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Equilibrium Constants of Chemical Reactions Involving Condensed Phases Pressure Dependence and Choice of Standard State Berta Perlmutter-Hayman The Hebrew University of Jerusalem, Jerusalem, Israel
Chemical equilibrium is a concept to which the student of chemistry is exposed at an early stage of his training. However, whereas the equilibrium between gases usually remains paper chemistry, problems of equilihria in condensed phases, particularly those involving solutes in dilute solution, are soon encountered in practical work. Yet the thermodynamics of these equilihria is a neglected area in a large numher of textbooks of chemical thermodynamics and of physical chemistry. The topic is often introduced for the first time in a chapter dealing with the electromotive force of electrochemical cells. We are informed that the thermodynamic equilibrium constant for the reaction
and that RTlnK.
= - AGO = zFE"
+ RTlnpi
(3)
where u"; . . uneauivocallv refers t o the eas in its standard state of unit partial pressure, and isa functionof temperntureonly. Since for an ideal eas.. uo, . .is thus eaual to G o ;(the standard molar Gihbs Free Energy), the expression &iuip"i for a chemical reaction (where the vi's are the stoichiometric coefficients of the participants, positive for the products and negative for the reactants) can he replaced by AGO without causing any misunderstanding. We can then write
-
A G O
= -RTlnK,
- -
with yiX 1as Zxi 0; yix is the "rational" activity coefficient and measures the deviation of the fuaacitv .. . of the solute from Henry's law, a law which is always obeyed ifthe solution is suffirientlv dilute. The meaning of u;O in this eauation is best left undefined. However, it is &m&nesdcscril;ed as the chemical ootrntial of the substance i in a ~olutinnin which x i = 1and which, a t the same time, is so dilute in i that the fu-. gacity of i follows Henry's Law. This is not a "hypothetical" state, as it is sometimes called, but a state which is a contradiction in terms. This has recently been pointed out very forcefully by Ben Naim ( I ) . Fortunately, however ("unfortunately," according to some texts (Z)),the concentration of a solute in dilute solution is usually expressed not in terms of mole fraction hut in terms of molality, m, or, less to he recommended, molarity, c. The chemical potential can then be written
(2)
where Eo is the standard ~otentialof the cell. and AGO is the change in the Cihhs ~ r e Energy e when unit reaction takes olace while all oarticioants are "in their stnndard states." For a proof df the &st part of eqn. (2) we are referred to an earlier c h a ~ t e r .This earlier chaoter often Drecedes the treatment of the thermodynamics of solution, and deals with rauilihria hrtween idpol eases-althoueh this h e r fact is not always stated exp~icitly.~ The chemical potential of an ideal gas is given by pi = poi
The chemical potential of a solute in a non-ideal solution can he written
(4)
Clearly, if eqn. (3) holds then all the quantities appearing in eqn. (4) are, by definition, independent of pressure, a fact which is emphasized in most textbooks. We shall now consider the question to what extent this statement is applicable to a reaction taking place in condensed phases in general, and between solutes in dilute solution in particular.
pi = fli0,'"
+ RTlnaim= pi'sm + RTlnm; + RTlnyi
(6)
where, y, is the "practical" activity coefl'icient and becomes numericdlv identical with ?,"when the solution issufficientlv dilute for xi and mi to be effectivelyproportional to each othe; We can define pio."' by (2)
and no hypothetical standard state has to he introduced. I t is to the activity as defined by eqns. ( 6 )and (7) that eqns. (1) and (2) refer. a fact that is often not clearlv stated or is explainid some pages alter thew equations are introduced. The uscof AG" in run. (2) issomewhat mislcadine. since u . O F Go. I t would be better (3)to replace AGO by %jpjo where, for the solutes involved, pio is &."'as defined by eqn. (7); should the solvent, or some solid, participate in the reaction, then the corresponding pio would refer to the pure suhstance. At this point, a decision has to he made about the pressure dependence of po in condensed phases. One possibility is to let both of the solutes, and pio of pure solids and liquids, refer to the temperature and pressure at which the experiment is carried out. According to this convention-let us call it convention (a)-u' of a Dure substance eauals the molar Gihhs Free Energy at the gi;m temperature aiid pressure. G. and not the standard mular Gibbs Free Enerev. Go. Since all the p0's are thus pressure dependent, it follok that for a reaction taking place in a condensed phase ~
~
...
(8)
where A T is the volume change for unit reaction, when all substances are in their standard states, i.e., when their chemical potentials equal posmfor solutes, and po for any pure substances taking part in the reaction. Another possibility is to define all standard states as referrine to the standard Dressure of one atmos~here.Let us call this conuention ( b ) : In this convention pd of a solid or liauid is identical with the correspond in^ standard molar ~ f b bFree s Energy, Go. As for its pre&e independence in this convention is usuallv explicitlv stated. Unfortunately, the fact chat two different conventions do, in fact, exist, is rarely mentioned in textbooks (see, however, ref. ( 4 ) ) .The drawbacks of convention (b) have been termed very nicely the "cloud of the silver lining" by Moore in the earlier editions of his well-known textbook (5). Whereas to the present author the cloud seems to overshadow the silver lining of making all standard states independent of pressure, these drawbacks are usually glossed over. True, K , of eqns. ( 1 ) and (2) becomes by definition independent of pressure-but a t a price! If po is pressure independent then the pressure dependence of p is of necessity thrown onto the activity coefficient, which becomes different from unity even when the solute follows Henry's Law at any given pressure, and the solution is thus an ideal dilute solution in theordinary sense of the word. Clearly, this must he very confusing for the student. Furthermore, if these ideas were carried their loeical conclusion. then an electrochemicalcell could be at its standard potential only a t unit pressure! Nevertheless, both conventions are used, and both are formally valid. This might lead to the conclusion that according to one convention, the thermodynamic equilibrium constant of a reaction in solution depends on pressure, and according to another convention, it does not. But an equilihrium constant is a measurable quantity, and its properties cannot depend on arbitrary conventions! The answer to this apparent paradox is that, except a t unit pressure, the two expressions for K , are, by definition, different, as are the activities of the participants in the reaction. What must be independent of the convention adopted is K,, the equilihrium ratio of the molarities. This quantity is often directly measurahle, e.g., by spectrophotometric measurements or, if the equilihrium is attained sufficiently slowly, hv conventional chemical analvsis. ' ~ r o mthe pressure dependence of p0 in convention (a) and that of the activitv coefficient in convention (b). it can easilv he derived that, in-both conventions, the press";; dependen& of K, is given by
where AVis the volume change when unit reaction takesplace between reactants not in their standard hut in their actual states. It might be argued that all this is of only theoretical interest, since the pressure dependence of the chemical potential is very
small. Nevertheless, most textbooks do find it necessary to present an equation for this dependence. And, how could osmoticpressure he explained if the pressure dependence of the chemical potential were so small as to he negligible? Although under ordinary laboratory conditions the influence of pressure on chemical equilibria in condensed systems is indeed unimportant, this influence forms the basis of the pressure-jump method for the investigation of the kinetics in solution (see for example, ref. ( 6 ) ) .Knowledge of this technique, together with other relaxation methods, is rapidly entering undergraduate programs. The pressure "jump" is usually achieved by the rapid release of a pressure of, typically, 40-60 atmospheres. This causes a "jump" in the equilihrium concentrations during a time interval which must he sufficiently short for the actual concentrations to remain virtually unchanged. They subsequently relax to their new equilihrium values with a relaxation time, or times, amenable to measurements. From an analysis of the relaxation times, conclusions about the reaction mechanism can he drawn. A student who bases himself on convention (h) might at first he quite unable to comprehend the theory behind this method. He would have to remind himself that he is not really concerned with the equilihrium constant, which is uninfluenced by pressure, hut rather with the equilihrium ratio of molalitie-
On the other hand, it is easilv demonstrated that, on the basis of convention (a), it does n i t matter very much whether we consider the influence of pressure on K , (eqn. . (8)) or on K,,, (eqn. (9))to get an estimate of the influence of pressure on the eauilihrium concentration. As an example, let us consider areaction between two ions to form a neutral molecule in aqueous solution. With AV around 1&15 cm3,we find from eqn. (8)that a typical pressure iump . . increases K. bv -3.370, a shift which is suitable for a relaxation experiment. Arbitrarily assuming the ionic strength to be -0.16 m, and using the Davies equation (7) for the evaluation of the activity coefficient, we find from a rough estimate of the pressure dependence of the dielectric constant ( 8 ) that, while K. increased by 3.3%, K,, increases by 3.1%. This shows that K,, and K, differ only little in their pressure dependence, whereas, a t any pressure, K , is smaller than K. by the non-negligible factor of -0.6. ~~
Literature Cited (1) Ben Naim, A , J Phys. Chem., 82,792 (1978). (2) Lewis. G. N.. Randall, M., pitzer, K., and Brewer, L., '"Thermodwamics,"2nd ed., McGraw-HillBwkCompany, Ine.,New Yark, 1961,Chap.20. (3) Denhigh, K.."The Pcinciplesof Chemical Equilibrium," 2nd ad.,Cambridge University Presa,Csmbridge, 1971,Chsp.lo. (41 Prigdiinc, I., and Defsy. R., "Chemical Thermodynamics,"Langmans and Green. New YorL. 1954.p. 87. (5) Mmre, W. J.,"PhysicaiChemistry,"2ndd., Prentiee-Hall, Englmmd Cliffs,NJ, 1955. p. 451.
(6) Csldin E. F.. "Paat Reactions in Solution,"Blackwdl Scientific Publication. Oxford. 1984. Chap. 4. 11) Davies, C. W.,"lonAssociation,"Buttenuorths,London. 1962. (81 Lsndult-Bumtein. " Z a h i e n m ~ r lund ~ Funkrion~n,"6th ed.; Springer. Berlin, Vol. IT. Part $5,p. fi87.
Volume 61 Number 9 September 1984
1959,
783
