Equilibrium frozen excess and volumetric properties of dilute solutions Michael J. Blandamer Department of Chemistry The University Leicester UK LE1 7RH The properties of aqueous solutions can be analysed in several ways leading to the identification of for example frozen equilibrium relaxational complex sophisticated delphic isodelphic ideal and excess contributions. These terms are examined and illustrated in terms of increasing orders of partial differentials of Gibbs energies volumes expansions and compressions. 1 Introduction The volume of a closed system is an important and indeed comprehensible thermodynamic function of state defined,1a for example for a single phase by the set of independent variables temperature T pressure p and amount of each chemical substance j nj; eqn.(1-1); equations are numbered according to the section in which they appear. V = V[T p ni] (1-1) We note that temperature T and pressure p are intensive variables. With reference to eqn. (1-1) the symbol ni represents the set describing the amounts of all chemical substances in the system. Chemists are challenged by eqn. (1-1) in several interesting ways. There is the problem of understanding the contributions made to volume V by each chemical substance j in the system. One way of tackling this problem probes the change in volume dV when dnj moles of substance j are added leading to the definition of a partial molar volume Vj. The consequence of switching from a closed to an open system in this formalism is not discussed here although it is an interesting point.1b For the most part we confine our attention to aqueous solutions prepared by adding nj moles of substance j (e.g.urea) to n1 moles of liquid water. We also confine our attention to solutes which do not undergo solvolysis reactions. In order to probe therefore the role of solute–solute solute–solvent and solvent– Professor Michael J. Blandamer graduated from the University of Southampton with BSc and PhD degrees in 1961. Following post-doctoral research at NRC in Ottawa (Canada) he joined the staff at the University of Leicester where he was appointed to a Personal Chair in 1990. He is Visiting Professor in the Department of Organic and Molecular Inorganic Chemistry at the University of Groningen The Netherlands.His research interests concern the thermodynamic and kinetic properties of solutes in aqueous solutions. Otherwise he is learning to play the piano the plan being to attain a standard which is likely to be a pale shadow of that demonstrated by the late Thelonius Monk. solvent interactions in determining the volume V(aq) of a solution a useful procedure identifies the corresponding volume V(aq;id) in the event that the properties of this solution are ideal. The difference [V(aq) 2 V(aq;id)] defines an excess volume. In the context again of eqn. (1-1) another challenge concerns understanding the dependences of volume V on temperature and on pressure. These dependences are described in terms of expansion E compression K expansivity a and compressibility k.Closely linked to this new set of properties are the corresponding instantaneous (frozen) and equilibrium properties. 2 Gibbs energies The set of independent variables [T p ni] used in eqn. (1-1) also defines the function of state called the Gibbs energy which for systems at fixed T and p is the thermodynamic potential. For a closed system at equilibrium the Gibbs energy is a minimum. Chemistry is based on the assumption that across the whole range of possible compositions (and organisations—see below) the minimum in Gibbs energy G is unique.2 (All experience based on experiment supports this assumption.) If the state defined by eqn. (1-1) is not at this minimum spontaneous chemical reaction/reorganisation takes place driven by the affinity for spontaneous change A the product of A and extent of chemical reaction being positive;1c A dx ! 0 De Donder’s inequality where A = 2 (dG/dx)T,p.Conceptually we freeze-frame the thermodynamic state defined by eqn. (1-1) in by definition state (I) where the affinity for spontaneous change is A(I) and the composition is x(I). Again conceptually we perturb the system into a nearby state by changes in pressure dp temperature dT and amount of substance j dnj. There are two interesting constraints which we impose on this perturbation. (A) Perturbation is to a near neighbouring state such that affinity A remains constant. (B) Perturbation is to a near neighbouring state at constant extent of reaction; i.e.at constant composition/organisation where x is constant. We are interested in the differential change in for example volume V under these two conditions. In fact we direct our attention to equilibrium states perturbed such that either (a) A = Aeq = 0 or (b) x = xeq. The keyword in the above two paragraphs is spontaneous. In a typical kinetics experiment ni moles of reactants are added to an aqueous solution at time t = 0 (at fixed T and p). Spontaneous chemical reaction driven by the affinity for change leads the system to a minimum in Gibbs energy. The rate at which this process occurs is examined using the formalism of chemical kinetics. Similarly when an aliquot of a solution containing micelles formed by an ionic surfactant is injected into water the micellar aggregates break up as the system spontaneously moves towards a minimum in Gibbs energy.3 3 Partial molar volumes The distinction between the two types of perturbation is illustrated using the following examples.A given closed 73 Chemical Society Reviews 1998 volume 27 1 eq aqueous solution of benzoic acid at specified temperature and pressure (close to ambient) contains at equilibrium (G = minimum A = 0 and x = xeq) n moles of water neq (PhCOOH) moles of benzoic acid neq (H+) moles of hydrogen ions and neq (PhCOO2) moles of benzoate anions. Then the analogue of eqn. (1-1) has the following form. eq,neq(PhCOO2),neq(H+)] V = V[T,p,n1 (3-1) This volume V is readily measured by direct measurements of density r and mass w.To this system we rapidly add dn(PhCOOH) moles of benzoic acid with consequent change in volume from V to (V + dV). The two limiting conditions identified in the previous section are now considered. The system is frozen such that the amount of benzoic acid in the new system is [neq(PhCOOH) + dn(PhCOOH)]. In other words the change in volume dV occurs at fixed extent of reaction xeq the perturbation being characterised by [dV/dn(PhCOOH)]T,p,xeq. The latter quantity is the frozen partial molar volume of PhCOOH in this aqueous solution; if j · PhCOOH the defined quantity is Vj,T,p,xeq (aq). By adding dn(PhCOOH) moles of benzoic acid the system is taken away from a minimum in G and the affinity for spontaneous acid dissociation increased.Spontaneous (and in this case fast) chemical reaction allows the system to regain after time Dt an equilibrium state where A is zero. So an alternative name for the frozen partial molar volume is an instantaneous partial molar volume. In the other class of perturbations the system responds such that in the new state chemical equilibrium is established the amounts of benzoate and hydrogen ions increasing. The increase in amount of PhCOOH in the solution when dn(PhCOOH) moles are added is moderated. [Moderation is not however universally true despite the widespread application of Le Chatelier’s principle—see ref. 1(d)]. In other words the change in volume dV occurs at constant affinity A actually at ‘A = 0’ so that the perturbation is characterized by the partial quantity [dV/dn(PhCOOH)]T,p,A = 0.This is the equilibrium partial molar volume of PhCOOH in this aqueous solution. One more example is relevant. Consider an aqueous solution at specified T and p prepared using n1 moles of liquid water (l) u moles of urea(s). The volume of this system is specified and n by eqn. (3-2); cf. eqn. (1-1). (3-2) V = V[T,p,n1 nu]eq The superscript ‘eq’ (plus conditions that G = minimum and A = 0) indicates that the organisation of the solution described by xeq characterising water–water water–urea and urea–urea interactions is unique to the system defined by eqn. (3-2). Conceptually we add dnu moles of urea producing a change in volume dV. The frozen (or instantaneous) partial molar volume of urea (dV/dnj)T,p,n1,xeq describes the change in volume where these intermolecular interactions remain unchanged.In contrast the equilibrium partial molar volume (dV/dnj)T,p,n1,A = 0 characterises the change in volume urea–urea urea–water and water–water interactions changing in order to hold the system at equilibrium where ‘A = 0’; see for example data in ref. 4. In the latter case although the Gibbs energies are different before and after addition of dnj moles of urea the Gibbs energies are at minima in both states. The above argument centres on volumes and partial molar volumes. But the question is raised as to which thermodynamic variables and related partial molar properties need be classified along similar lines namely frozen (instantaneous) and equilibrium.4 Gibbs energies and potentials Returning to eqn. (1-1) we replace variable V by the Gibbs energy G; eqn. (4-1). (4-1) G = G [T,p,ni] We assert that eqn. (4-1) is valid over the range of compositions/ organisations described by the variable x. Chemical Society Reviews 1998 volume 27 74 The system described by eqn. (4-1) is perturbed by a change in pressure dp. Two limiting perturbations of the Gibbs energy are envisaged; (i) at constant affinity A and (ii) at constant x linked by the following calculus operation. At constant temperature eqn. (4-2) holds. p p p æ è ç ¶G ¶ A Fig. 1 Functions of state ø ÷ ö = æ è ç ¶Gö ø ÷ - æ è ç ¶Aö æè ç ¶x öø ÷ æ è ç ¶Gö ø ÷ ¶ ø ÷ ¶ ¶ ¶ i A nj G nj A ¶G æ x ¶A p x x The latter equation applied to the state where G is a minimum (i.e.at equilibrium) and ‘A = 0’. But1e the affinity A equals T . In other words at equilibrium (dG/dx)T,p is zero. T equals volume V. G ¶nj 2(dG/dx) Moreover the partial differential (dG/dp) Hence eqn. (4-3). V(A = 0) = (¶G/¶p)T,A = 0 = (¶G/¶p)T,xeq = V(xeq) (4-3) Therefore the equilibrium and frozen volumes are the same which is not unexpected because volume V is a function of state being a property which is not path dependent; ‘volume’ is ‘volume’. Similar transformations [cf. eqn. (4-2)] with respect to the dependences on temperature of G and G/T (at fixed pressure) confirm that (a) S(A = 0) = S(xeq) and (b) H(A = 0) = H(xeq); thus entropy S and enthalpy H are functions of state;4 Fig.1. An important first differential of the Gibbs energy G is with respect to the amount of substance j namely (dG/dnj) at fixed T p and ni ­ j. This important partial differential is the chemical potential of chemical substance j. We return to eqn. (4-1) and consider a freeze-frame description of the system at defined T and p prepared with composition n0; the superscript ‘0’ refers to time zero. Spontaneous chemical reaction occurs driven by the affinity for spontaneous change. We freeze-frame the system where the composition is x and the affinity equals A. This system is perturbed by adding dnj moles of substance j one of the i-substances. The analogue of eqn. (4-2) has the following form (at defined T and p) eqn.(4-4). æ ¶ ö è ç ¶ ø ÷ æ ¶ è ç ¶ ö ø ÷ - ö ø ÷ = ¶ æ è ç We consider the case where the system being perturbed was at equilibrium where ‘A = 0’ G is a minimum and significantly (dG/dx) is zero. Therefore eqn. (4-5) holds. mj(A = 0) = (¶G/¶nj)A = 0 = (¶G/¶nj)xeq = mj(xeq) (4-5) In other words the chemical potential mj for substance j in this system at ‘A = 0’ equals the chemical potential mj for the condition x = xeq. Thus frozen (instantaneous) and equilibrium chemical potentials are equal placing chemical potentials on a par with the function of state H V and S in the thermodynamic hierarchy these being first derivatives of the Gibbs energy; Fig. 1. Hammett reached the same conclusion by noting in his terms that ‘sophisticated’ and ‘primitive’ chemical potentials5 are equal.In the context of describing the properties of solutions Grunwald uses the terms6 isodelphic to describe the partial molar properties of substance j when addition of dnj moles of solute j does not change the organisation of the solvent network; lyodelphic describes the difference between isodelphic and equilibrium partial molar properties. The lyodelphic contribution to the chemical potential of solute j is zero [cf. eqn. (4-4)]. In other words the equilibrium chemical potential of solute j is equal to the isodelphic chemical potential.7 In a similar context Ben-Naim8 uses the terms (4-2) x A p p (4-4) è ç ¶x ¶ ö æ ø ÷ è ç ö ø ÷ nj x x ‘freeze’ and ‘release’ in treating the thermodynamics of aggregation.5 Chemical equilibria Using an approach based on Henry’s Law the chemical potential of a neutral solute j [e.g. urea(s)] in aqueous solution is related to the molality mj using eqn. (5-1) where gj is the activity coefficient; m0 = 1 mol kg21. mj(aq;T;p) = mj(aq;id;mj = m0;T;p) + RTln(mjgj/m0) (5-1) By definition limit (mj ? 0; mi ? 0) gj equals 1.0 at all temperatures and pressures; mj(aq;id;mj = m0;T;p) is the chemical potential of solute j in an ideal solution (gj = 1.0) having unit molality. Consider an aqueous solution prepared using n0 Granted that mj(aq;T;p) is the same for frozen xeq and equilibrium properties the question arises—does this same condition apply to mj #(aq;T;p) [·mj(aq;id;mj = m0;T;p)]? x moles of chemical substance X where again superscript ‘0’ indicates at time zero.Experimental evidence indicates that two chemical substances X(aq) and Y(aq) exist in chemical equilibrium (at G = minimum where A = 0 at defined T and p). Such experimental evidence often arises because the two solutes have quite different UV–VIS absorption spectra. At equilibrium eqn. (5-2) holds. (5-2) meq X (aq) = mY eq(aq) Eqn. (5-2) offers the link with the argument developed in conjunction with the chemical potential of substance X. For this aqueous solution we would record the same increment in Gibbs energy G when dnX moles of substance X are added irrespective of whether only substance X is present in the solution or substance X is in equilibrium with substance Y.Using Hammett’s terminology5 the primitive and sophisticated chemical potentials of solute X are identical. Using the primitive description of the above system the chemical potential of substance X(aq) is related to the composition of the solution using the following equation [cf. eqn. (5-1)] for fixed T and p eqn. (5-3). m Y X X(aq) = mX #(aq;prim) + RTln[m0 XgX(prm)/m0] (5-3) Using the sophisticated description of the solution wherein the composition is described by molalities meq(soph) and meq- (soph) combination of eqn. (5-1) and (5-2) yields eqn. (5-4) for the solution at fixed T and p. X m X #(aq;soph) + RTln[meq(soph)g X eq(soph)/m0] = m # Y(aq;soph) + RTln[mY eq(soph)g r eq/m0] eq(soph) (soph)g X X eq Y K = m (soph)g Y eq(soph)/meq (5-4) By definition equilibirum constant K is given by eqn.(5-5). (5-5) = {1 + X X X X X Y But m0 = [mY eq(soph) + meq(soph)] so that m0 [Kg X eq(soph)/g eq(soph)]} meq(soph)g eq(soph). Then using the formulations for the chemical potential of substance X in solution we obtain eqn. (5-6). X mX (5-6) # (aq;prim) = m# (aq;soph) 2RTln[{1 + [Kg eq(soph)/g eq(soph)]}/gIRX(prim)] Y X In the limit that the solutions have ideal properties under both descriptions we obtain eqn. (5-7). (5-7) #(aq;prim) = m#(aq;soph)2RTln[1 + K] X mX Therefore the reference chemical potentials for a given solute are not identical. Consequently all other reference partial molar quantities (e.g.limiting partial molar volumes and limiting partial molar enthalpies) characterising solutes in sophisticated and primitive descriptions also differ. 6 Limiting partial molar properties and excess properties According to eqn. (5-1) the chemical potential of solute j in a solution molality mj is related to the activity coefficient gj and j a reference chemical potential m#(aq) at fixed temperature and pressure. In the event that the solution is ideal such that there are no solute–solute interactions gj = 1.0 and the chemical potential for solute j mj(aq;id) is given by eqn. (6-1). (6-1) mj(aq;id) = mj #(aq) + RTln (mj/m0) Interestingly in the limit that mj tends to zero mj(aq;id) [and mj(aq) for real solutions] tends to minus infinity.9 This means that solute j is increasingly stabilised as the solution becomes more dilute.This is the thermodynamic reason for the problems faced by industries which require solvents having very high purity. To remove the last trace of solute presents an awesome task. Eqn. (5-1) and (6-1) are important equations because they provide the basis for equations which describe the dependences on composition of other partial molar properties. For example the partial molar volume Vj(aq) of solute j is given by the partial derivative (¶mj/¶p)T. Thus from eqn. (5-1) for simple solute j we obtain eqn. (6-2). (6-2) V j j j(aq) = Vj #(aq) + RT(¶lngj/¶p)T Therefore from the definition of gj limit (mj?0) Vj(aq) equals V#(aq) which we may also write as VH(aq) the limiting partial molar volume.We make one further point in this connection because it is always advisable to examine these and comparable equations in terms of what happens in certain limits.9 For example the partial molar entropy Sj of solute j is given by the partial derivative 2(¶mj/¶T)p. Then eqn. (6-3) holds. (6-3) Sj(aq) = Sj #(aq)2Rln (mjgj/m0)2RT(¶lngj/¶T)p j Therefore from the definition of gj limit (mj?0) Sj(aq) equals + H. In other words the term SH(aq) has no practical meaning whereas Vj H(aq) does. As preparation for some of the subject matter discussed in the following sections we set down the basis of a definition for the excess Gibbs energy of an aqueous solution containing the single solute chemical substance j.For a solution with ideal properties (at fixed T and p) the chemical potential of solute j is given by eqn. (5-1) with gj = 1.0. Then eqn. (6-4) holds. (6-4) E(aq) = mj(aq)2mj(aq;id) = RTlngj j (6-5) m1(aq) = m1 *(†)2fRTM1mj (6-6) 7 Volumes V A V x ¶ (7-1) m For the solvent water in this solution the chemical potential m1(aq) is related to mj and the practical osmotic coefficient f using eqn. (6-5) where m1 *(†) is the chemical potential of water at the same time T and p eqn. (6-5) applies. For an ideal solution f = 1 and gj = 1. The excess Gibbs energy is defined by eqn. (6-6) for a solution prepared using 1 kg of solvent and mj moles of solvent. GE = mjRT[12f + lngj] GE is an intensive property of the solution because it refers to a thermodynamic property of a solution prepared using a fixed mass of solvent.The link between the dependence of f and gj on molality mj is provided by the Gibbs–Duhem equation. We consider a solution prepared using n In Section 3 we commented on the significance of equilibrium and frozen partial molar properties using volumetric properties of solutions as examples. Here we take up the story again but develop the argument along the lines given in Section 5. The broad sweep of our analysis is set out in Fig. 2 starting with the function of state volume V. j moles of substance 1 moles of water (†). The analogue of eqn. j (e.g. urea) and n (4-2) describes the change in volume dV when dnj moles of chemical substance j are added to the solution (i) at constant affinity A and (ii) at constant extent of reaction x both at constant T p and n1 eqn.(7-1). V j ¶ ¶A ¶ ¶n ¶ ¶n ¶ ¶n ¶x j j ö ø ÷ æ è ç ö ø ÷ ö æ ø ÷ è ç ö æ ø ÷ - è ç ö æ ø ÷ = è ç A x x nj nj æ è ç Chemical Society Reviews 1998 volume 27 75 Fig. 2 n But (¶V/¶x) j is not zero and so the equilibrium partial molar volume of substance j Vj(A = 0) is not equal to the frozen partial molar volume Vj(xeq). We recall our discussion in Section 2 of the volumetric properties of urea(aq). The volume of an aqueous solution comprising n1 moles of water and nj moles of solute j is related to the equilibrium partial molar volumes Vj(aq) and V1(aq) using eqn.(7-2). (7-2) V(aq) = n1V1(aq) + njVj(aq) However an alternative form of eqn. (7-2) defines V(aq) in terms of the apparent molar volume10 f(Vj) eqn. (7-3). (7-3) (7-4) V(aq) = n1V1 *(†) + njf(Vj) If the properties of the solution are ideal eqn. (7-4). V(aq;id) = n1V1 *(†) + njf(Vj)H Here f(Vj)H [ = Vj H(aq)] is the limiting (indinite dilution) partial molar volume of solute j. Another form of eqn. (7-3) describes the volume of a solution prepared using 1 kg of water. So we have eqn. (7-5) where eqn. (7-6) holds. (7-5) (7-6) V(aq) = (1/M1)V1 *(†) + mjf(Vj) V(aq;id) = (1/M1)V1 *(†) + mjf(Vj)H Therefore the excess volume VE is given by eqn. (7-7). (7-7) VE(aq) = mj[f(Vj)2f(Vj)H] An advantage of the latter equation is that VE is an intensive variable.Eqn. (7-3) forms the basis of the experimental j). The mass of the aqueous solution w determination of f(V equals (w1 + wj) or (w1 + nj Mj) where Mj is the molar mass of the solute. The densities of the solution and the solvent are r(aq) 1 *(†) respectively. Hence11a eqn. (7-8) [ = w/V(aq)] and r holds. (7-8) f(Vj) = (mj)21[r(aq)212r1 *(†)21] + (Mj/r) Eqn. (7-8) does not determine the dependence of f(Vj) on mj; the dependence is characteristic of the solution. In fact the form of Chemical Society Reviews 1998 volume 27 76 2 this dependence12 is not defined by thermodynamics although in many cases the dependence of f(Vj) on mj for dilute solutions is linear.[For salt solutions the Debye–H�uckel limiting law prompts analysis in the form of a dependence of f(Vj) on (mj)1.] 8 Isochoric conditions An interesting set of independent variables defines the Helmholtz energy of a system F; cf. eqn. (1-1) eqn. (8-1). (8-1) F = F[T,V,ni] Then all spontaneous processes under isochoric–isothermal conditions lower spontaneously the Helmholtz energy of a closed system. We do not develop this point further except to note that in contrast to the set of independent variables given in eqn. (1-1) and (4-1) the set in eqn. (8-1) involves two extensive variables volume and amounts ni. The isochoric condition has aroused controversy13–16 in analysis of kinetic data with respect to the calculation of isochoric activation parameters.14,17,18 The controversy centres on answers to the simple question—which volume is held constant?13 The issue remains unresolved.9 Isobaric expansions and isobaric expansibilities We return to a consideration of thermodynamic variables defined by the set of independent variables specified in eqn. (1-1) and (4-1). The volume of an aqueous solution having defined composition (e.g. n1 moles of water and nj moles of solute j) depends on temperature at fixed pressure. There are two limiting ways in which the volume of this solution may change as a result of a change in temperature; (a) at constant affinity A and (b) at constant extent of reaction/organisatn x. These two limiting changes are related (at defined T and p) eqn. (9-1).(9-1) A T T æ V ö ¶ è ç ¶T A T x x T æ ¶ ¶ ö ¶ ö æ ø ÷ = æè ç ¶ V öø ÷ - æè ç ¶ Aöø ÷è ç ¶ x ø ÷ è ç V ø ÷ ¶ ¶x In particular case of a system at equilibrium we identify two limiting expansions; the equilibrium isobaric expansion Ep(A = 0) [ = (¶V/¶T)p;A = 0] and the frozen isobaric expansion Ep(xeq) [ = (¶V/¶T)p;xeq]. Further for the condition ‘A = 0’ then (¶V/¶T)p;x equals1f T (¶H/¶x)T,p. Hence eqn. (9-2). Ep(A = 0) = Ep(xeq)2T21 (¶x/¶A)T,p (¶V/¶x)T,p (dH/dz)T,p (9-2) Equilibrium isobaric thermal expansions of aqueous solutions can be directly measured dilatometrically.19 The analogue of eqn. (7-1) in which enthalpy H replaces volume V is a key equation with respect to the temperature-jump fast reaction technique.20 Although at equilibrium (¶A/¶x)T;p is negative the sign of the product (¶V/¶x)T;p (¶H/¶x)T;p is not predetermined.Therefore the sign of Ep(A = 0) is not fixed. In fact for water below 277 K at ambient pressure the temperature of maximum density (TMD) Ep(A = 0) is negative;21 a similar feature is shown for many aqueous solutions22,23 in the region of 277 K. As written in eqn. (9-2) Ep(A = 0) and Ep(xeq) are extensive variables but not functions of state because they characterise pathways. We divide by the volume an extensive function of state (see above) to define an isobaric equilibrium expansibility a(A = 0) [ = V21 Ep(A = 0)] and an isobaric frozen expansivity a(xeq) [ = V21 Ep(xeq)] two volume intensive properties of a solution eqn.(9-3). a(A = 0) = a(xeq)2(VT)21(¶x/¶A)T,p(¶V/¶x)T,p(¶H/¶x)T,p (9-3) In these terms a(xeq) represents the volumetric response of a system to a thermal shock as the temperature is increased in an infinitesimal time. The extended product term describes the relaxation of the system24 to the state characterised by a(A = 0). For the aqueous solution described in eqn. (7-4) the isobaric thermal expansion is described by eqn. (9-4). N 0) E (A 1 p = = æè ç ¶V öø ÷ ¶T P;A=0 = = m EEp 1)E1 *(†;A = 0) + mjf(Ej)H j[f(Ej)2f(Ej)H] j[f(Vj)/V(aq)][f(Vj)]21f(Ej) ap(A = 0)V = n1V1 *(†)a1 *(A = 0) + njf(Ej) ö ø ÷ We define an apparent equilibrium molar isobaric expansion of solute j f(Ej) by the partial differential (¶f(Vj)/¶T)p;A = 0 an intensive property of solute j.Similarly for the solvent water E1 *(†;A = 0) equals [¶V1 *(†)/¶T]p;A = 0. Hence for a solution prepared using 1 kg of solvent where both E1 *(†;A = 0) and f(Ej) are intensive variables we have eqn. (9-5). Ep(A = 0;aq;w1 = 1 kg) = (1/M1)E1 *(†;A = 0) + mjf(Ej) (9-5) For the corresponding ideal solution we have eqn. (9-6). Ep(A = 0;aq;w1 = 1 kg;id) = (1/M The equilibrium partial molar expansion of solute j Ej(aq) and of solvent water E1(aq) are defined by [¶Vj(aq)/¶T]p and [¶V1(aq)/ ¶T]p. These quantities are normally calculated from measured dependences of Vj(aq) and V1(aq) [cf. eqn. (7-8) using f(Vj)] on composition at a series of fixed temperatures. For urea(aq) f(Ej) and f(Ej)H are positive4 at ambient pressure over the range 0 @ mj @10.0 mol kg21.For 2-methylpropan-2-ol(aq) in very dilute solutions,12 the dependence of Ej(aq) on mj and temperature is complicated. The second differential [¶2VH(aq)/¶T2]p has been used to classify solutes on the basis of their effect on water– water interactions.25 An (intensive—based on fixed mass of solvent) excess equilibrium isobaric expansion EE is defined by eqn. (9-7); cf. eqn. (6-6). EEp (A = 0) = Ep(A = 0;aq;w1 = 1 kg) 2Ep(A = 0;w1 = 1 kg;aq;id) Therefore eqn. (9-8) holds. In other words the excess expansion E (A = 0) is a welldefined property given by the product of solute molality mj and a difference in real and ideal apparent molar expansions. However a similar clear definition does not emerge if we turn our attention to expansibilities.The starting point is the definition given above for aT(A = 0) [ = V21Ep(A = 0)]. Then using eqn. (9-4) we have eqn. (9-9). ap(A = 0) = n1[V1 *(†)/V(aq)][V1 *(†)]21E* p(†;A = 0) + n If for the pure solvent ap *(A = 0) equals [V1 *(†)]21Ep *(A = 0;†)] then the isobaric expansibility of the solution ap *(A = 0) is given by eqn. (9-10). ap(A = 0) = n1[V1 *(†)/V(aq)]ap *(†) + [nj/V(aq)]f(Ej) (9-10) The property of the solution ap(A = 0) is given by an equation which only partly resembles eqn. (7-3) with the added complexity of a volumetric ratio [V1 *(†)/V(aq)]. There is no obvious quantity which could be described as an apparent molar isobaric expansivity of the solute j.Consequently there is no obvious route leading in an elegant manner to an excess isobaric expansibility of the solution; cf. eqn. (9-8). We end this section by returning to eqn. (9-9) written as eqn. (9-11). The latter equation forms the starting point for the derivation of eqn. (9-12) which shows how f(Ej) is calculated using measured p(A = 0) for a solution molality mj. a f(Ej) = [mjr(aq)r1 *(†)]21 {[r1 *(†)ap(aq;A = 0)] 2[r(aq)a1 *(†;A = 0)]} + [a(aq)Mj/r(aq)] This equation closely resembles eqn. (7-9) and is a member of the same family of volumetric equations (see below). (V ) f ¶ (l) j * 1 + nj ¶V ¶T ¶T æ è ç p;A+0 P;A=0 j æ è ç p Ep ö ø ÷ (9-4) (9-6) (9-7) (9-8) (9-9) (9-11) (9-12) 10 Isothermal compression and compressibility In addition to isobaric expansion the other major component of Fig.2 describes compressions and compressibilities of solutions. There are two limiting ways in which the volume of a solution may change as a result of a change in pressure (at fixed temperature); (i) at constant affinity A and (ii) at constant extent of reaction/organisation x. Thus at fixed temperature V (10-1) ¶A ¶p ¶p æ ¶V è ç ¶ p p A x p The partial differential (¶A/¶p)T;x equals1g 2(¶V/¶x)T;p. In the case where the system was at equilibrium the partial differential 2(¶V/¶p)T;A = 0 is the equilibrium isothermal compression KT(A = 0) whereas 2(¶V/¶p)T;xeq is the frozen compression K K (10-2) k (10-5) x T(xeq).Hence from eqn. (10-1) we have eqn. (10-2). T(A = 0) = KT(xeq)2(¶x/¶A)T;p eq [¶V/¶x)T;p eq ]2 The compressions KT(A = 0) and KT(xeq) are extensive properties of a solution. The volume intensive properties are isothermal compressibilities kT(A = 0) and kT(xeq) defined by eqn. (10-3) and (10-4). T(A = 0) = 2(1/V)(¶V/¶p)T;A = 0 = + V21KT(A = 0) (10-3) (10-4) T;A=0 ÷ è æ ¶V*(l)ö = -n ç j (10-6) x ø ÷ ö = æ ¶V ö ø ÷ - æ ¶Aö æ ¶ ö æ ¶ ö è ç ø ÷ è ç è ç ø ÷ è ç ø ÷ ¶x 1 1 ¶p ø T;A 0 = = k T(xeq) = 2(1/V)(¶V/¶p)T;xeq = + V21KT(xeq) Then from eqn. (10-2) we have (10-5). kT(A = 0) = kT(xeq)2(¶x/¶A)T;p eq [(¶V/¶x)T;p eq ]2 But (¶A/¶x)T;p is negative for all stable phases.Hence irrespective of the volume of reaction (¶V/¶x)eq T;p kT(A = 0) is always greater than kT(xeq). In fact eqn. (10-5) is the key equation for the pressure-jump fast reaction technique,20 the second (large) term on the right-hand side of eqn. (10-5) being the relaxational term. For an aqueous solution the isothermal equilibrium com- T(A = 0) is related to the isothermal differential of pression K eqn. (7-3) with respect to pressure eqn. (10-6). K (A T = 0) = -( ¶V / ¶p) K = m E T - nj T (10-9) (10-10) æ ¶f(V )ö è ç¶p ø ÷ T;A 0 Thus KT(A = 0) is an extensive property of a solution. It is convenient to define an equilibrium apparent molar isothermal compression f(KTj) { = 2(¶f(Vj)/¶p)T;A = 0}.For an aqueous solution prepared using 1 kg of solvent we relate the intensive compression KT(A = 0;aq;w1 = 1 kg) to the composition using eqn. (10-7). KT(A = 0;aq;w1 = 1 kg) = (1/M1)K* 1T(†;A = 0) + mjf(KTj) (10-7) For the solution whose properties are ideal eqn. (10-8) applies. KT(A = 0;aq;w1 = 1 kg) = (1/M1)K* 1T(†;A = 0) + mjf(KTj)H (10-8) Then the excess compression KE is defined by eqn. (10-9). j[f(KTj)2f(KTj)H] Interestingly the form of eqn. (10-9) resembles those for the excess volumes [eqn. (7-8)] and excess isobaric expansions [eqn. (9-8)]. The general form of eqn. (10-7) describes the extensive compression of a solution containing nj moles of solute j and n1 moles of water eqn. (10-10). KT(A = 0) = n1K1T(†;A = 0) + njf(KTj) Calculation of f(Ej) from compressibilities uses eqn.(10-11) which is derived in a manner analogous to that used to obtain eqn. (9-12). Chemical Society Reviews 1998 volume 27 77 f(KTj) = [mjr(aq)r1 *(†)]21{[r1 *(†)kT(aq;A = 0)] 2[r(aq)k1 *(†);A = 0)]} + [kT(aq);A = 0)Mj/r(aq)] (10-11) Further just as for expansibilities an elegant equation similar to that used for excess volumes [eqn. (7-8)] cannot be used to define an excess isothermal compressibility. Similarly there is no analogue of a partial molar volume which can be identified as a partial molar compressibility. Direct measurement of isothermal compressibilities of aqueous systems is nevertheless not straightforward bearing in mind that k(aq) depends on composition temperature and pressure.11 Isentropic compression and isentropic compressibilities Data describing equilibrium isothermal compressibilities of aqueous solutions are not extensive. It may at first sight seem surprising therefore that information concerning equilibrium isentropic compressibilities (i.e. compression at constant entropy kS) is more extensive. The reason for this state of affairs is the Newton–La Place equation relating isentropic compressibility to the density of a solution r and the velocity of sound u in the solution; kS = (u2r)21. If the frequency of the sound wave is low (e.g. in the MHz range) the calculated quantity is the equilibrium isentropic compressibility kS(A = 0). In other words each microscopic volume of a solution is compressed at constant entropy.Moreover these compressibilities can be precisely measured.26,27 The isentropic condition raises problems from the standpoints of both thermodynamic theory and the interpretation of derived parameters. From the outset we have to change the basis of the thermodynamic treatment. Thus in reviewing equilibrium isobaric expansions Ep(A = 0) and isothermal compressions KT(A = 0) the analysis developed in a straightforward manner from the function of state called volume defined using eqn. (1-1). In the latter case our interests centred on the T–p composition domain for which the thermodynamic potential function is the Gibbs energy. In order to understand the significance of isentropic compressibilities kS and isentropic compressions KS we switch interests into the S-p-composition domain.This is not a trivial switch. All spontaneous processes in closed systems at fixed entropy and pressure lower the enthalpy H of a system such that thermodynamic equilibrium corresponds to a minimum in enthalpy where the affinity for spontaneous change is zero. Thus the enthalpy of a closed system is defined1e by eqn. (11-1). (11-1) H = H [S p x] The volume of the system is given by the partial differential (¶H/¶p)S;x. The analogue of eqn. (1-1) is eqn. (11-2). (11-2) V = V [S p ni] In other words we have switched the set of independent variables defining the volume from [T p ni] to [S p ni] a switch from ‘Lewisian’ to ‘non-Lewisian’ independent variables. 28 The key point to note is that in one set T and p are both intensive variables whereas in the other set [cf.eqn. (11-1) and (11-2)] the independent variable S is extensive making two extensive variables in this definition. This contrast between the two sets is not trivial indicating that the choice of independent variables is more than a matter of convenience. Two equilibrium quantities are of interest in this section; (i) the equilibrium isentropic compression KS(A = 0) and (ii) the equilibrium isentropic compressibility kS(A = 0). The corresponding instantaneous properties are KS(xeq) and kS(xeq). The difference between these equilibrium and instantaneous properties is at the heart of the ultrasonic fast reaction technique.20,29 At low rates of compression (i.e.change in pressure) at constant entropy solvent–solvent solvent–solute and solute–solute interactions within each microscopic volume of an aqueous solution change in order to keep the system at a minimum in the enthalpy. The differential dependence of volume V on pressure at constant temperature and affinity is related to the differential Chemical Society Reviews 1998 volume 27 78 dependence of volume V on pressure at constant entropy and affinity using eqn. (11-3). V T S æ V (11-3) ¶ ¶T ¶ ¶p p ¶S ¶p ö ø ÷ ¶ ö ø ÷ æ è ç ö ø ÷ - æ è ç ö ø ÷ ö ø ÷ p;A p;A T;A T;A S;A æ è ç T;A = 0 (11-8) = æ V ¶ è ç p(aq;A = 0)V(aq)]2T/Cp(A = 0;aq)} (11-9) ¶ è ç¶ We use eqn. (11-3) with the constraint that constant affinity A is at ‘A = 0’.The last partial derivative in eqn. (11-3) is therefore the isobaric equilibrium expansion Ep(A = 0); cf. eqn. (9-2). A Maxwell relationship1g shows that (¶S/¶p) equals 2(¶V/¶T)p;A = 0 which is 2Ep(A = 0). Further the partial differential (¶S/¶T)p;A = 0 equals1h the ratio of the equilibrium isobaric heat capacity to the temperature Cp(A = 0)/T. Moreover Ep(A = 0) equals ap(A = 0)V; cf. eqn. (9-3). Therefore,30 KS(A = 0) = KT(A = 0)2{[ap(A = 0)V]2T/Cp(A = 0)} (11-4) Thus by definition the equilibrium isentropic compression KS(A = 0) equals 2(¶V/¶p)S;A = 0. In eqn. (11-4) KS(A = 0) KT(A = 0) V and Cp(A = 0) are extensive properties of the solution. The ratio [Cp(A = 0)/V] is the equilibrium isobaric heat capacity per unit volume of the solution s(A = 0).Then we have eqn. (11-5). KS(A = 0) = KT(A = 0)2{[ap(A = 0)]2VT/s(A = 0)} (11-5) Similarly in terms of isentropic compressibilities eqn. (11-6). kS(A = 0) = kT(A = 0)2{[ap(A = 0)]2T/s(A = 0)} (11-6) In the next stage we return to an equation for the volume of an aqueous solution prepared using n1 moles of solvent water and nj moles of solute j; eqn. (7-3). The isobaric compression KT(A = 0) is given by eqn. (10-6) in terms of the equilibrium isobaric compression of the solvent K* 1T(†) and an equilibrium partial differential isothermal dependence of f(Vj) on pressure. Thus we have eqn. (11-7). KT(aq;A = 0) = n1K* 1T(†;A = 0)2nj[¶f(Vj)/¶p]T;A = 0 (11-7) The task at this stage is to write down a satisfactory equation for the isentropic equilibrium compression of the solution.In the context of treating the thermodynamic properties of binary liquid mixtures the way forward was signalled by Benson and Kumaran,31 by Reis28 and by Douh�eret Moreau and Viallard32 particularly in developing equations describing excess compressibilities and excess compressions. Here we comment on the equilibrium isentropic compressions of aqueous lutions. Thus it follows from eqn. (11-4) that for the pure solvent water eqn. (11-8). K* 1T(†;A = 0) = K* S1(†;A = 0) + {[a* 1p(†;a = 0)V* 1(†)]2T/C* p1(†;A = 0)} For the solution according to eqn. (11-4) we have eqn. (11-9). KT(aq;A = 0) = KS(aq;A = 0) + {[a In eqn.(11-8) K* 1T(†;A = 0) and K* S1(†;A = 0) refer to a mole of liquid water whereas in eqn. (11-9) V(aq) and Cp(A = 0;aq) are extensive properties of the solution described in eqn. (11-7) having compression KT(aq;A = 0). The analytical problem is highlighted by eqn. (11-8) and (11-9). The isentropic condition on K* S1(†;A = 0) is understandable in terms of the condition associated with compression by the sound wave. The isothermal conditions on K* 1T(†;A = 0) and KT(aq;A = 0) are understandable in terms of the isothermal condition associated with measuring the dependence of volumes on pressure. Thus we can arrange experimentally for these two temperatures to be the same in order to compare the isothermal compressions of water and an aqueous solution having defined molality.Unfortunately we cannot ensure in a comparison of K* 1S(A = 0) for water and KS(aq;A = 0) for an aqueous solution that the two S entropies are the same. In each case the compression is isentropic (cf. the Newton–Laplace equation) but we cannot be sure that the properties of solution and solvent are compared at the same entropy. Furthermore in examining the dependence of KS(aq;A = 0) on the composition of a solution (e.g. on molality of solute j) we cannot be sure that comparisons can be made of the properties of these solutions at the same entropy. There is merit in defining an apparent equilibrium isentropic compression of solute j f(K j) in terms of 2[¶f(Vj)/¶p]S;A = 0. Similarly f(KTj) = 2[(¶f(Vj)/dp]T;A = 0.Granted therefore that experiment yields kS(aq;A = 0) for an aqueous solution (at defined T and p) and k* S1(†;A = 0) for the pure solvent water eqn. (11-10) yields the apparent molar isentropic compression of the solute f(KSj). (11-10) f(KSj) = [1/mjr* 1(†)][kS(aq;A = 0)2k* S1(†;A = 0)] + kS(aq;A = 0)f(Vj) Indeed this is a classic equation used by many authors who cite as the key reference the monograph by Harned and Owen,12b confidence being boosted by the strong similarity with equations for f(Vj) f(Epj) and f(Kpj) as described in the previous sections. So common practice has been to use eqn. (11-10) as a method of determining the apparent property of f(KSj) from measured kS(aq;A = 0) for a solution molality mj. S Then the patterns which emerge are discussed in the conceptually simpler context of isothermal compression.Lara and Desnoyers note33 that for 2-butoxyethanol(aq) at 298 K ‘isothermal and isentropic compressibilities are quite similar and reflect the same kind of interactions’. Franks and coworkers34 discuss the dependence of f(Kj) on composition and structure for various sugars in aqueous solutions in terms of solute-hydration and solute–solute interactions although the measured quantity was f(K j). In the context of aqueous salt solutions for example Criss and co-workers35 compare f(KSi)H for ion i with the trend predicted by the Born equation for the isothermal property. Quite generally therefore measured isentropic properties are understood in terms of models based on isothermal properties.This approach has obvious practical merit. Nevertheless this author has reservations concerning what seems a somewhat cavalier approach to the isentropic condition. Perhaps there is a need for a new approach to the task of understanding the significance of isentropic compressibilities of solution. 12 Acknowledgements I thank Professors H. Høiland (University of Bergen) and J. B. F. N. Engberts (University of Groningen) for valuable discussion. Also my long-suffering Secretary Vikki who has seen this same paper (with alterations) more times than she cares to remember. 13 References 1 I. Prigogine and R. Defay Chemical Thermodynamics trans. D. H. Everett Longmans Green London 1954 (a) p. 3; (b) p. 67; (c) p.38; (d) p. 266; (e) p. 52; (f) p. 59; (g) p. 54; (h) p. 48. 2 F. Van Zeggeren and S. H. Storey The Computation of Chemical Equilibria Cambridge University Press Cambridge 1970. 3 J. Bach M. J. Blandamer J. Burgess P. M. Cullis L. G. Soldi K. Bijma J. B. F. N. Engberts P. A. Kooreman A. Kacperska K. C. Rao and M. C. S. Subha J. Chem. Soc. Faraday Trans. 1995 91 1229. 4 R. H. Stokes Aust. J. Chem. 1967 20 2087. 5 L. P. Hammett Physical Organic Chemistry McGraw-Hill New York 1970 2nd edn. p. 16. 6 E. Grunwald J. Am. Chem. Soc. 1984 106 5414; 1986 108 1361; 5726. 7 M. J. Blandamer J. Burgess A. W. Hakin and J. M. W. Scott Water and Aqueous Solutions ed. G. W. Neilson and J. E. Enderby Colston Papers No. 37 Adam Hilger Bristol 1986. 8 A.Ben-Naim Hydrophobic Interactions Plenum Press New York 1980 p. 130. 9 J. E. Garrod and T. M. Herrington J. Chem. Educ. 1969 46 165. 10 See for example G. N. Lewis and M. Randall Thermodynamics revised by K. S. Pitzer and L. Brewer McGraw-Hill New York 1961 2nd edn. 11 H. S. Harned and B. B. Owen The Physical Chemistry of Electrolytic Solutions Reinhold New York 1958 3rd edn. (a) p. 358; (b) p. 376. 12 F. Franks and H. T. Smith Trans. Faraday Soc. 1968 64 2962. 13 M. J. Blandamer J. Burgess B. Clark R. E. Robertson and J. M. W. Scott J. Chem. Soc. 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Høiland and E. Vikingstad J. Chem. Soc. Faraday Trans. 1 1976 72 1441. 28 J. C. R. Reis J. Chem. Soc. Faraday Trans. 2 1982 78 1595. 29 M. J. Blandamer Introduction to Chemical Ultrasonics Academic Press London 1973. 30 G. Douh�eret and M. I. Davis Chem. Soc. Rev. 1993 43. 31 G. C. Benson and M. K. Kumaran J. Chem. Thermodyn. 1983 15 799. 32 G. Douh�eret C. Moreau and A. Viallard Fluid Phase Equilibria 1985 22 277 289; 1986 26 221. 33 J. Lara and J. E. Desnoyers J. Soln. Chem. 1981 10 465. 34 F. Franks J. R. Ravenhill and D. S. Reid J. Soln. Chem. 1972 1 3. 35 J. I. Lankford W. T. Holladay and C. M. Criss J. Soln. Chem. 1984 13 699. Received 20th June 1997 Accepted 5th August 1997 79 Chemical Society Reviews 1998 volum