I. Chem. SOC., Faraday Trans. 1, 1986,82, 2557-2564 Remarks on Dependence on Temperature ‘ at Constant Volume ’ Patrick G. Wright Department of Chemistry, The University, Dundee DDl 4HN It is sought to extend recent treatments of the dependence of equilibrium constants on temperature ‘at constant volume’, with explicit attention to the behaviour of equilibrium constants based on various different types of specification of concentration. It is shown that for reactions in solution, if ‘at constant volume’ means ‘at constant molar volume of the pure solvent’, then the dependence on temperature is indeed rclated to a certain type of AU in precisely the way that has for a long time been taken to be the case. If, however, ‘at constant volume’ means ‘at constant total volume of an actual reacting system’, then the dependence on temperature may be incompletely specified except in the limit of infinite dilution, when it becomes the same as at constant molar volume of the pure solvent.In contrast to the situation which obtains with the dependence on temperature at constant pressure, the use of amounts of solute per unit volume involves no complications in the dependence at constant molar volume of the pure solvent. 1. Introduction In a recent paper, Blandamer et a1.l have clarified some aspects of the dependence of equilibrium constants on temperature ‘at constant volume ’. Some extension of the argument seems capable of attaining rather greater definiteness in some of the conclusions. To do so, however, it is convenient to use a notation which goes beyond the normal set of superscripts.For example, the symbol AUm as normally used means where the partial derivative relates to a closed system and the limiting process envisaged is an approach to infinite dilution. In the argument which follows, however, a symbol is needed not only for this quantity but also for the quite distinct quantity 2. Notation to be used A sufficient symbolism for the present purpose can be provided by the following. Homogeneous systems alone will be considered : further complications would arise if attention were to be directed to such situations as, say, partition of a solute between two solvents. ( a ) Let { X ) denote the measure-number of a physical quantity X in terms of some appropriate unit. Let { { X } } denote the ratio of a physical quantity X to some constant having the same dimensions as X .This constant may or may not be equal to the unit that is used. ( X } depends not only on X but also on the unit that is chosen to be used. On the other hand, ( ( X } } depends on X and on the constant, but is independent of the choice of unit. For example, if Xis the pressurep, then ifp = 64.17 kPa and the constant is 101.325 kPa, then ( p } is 64.17 in terms of the unit kPa, 6.417 x lo4 in terms of the unit Pa, 0.6333 in terms of the unit atm, 481.3 in terms of the unit mmHg and so forth, but ( { p ) } is 0.6333 whichever unit of pressure is used.2558 Dependence on Temperature ‘ at Constant Volume ’ (b) Let ‘concentration-variable’ be used as a generic term for such quantities as: (i) the amount of a gas or a solute per unit volume, (i’) the mass of a gas or a solute per unit volume, (ii) the formal partial pressure x i p of a gas, (iii) the amount of a solute per unit amount of solvent, (iii’) the mass of a solute per unit amount of solvent, (iii”) the amount of a solute per unit mass of solvent, (iii”’) the mass of a solute per unit mass of solvent, (iv) the mole fraction of a substance in a solution, (iv’) the mass fraction of a substance in a solution. Some such variables are dimensionless, and some are not.(c) Both for dilute gaseous systems and for dilute solutions, some quantities exhibit a logarithmic asymptotic dependence on concentration-variables. For example, for any particular concentration-variable q, there are asymptotic dependences pi N ‘constant’ + RT In ({qi)) (1) 2: ‘constant’+Z vi RT In { { q i } ] i 2: ‘constant’ -C vi R In ((qi)).( 1 ”1 a Let tX denote what will here be called the ‘non-logarithmic part of’ a quantity X , i.e. the quantity which remains after the logarithmic terms, from it. Thus, for example: t ~ i = pi - RT In {(rill and but if any, have been subtracted is equal to (i3H/2t)T, itself. ( d ) For any extensive thermodynamic quantity 2, let and A211 denote l i m t - (3, p (2”) (3) (3’) (where V is the actual volume of an actual closed system), and for a reaction in solution let A2?l denote l i m t (3’’) (where V i is the molar volume of the pure solvent-species A); with ‘lim’ everywhere referring to the limit of infinite dilution. If 2 is U or H , the limiting values could also be specified as the appropriate sorts of lim(aZ/ac), but if 2 is S, A ( = U - TS), G, J(= - A / T ) or Y( = - G / T ) , then the specification has to be in terms of a non-logarithmic part.If Z is U or H, and to some extent for all sorts of 2 in the case of the special stoichiometry xi vi = 0, the limiting values AZ11, AZ)) and A211 are unique. However, if Zis S, A , G, J o r Y, and Ci vi # 0 (or i f x i vi = 0 and ‘mass-based’ concentration-variablesP. G. Wright 2559 are considered), then the limiting values AZll, A D ) and A211 are affected by which particular concentration-variable r , ~ is used, and by the magnitude of the constant q e by which q is divided in obtaining ({q)). To distinguish which particular concentration-variable has been used, it would be possible to distinguish different sorts of (say) AG]] as: AGill, AGi’ll, AGiill, AGiiiJ1, AGiii’ll, .. . where the numerals refer to the quantities listed in section 2(b). For reactions of gases, the ordinary A G e is AGiiIl. A g l , when it exists (i.e. when there is a solvent), is (unless the solvent is incompressible) automatically equal? to AZll. For and for constant T the molar volume F/; of the pure solvent is constant if p is constant; and thus It thus suffices to consider only AZIl’s and AZ))’s; the latter of which are related to partial derivatives (aZ/o?t),, r , ~ $ always exhibits an ‘ asymptotic constancy’, whichever particular concentration-variable is considered. In such terms, it is possible to write at constant actual volume of an actual reacting system.(e) The equilibrium value of the quotient of powers of concentration-variables lim (n ~ { i ) , ~ = K,, (4) i where K,,, especially if Zi vi # 0, depends on which of the concentration-variables is used. For reactions of gases (for the moment ignoring non-ideality), K,, is K, if ri is the amount per unit volume n i / V ; K p if qi is the partial pressure x i p ; and K e if qi is the related quantity x,p/p*. 3. Relations for Reactions of Gases and for Reactions in Solution For reactions of gases, AG]] and so forth depend on T but not on p . The dependence of any Kq on T can be inferred from either of the equivalent relations (which apply whichever sort of q is used), but while AG]] = - RT In {{K,,}}, AY]] = R In ({K,,)} ( 5 ) dAGiill/dT = - A p i l l , d A p i l l / d T = + A H i i l l / p and thus d In ({Kp})/d T = AH11/RT2 no such relation applies to K,. As has often been pointed out (if not quite in the present notation), if E i v i # 0, then: dAGi’l/dT # - AF]], dA Yill/dT # AHi]]/ T 2 and d In ((K,})/dT = AUlI/RT2 # AH11/RT2.(7) It may be noted in passing that for reactions of gases: AH]] = AH) and AU]] = A U ) t Equal to AZ]], and not (as might momentarily have been conjectured) to A Z ) .2560 Dependence on Temperature ' at Constant Volume' and that, since the Kq's just mentioned exhibit only a dependence on T and not also a dependence on p or V, the dependence on T is the same at constant V as it is at constant p . Well known relations2-6 for reactions in solution include, in the present notation: 8 ln((K'I}} AH]] ( aT )P = a 1 n W >> - Avll types (iii) and (iv) ( ap'I IT- RT for concentration-variables of butt (9) (10) for concentration-variables of type (i) (where a; is the coefficient of thermal expansion of the pure solvent, and isothermal compressibility). is its The dependences on T at constant V l are readily inferred, for For concentration-variables of types (iii) and (iv), this becomes and the limiting values of these four partial derivatives are, respectively: AU]], AU), (aU;/aV&, and A V ] .It follows that the above expression for reduces to the simple forms AU) RT2 t This is a point at which further complications can enter if attention is not restricted to homogeneous 3 Z.e. the A I ~ of Blandamer et al.' is equal to AU)).What basically is involved is the relation systems. AH']- TAVll(ap/aT)V~ = AU" equations equivalent to which have frequently appeared in the literature (but without full recognition that what is involved is the dependence of K,, on T at constant molar volume of the pure solvent).P. G. Wright 2561 This derivation relates to concentration-variables of types (iii) and (iv) and it will now For concentration-variables of type (i) : be shown that precisely the same result holds for concentration-variables of type (i). and this expression differs from the preceding one [eqn (1 l)] by: which is equal to zero, since A basically simple relation can be discerned, that: For all of the sorts of Kq most usually used? for reactions in solution, a relation holds which resembles that for the effect of temperature on K, for reactions of gases.This may be contrasted with the less simple but more familiar proposition that: For a K for reactions in solution that is based on amounts of solute per unit mass of solvent, or on mole fractions, but not for one that is based on amounts per unit volume, a relation AH]] = lim (g) i2 ln({K,)) - AH]] ( aT )P-w, T, P holds which resembles that for the effect of temperature on Kp or K* for reactions of gases. The simplicity of the result appears only in terms of A V ) . In terms of AU]] [ E lirn(aU/ac),, the relation which holds is (but even this has the feature of being the same for all of these sorts of KJ. A partial derivative (a In {{Kv}}/i3T)v, with V denoting the actual volume of an actual reacting system, is for many sorts of KV not a properly defined quantity.(For any KV which depends on both T and p , the derivative would be properly defined if p were completely determined by T and V. Normally, however, this is not the case: p would be completely determined by T, V and amounts of substances, but not by T and V alone.) In the special case of the limit of infinite dilution, (a ln((K,>}/aT), does become a properly defined quantity, but it is then equal to (a ln{(K,J)/C)T)v;. Consequently, no further relations arise. The one special feature of V, as distinct from V l , in the dilute limit is that '@V/a<),, v' is identically zero, but (i3P'/ac)T, v; is equal to (aV/2<),, P . As Blandamer et aZ.l point out, relations amounting to (8 ln((K,))/aT), = A U ) / R T 2 t A K based on volume fractions can exhibit special features.This question is treated in a further communication.2562 Dependence on Temperature ' at Constant Volume ' have frequently appeared in the literature without explicit statements that they actually relate to constant Vl; or, if to constant actual volume, to constant actual volume in the limit of infinite dilution. Such relations have, moreover, often appeared without clear indication of what sort or sorts of Kv are being considered. 4. Relations Involving a AC, Blandamer et a1.l raise a related point, concerning the dependence of AU)) on temperature. A clear-cut and simple relation holds for the dependence at constant V i (though not for that at constant actual volume, save in the limit of infinite dilution).The quantity @A w / a T),; This second derivative whose limit has to be taken will necessarily satisfy the equation From the relations: it follows that which implies that = 0. Therefore, eqn (14) implies that By using limit (13), the right-hand side of this equation is equal to (aA U)/a T ) , . The left-hand side is equal to which is equal to AC)?. Therefore, it is now seen that t Here, and elsewhere, all double limits are taken to behave 'non-pathologically' (as they will, unless a gas/liquid critical point, critical point for partial miscibility, or other critical point, is involved).P. G. Wright 2563 As Blandamer et a1.l indicate, this simple relation holds for a dependence at constant molar volume of the pure solvent, and not for a dependence at constant pressure.A corresponding relation for constant pressure is, however, easy to infer. Since and it can be seen that eqn (15) implies that: The right-hand side of this equation could easily differ appreciably from ACV. Typical values of (ap/aT),; are ca. I MPaK-l and, judging by analogy with experimental observations on the dependence of AH]] values and activation energies on pressure, values of (aAU’))/C3p)T of ca. m3 mol-l do not seem impossibly large (at least for some reactions). These values would correspond to (aAU))/alJ, differing from ACI by ca. 10 J mol-l K-l; and if (aAV)/i3p)T were, rather, ca. 10+ m3 mol-l, then the difference would still be ca. 1 J mol-l K-l. 5. Dependence on Pressure at Constant Molar Volume of the Pure Solvent It is evident that and it follows that (and this holds for a Kv based on amounts per unit volume just as much as for one based on mole fractions or on amounts of solute per unit mass of solvent).Otherwise expressed, the Ad of Blandamer et a1.l is equal to - AU))/T(ap/aT),;. This ‘ dependence on pressure ’ reduces virtually to a dependence on temperature. 6. Concluding Remarks It is hoped that the present analysis will provide a conclusive identification of various ‘ isochoric ’ dependences on temperature, frequently contemplated in the past, in the light of specific matters which Blandamer et a1.l have called into question. The equilibrium relations relied upon by such authors as Brummer and Hills’ and WhalleyR are indeed correct, but as Blandamer and coworkers have pointed out in their detailed analysis of this subtle problem, there has been an overriding need for clarification of the precise significance of ‘isochoric’ in these relations.Past ambiguities are perhaps related to the fact that each of the definitive relations (a ln((K,>)/c7T),~ = AU))/RT22564 Dependence on Temperature ‘at Constant Volume ’ and (aAU)/aT),; = AC; involves two senses of‘ isochoric’, and not just one. The explicit partial derivative relates to constancy of the molar volume of the pure solvent, while A@) and AC$ are quantities of the type formed from a partial derivative which, on the contrary, relates to constancy of the total volume of an actual system. It may finally be stressed that every sort of K,, considered here is equal to a limiting value lim (az/at I T , v lim (n r& i and is not the quotient of concentration-variables in some actual system which is not necessarily particularly dilute. There is no necessity for RT2@ In cccn rSi),,>>/~T)v: i to be equal either to AU) = lim (aU/at),, , or to the value of (aU/at),I, , for the actual experimental situation.Similarly, it has often been realised (and unfortunately, sometimes has not been realised) that in cases where there is no necessity for (a In {(K,)}/eT)p = AH11/RT2 to be equal either to AH]] = lim(C7H/a<)T,. conditions of the actual experimental situation. A similar remark applies to or to the value of (i3H/at)T, for the Afl] 3 lim(aV/ag),, and the value of (aV/ay),, for the actual experimental situation. References 1 M. J. Blandamer, J. Burgess, B. Clark and J. M. W. Scott, J . Chem. SOC. Faraday Trans. 1, 1984, 80, 2 E. A. Guggenheim, Trans. Furuduy SOC., 1937,33,607; Thermodynamics (North Holland, Amsterdam, 3 S . D. Hamann, Physico-Chemical Efects of Pressure (Butterworths, London, 1957). 4 G. S. Kell, J. Chem. Eng. Data, 1975, 20, 97. 5 J. B. Rosenholm, T. E. Burchfield and L. G. Hepler, J . Colloid Interface Sci., 1980, 78, 191. 6 L. G. Hepler, Thermochim. Acta, 1981, 50, 69. 7 S. B. Brummer and G. J. Hills, Trans. Faraday Soc., 1961, 57, 1816; 1823. 8 E. Whalley, Adu. Phys. Org. Chem., 1964, 2, 93. 3359. 3rd edn, 1957) p. 319. Paper 5 / 1786; Received 15th October, 1985