Ind. Eng. Chem. Fundam. 1984, 2 3 , 129-130
129
Calculation of Chemical Thermodynamic Properties of Isomer Groups Containing Fixed-Ratio Subgroups Chemical thermodynamic properties of isomer groups can be calculated by using the properties of subgroups such as racemic mixtures or mutually exclusive groups of molecules with different numbers of branches or other characteristics. Fixed-ratio subgroups of species with different molecular formulas can also be Included in the calculation of the chemical thermodynamic properties of an isomer group. These options can lead to significant simplifications in equilibrium calculations.
In making equilibrium calculations on organic systems with isomers, B. D. Smith (1959) recommended that groups of isomers be treated as a single species by use of an equilibrium constant KU,which is the sum of the equilibrium constants Kfifor the formation of the individual isomers from the elements.
The number of isomers in a group is represented by NP After calculating the equilibrium mole fractions y I of isomer groups, the equilibrium mole fractions yi of individual isomers may be calculated by Yi = -Kfi = ri
YI
Kg
where ri is the mole fraction of i within the isomer group. Since equilibrium calculations are usually made with standard Gibbs energies of formation, these two equations are more frequently (Dantzig and DeHaven, 1962;Duff and Bauer, 1962; Smith and Missen, 1974) written as NI
AGOU = -RT In [Cexp(-AGofi/RT)] i=l
ri = exp[(AGou- AGofi)/RT]
(3)
(4)
It has recently been shown (Alberty, 1983) that eq 3 may be written in the form N,
N,
i=l
i=l
AGO^ = C r i ~ ~ o+ fRT& i
In ri
(5)
and that the corresponding relations for AHoU,SoI,and eopI for isomer groups may be obtained by differentiation of eq 3. The purpose of this communication is to show that (1) the chemical thermodynamic properties of an isomer group may be considered to be made up of contributions of subgroups calculated using these equations for isomer groups, and (2) fixed-ratio subgroups of molecules with different molecular formulas can be included in the isomer group corresponding to the average composition of the mixture. The exponential terms in eq 3 may be separated into groups for various reasons. For example, stereoisomers must be included in the summations, and since D and L isomers have identical chemical thermodynamic properties, their terms make equal contributions. In making thermodynamic tables for the alkanes, Scott (1974) chose to tabulate the properties of DL mixtures by adding R In 2 to the calculated standard entropy and -RT In 2 to the calculated standard Gibbs energy of one of the forms at each temperature. The two terms for the D and L forms in eq 3 may actually be calculated by putting in one term for the racemic mixture, as calculated by Scott. The standard Gibbs energy of formation of the racemic mixture is given by 0 196-431318411023-0129$0 1.ti010
AGOmL
=
1
1
-mom + -AGO, 2 2
;:
+ RT ( ;- In - + - In -;)
(6)
Since AGOm = AGOn, AGOmL = AGOm - R T In 2 so that e-AGam,/RT = e-AGemfRTeh2 = e-AG'm/RT + e-AG*a/RT (7) Scott's tables and the API Project 44 revisions (1974), which give the IUPAC designations of the stereoisomers, also give properties for equilibrium DL-meso mixtures and mixtures of two diastereomeric pairs. An isomer group may be considered as being made up of subgroups with certain characteristics, and the chemical thermodynamic properties of these subgroups may be calculated by using eq 3 and equations derived from it. If the subgroups G1, G2, ..., Gn are mutually exclusive and represent the entire isomer group, eq 3 in the following form may be used to calculate the standard Gibbs energy of formation of the whole isomer group. AGOU = -RT In [exp(-AGoGl/RT) + exp(-AG0fc2/RT) + ... + exp(-AGo~,/RT)] (8) Other chemical thermodynamic properties of an isomer group may be calculated by using contributions of mutually exclusive subgroups which include all NI isomers. For example, the alkenes and cycloalkanes of the same carbon number are isomers, and standard chemical thermodynamic properties can be calculated for alkenes and cycloalkanes separately and used in calculations on systems where only one subgroup or the other is present. The properties of the combined group can be calculated by using eq 8. As another example, only isomers with a certain type of branching may be produced in a catalytic process. If only normal alkanes and species with one methyl branch appear, only terms for these species would be used to calculate the standard chemical thermodynamic properties for that subgroup. Isomers with a given formula may be divided into several mutually exclusive subgroups according to the number or types of branches, and the chemical thermodynamic properties may be calculated for each of the subgroups. The standard Gibbs energy of formation of a group of subgroups or the whole isomer group may be calculated by using eq 8. An isomer group may include a fixed-ratio subgroup of molecules with different molecular formulas providing that their average composition corresponds with that of the subgroup. For example, if ethane and benzene are produced in a 3:1 molar ratio and propene is present at equilibrium, an isomer group can be constructed from C3H, and 3/4C2& + 1/4CB&. This type of example is suggested by the production (Chang and Silvestri, 1977; Weisz, 1980) of hydrocarbons from methanol since element balances require that 3 1 CH30H = H2O + CnH2n+2 + 3n + m 3n + m ~InHZnI-6 (9) 0 1984 American Chemical Soclety
130
Ind. Eng. Chem. Fundam., Vol. 23,No. 1, 1984
The formation reactions from the elements are
+ 3H2 = C3HG K1 3 1 3C + 3H2 = -CzHs + -C&5 4 4 3C
(10) Kz
(11)
Note that the sum of the numbers of moles in the fixedratio subgroup must be unity. The partial pressure exerted by the isomer group PI can be expressed in terms of the partial pressure of hydrogen and the equilibrium constants for reactions 10 and 11.
The partial pressure of the isomer group is also given by PI = KUPH;. Since equilibrium calculations are usually made using standard Gibbs energies of formation r
AGO, = -RT In
ex.(
AG0fc& 7 + )
where the standard Gibbs energy of formation of the fixed-ratio subgroup 3/4C2H6 1/4C6H6is calculated using eq 5.
+
AGOWRG
=
Thus the standard chemical thermodynamic properties of this C3H6isomer group contains contributions from two exponential terms, one for propene and the other for the fEed-ratio subgroup. After this AGOffis used to calculate the equilibrium mole fraction yI of the C3H, isomer group, the equilibrium mole fractions of propene, ethane, and benzene can be calculated by Y C ~ H=~YI
1
exp[(AGou- AG0c3~J/R7'l
Y C ~ H=~ ~ Y exp[(AG0n I -
AG'WRG)/RTI
(15)
(17)
In this case the components of the fixed-ratio group are single species, but with higher alkanes and arenes the
components could be isomer groups. Thus a C7H14 fixed-ratio subgroup of 3/4c$H14+ !'&H14 could be made up of 5 isomers of hexene and 3 isomers of tetramethylbenzene, assuming only mebhylbenzenesare to be included. There is an advantage in using such isomer groups in a calculation, but there is the disadvantage that the calculated mole fractions within the fiied-ratio group must have the assumed ratio. Bjornbom (1975,1977,1981),W. R. Smith and Missen (19821, and Happel and Sellers (1982) have emphasized that certain products in a reaction may arise in fixed ratios, although this is not required by the element balances. These restrictions arise in the mechanism of the reaction. When restricted chemical stoichiometry is encountered in the products of a reaction, this fact may be incorporated in equilibrium calculations by use of fixed-ratio groups, that is by using AGOfiRG in the calculations or by incorporating the fued-ratio group in an isomer group and using iiGofp Nomenclature AGOfi = standard Gibbs energy of formation of one mole of isomer i A G O ~ = R standard ~ Gibbs energy of formation of one mole of a fixed-ratio group AGOGi = standard Gibbs energy of formation of one mole of a group of isomers AGOfI = standard Gibbs energy of formation of one mole of isomer group I Kfi= equilibrium constant for the formation of isomer i from elements Ku = equilibrium constant for the formation of isomer group I from elements NI = number of isomers in a group ri = mole fraction of species i in an isomer group y i = mole fraction of species i y1 = mole fraction of isomer group I Literature Cited Aiberty, R. A. Ind. Eng. Chem. Fundem. 1083, 22, 318. American Petroleum Research Institute Project 44, Texas A&M University, College Station TX. 1974. Bjombom, P. Ind. Eng. Chem. Fundam. 1075, 14. 102. Bjornbom, P. Ind. Eng. Chem. Fundam. 1077, 23, 285. Bjornbom, P. Ind. Eng. Chem. Fundem. 1981, 20, 161. Chang, D. C.; Slhrestri, A. J. J. Catel. 1977, 249. Dantzlg. G. B.; DeHaven, J. C. J. C t " . Phys. 1062, 36, 2620. Duff, R. E.; Bauer, S.H. J. Chem. Phys. 1982, 36. 1754. Happel, J.; Sellers, P. H. Ind. Eng. Chem. Fundam. 1982, 27, 67. Scott, D. w. J. them. m y s . i m , 6 0 , 3 1 4 . Scott, D. W. "Chemical Thermodynamic Properties of Hydrocarbons and Related Substances: Properties of the Alkane Hydrocarbons, C, through C,o, in the Ideal Gas State from 0 to 1500 K", Bulletin 666, US. Bureau of Mines, Washington, DC, 1974. Smith, B. D. AIChE J. 1959, 5, 26. Smith, W. R.; Missen, R. W. Can. J. Chem. Eng. 1074, 52, 280. Smith, W. R.; Missen. R. W. "Chemical Reaction Equilibrium Analysis: Theory and Algorithms"; Wlley-Intersclence: New York, 1982. Welsz, P. B. Pure App//ed Chem. 1980, 52, 2091.
Department of Chemistry Massachusetts Institute of Technology Cambridge, Massachusetts 02139
Robert A. Alberty
Received for review April 26, 1983 Accepted September 20,1983
