/ p) TA,. Returning to the

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1 Toic2610 Proerties; Equilibrium and Frozen A given closed system having Gibbs energy G at temerature T, ressure, molecular comosition (organisation ) and affinity for sontaneous change A is described by uation (a). [ ] G G T,, (a) In the state defined by uation (a), there is an affinity for sontaneous chemical reaction A. Starting with the system in the state defined by uation (a) it is ossible to change the ressure and erturb the system to a series of neighbouring states for which affinity remains constant. The differential deendence of G on ressure for the original state along the ath at constant A is given by ( G / ) TA,. Returning to the original state characterised by T, and, we imagine that it is ossible to erturb the system by a change in ressure in such a way that the system remains at fixed extent of reaction,. The differential deendence of G on ressure for the original state along the ath at constant is given by ( G / ) T,. We exlore these deendences of G on ressure at fixed temerature and at (i) fixed comosition, and (ii) fixed affinity for sontaneous change, A. The rocedure for relating ( G ) ( ) TA, and G T, is a standard calculus oeration. At fixed temerature, G G A G A A (b) This interesting uation shows that the differential deendence of Gibbs energy (at constant temerature) on ressure at constant affinity for sontaneous change does NOT ual the corresonding deendence at constant extent of chemical reaction. This inuality is not surrising. But our interest is drawn to the case where the system under discussion is, at fixed

2 temerature and ressure, at thermodynamic uilibrium where A is zero, d dt is zero, Gibbs energy is a minimum AND, significantly, ( G ) is zero. G G Hence V TA, 0 T, (c) The deendence of G on ressure for differential dislacements at constant A 0 and are identical. We confirm that the volume V of a system is a strong state variable. These comments seem trivial but the oint is made if we go on to consider the volume of a system as a function of temerature at constant ressure. We use a calculus oeration to derive uation (d). V V A V T T T A (d) A T T Again we are not surrised to discover that in general terms the differential deendence of V on temerature at constant affinity does not ual the differential deendence of V on temerature at constant comosition/organisation. Indeed, unlike the simlification we could use in connection with uation (b), {namely that at uilibrium ( G ) assume that the volume of reaction, ( V ) is zero} we cannot is zero at uilibrium. In other words for a closed system at thermodynamic uilibrium at fixed T and fixed {when A 0, and d dt 0 }, there are two thermal exansions, at constant A and at constant. We consider a closed system in uilibrium state I defined by { } the set of variables, TI [], A,, [] I 0. The uilibrium comosition is [] I at zero affinity for sontaneous change. This system is erturbed to two nearby states at constant ressure.

3 (a) State I is dislaced to a nearby uilibrium state II defined { } by the set of variables, TI [] + δt,a,, [ II] 0. This uilibrium dislacement is characterised by a volume change; ( ) [ ] [ ] V A 0 VII VI (e) At constant ressure we record the uilibrium thermal exansion; [ ] V[ I] V II E ( A 0) T (f),a 0 The uilibrium isobaric exansibility, ( A 0) E ( A 0) V α (g) In order for the system to move from one uilibrium state, I with comosition [] I to another uilibrium state, II with comosition [ II ], the system changes by a change in chemical comosition and/or molecular organisation. (b) State I, comosition [] I, is dislaced to a nearby state II where the comosition and/or molecular organisation is identical, [ II ] [] I. In other words the comositionorganisation is frozen in that of state I. But state II is not an uilibrium such that the affinity for sontaneous change A is not zero. The recorded volume change, [ 2,,, 0] [ 1,,, 0] V V T A V T A (h) Hence we define the frozen isobaric exansion, E ( fixed). An alternative name is the instantaneous exansion because, ractically, we would have to change the temerature at a such a high rate that there is no change in molecular comosition or organisation in the system. V E ( fixed) (i) T,

4 Further α ( ) 1 V fixed (j) V T Similar comments aly to isothermal comressibilities, κ T ; there are two limiting quantities κt ( A 0 ) and κt ( ). In order to measure κ ( ), T we have to change the ressure also in an infinitely short time. The entroy S is given by the artial differential, ( G T),. At uilibrium where A 0, S-( G T). A, 0 We carry over the argument described in the revious section but now concerned with a change in temerature. We consider the two athways, constant A and constant. ( G T),A ( G T), ( A) T, T), ( G ) T, But at uilibrium, A which uals ( G ) ( ) ( ) SA 0 uals S (k) is zero, and so. Then just as for volumes, the entroy of a system is not a roerty concerned with athways between states; entroy is a strong function of state. Another imortant link involving Gibbs energy and temerature is rovided by the Gibbs-Helmholtz uation. We exlore the relationshi between changes in ( GT ) at constant affinity A and at fixed, following erturbation by a change in temerature. [ ( G T) T],A [ ( G T) T], ( 1 T) ( A) T, T), ( G ) T, But at uilibrium, A which uals ( G ) ( ) ( ) (l) is zero. Then HA 0 H. In other words, the variable enthaly is another strong function of state. This is not the case for isobaric heat caacities.

5 ( H T ),A ( H T), ( A) T, T), ( H ) T, (m) We cannot assume that the trile roduct term in the latter uation is zero. Hence, there are two limiting isobaric heat caacities; the uilibrium isobaric heat caacity C ( A ) and the frozen isobaric heat caacity C ( ) 0. In other words, an isobaric heat caacity is not a strong function of state because it is concerned with a athway between states. Unless otherwise stated, we use the symbol C to indicate an uilibrium transformation, C ( A ) 0.