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J. Phys. Chem. 1989, 93, 8240-8243
Theory of Enthalpy-Entropy Compensation H. M. J. Boots* and P.K. de Bokx Philips Research Laboratories, P.O. Box 80000, 5600 J A Eindhoven, The Netherlands (Received: January 1 I , 1989; I n Final Form: June 2, 1989)
The ubiquitous nature of enthalpy-entropy compensation calls for a general model. The present model, which is formulated with compensation in ion exchange in mind, is an extension of an older general approach in that the molar fractions of the species that are exchanged in the process enter explicitly. In a different way, it is an extension to the theory of conformal solutions, which only leads to compensation under the condition of conformity. The present model allows one to decide from experiment whether interactions between exchanging species or interactions with third species (or external fields) are responsible for the compensation effect.
1. Introduction Enthalpy-entropy compensation has been found in a wide variety of processes and reaction equilibria: heterogeneous catalysis; diffusion in metals, ionic crystals, and amorphous polymers; conduction in amorphous semiconductors; and phase equilibria between hydrophobic and hydrophylic phases and between multicomponent fluid phases are but a selection. The term is used to indicate that the compensation temperature T,, defined by
T, = AH/AS
(1)
is system independent for a class of similar experimental systems. Here AH and AS are suitably defined’ enthalpy and entropy differences. The usage of the word compensation in this connection originates from the simplest case, when enthalpy and entropy changes are independent of temperature T i n a range including not only the experimental temperatures used to determine the compensation temperature, but also the compensation temperature itself. Then the Gibbs free energy change AG = AH - TAS vanishes’ at the same temperature T = T, for all systems in the class. If, however, AH and AS do depend on temperature, the compensation temperature is a function of temperature, too. Then T, cannot be interpreted as a characteristic temperature. This is, however, not essential for the compensation effect; but it is essential that similar systems are described by the same function TC(7-l. Numerous explanations for the compensation effect have been given. Often these models rely on the specifics of the process under consideration, which severely limits their range of applicability. A brief guide to the literature on compensation models is given in Appendix A. We will focus on a general approach for the following reasons. In the first place the ubiquitous nature of the phenomenon asks for a general model. Secondly, the microscopic processes leading to compensation are not well understood in most cases, including the case of ion exchange, which in the accompanying paper2 is shown to exhibit the compensation effect. In section 2 we will reformulate a general approach which has been used in the p a ~ t , ~but - ~which did not receive sufficient attention in part of the more recent literature. In that approach eq 1 is replaced by an equivalent statement on the form of a suitably defined Gibbs free energy change AG. Indeed, the Gibbs free energy is the relevant quantity for both activated processes and reaction equilibria. It is easily shown that AG is of the compensation form if it is justified to expand G to linear order in a parameter which describes the difference between the different (1) Often a constant term appears on the right-hand side of eq 1 . This implies an extra term, independent of T and M,on the right-hand side of eq 2. Here we choose to define a AG such that this term does not appear. (2) de Bokx, P. K . ; Boots, H. M. J . J . Phys. Chem., following paper in this issue. (3) Hammet, L. P. Physical Organic Chemistry; McGraw-Hill: New York, 1940; p 186. (4) Leffler, J. E.; Grunwald, E. Rates and Equilibria of Organic Reactions; Wiley: London, 1973. (5) Leffler, J . E. Nature 1965, 205, I101.
0022-3654/89/2093-8240$01.50/0
systems. Neglect of higher order terms reflects the experimental fact that systems having the same compensation temperature are similar. This is a natural and simple way to derive a compensation law. The purpose of the present paper is to extend this general approach to multicomponent systems where the component concentrations play a nontrivial role, as is the case in multicomponent liquid-vapor phase equilibria6 and in ion exchange.2 This is done in section 3. In order to find a compensation law containing concentrations, we use statistical thermodynamics and suppose that similar systems have similar potential energies. Our approach closely parallels the work by Longuet-Higgh6 However, his model is not applicable to ion exchange, in contrast to the more general model presented in the third section. If the dominant difference between two systems of a class is in the interaction potentials, the model results in chemical potentials which only differ in parametersf, describing the interaction between particles of type r and s. Knowledge of the interaction parameters from some experiments allows the prediction of chemical potentials for other experiments. More importantly, if we interpret experiments on ion exchange2 according to the model, we find conditions that any detailed molecular model of that process must fulfill. In that way this work is a step in the direction of a model for the origin of selectivity. 2. Compensation Enthalpy-entropy compensation is equivalent to the following form of a suitably defined Gibbs free energy change’ as the product of a temperature-dependent, system-independent factor y ( r ) and of a temperature-independent, system-dependent factoP5
~(w)
AG = Y ( T ) 4v)) (2) Here the system dependence is described by a collection of paWe shall only prove the validity of this formulation rameters of compensation starting from the formulation in the introduction; then the derivation of the first formulation of compensation from the second one is trivial. Instead of eq 1, one may write
w.
(3) where the compensation temperature is independent of v). Mostly measurements are restricted to a region where AH and AS are independent of temperature. In that case the compensation temperature is a constant and eq 3 is of the form eq 2 with y(T) = T, - T and r(v))= AS(y)). (The splitting is unique apart from If, however, AH and AS do a factor independent of T and depend on temperatue, we use AS = -dAG/aT in eq 3 and solve the resulting differential equation. Thus we find that AG is of the form eq 2 with
w.)
(4)
( 6 ) Longuet-Higgins, H. C. Proc. R. SOC.A 1951, 205, 247.
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 25, I989 8241
Theory of Enthalpy-Entropy Compensation and
4 v ) ) = AG(To9v))
(5) where To is arbitrary. Equation 4 is easily inverted to express the compensation temperature in y ( T ) Tc(V = -Y(T)/W/dT) + T (6) Equivalently, one may use a separability condition analogous to eq 2 on A3! or AH. Ritchie and Sager' use in fact a separability condition on the specific heat. We prefer eq 2 as the more fundamental one in view of the development in the next section. Up to this point we have only introduced other, equivalent formulations of the compensation law. Now we use the fact that systems having the same compensation temperature (belonging to the same compensation class) are very similar as a weak condition for compensation. Let the Gibbs free energy change of interest be given by Accfafb) = G c f ) (7) where the temperature dependence is not indicated explicitly. The similarity of the systems means that the values of Cy) are nearly equal for all systems in the compensation class. This is the motivation to expand the relevant Cy) for all systems around the same reference Ccfo). Then Accfafb) = (dC/df)f-locfa-fb) (8) This equation is of the compensation form given in eq 2, so that here we have a quite general model for compensation. Due to its generality, it cannot be very rich. However, we see immediately that if G depends on more than one parameter, compensation is not guaranteed. For example, if the parameters are f and g AG(fafbXa&J = (ac/af!)/=/,cfa -fb) + (aG/afZ)g=go(ga - gb) (9) is only consistent with compensation (eq 2) if the derivatives in eq 7 differ by a factor which does not depend on temperature (but which may depend on the reference valuesfo and go). This point has been stressed before4qsand has been studied in detail by Hine.* In the next section we will encounter the possibilities that the compensation effect arises from a one-particle and from a twoparticle potential. In view of the discussion above, these possibilities will be mutually exclusive if the derivatives of AC to the corresponding structural parameters have different temperature dependences. In fact, eq 8 is sufficient to describe compensation if we can treat the system by considering only one type of particles. However, the generalization to a system with more types of particles as given, e.g., in eq 26 is not straightforward. The reason for the slow variation of AG will mostly be in the molecular potential. Though we do not aim at a particular model for the potential, we shall relate AG to small variations in the one- and two-body potentials.
3. The Compensation Model It is natural to ascribe the differences between similar systems to differences in the potential energy, which may be written as a sum of terms involving one particle and interaction terms involving two or more particles U = CUi(?j)+ cui,(7i,7j, + ... (10) i
i
