A thermodynamical substitution group

Physica XlII, no 6--7

A u g u s t u s 1947

A THERMODYNAMICAL

SUBSTITUTION

GROUP

by J. A. PRINS Lab. v. Techn, Physiea, T.H. Delft

An infinite number of thermodynamical relations for a system with two degrees of freedom (e.g. the pressure P and volume V or temperature T) follows from the mere existence of an entropy S and energy U satisfying T d S = d U + P d V . Introducing no new physical principles but only three mathematical auxiliary functions, F, W, G defined below, we get the following well known starting point : Energy U Enthalpy W = U + PV Free energy F = U - - T S FreeenthalpyG = U + PV--TS

dU dW dF dG

= TdS ~ PdV = TdS + VdP =--SalT--PdV ----SdT + VdP

|

i

(1)

Of the four differential equations three' are dependent on one to be chosen arbitrarily, as a consequence of the left-hand definitions. These definitions nfight have been written with any of the symbols G, W, F in stead of U as the fundamental symbol, e.g. u = F + T S and so on. In this way the complete system of functions and differentials exhibits a kind of symmetry which is no doubt, at least subconsciously, known to most workers on ,this subject, but which ~I have nowhere found stated explicitly in a precise form. See however note 2). On disregarding the signs we m a y express the symmetry by remarking that the four variables P, V, T, S can be interchanged, provided they are kept together in "conjugated" pairs P, V and T, S. These substitutions of the "primary" variables P, V, T, S induce simultaneous substitutions of the "secundary" variables U, G, W, F. It is the object of this note to formulate the complete group of these allowed thermodynamical substitutions. --- 4 1 7 Physica XIII

27

418

j . A . PRINS

The matter is complicated a little by the fact that we should not, of course, disregard the changes in sign. We have to include them in the complete group, wherein they form the subgroup : E.... A .... B .... C .... a

. . . .

.

.

.

UG

++

++ ++ + + + + + + + + + + + +

++ ++

++

b .... a

P V T S

.

. . . . .

--+

--+

+---+

+-+--

+

WF

(2)

+

The meaning of this table is e.g. in the case of substitution A that P and V m a y be changed simultaneously in sign without changing the original set of relations (l). In the case of a we have not only to change the signs of P and T but also of U, G, W and F to leave (1) invariant. As a consequence of leaving (1) invariant all these substitutions also leave true any true thermodynamical relation, as such relations m a y all be deduced from (1). This remark gains in importance because a large number of interchanges of variables also leaves invariant the original set (1). All these substitutions together form a group of 64 elements. The complete group may however be generated from those above and the eight below b y multiplications of type A I I I : E .... I ....

II . . III . . IV V ....

VI . . VII.

P --V P

V P V

T T --S

S S T

U G WF F W

WF U G G U

--V

P

--S

T

S T ST

--T --T

S S

G U

F

VP --PV

G U FW

WF UG

W

V P --P V

W F U G

G U F W

(3)

It m a y be noted that the right-hand variables keep together in pairs as well as the left-hand variables. If the group (2) in generating with (3) is restricted to E, A, B, C then the result is 32 substitutions which are all "even" with Iespect tot U, G, W, F., i.e. leave their signs positive. If we generate with a, b, a, 15 only the re-

A T H E R M O D Y N A M I C A L S U B S T I T U T I O N GROUP

41 ~)

sults will all be "odd". These 32 odd sttbstitutions do not form a group, as the 32 even substitutions do, but only a so called Cauchy set. As principal odd substitution we may consider: VIII

T S

P

V

--U--G

--F

--W

being the interchanging of the two pairs of conjugated primary variables P, V and T, S inducing a more complicated substitution of the secundary variables. Incidentally: viii--product aVlI. As an illustration of the substitution group we mention that all its operations transform the four equations of M a x w e 11:

into each other. A more substantial example is the following: If we have proved the formula for the Joule-effect :

U~

CV

we m a y apply e.g. substitution I and thus consider as also proved the equally well known formula for the Joule-Kelvin-effect:

--V+T-~p

The definition of the heat capacity C in these formulae must, of course, be taken to be TOS/OT. The substitutions might even be applied more or less to the equation of C 1 a p e y r o n for a heterogeneous system:

dP S' ~ S" dT-- V'-- V"' but this case requires some care as primed variables do not enter in our original conception (1). As a more general result of these considerations we remark that the distinction between "extensive" variables (V, S) and "intensive" variables (P, T), though physically important, is irrelevant for all therm0dynamicai calculations, as operations I - V I interchange the two kinds. This seems already to have been noted

420

j.A.

PRINS

for the magnetic variables/-/and J, the energy somethnes appearing as HdJ, sometimes as JdH. As is well known, our starting points

V ,

M i i

. . . . . p____~ ............

,

T

---1

0 O

Fig. 1. R e p r e s e n t a t i o n of the elements E, I, I I . . . . V I I b y fine segments • O

Fig. 2. R e p r e s e n t a t i o n of the complete t h e r m o d y n a m i c a l s u b s t i t u t i o n group. For the odd s u b s t i t u t i o n s • a n d O m u s t be i n t e r c h a n g e d .

A T H E R M O D Y N A M I C A L S U B S T I T U T I O N GROUP

42'1

m a y be extended to include magnetic effects b y adding a term HJ to the characteristic functions 1). These terms, as well as electric and other analogous terms, also extend our group in an obvious way. It should be noted that by transformation I V - V I I I the distinction between thermal and mechanical (or electrical or magnetic) variables is irrelevant as regards the thermodynamical formulism. Another general conclusion from our considerations is that all thermodynamical expressions m a y be classified according to how many different expressions arise from them when all substitutions of the group are applied. M a x w e 11 s equations for instance are four-valued equations, but the expressions figuring in them have more values. This point is to m y opinion of primary, imp6rtance for the question of a systematic nomenclature in thermodynamics. A labelling of magnitudes corresponding to the subgroups that leave them invariant seems to me possible and recommendable. The existence of a group not only puts this question of the functions associated with its subgroups but also the equally general question of the best way of "representing" the group..The matrixrepresentation of our group contains skew matrices as well as symmetrical ones and mixed ones, diagonal matrices as well as nondiagonal matrices. The reprentation of the substitutions b y cycles shows a remarkable number of interchanges. A geometrical representation, with the drawback that the composition rule between the elements is not obvious in it, m a y be found b y giving P and V as cartesian coordinates to a point • and T and S to a point ©. Every element of the substitution group is then represented b y a pair of these points or b y the line segment • © (fig. ~ and 2). T h i s representation is concerned with the primary variables only. An analogous representation of the secundary variables would be restricted to the first and third quadrants. The secundary gr6up is 4, l-meromorphous with the primary group, as m a y be seen from (2). In (3) it froms D 4 (transformations of a quadrate into itself). Received March 18th, 1947. REFE. RENCES 1) See for instance Herzfeld, Handb. d. Physik Geiger-Scl~eel 9, 58. 2 ) , (Note added after correction:) An article by F. O. Koenig in J. chem. Phys. 3, 29 (1935) is so much like my present article that I regret not to have seen it before. He gives the principal group D4 of 8 elements contained in our group but is not so clear in the necessary extension to 64 (or at least 32) elements.