William F. Bailey and Audrey S. Monahan Department of Chemistry and Institute of Materials Science The University of Connecticut Storrs. Connecticut 06268
II
Statistical Effects and the Evaluation of Entropy Differences in Equilibrium Processes Symmetry corrections and entropy of mixing
Evaluation of entropy differencesin terms of the symmetry properties o f the species involved in the equilibrium is not only a pedagogically useful exercise, it is of fundamental importance in the evaluation of empirical thermodynamic data. A renaissance in the investigation of equilihrium proceases has been occasioned by the development of instrumental techniques, such as ion cyclotron resonance spectroscopy (1 ), which allow for the studv of eas ohase eauilihria. The thermodynamic data which iesug from snch'studies are of fundamental importance to the chemist since they indicate the manner in which structural changes affect equilibria The fact that empirical equilihrium constants and standard free energy differences contain entropic contributions from purely statistical effects cannot he overlooked in the interpretation of such data since the entropic effects can totally mask the intrinsic chemistry involved in an equilihrium. Thus, if the goal is to relate thermodynamic data to structural parameters such as inductive and resonance effects, steric effects.. ete.,. a correction must he applied to the em$rical data to remove contributions to the standard entropy change (ASO) arising from symmetry differences and entropy of mixing. Such statistical contrihutions to AS0 are easily evaluated from consideration of the symmetry elements (2) present in the species involved in an equilibrium. Moreover, an evaluation of the statistical contrihution to AS0 often suffices to establish both the sign and magnitude of A S O . The evaluation of statistical contributions t o AS0 and the correction of empirical thermodynamic data is of particular imoortance in oreanic chemistrv where laree molecules of high symmetry are i&olved. Most e1ementar;organic texts o i i t discussion of the factors involved in evaluation of entroov differences and, with a few notable exceptions (3-5),most advanced oreanic texts treat the tooic in a cursorv manner. I t is unfortuiate that the principles involved here have not been more widely discussed since the topic is of more than academic interest. The evaluation of entropy differences in terms of the svmmetrv oro~ertiesof the soecies involved in an equilihriu& is not &iy a-pedagogically iseful exercise hut it is of fundamental imoortance in the evaluation of emoirical thermodynamic data. In order to establish the orincioles involved in the evaluation ol'statistical wntrihutions, it is instructive toeniploy the formalism of s t a t i s t i d mevhanics. The fdlowinp discussion is based on the treatment presented by Benson (6). Entropy Contributions Due to Symmetry Number Consider the following equilibrium involving the species A, B, C, and D A+BaC+D The eauilibrium constant can he written in terms of the enerev change for the reaction a t absolute zero ( m o o ) and the partition functions (Q)of the species involved. Thus QcQo K , = --e-bEoQ/RT QaQn where R is the gas constant and T is the absolute temperature. In most applications of interest to the organic chemist, the total partition function can be written, t o a good approxima-
tion, as a product of translational, overall rotational, internal rotational, vibrational, electronic, and nuclear components (3).At ordinary temperatures, contributions to the thermodynamic functions from the vibrational, electronic, and nuclear partition functions tend to cancel in the reactants and products and, to a fmt approximation, may he neglected.' The magnitude of the translational partition function for a species deoends on its mass and is indenendent of the svmmetrv properties of the molecule:Thus, since we are interested in statistical contrihutions to the thermodvnamic functions. we need he concerned only with the rotational partition f&ctions. The partition functions descrihing rotation of the molecule as a whole (overall rotation) and rotation about bonds (internal rotation) depend upon the symmetry properties of the molecnles under consideration. Each of the rotational partition functions contains in its denominator a factor known as the "symmetry numher" (7): the external symmetry number, a, in the denominator of the partition function describing overall rotation and the internal symmetry number, n, in the denominator of the partition function descrihing internal rotation. The nature of the symmetry numbers (both a and n ) are discussed helow and for our present purpose it is sufficient to note that the symmetry number represents the numher of permutations oflike a&s in a molecde which can be accomplished by changes in the rotational coordinates alone (8).Thus, the external symmetry number, a, reflects the overall symmetry of a molecule and the internal symmetry numher, n, depends on internal symmetry properties of bonds2 If we. factor the symmetry numhers from the rotational oartition functions we can remove the statistical dependence on symmetry properties of the molecule and define a new overall nartition function.. 0'. . . which is svmmetrv independent. ~ l k s
.
~
~~~~d
~~A
'We now that the el~etn,nirpart~timfunrtion must be e~plmtly eons~deredwhen the reactants and/or products have unpanrrd electrons since there is a degeneracy due to the electron spin. If the degeneracy differs in reactants and products, the electronic partition functions may contribute significantly to the thermodynamic functions. The reader is referred to anv standard tent on statistical thermodynamics for a more complete discussion (for example, see reference (7)). 2To OUT knowledge there is no consensus as to the symbols to be used to represent symmetry numbers. Reference to "symmetry number" in the older literature may be to the external symmetry number, the internal symmetry number, or to the product of the two (for example, see reference (6)).To avoid possible confusion,we note that we have adopted the symbols c and n,used by Hine (3)and others (7), to represent the external and internal symmetry numbers, respectively. Our choice of K , to represent the ratio of the products of both external and internal symmetry numher of molecules involved in an equilibrium differsfrom the symbol employed by Benson (6) and ~ i n (3) e hut avoids unnecessary ambiguity.
volume 55, Number 8, August 1978 1 489
The equilibrium constant is then given by
The statistical effects reflected in o A d n may be removed from K, by defining a "symmetry corrected" equilibrium constant which Benson (6) has denoted as Kchem
External and Internal Symmetry Numbers The external symmetry number is the number of nonidentical but indistinguishable positions into which a molecule may be turned by simple rigid rotation(s). For water one finds that there are two non-identical but indistinguishable positions that can he reached by simple rigid rotation about the C2 axis: a = 2
The ratio of the product of the symmetry numbers2 we will represent as K, Similarly, for the chair conformation of cyclohexane, o = 6. Rotation about the CPaxis eives three eauivalent structures I t follows that anv constant can he written as the . eauilibrium . product of two cmstants: m e , K,,, reflecting the symmetry of the mdecules involved in the ecluilibrium and pmperties . . the other, Kchem,from which symmetry (or statistical ( 6 ) ) effects have been removed. Thus
.
K = K," Kchem
(2)
Since K,, is a temperature independent ratio of the product of symmetry numbers, it follows that its contribution to the overall equilibrium constant is entropic in origin. By analogy with our treatment of the equilibrium constant, it is instructive to consider the overall entroDv difference.. A S O .. as the sum of two factors. The first, which we represent as AS,,, is the entropy difference due solely to differences in symmetry numbers between reactants and products and the second term, ASo,h., contains all other contributions t o ASO. Thus
Also, rotation of structure (I) about the C2 axes shown in (IV-VI) gives three more equivalent but non-identical structures
.-
AS0 = AS",h.,
+ AS,
(3)
where AS, is independent of our choice of standard states (9) and is given by AS,, = RlnK,. = Rln ( o n ) ' e s c ~ * (dPrnd"
