J. Chem. SOC., Faraday Trans. I, 1986, 82, 2565-2568 Special Features of Equilibrium Constants that are based on Volume Fractions Patrick G. Wright Department of Chemistry, The Uniuersity, Dundee DDI 4HN Attention is directed to special features of the effect of temperature and pressure on equilibrium constants that are based on certain kinds of volume fractions. These features are akin to the well known complications that arise with equilibrium constants that are based on amounts per unit volume. ~- ~~ Equilibrium Constants based on Volume Fractions Dependence on Temperature Relations for chemical equilibrium are sometimes1$ expressed in terms of quotients of volume fractions; and it is the purpose of this note to point out that such kinds of equilibrium constant can have a dependence on temperature that involves complications similar in origin to the complications which arise3-' with equilibrium constants based on amounts per unit volume.Let K4 denote an equilibrium constant equal to where di is the volume fraction of species i in a solution in a solvent A, and the limiting process envisaged is one of approach to infinite dilution. Then there will be four cases of the dependence of equilibrium constants K4 on temperature, one for each of four distinct meaningss of 'volume fraction'. (a) A first type of volume fraction q5i is the quantity xi q/ xi q all i where xi is the mole fraction of i in the solution, and In this case, di exhibits an asymptotic behaviour is the molar volume of pure i. di xi T/V"A (3) (PA being the molar volume of the pure solvent) and it follows that: (Kz = lim [n ~ ; i ] ~ ~ ) .i By logarithmic differentiation, it follows that : which implies that: A HI] r+) = -+x viaF-ai c vi RT2 i ( 5 ) 25652566 where (using the notation of the previous paperg) AH]] is an ordinary AHm equal to Equilibrium Constants based on Volume Fractions lim (awac)?., p and a: is the coefficient of thermal expansion of pure i. The term -a: C vi is one whose presence was long ago shown3 to be required for equilibrium constants based on amounts per unit volume. The somewhat similar term i via: i is a special feature arising with the present sort of K4, whose dependence on temperature at constant pressure thus involves the thermal expansion of solutes as well as that of the solvent. It is also possible to consider an ‘isochoric’ dependence of Kd, as expressed1* by a partial derivative at constant molar volume of the pure solvent.From eqn (4), it is seen that: (with no third term, since a partial derivative at constant ITA is involved). Now: A U)) v: R P where AU)) is a sort of ‘ AUm ’ equal to lim ( W a O T , v * Further : and therefore it follows that: In contrast to what happens9 with all of the various sorts of K not based on volume fractions, the right-hand side does not reduce to AU))/RT2. (b) A second type of volume fraction bi is the quantity where is the partial molar volume of species i in the solution. In this case, there is an asymptotic behaviour (bi N xi tp/voA (3’) and, by following through algebra exactly analogous to that just given, it is seen that for this sort of K4: andP.G. Wright (c) A third type of volume fraction is the quantity 2567 where qi is a constant (commonly corresponding to some fixed number of sites in a lattice model). In this case, there is an asymptotic behaviour bi 2 xi (constant) and so K4 is equal to the product of Kx and a constant; and therefore K4 has precisely the same dependence on temperature as Kx. ( d ) A fourth type of volume fraction is the quantity xi C(T,P~)/'X xi K(T',PJ i where all the molar volumes relate to a particular fixed temperature and pressure and not to the temperature and pressure of the conditions of an actual experiment. This case8 amounts to a sub-case of the preceding one; and so again corresponds to a K4 which has precisely the same dependence on temperature as Kx.(Cf. a remark by Isaacs. 11) Dependence on Pressure at Constant Temperature (a) From eqn (4) above, it follows by logarithmic differentiation that: Since it follows that for this sort of K4: (a In KJap), = - A ml/RT The isothermal compressibility of solutes is involved, as well as that of the solvent. (b) Similar reasoning shows that for the second sort of K4: (c) and ( d ) For the same reasons as with the dependence on temperature, the third and fourth sorts of K4 exhibit precisely the same dependence on pressure as Kx. References 1 Y. Marcus, Introduction to Liquid State Chemistry (Wiley, New York, 1977), pp. 189-190. 2 Yu. V. Kazakevich and Yu. A. Eltekov, Zh. Fiz. Khim., 1980, 54, 154 (Russ. J . Phys. Chem. Engl. 3 E. A. Guggenheim, Trans. Faraday Soc., 1937,33,607; Thermod-vnamics (North Holland, Amsterdam, 4 S. D. Hamann, Physico-Chemical Efects of Pressure (Butterworths, London, 1957). 5 G. S. Kell, J. Chem. Eng. Data, 1975, 20, 97. 6 J. B. Rosenholm, T. E. Burchfield and L. G. Hepler, J . Colloid Interface Sci., 1980, 78, 191. 7 L. G. Hepler, Thermochim. Acta, 1981, 50, 69. 8 J. H. Hildebrand and R. L. Scott, Solubility of Non-electrolytes (Reinhold, New York, 3rd edn, 1950), Transl., 1980, 54, 82). 3rd edn, 1957), p. 3 19. p. 133; Regular Solutions (Prentice Hall, New Jersey, 1962), p. 11.2568 Equilibrium Constants based on Volume Fractions 9 P. G. Wright, J . Chem. Soc., Faraday Trans. 1, 1986, 82, 2557. 10 M. J. Blandamer, J. Burgess, B. Clark and J. M. W. Scott, J . Chem. SOC., Faradav Trans. 1, 1984, 80, 1 1 N. S. Isaacs, Liquid-Phase High Pressure Chemistry (Wiley, New York, 1981), p. 187. 3359. Paper 511787; Received 15th October, 1985