T H E K 11 L C0 E F F I CI E S TS
BY J. E. T R E V O R
Introduction
In the therinod!mainics of isotropic one-phase .s;!.steiiis of constant mass atid composition, it has long been knon-n that the thermal coefficients YIand Y2 of tlie heat equation
d'Q ::: Y , d . y , -- Y,.d.v:, in which .r1and .I-* represent the independent state-variables of the system, can be expressed as functions of certain fundaaiental functions - the energ!., the ' characteristic functions ' of Massieu, and the X-function of Gibbs -and the partial derivatil-es of these functions. fe\v of these formulas are well kno\vii. T h e object of tlie present paper is to derive them all, and to point out the analytical relatioilship that obtains between them. If we take the volume is and the thermal potential, the thermodynamic temperature, 6 ' of oiii systeni as indepeiidently variable. the pressure &J and the entropy 17 will be deteriniiied by these quantities; aiid n-e shall have two equations
of unkno~viiforms, between these fonr variables, aiid such that by elimination we ma)- obtain the two further equations
~ ( pe , .T ) a ( $ ), (201, i
1s'
'
19)
( 20
1
Rs ii~' Cz ti6
Odq
-18 dj5
=
B, n';
- C1.
tie
d,.
we deduce for the tlierinal coefficients tlie forinulas
Xe
6(;;)
2s
Each of the fundainental fiuictioiis E, F, G, H,b!. means of which these several therinal coefficients are to lie expressed, contains two independent variables, sa!. .I- and -1'. 111the case of the energy these variables are i~ and 77. S o w , in an?. case, the inclicated deril-atil-es in certain of the equations ! 2 7 ) to ( 3 2 ) exhibit these 1-ariables, and a dependent variable c. In the first eciiiatioii of the set, for esaiiiple, n e find the independent variables
;'.T of the energ!. equation, aiitl a depeiideiit \.nriahle j. Iu these indicated deri\.ati\.rs, h t h or 1;ut otic o f tlie t i e n independent \-ariai)les iiiah. occur xvithiti the Iiarentlieses : i n mine cases \ve shall have
in others
It1
tlie first o f tlie5e cn\ey \ye l i a ~ .for ~ , tlir thertiial coefficient,
either of \vIiicIi rediices to tile desired resitit on effecting tlle differentiatioii of the dependelit \.arinhle ?, and replacin:: 6 t)!, its value. when H is not
