-
-
This can be verified by evaluating T (aUlaS), and P -(aU/aVjs and comparing the results with familiar expressions. Irreversibility may arise in many different ways-irreversible beat flow, friction in the system, redistribution of energy (relaxation),etc., and toall the same magical inequality (eq 1)applies. Illustrations of this and of the difference between du and d U are afforded by the following: Consider a grandfather clock, with its weights pulled high, as a material system in isothermal quasi-equilibrium with its surroundings; its volume is constant (dV = 01, and we assume it to run at constant temperature so that its entropy remainsconstant (dS = 0). The clock is not inits equilibrium state, however, and as it irreversibly ticks away, sounding the passing hours, its actual energy u decreases and eventually reaches its minimum while the state function U(S,V) (which includes neither the gravitational excess potential energy of the weights nor the vibrational kinetic and potential energies of the pendulum) remains constant. The correct inequality, du < T*dS - P*dV = 0, is satisfied (du < 01, hut the incorrect inequality, d U < TdS - P d V is nut, because both sides are zero. Under certain conditions of hardness, strength, and thermal conductivity, a rock rolling down an isothermal mountain side would represent a similar situation. Both clock and rock offer possibilities for differing definitions of u and U that we shall only mention. We have taken u to be the total energy, including the bulk kinetic energy and the excess of gravitational potential energy above the minimum of the final equilibrium state, whereas U has been taken as the internal energy (cf, e.g., Planck,I p 48). Possible different choices are whether or not to include the gravitational potential andlor the hulk kinetic energy in u and/or U; correct, but in general different, expressions result that, however, always include the inequality for the du but sometimes (as ahove) do not for dU. Finally, consider the isothermal&ansion of a perfect gas from P I . V , toP1.V2 at constant temperature T'. The change in S will be R In V d V , and both u and C'will be constant, no matter whether the ex~aasionis rewrsible (with P = P' and varying throughout thk process) or not. As an example of an irreversible expansion, let there be a partition inside a box, all the gas being initially on the left side of the partition (volume V,) and gradually, through a tiny hole in the partition, filling the entire box (volume VZ);Pleftgradually falls from PI to Pz while Pright rises from 0 to Pz; the system volume (Vz) does not change, so that P* is immaterial. The inequality du < T*dS - P'dVis then correct, with du = 0, T*dS > 0, and P X d V= 0, but the statement d U < TdS PdV is devoid of meaning, since P and U(S,VJ refer to a homogeneous state, whereas in intermediate stages the gas is heterogeneous and (for the whole system) neither T nor P is defined. The usual expressions
to which we now turn, are similarly a t fault whenever the inequality prevails: wrong because T and P m u s t give way to T* and P*,and ambiguous and confusing because it must be understood that dH, dA, and dG are not the differentials of the state functions H U PV, A U - TS, and G U- T S PV, but rather merely increments of h u P*V, a u T*S, and g I u - T*S P*V, which are no longer functions of equilibrium thermodynamic state, both because u isnot U and because T* andP* refer to the surroundines rather than the system. The correct ex~ressionfor the inexact differentials du. dh. da, and dg are
+
-
+
+
- - -+
-
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Journal of Chemical Education
du 5 T'dS - P*dV
(5)
dh = du
+ P*dV + VdP* 5 T * ~ +S VdP*
(6)
da = d v
- T'dS - SdT* 5 -SdT* - P d V
(7)
dg = da + P*dV
+ VdP'
5 -SdT'
+VdP
(8)
With the indicated restrictions, these expressions can be integrated: Au 5 0
S,V const.
(9)
Ah = Au
+ P'AV
50
SP' const.
(10)
Aa = Au
-PAS 50
T*,V const.
(11)
Finally, in this listing of thermodynamic inequalities with proper attention to the distinctions that must be drawn between T and T*, etc., and between U and u, etc., we include the following important standard relations: u [ S , v t U(S,V), h[S,P] t H(S,P), o [ T , v t A(T,V), g[T,P] t G(T.P)
Here such matters as including gravitational energy and bulk kinetic energy in u and Umust be handled consistently, so that for equilibrium states the equalities u = U, h = H, etc., will be preserved. The brackets, for example in h[S,P], are intended to specify that h is not a function of S and P* alone, with P* = P , but depends on them and on all the possibly enormous number of further parameters that would be needed to specify the actual nonequilihrium state of the system. (How marvelous is the equilibrium state!) Further, if work w' other than P,V work is done on the system (dw = -P*dV dw'), the right-hand sides of relations 5-8 are each increased by dw'and of relations 9-12 by w' to give, finally,
+
S,V const.
(13)
M=Au+P'AV5w'
S,P* const.
(14)
Aa = Au - F A S 5 w' = w
T*,V const.
(15)
Ta,P* const.
(16)
Au 5 w' = w
&=ALL-T*AS+PAVSw';
The conditions 11and 12 reduce to AA 5 0 and AG 5 0 if theinitialand final states are both equilibrium states with Ti = Ti = T* and with Ti = Tr = T* and Pi = Pi = P*, respectively. This is a correct derivation of these familiar criteria of thermodynamic possibility, in contrast to such usual derivations as incorrectly assume the system temperature to he uniform and constant, or both the system temperature and the system pressure to be uniform and constant, when in fact all that can properly be assumed are the conditions on T* or on T* and P* specified above. Indeed, for irreversible processes it is almost certain that the system temperature and pressure cannot be held uniform and constant, and sometimes the deviations will be extreme. We have made these points also in connection with a simpler derivation and discussion4 of relations 15 and 16. One may well doubt that h, a, andg (or the similar quantities that can he defined if the system is such that the expression for w includes other terms than just - P d V are ever really well suited to thermodynamic discussions of irreversible processes, that is, to discussions in which the familiar ineaualities occur. or whether we should s i m.~.l vuse u. U, S, T*,'P*, etc., instead. Clear notation is essential to clear thinking and understanding. Inadequate notations and consequent unclear writing afflict many presentations of thermodynamics in chemical textbooks.
