Edward L. King University of Colorado
Boulder
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Standard Entrow .- Changes in Composite Reactions
M a n y chemical reactions involve reactants or products which exist as mixtures of closely related species. The closely related species may be isomers, species with different extents of solvation, "intimate" ion-pain, and "solvent-separated" ionpairs, etc. For reasons of convenience in analysis or because this facet of the system is not understood, thermodynamic quantities are measured for the composite reaction, not the simple reactions. The purpose of this paper is derivation of the relationship of these quantities for the composite reaction and those associated with the simple reactions making up the composite reaction. Of particular interest is the manner in which the entropy of mixing of the related species comes into the equation for the standard entropy change of the composite reaction. Let us consider the composite reaction represented
values associated with reactions 1 and 2. One mole of C produced in the composite reaction is made up of r/(l r ) moles of C" and 1/(1 r) moles of C'. Therefore, the enthalpy change associated with formation of this one mole of C is
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S i c e [ C ] = [C'] [C"], the equilibrium constant for the composite reaction
is the sum ( K 1
+ Kz), and
AGO = - R T In ( K ,
+ K*) = AHo - TASa
Solving for entropy change of the composite reaction, we have
A + B = C
The produce C exists in two forms, C' and C" ( [ C ] = [C'] [C"])with a relative stability governed by the equilibrium constant:
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which upon substitution of (ASP - R In K , ) for AHP/T (i = 1 3 becomes
IC"1 = ~IC'I
Each of the simple reactions A+B=C' A+B=C"
(1) (2)
has associated changes of thermodynamic quantities which are not composites. The values of AH0, AGO, and ASU for the composite reaction are related to the
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Journal o f Chemical Education
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The quotients K1/(K1 K 2 ) , [equal to 1/(1 r ) ] ,and K Z / ( K 1 K z ) , [equal to r/(l r ) ] ,are the fractions of C which are C' and C",respectively, and the last term in
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the expression for the entropy change of the composite reaction is an entropy of mixing term. The entropy change associated with mixing 1/(1 r) moles of C' and r/(l r) moles of C" is
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Therefore, the entropy change in the composite reaction is the weighted average of entropy changes of the constituent simple reactions plus the entropy of mixing of the product species which are represented as a single species in the composite reaction. If there are two species being represented as a single species, the maximum value possible for the entropy of mixing term corresponds to r = 1 (equal amounts of the two species) which is R In 2 = 1.4 cal mole-' deg-'. If a larger number of species is pooled together, the contribution of the entropy of mixing to the entropy change of the composite reaction will be much larger. For one mole of mixture consisting of (l/n)th of a mole each of n species, the entropy of mixing term is R in n; this quantity has the values 2.2, 2.8, 3.2, and 4.6 cal mole-' deg-I forn = 3 , 4 , 5, and 10, respectively. The entropy of mixing term is relevant in understanding the statistical contribution to ASO (and AGO) for composite reactions involving isomeric species. An interesting result is obtained if the isomeric species are present in statistically expected relative amounts. The relative concentration of each isomer expected on statistical grounds is inversely proportional to its symmetry number,' a, which is the number of permutations of atoms ohtained by rigid rotations of the molecular species which leave the appearance of the species un~ h a n g e d . ~This inverse proportionality arises because a-I appears in the rotational partition function for a molecular ~pecies.~I n a mole of m isomeric species, P', the statistically expected amount of an isomer (P')of symmetry number a, is, therefore,
(where n, is the fraction of P which is Pf)plus the entropy of mixing term Substitution for n, in terms of symmetry numbers in the entropy of mixing term, and then combining terms gives for the statistical contribution to ASO:
since Zn, = 1. If the reactant is a mixture of isomers, a, in the above equation is replaced by an analogous summation: aa = (21/n)-l. A specific example will now be discussed, namely the formation of isomeric octahedral MA4B2 species from MASB: MAsB MAsB
+ B cis MA4B2 + A + B = tmns MA4& + A =
The statistically expected relative amounts of cis and trans isomers of MA4B2 are 4 parts cis/l part trans. The statistical contribution to ASo for the reaction forming the cis isomer is R In (&/*) (R in U,/U,, where cR and up are the symmetry numbers of reactant and product species, respectively), and the statistical contribution to ASo for the reaction forming the trans isomer is R In 4 / 8 . The sum of these statistical contributions to the entropy of formation of 4/5 mole of cis isomer and mole of trans isomer plus the entropy of mixing of the statistically expected relative amounts of the two isomers is
This is equal to: (For a mixture of two isomers, this becomes This is the result obtained from our general equation arid for a mixture of three isomers,
etc.) The statistical contribution to the standard entropy change in a reaction producing one mole of this mixture of m isomeric species from a reactant which is a single species of symmetry number aBis the sum of m terms
' BENSON,S. W., J. Am. C h m . Soe., 80, 5151 (1958). ¶The symmetry number of a species is also obtainable from the appropriate character table. I t is the sum of the number of proper rotations plus one for the identity operation, and is, therefore. the sum of the coefficientsfor these onerations found as eolumh headings in the table (e.g., for tetrahehrd symmetry, Ts: F, 8Ca, 3C2; 1 8 3 = 12.) =DOLE,M., "Introdueti~n to Statisticd Thermodynamics," Prentice-Hall, Inc., Englewoad Cliffs, N. J., 1954, p. 111.
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This quantity is exactly the statistical contribution to the value of ASO which would be ohtained if one did not recognize the existence of isomeric MA4B2species. In such a case, the statistical contribution to the value of AS0 is R times the logarithm of the ratio of the number of sites in the reactant which are available to B in the forward reaction (five) t o the number of sites in the product which are available to A in the reverse reaction (two) :
This calculation is based on the assumption that B has no relative preference for cis or trans sites, which is the same assumption that would give rise to statistically expected relative amounts of cis and trans isomers. The author wishes to acknowledge clarifying discussions of this topic with W. K. Wilmarth. Volume 43, Number 9, September 1966
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