pieces of atoms at the corners. There are eight corners and each one contributes one-eighth of an atom. To compute the packing fraction, we need to relate the box length, L, to the diameter, σ. Note that all the corner atoms are touching the atom in the center. Therefore, a diagonal line from the lower left to the upper right corner cuts through 2σ. In Cartesian coordinates, this same distance represents (L2+ L2+ L2)½ = L(3)½. Therefore, L = 2σ/3½ and the packing fraction is 2πσ3/(6L3) = 0.68 = ηPbcc. Liquids at typical conditions cannot pack this efficiently, so typical packing fractions for liquids are 0.25–0.45.
11. The principles are simple for extending molecular simulation to non-spherical molecules. These are described along with a number of implementation resources in an online supplement.
12. For example, see the discontinuous molecular dynamics (DMD) module at Etomica.org.
14. The 2D perspective is not convenient for a general program. Therefore, the online supplement includes formulas for a general methodology in three dimensions, handling attractive collisions, extensions to multi-site molecules, and resources for implementation, all in vector notation.
15. Spiegel, M.R. 1968. Schaum’s Mathematical Handbook of Formulas and Tables, New York:McGraw-Hill, p. 36.
16. Erpenbeck, J.J., Wood, W.W. 1984. J. Stat. Phys. 35:321.
17. Alder, B.J., Wainwright, T.E. 1959. J. Chem. Phys. 31:459.
18. The configurational energy is that energy due solely to the intermolecular interactions at given distances, hence the adjective configurational.
19. Hansen, J-P., McDonald, I.R. 2006. Theory of Simple Liquids, 3rd ed. New York:Academic Press, p. 32.
20. The exception to this discussion occurs very near the critical point, but addressing this problem is beyond the scope of this text.
22. Dymond, J.H., Smith, E.B. 1969. The Virial Coefficients of Pure Gases and Mixtures, New York: Oxford University Press.
23. Vargaftik, N.B. 1975. Handbook of Physical Properties of Liquids and Gases, 2nd ed. New York: Hemisphere.
24. Soave, G. 1972. Chem. Eng. Sci. 27:1197.
25. Elliott, J.R., Suresh, S.J., Donohue, M.D. 1990. Ind. Eng. Chem. Res. 29:1476.
26. Alder, B.J., Wainwright, T.E. 1959. J. Chem. Phys. 31:459.
27. Dymond, J.H., Smith, E. B. 1980. The Virial Coefficients of Pure Gases and Mixtures. New York: Oxford University Press.
1. Generalization of the ideal gas state is also possible beyond the two choices discussed here. For a discussion, see Reid, R.C., Prausnitz, J.M., Poling, B.E. 1987. The Properties of Gases and Liquids. 4th ed. New York: McGraw-Hill.
2. Starling, K.E. 1973. Fluid Thermodynamic Properties for Light Petroleum Substances, Houston, TX: Gulf Publishing.
1. Once the system volume is decreased below a volume where V < nVsatL, we are compressing a liquid, and the pressure could become quite high. We would need to compute how high using an equation of state. An analogous discussion could be developed for expansion of system volume showing that only vapor will exist for V > nVsatV. The key to notice is that values of q are only physically meaningful in the range 0 < q < 1.
2. What value of C would be common if the Clausius-Clapeyron equation was exact? Compare that value to tabulated C values.
3. Gmehling, J. 1977-. Vapor-liquid Equilibrium Data Collection. Frankfurt, Germany: DECHEMA.
4. Naturally, the accuracy of our calculation is dependent on the accuracy of predicting Z, so we must use an accurate equation of state or correlation.
5. Eubank, P.T., Elhassan, A.E., Barrufet, M.A. 1992. Whiting, W.B. Ind. Eng. Chem Res. 31:942. Tang, Y., Stephenson, G., Zhao, Z., Agrawal, M., Saha, S. 2011. AIChE J. 57:3333.
6. It is possible to program conditionals to avoid unstable roots, but due to the importance of chemical engineers understanding the conditions, we require the users to make the determination. Can you see how to program the conditionals?
7. Frequently, we arbitrarily set SR = 0 and either HR or UR = 0 at our reference states. For consistency in our calculations, GR = HR – TRSR. As a result, the calculated value of S at a given state depends on our current state relative to the reference sate. Calculated S values may be positive or negative due to our choice of SR = 0, and Gibbs energy thus calculated may increase or decrease with temperature. Entropy does not actually go to zero except for a perfect crystal at absolute zero, and entropy of all substances at practical conditions is positive. The fact that our calculations result in negative numbers for S is purely a result of our choice of setting SR = 0 at our reference state (to avoid more difficult calculation of the actual value relative to a perfect crystal at absolute zero). See third law of thermodynamics in Subject Index.
1. There are many variations of the diagrams, and this discussion is meant to introduce only the most commonly encountered types of diagrams. More complex diagrams are introduced gradually, and are discussed in depth in Chapter 16. These diagrams are cross sections of three-dimensional diagrams which are discussed in Chapter 16.
2. This is a convenient manipulation to visualize both diagrams if only one diagram is available.
3. Note that when at 100% liquid, x = z and at 100% vapor, y = z. In some cases, such as formulas for an equation of state, we discuss a generic phase which may be liquid or vapor, and thus use either x or y.
4. An interesting perspective on the contributions of Raoult is available in Wisniak, J. 2001. “François-Marie Raoult: Past and Modern Look”. The Chemical Educator 6 (1): 41–49. doi:10.1007/s00897000432a.
5. If a supercritical component is present in significant quantity, the user must beware, because the shortcut K-ratio may falsely predict a liquid phase due to extrapolation of the vapor pressure. An interesting problem arises when we must calculate the VLE K-ratio for a component in the liquid phase but above its critical temperature. Carbon dioxide (Tc = 31°C) in soda pop on a 32°C day would be a common example. Since the saturation pressure of CO2 does not exist above the critical temperature, and pure CO2 cannot condense, we might consider that CO2 would not exist in the liquid phase and that Raoult’s law might indicate an infinite value for the K-ratio. Experience tells us that this component does exist in the liquid phase over a portion of the composition range. Remarkably, the extrapolated vapor pressures in the above formula give reasonably accurate results at small liquid phase concentrations of noncondensable components. Of course, it is more accurate for components that are only slightly above their critical temperature, because then the extrapolation is slight. Calculations with supercritical components are best done using a Henry’s law, or a hypothetical liquid fugacity (Section 11.12 on page 443) or an equation of state (Example 15.9 on page 599).
6. Rachford Jr., H.H., Rice, J.D. 1952. J. Petrol. Technol. 4(sec. 1):19, 4(sec. 2):3.
7. Note that Eqns. 10.20–10.22 can be used to estimate the bubble and dew points regardless of whether the components are supercritical or whether vapor and liquid phases are indeed possible. We will see in the discussion of equations of state that mixtures can have critical points, too, and this leads to a number of subtle complexities.
8. OAQPS, Control of Volatile Organic Compound Emissions from Batch Processes–Alternate Control Techniques Information Document, EPA-450/R-94-020, Research Triangle Park, NC 27711, February 1994.
9. OAQPS, Control of Volatile Organic Emissions from Manufacturing of Synthesized Pharmaceutical Products, EPA-450/2-78-029, December 1978.
10. U.S. E.P.A., Compilation of Air Pollution Emission Factors-Volume 1, (1993) EPA Publication AP-42.
11. Even though μL = μV at equilibrium, the dependency of μV on composition will be quite different from the dependency of μL on composition because the molecules are arranged very differently.
12. The concept of fugacity becomes especially useful when we begin to discuss phase equilibrium in mixtures. In that case, it is conceivable that we could have some supercritical component dissolved in the liquid phase despite its high escaping tendency, (e.g. CO2, in a carbonated beverage at 100°F). The possibility of a component that cannot be a liquid still dissolving in a liquid requires a very general concept of escaping tendency because the pure-component vapor pressure does not exist at those conditions. The definition of fugacity provides us with that general concept.
13. OSHA may change these limits at any time.
1. In polymer-solvent systems the activity coefficient of the solvent at high solvent concentrations is typically > 1, but the activity coefficient of the polymer is <<1. However, the overall system has positive deviations from the perspective of bubble pressure. We focus above on the behavior in systems of molecules of approximately the same size.
2. Another common characterization of the Margules one-parameter model is: GE = A12x1x2. This results in an explicit temperature dependence, RTlnγi = A12(1–xi)2. Use care when using parameters from various sources.
3. Kamlet, M. J., Abboud, J.-L.M., Abraham, M.H., Taft, R.W. 1983. J. Org. Chem. 48:2877.
4. Elliott, J.R. 2010. Chem. Eng. Ed. 44(1):13–22.
5. Lazzaroni, M.J., Bush, D., Eckert, C.A., Frank, T.C., Gupta, S., Olson, J.D. 2005. Ind. Eng. Chem. Res. 44:4075.
7. We ignore the pressure dependence of activity coefficients since most models ignore the effect, when the standard state pressure is the system pressure.
8. Lorenzana, T., Franjo, C., Jiménez, E., Fernández, J., Paz-Andrade, M.I. 1994. J. Chem. Eng. Data 39:172.
9. Redlich, O., Kister, T. 1948. Ind. Eng. Chem. 40:345–348.
10. If RT is omitted from the excess Gibbs energy, then explicit temperature dependence of the γi results, and the “ln” terms of Eqn. 11.37 are preceded with RT. Use care when comparing parameters from various sources because both conventions are used.
11. Technically, this would not be true for a double azeotrope, but these are extremely rare.
12. Gmehling, J. 1991. Azeotropic Data. Frankfort, Germany: DECHEMA Press; Weast, R.C. 2001. Handbook of Chemistry and Physics. Boca Raton, FL: CRC.
13. Anderson, T.F., Abrams, D.S., Grens, E.A. 1978. AICHE J. 24:20.
14. Prausnitz, J., Anderson, T., Grens, E., Eckert, C., Hsieh, R., O’Connell, J. 1980. Computer Calculations for Multicomponent Vapor-Liquid Equilibria, Upper Saddle River, NJ: Prentice-Hall.
15. Van Ness, H.C., Byer, S.M., Gibbs, R.E. 1973. AIChE J. 19:238.
16. Examples of techniques that measure only one activity are osmotic pressure, partial pressure of solvent over a non-volatile polymer solution, the isopiestic method for measuring solvent activity in electrolyte systems, and electrochemical emf techniques in liquid metal solutions.
17. Do not be confused that we discuss Henry’s “law,” Raoult’s “law,” and the Lewis-Randall “rule.” The designations as “law” or “rule” are purely historical names, and are simply different perspectives on modeling the real solution.
18. The reason that the Henry’s law constant appears to be inverted is because the solubilization is represented as an equilibrium constant for a “reaction” where the dissolved species is the “product” and the vapor phase species is the “reactant.”
19. Chao, K. C., Seader, J. D. 1961. AIChE J. 7:598; Grayson, H.G., Streed, C.W. Paper 20-PO7, 6th World Petroleum Conference, Frankfurt, June 1963; Prausnitz, J.M., Shair, F.H. 1961. AIChE J. 7:682.
20. Tester, J.W., Modell, M. 1997. Thermodynamics and Its Applications, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, p. 469.
21. Connors, K.A. 2002. Thermodynamics of Pharmaceutical Systems. Hoboken, NJ: Wiley.
22. As you might imagine there are reasons why “normal saline” for medical IV has other salts dissolved. It is important to maintain balance of many specific electrolytes, and therefore we must be careful which species are delivered, though from a superficial level, any species can be used to make the solution isotonic.
23. Yound, H.D., Nelson, O.A. 1932. Ind. Eng. Chem. Anal. Ed. 4:67.
24. Suggested by O’Connell, J.P. 2010. NSF BioEMB Workshop, San Jose, CA.
1. Quoted by Cor Peters on the occasion of his Area 1a lecturer award, 2010. AIChE National Meeting, Salt Lake City, UT.
2. The variable x is customarily used as a generic composition variable for the mixing rule, whether applied to vapor or liquid roots.
4. The actual probability for a 1+1 interaction is , but when N is large it is equal to . Likewise, for a 2+2 the probability is . For a 1+2 it is , which is effectively x1x2.
5. Readers should be aware that sometimes Eqns 12.11 and 12.12 are written without the RT terms. With that parameterization, then the “ln” term in Eqn. 12.15 is multiplied by RT and the activity coefficients have explicit temperature dependence. Use care when using literature parameters.
6. Huggins, M.L. 1941. J. Phys. Chem., 9:440; and 1942. Ann. N.Y. Acad. Sci. 43:1.
7. Blanks, R.F., Prausnitz, J.M. 1964. Ind. Eng. Chem. Fundam, 3:1.
8. Hansen, C.M. 2007. Hansen Solubility Parameters: A User’s Handbook. Boca Raton, FL: CRC Press, Inc.
9. Lazzaroni, M.J., Bush, D., Eckert, C.A., Frank, T.C., Gupta, S., Olson, J.D. 2005. Ind. Eng. Chem. Res. 44:4075.
10. Kamlet, M.J., Abboud, J.-L.M., Abraham, M.H., Taft R.W. 1983. J. Org. Chem., 48:2877. For solutes in liquids see Abraham, M.H., Andonian-Haftvan, J., Whiting, G.S., Leo, A., Taft, R.W. 1994. J. Chem. Soc. Perkin Trans. 2:1777.
11. 1998. Fluid Phase Equil. 144:191.
1. Scott, R.L. 1956. Annu. Rev. Phys. Chem 7:43.
2. Wilson, G.M. 1964. J. Am. Chem. Soc. 86:127.
3. Advanced readers may note that our definition of local compositions differs slightly from Wilson’s. Wilson’s original derivation combined the two-fluid theory of local compositions with an ad hoc “one-fluid” Flory equation. The same result can be derived more consistently using a two-fluid theory. The difference is that the local compositions are dependent on size as well as energies as defined by Eqns. 13.1, 13.2, and 13.18. This gives xij/xjj = (Φi/Φj)exp(–Aji/(RT)) where the original was xij/xjj = (xi/xj)exp(-Aji/(RT)).
4. Renon, H., Prausnitz, J.M. 1969. Ind. Eng. Proc. Des. Dev. 8:413.
5. Abrams, E.S., Prausnitz, J.M. 1975. AIChE J. 21:116.
6. Note that these assumptions create local compositions of the form xij/xjj = (θi/θj)exp(–aij/T). Compare this with the form of Wilson’s equation (footnote page 505). Note that the use of the subscripts for the local composition energetic parameters τ and a are switched for the UNIQUAC relative to the Wilson equation λ and A.
7. Maurer, G., Prausnitz, J.M. 1978. Fluid Phase Equil. 2:91.
8. Staverman, A.J. 1950. Recl. Trav. Chem. Pays Bas. 69:163.
9. Guggenheim, E.A. Mixtures, 1952. Oxford, England: Oxford University Press.
10. Lichtenthaler, R.N., Abrams, D.S., Prausnitz, J.M. 1973. J.M. Can. J. Chem. 51:3071.
11. The group parameters are based on the relative van der Waals volume and surface area of sites, Rk = Vk (cm3/mol)/15.15, Qk = Ak(cm2/mol)/2.5E9; see Abrams, Prausnitz, 1975. AIChE J. 21:116. Even though the reducing parameters are based on -CH2-, the values of Rk and Qk are nonunity. See the reference for details. The unity value for -OH volume is a coincidence.
12. Bondi, A. 1968. Physical Properties of Molecular Crystals, Liquids and Glasses. Hoboken NJ: Wiley.
13. Fredenslund, Aa., Jones, R.L.; Prausnitz, J.M. 1975. AIChE J. 21:1086.
14. A model exists in the literature that is called ASOG, which is different from the UNIFAC approach, but also uses functional groups.
15. Klamt, A. 1995. J. Phys. Chem. 99:2224. Klamt, A., Jonas, V., Buerger, T., Lohrenz, J.C.W. 1998. J. Phys. Chem. 102:5074. Klamt, A., Eckert, F. 2000. Fluid Phase Equil. 172:43.
16. Klamt, A., Schurmann, G.J. 1993. J. Chem. Soc. Perkin Trans. 2:799.
17. Lin, S.T., Sandler, S.I. 2002. Ind. Eng. Chem. Res. 41:899. DMOL3 was the original basis of Klamt’s work and is still supported.
19. Each discretized value of σ represents the center of the bin for that range of σ.
21. Sandler, S.I., Lee, K-H. 1986. Fluid Phase Equil. 30:135.
22. Larsen, B., Rasmussen, P.S., Fredenslund, Aa. 1987. Ind. Eng. Chem. Res. 26:2274.
23. Gmehling, et al., 1994. Azeotropic Data, NY: VCH.
24. Gmehling, J., Onken, V., Arlt, W. 1977. Vapor-Liquid Equilibrium Data Collection, Frankfurt, Germany: DECHEMA.
1. To begin the calculations, we must specify a reference state. In any thermodynamic analysis, we must have only one reference state for each chemical species; for our example here, the reference state for water must be the same whether the water is solid or liquid. See Section 9.12 on page 361 and footnote therein.
2. Hansen, J.H., Fredenslund, Aa., Pedersen, K.S., Ronningsen, H.P. 1988. AIChE J. 34:1937.
3. Won, K.W, 1986. Fluid Phase Equil. 30:265. See also Won, K.W. 1989. Fluid Phase Equil 53:377.
4. Pedersen, K.S. 1995. SPE Prod. and Fac. Feb:46.
5. Hiemenz, P.C. 1986. Principles of Colloid and Surface Chemistry, 2nd ed. New York, NY: Marcel Dekker, NY.
6. ... and reasonable estimate when
7. Tester, J.W., Modell, M. 1997. Thermodynamics and Its Applications, 3rd ed. Upper Saddle River, NJ: Prentice-Hall.
8. 1998. J. Chem. Eng. Data 43:72.
9. Ward, H. 1926. J. Phys. Chem. 30:1316.
10. Weissenberger, G. 1927. Z. Agnew. Chem. 40:776.
11. Hildebrand, J. 1920. J. Am. Chem. Soc. 42:2180.
12. Sunier, A. 1930. J. Phys. Chem. 34:2582.
13. Shalmashi A., Eliassi, A. 2008. J. Chem. Eng. Data 53:199-200.
14. Connors, K.A, 2002. Thermodynamics of Pharmaceutical Systems: An Introduction for Students of Pharmacy, Hoboken, NJ: Wiley. p. 131.
1. Reid, R., Prausnitz, J.M., Poling, B. 1987. The Properties of Gases and Liquids. 4th ed. New York, NY: McGraw-Hill, p. 133.
2. Reid, R., Prausnitz, J.M., Poling, B. 1987. The Properties of Gases and Liquids. 4th ed. New York, NY: McGraw-Hill, p. 83.
3. Wong, D.S.H.; Sandler, S.I. 1992. AIChE J. 38:671.
4. Gmehling, J., Onken, U., Arlt, W. 1977– Vapor-Liquid Equilibrium Data Collection, Frankfurt: DECHEMA.
1. Much of this section has been published in Lira, C.T. 1996. “Thermodynamics of Supercritical Fluids with Respect to Lipid-Containing Systems,” in Supercritical Fluid Technology in Oil and Lipid Chemistry, King, J.W., List, G.R., eds., Champaign, IL: AOCS Press. Reproduced with permission.
2. Luks, K.D. 1986. Fluid Phase Equil. 29:209–24.
3. Jangkamol-kulchai, A., Lam, D.H., Luks K.D. 1989. Fluid Phase Equil. 50:175–187.
4. Estrera, S.S., Arbuckle, M.M., Luks, K.D. 1987. Fluid Phase Equil. 35:291–307.
5. McHugh, M.A., Krukonis, V.J. 1986. Supercritical Fluid Extraction: principles and practice, Stoneham, MA: Butterworths.
6. Miller, M.M., Luks, K.D. 1989. Fluid Phase Equil. 44:295–304.
7. Schneider, G.M. 1991. Pure and Applied Chem. 63:1313–1326. Also published as Schneider, G.M. 1991. J. Chem. Therm. 23:301–26; Schneider, G.M. 1970. Adv. Chem. Phys. 1:1–42.
8. White, G.L., Lira, C.T. 1992. Fluid Phase Equil. 78:269; White, G.L., Lira C.T. 1989. in Supercritical Fluid Science and Technology. Johnston, K.P., Penninger, J.M.L. eds., Washington, DC: American Chemical Society, pp. 111–120.
9. Schneider, G.M. 1978. Angew. Chem. Int. Ed. Engl. 17:716–27.
10. van Konynenburg, P.H., Scott, R.L. 1980. Phil. Trans. Roy. Soc. London, Ser. A 298(1442):495–540.
11. Streett, W.B. 1983. in Chemical Engineering at Supercritical Fluid Conditions. Paulaitis, M.E., Penninger, J.M.L., Gray, R.D., Davidson, P. eds., Ann Arbor, MI: Ann Arbor Science, pp. 3–30.
12. Modell, M., Reid R. 1983. Thermodynamics and Its Applications, 2ed. Upper Saddle River, NJ: Prentice-Hall, pp. 259–264; Denbigh, K. 1981. The Principles of Chemical Equilibria, 4ed. Cambridge, UK: Cambridge University Press, pp. 188–190.
13. Cismondi, M., Michelsen, M. 2007. J. Supercrit. Fluids 39:287. gpec.efn.uncor.edu. accessed 11/2011.
15. Eubank, P.T., Elhassan, A.E., Barrufet, M.A., Whiting, W.B. 1992. Ind. Eng. Chem. Res. 31:942.
16. Peters, C.J., Lichtenthaler, R.N. de Swaan Arons, J. 1986. Fluid Phase Equil. 29:495–504.
17. Enick, R., Holder, G.D., Morsi, B.I. 1985. Fluid Phase Equil. 22:209–24.
18. Hildebrand, J.H., Prausnitz, J.M., Scott, R.L. 1970. Regular and Related Solutions. Van Nostrand.
19. Fall, D.J., Fall, J.L., Luks, K.D. 1985. J. Chem. Eng. Data 30:82–88.
20. Fall D.J., Luks, K.D. 1984. J. Chem. Eng. Data 29:413–417.
21. For comparative purposes the reduced temperature is calculated based on the critical temperature of ethane or CO2 for the respective systems.
22. King, M.B., Bott, T.R., Barr, M.J., Mahmud, R.S. 1987. Sep. Sci. Techn. 22:1103–20; Brunetti, L., Daghetta, A. Fedeli, E., Kikic, I., Zanderighi, L. 1989. J. Am. Oil Chem. Soc. 66:209–17; Chrastil, J. 1982. J. Phys. Chem. 86:3016–21.
23. Goncalves, M, Vasconcelos, A.M.P., Gomes de Azevedo, E.J.S., Chaves das Neves, H.J., Nunes da Ponte, M. 1991. J. Am. Oil Chem. Soc. 68:474–80.
24. Nilsson, W.B., Gauglitz, E.J., Hudson, J.K. 1991. J. Am. Oil Chem. Soc. 68:87–91.
25. Bamberger, T., Erickson, J.C., Cooney, C.L., Kumar, S.K. 1988. J. Chem. Eng. Data 33:23.
26. Czubryt, J.J., Myers, M.N., Giddings J.C. 1970. J. Phys. Chem. 74:4260–66.
27. Hixson, A.W., Bockelmann, J.B. 1942. Trans. Am. Inst. Chem. Eng. 38:891–930; Drew, D.A., Hixson, A.N. 1944. Trans. Am. Inst. Chem. Eng. 40:675–694; Hixson, A.N., Miller, R. 1940. U.S. Pat. 2,219,652; 1944. U.S. Pat. 2,344,089; 1945. U.S. Pat. 2,388,412.
28. Hixson, A.W., Hixson, A.N. 1941. Trans. Am. Inst. Chem. Eng. 37:927–957.
29. Bogash, R., Hixson, A.N. 1949. Chem. Eng. Progress 45:597–601.
30. Coorens, H.G.A., Peters, C.J., de Swaan Arons, J. 1988. Fluid Phase Equil. 40:135–151.
31. Passino, H.J. 1949. Ind. Eng. Chem. 41:280–287, Dickinson, N.L., Meyers, J.M. 1952. J. Am. Oil Chem. Soc. 29:235–39.
32. Stahl, E., Quirin, K.-W., Gerard, D. 1988. Dense Gases for Extraction and Refining, New York, NY: Springer-Verlag.
33. Eisenbach, W. 1984. Ber. Bunsenges. Phys. Chem. 88:882; Nilsson, W.B., E.J. Gauglitz, Jr., J.K. Hudson, V.F. Stout, J. Spinelli 1988. J. Am. Oil Chem. Soc. 65:109–117; Nilsson, W.B., Gauglitz, E.J., Hudson, J.K. 1989. J. Am. Oil Chem. Soc. 66:1596–1600.
34. Chai, C.-P., 1981. “Phase Equilibrium Behavior for Carbon-Dioxide and Heavy Hydrocarbons,” Ph.D. dissertation, University of Delaware.
35. Frank, T.C. 1997. Chem. Eng. Prog., 93(4):52–63.
36. Interested readers may find a review article helpful. See Fien, G.-J.A.F., Liu, Y.A. 1994. Ind. Eng. Chem. Res. 33:2505. The topic is also available in chemical engineering textbooks intended for separations courses.
37. This guideline may be overcome when the region boundary is highly curved and the feed is near the boundary on the concave side; however, such application is rarely economical.
38. The products of a two-feed column may lie outside the bow-tie region drawn from the overall feed, a principle exploited in extractive distillation. See Wahnschafft, O.M., Westerberg, A.W. 1993. Ind. Eng. Chem. Res., 32:1108.
39. Barnicki, S.D., Siirola, J.J. 1997. “Separations Process Synthesis,” and Doherty, M.F., Knapp, J.P. 1997. “Distillation, Azeotropic and Extractive,” in Kirk-Othmer Encyclopedia of Chemical Technology, 4th ed., New York, NY: Wiley.; also Van Dongen, D.B., Doherty, M.F. 1985. Ind. Eng. Chem. Fundam. 24:454.
40. Experimental data are available in Leder, F., Irani, C.A. 1975. J. Chem. Eng. Data 20:323.
41. It is possible to obtain acetone distillate from a two-feed column if an equimolar acetone(1) + methanol(2) is fed near the center of the column and water is fed near the top of the column, however such separations are not predicted by the bowtie approximation. A separation of this type is called an extractive distillation.
1. An alternative method of arriving at the same result is to write at fixed T, P, . The derivative is mathematically found by using the product rule for differentiation resulting in and the last sum is zero by the Gibbs-Duhem equation.
2. By comparison, CO pyrolysis is less favored above 700 K, owing to its smaller heat of reaction.
3. O’Connell, J.M., Fernandez, E., Komives, C. July, 2010. NSF BioEMB Workshop on Thermodynamics, San Jose, CA.
4. Fogler, H.S. 1999. Elements of Chemical Reaction Engineering, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, pp. 340ff.
5. The entropy of formation can be obtained by inserting formation values into S = (H – G)/T.
6. Hamilton, T.B., Borel, F., Romaniuk, P.J. 1998. Biochemistry, 37:2051–2058.
7. Suggested by O’Connell, J.P. July 2010. BioEMB Workshop, San Jose, CA.
8. Cooper, A.; Eyles, S.J.; Radford, S.E.; Dobson, C.M. 1992. J. Mol. Biol., 225:939–943.
9. Different values of ΔV are given by Li, T.M., Hook. J.W., Drickamer, H.G., Weber, G. 1976. Biochem, 15:5571–5580. Samarasinghe, S.D., Campbell, D.M., Jonas, A., Jonas, J. 1992. Biochem 31:7773–7778.
1. Experimental data from Young, T.F., Maranville, L.F., Smith, H.M. 1959. The Structure of Electrolyte Solutions, p. 35; Clegg, S.L., Rard, J.A., Pitzer, K.S. 1994. J. Chem. Soc., Faraday Trans. 90:1875; Clegg, S.L., Brimblecombe, P. 1995. J. Chem. Eng. Data 40:43; Walrafen, G.E., Yang, W.-H., Chu, Y.C., Hokmabadi, M.S. 2000. J. Solution Chem. 29:905.
2. Wang, P.; Anderko, A.; Springer, R.D.; Young, R.D. 2006. J. Mol. Liq, 125:37–44. Another model using a symmetrical convention is available, Que, H; Song, Y.; Chen, C.-C. J. 2011. Chem Eng. Data 56:963–977.
3. van’t Hoff, J.H., 1902. “Raoult Memorial Lecture.” J. Chem. Soc., Trans. 81: 969–981. doi:10.1039/CT9028100969
5. For example, the apparent moles would be computed from the mass of acetic acid divided by the molecular weight of an undissociated monomer of acetic acid. We discuss the dimerization of acetic acid in Chapter 19.
6. The mole fraction scale is least ambiguous, and more consistent with previous chapters. The electrolyte literature for dilute solutions historically uses molality or molarity, and thus we follow those conventions. Molarity has the disadvantage of requiring a density calculation which often must be based on a model. Process simulators typically use the mole fraction scale.
8. A common notation in the older literature is to use f for the rational (Henry’s law) activity coefficient, which is very confusing with the use of f as fugacity.
9. A more rigorous value is 10–13.995, but the value of 10–14 will be used for casual calculations.
10. Butler, J.N. 1998. Ionic Equilibria. New York: Wiley, p. 8.
11. As a practical note, water is important to include in aqueous systems, but ionic liquids and some other species (HF) may exist dissociated or partially dissociated when pure.
12. Butler, James N. 1998. Ionic Equilibria. New York: Wiley. pp. 20–35.
13. A widely-accepted method for calculating infinite dilution properties uses the Helgeson-Kirkham-Flowers (HKF) EOS. This equation of state is specifically developed for the infinite dilution properties as a function of temperature and pressure. The EOS is extremely detailed, as explained in four papers: (1) Helgeson, H.C., Kirkham, D.H. 1974. Am. J. Sci. 274:1089–1198; (2) Helgeson, H.C., Kirkham, D.H. 1974. Am. J. Sci. 274:1199–1261; (3) Helgeson, H.C., Kirkham, D.H. 1976. Am. J. Sci. 276:97–240; (4) Helgeson, H.C., Kirkham, D.H., Flowers, G.C. 1981. Am. J. Sci. 281:1249–1516. A database of parameters is available at www.predcent.org in the SUPCRT and OBIGT (includes some extensions) databases. A windows interface is provided to use with some of the data, though the tables can be used directly for calculations at the reference state of 298.15 K and 1 bar. Application of the EOS is illustrated in Wang, P., Anderko, A., Young, R.D. 2002. Fluid Phase. Equil. 203:141–176 and references 26–30 therein include parameters.
14. Fogh-Andersen, N., Bjerrum, P.J., Siggaard-Andersen O. 1993. “Ionic Binding, Net Charge, and Donnan Effect of Human Serum Albumin as a Function of pH.” Clin. Chem. 39:48–52.
16. Readers should be careful when using other resources to note if the IUPAC convention is followed.
17. The standard potentials are sometimes called “redox” potentials, which can be confusing as to whether they refer to reduction or oxidation. Here we follow the IUPAC convention and always tabulate the “reduction” potentials.
18. Oxidation states are distinct from oxidation numbers. They are usually, but not always, the same in complex molecules.
19. Roels, J.A. 1987. “Thermodynamics of Growth.” in Basic Biotechnology, Bu’lock, J., Kristiansen, B., eds., Academic Press, pp. 57–74.
20. For special nitrogen sources see Roels, J.A. 1983. Energetics and Kinetics in Biotechnology, New York: Elsevier Biomedical Press, p. 40.
21. Grethlein, A.J., Worden, R.M., Jain, M.K., Datta, R. 1990. Appl. Biochem. Biotechnol. 24/25:875.
22. Shuler, M.L.; Kargi, F. 2006. Bioprocess Engineering: Basic Concepts, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, pp. 209–216.
23. The binding polynomial is an analogy to a partition function in statistical mechanics. It represents a normalization constant given by the sum of the proportionality function for the probability of each binding event. The average binding number is analogous to the expectation value.
24. Note that the actual charge on NAD varies depending on protonation of the NH2 and phosphates. The usual convention is to indicate the oxidized form as NAD+, indicating the charge on nitrogen, not the overall charge. The reduced form is conventionally represented as NADH, again, not necessarily reflecting the overall charge on the molecule.
25. Various symbols are used in the literature for the molal standard state and molal activity coefficient and there is not a standard convention. We use the square () to designate the 1 molal standard state and use it throughout for clarity. It is quite common in the electrolyte literature that the special symbol be omitted on the activity coefficient, obfuscating the implemented standard state.
26. Zemaitis, J.F., Clark, D.M., Rafal, M., Scrivner, N.C. 1986. Handbook of Aqueous Electrolytes. New York: AIChE-DIPPR. p. 595.
27. Tester, J., Modell, M. 1999. Thermodynamics and Its Applications, Upper Saddle River, NJ: Prentice-Hall.
28. In addition to the mathematical approximations, other common approximations are included in Eqn. 18.102. The ionic strength should technically be written in terms of ion concentration rather than molality which results in the solution density rather than solvent density in Eqn. 18.103 which makes little significance for dilute solutions.
29. The activity coefficient for Eqn. 18.106 is rigorously γ* rather than γ□. These differ by a factor of xH2O (see Eqn. 18.163). However, when ion molality is 0.1 m, xH2O = 0.998. At higher concentrations the difference is more important. At ion molality of 6 m, xH2O = 0.9. For more discussion of this detail see Lee, L. 2008. Molecular Thermodynamics of Electrolyte Solutions. Hackensack, NJ: World Scientific Publishing, and Robinson, R.A.; Stokes, R.H. 1959. Electrolyte Solutions, 2nd Ed., Butterworths. p. 229.
30. Alberty, R.A. 2001. J. Phys. Chem. B 105:7865–7870.
31. Alberty, R.A. 2003. Thermodynamics of Biochemical Reactions. Hoboken, NJ: Wiley, p. 69. Note that updated values for ATP, ADP, AMP, and transformations with Mg are explained in ref. 32.
32. Alberty, R.A. 2003. J. Phys. Chem. B 107:12324–12330.
33. Alberty, R.A. 2006. Biochemical Thermodynamics: Applications of Mathematica. Hoboken, NJ: Wiley.
35. Iotti, S.; Sabatini, A.; Vacca, A. 2010. J. Phys Chem B. 114:1985–1993.
36. The electrolyte NRTL model, available in the ASPEN Plus simulator, is a common example. Originally developed as an unsymmetric model, a symmetric version is now available. Song, Y.; Chen, C.-C. 2009. “Symmetric Electrolyte Nonrandom Two-Liquid Activity Coefficient Model.” Ind. Eng. Chem. Res. 48:7788–7797.
37. personal communication with Professor Kaj Thomsen of Department of Chemical and Biochemical Engineering, Technical University of Denmark is acknowledged for development of material in this section.
38. “Rational” means based on mole fraction.
39. 1941. Ind. Eng. Chem. 33:741.
40. Acetic acid can be metabolized to CO2 and water. Ethylene glycol is toxic because the oxidation products are glycolic acid and oxalic acid. Isopropanol is toxic because it produces acetone. Methanol is toxic because it is metabolized to formic acid.
2. Campbell, A. N., Kartzmark E. M. 1960. Can. J. Chem. 38:652.
3. The apparent number of moles is given by the species mass divided by the molecular weight of the monomer. On an apparent basis, we ignore the effect of association to dimers, timers, and so on.
4. Hales, B. J., Bertrand, G. L., Hepler, L.G. 1966. J. Phys. Chem. 70:3970.
5. Kamlet, M. J., Abboud, J.-L. M., Abraham, M. H., Taft, R. W. 1983. J. Org. Chem. 48:2877. For solutes in liquids see Abraham, M. H., Andonian-Haftvan, J., Whiting, G. S., Leo, A., Taft, R. W., 1994. J. Chem. Soc. Perkin Trans. 2:1777.
6. Lazzaroni, M.J., Bush, D., Eckert, C.A., Frank, T.C., Gupta, S., Olson, J.D. 2005. Ind. Eng. Chem. Res. 44:4075.
7. We restrict the discussion to binaries simply because the notation becomes unwieldy in multicomponent solutions, and the reader should recognize that the concepts and proofs extend to additional components.
8. Harris, H. G., Prausnitz, J. M. 1969. Ind. Eng. Chem. Fundam. 8:180.
9. Coleman, M. M., Painter, P. C., Graf, J. F. 1995. Specific Interactions and the Miscibility of Polymer Blends. Boca Raton, FL: CRC Press.
10. Wertheim, M. S. 1984. J. Stat. Phys. 35:19.
11. Heidemann, R. A., Prausnitz, J. M. 1976. Proc. Nat. Acad. Sci. 73:1773.
12. Suresh, S.J., Elliott, J.R. 1992. Ind. Eng. Chem. Res. 31:2783.
13. The ‘C’ denotes a ‘C-type’ dimerization of identical molecules. Later we use ‘AD’ to denote acceptor/donor association.
14. You might wonder what happens when the bond energy approaches that of a covalent bond. See the section on Wertheim’s theory of polymerization.
15. They also compared to ΔCp/R = –1, an alternative that yields a similarly compact expression for Ka. Suresh, S.J., Elliott, J.R. 1992. Ind. Eng. Chem. Res. 31:2783.
16. Wertheim used the symbol “assoc” instead of “C.” That was appropriate to his analysis because he was treating dimerization of a single component. His theory has been widely adapted to all forms of complexation, however, so we feel that confusion can be minimized in the long term by using the letter “C.”
17. This introductory derivation is adapted from a presentation by W.G. Chapman at EquiFase, October 13–19, 2002.
18. An alternative perspective would allow C to interact with A, C, or D. This would not change Eqn 19.77, but would require terms like ΔCA and ΔCD. cf. Muro-Suñé N, et al. 2008. Ind. Eng. Chem. Res. 47:5660–5668.
19. Here we choose to use subscript o to clearly distinguish the notation for apparent moles, even though it would be the quantity normally reported from a macroscopic experiment.
20. Michelsen, M.L., Hendriks, E.M. 2001. Fluid Phase Equil. 180:165.
21. Elliott, J.R. 1996. Ind. Eng. Chem. Res. 35:1624. To translate notation, in this case. The SRCR formulation of 1996 provides a compact theory, but cannot describe strong solvation as in amine+alcohol mixtures.
22. Wertheim, M.S. 1986. J. Stat. Phys. 42:459.
23. Mansoori, G.A., Carnahan, N.F., Starling, K.E., Leland, T.W. 1971. J. Chem. Phys. 54:1523.
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28. Algebraically, 4·0.90476/1.90476=1.900. The coefficient 1.90476 was inferred originally in a very different way, but it is entirely consistent with Wertheim’s theory when g(σ)=1/(1–1.9ηP).
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