The SECOND LAW and ENTROPY 111. The Thermodynamic Functions
R. C . CANTELO University of Cincinnati, Cincinnati, Ohio
+
The criteria proposed i n the previous &be?, for reversible and irreversible processes, are transformed by simple algebra into equations and inequalities which involve the thermodynamic functions: the work function and the free energy function.
I
N THE second paper of this series' the writer proposed the following criteria: for a reversible process, the relation T d S = dQ; and for an irreversi' CANTELO, "The second law and entropy. 11. Irreversible
processes," J. Cmaa. EDWC., 10, 45-6 (Jan., 1933).
'
ble one, T d S = 69 E. That is, in an irreversible change the heat absorbed from the surroundings is less than corresponds to the increase in entropy. It is the purpose of the present paper to show how criteria for natural spontaneous (irreversible) changes can be obtained, for isothermal processes, in terms of certain derived functions of the entropy, namely, in terms of the so-called thermodynamic functions. The condition for a natural spontaneous change of a system from state A to state B may be written:
For an isothermal change, - this becomes: Q- = T ( S s - SA) Tn
-
(2)
Q- is the total quantity of heat absorbed during the change in state, and Q- = ( U s
-
UA)
+ W+
thermal vrocess the decrease in free enerw is greater than thework, E, obtained from the proce%. E = W - p AV, where W is the total work done by the system. H ~ in anv ~ natural ~ Drocess, ,
(3)
If the process is one in which the work done is due ( U , - U,) is the increase in the internal energy of the entirely to a change in volume at a constant pressure, system undergoing the irreversible change in state, and W = p AV, and W+ is the quantity of work given to the surroundings. -(AF)T.. > 0 (14) We have, now, the relation: We can evaluate - AA and - AF for any natural (UB-UA)-T(SB-SA)+W+=-TC (4) isothermal process, if we can devise a reversible isoor thermal process by means of which we can pass from (UB-TSs)-(UA-TS")+W+=-Tn (5) the initial to the final state of the system. For such The expressions in the brackets are evidently completely a reversible process, TdS = dQ, and the quantity Tc defined by the variables which determine the states vanishes from equation (2). Therefore, for a finite B and A . We shall write, therefore, the symbol, A, reversible process, from (7) and ( S ) , for U - TS. Then,
+ W+ = -Tu
( A e - A&
We saw in the previous paper that a is a positive quantity for all real changes. Therefore, equation (6) shows that for all real changes
+
AAT
that is
W+
w+ (8) The function, U - TS, is called the wmkfunctwn, A .
A is determined solely by the given state of the system, so that when we wish to regard A solely as a property of the system, we shall call it the work content. The inequality (8) shows that in any natural isothermal process the work done by the system is less than the decrease in work content. When the system does work by a change in volume against a constant external pressure, the work is equal to p(V, - VA),and this quantity, p(V, - V.J, depends solely upon the initial and final states of the system. Hence, when the system does mechanical work due to a change in volume against a constant pressure, and, in addition, does other kinds of work, as for example, electrical work, we can write for W+, p(VB - V A ) E, where E measures the other kinds of work. Then we can write for equation (5),
+
-
( U s - TSB) - ( U A TSn) + ~ ( V B - V A ) + E = -Te or (9) ( U B -TSB f PVB) - ( U A - TSA f PVA) E = -Tu (10)
-AAT = W
(15) (16)
This means that the decrease in the work content of a system undergoing any reversible isothermal change in state is equal to the maximum work done by the system in going from its initial to its final state. Since - AA, is defined solely by the initial and final states, the decrease in work content will always be the same for a given change in state, whether the change in state be brought about by a reversible path or by an irreversible path. Similarly, we can write, for a reversible change in state,
+
AFT,^ E = o AFT,^ = E = En
(17) (18)
This means that the decrease in the free energy of a system in going from an initial to a final state is always equal to the reversible work done by the system during the change in state, other than the work due to a volume change against a constant pressure. Thus, AFT,^ = W E
or -AFT, =
-
- AAT
PAV
(19)
- fi AV
(m)
If the work done by the system be entirely reversible mechanical work at constant pressure,
+
+
The function U - TS pV is defined solely by the variables defining the state of the system, for U, S, V are so defined and T and p are constant quantities. This function is called the free energy function, F, and when we wish to look upon it as a property of the system, we shall speak of it as thefree energy of the system. Then (FB FA^ f E = -Tm (11)
-
and for all real changes in a system -(AF)T,,
>
E
(12)
The inequality (12) shows that for any natural iso-
~ i h c e- AF,,, is defined solely by the initial and final states, the decrease in the free energy of a system undergoing a change in state will always be the same for the given change in state, whether the change in state be brought about by a reversible path or by an irreversible Even for a process in which no work whatsoever is done, - AA, and - AFT,, have exactly the same values as they have when the change in state is brought about reversibly. This is because A and F are determined exactly by the variables, ( X I ,X z , . . .X,)I and ( X I , Xe, .. . X J z , which define the initial and final states of the system.
