R E L A T I O N S H I P S B E T W E E N THERMODYNAMIC F U N D A M E N T A L FUNCTIONS
BY J. E. TREVOR
I. Introduction
In reversible compression thermodynamics, two familiar differential properties of the thermodynamic potential H ’ are
in which equations p and B denote the equilibrium values of the pressure and thermodynamic temperature, WTzand Q,, the work and heat absorbed by a body or system of bodies-a system ’in passing from a state I to another state 2, and H the thermodynamic potential
H(p, 0) = E +Pv - 97, in which equation E, u,7 denote, respectively, the energy, volume, and entropy of the system. T h e subscripts p , B indicate that the change in question is effected at constant p , B ; and it is understood that no mass is added to or removed from the system during the operation. T h e question arises, whether there are other relations of the form of (I), and if so what they are. 11. Closed systems
In the reversible compression thermodynamics of a ‘ closed system ’ of phases, i. e., of a system of given masses, the differential of the energy of the system is given by the equation d E = --#dv+
Hdq ;
which relation, by means of the definition equations F-E-07
7
G=E+& H = E L P V- 07,
1
I
(2)
may be expressed in the different ways : dE(v, 7) = - j d z J d F ( v , 0) = - j d v
7-0dv -
+
vd0
7
1
(3)
d G ( P , 7) = VdP edv I d H ( p , 0) = ~ d p - vd0. J T h e answer to our question is to be sought by expressing each of the ftindamental functions ’
E, F, G, H, in terms of each of the others, and applying the results to changes between two different states of the closed system. We are, accordingly, to find the equations E =E(F)
F -- F(E)
G = G(E)
H =H(E)
= E(G)
= F(G)
= G(F)
= H(F)
=E(H), = F(H), =GW), = H(G); in which the indicated function in each case is to be understood 3s a function of the independently variable function, and of its derivatives and independent variables. T o this end we have, from the definition equations (2), the equations of the first column below ; which become, bysubstitution of the partial derivatives that appear in the differential equations ( 3 ) , the corresponding equations of the second colnnin.
J. E. Trevov
572
= H - j - aH
aP
I t should be noted a t this point that, because of the relation
E+H=F+G, the long equation of each of the above four sets is equivalent to the sum of the other two that contain the same derivatives. These twelve equations can be written more compactly. Each one has one or the other of the forms
av u=v--x-, ax av u=v-x----yax
which may be written
av aY
The~iizodyiza;micFit izda nze~zta I Fzt izct ioizs
573
By means of these transformations, our equations take the forms:
E
=-
a~ e'-.ae e
Discarding the redundant long equation of each set, and arranging the others with regard to the functions that are differentiated] we find the following general results :
On effecting the indicated differentiations] we obtain, of course, our initial equations (4a)to (44; of which the present equations are interesting forms. T h e general results to (H), when applied to changes in which the explicitly appearing independent variable is maintained constant in each case, become
(a)
574
J. E. Trevor
T h e correctness of these equations may be tested by executifig the indicated differentiations. T h e first equation ( E a ) , for example, becomes
+ ed,
d~ = -Jdp
for constant volume. In this way, the eight equations are seen to give the corresponding integrations of the equations (3) for the differentials of the four fundamental functions.' From the equations (F) to (H), further, we can derive four similar equations, when we note that a closed system can liave different thermodynamic states with different 77 at the same v , 0 ' I
'z?
( 1
$37)
I (
( 1
Ii
77
P,6 P,
e.
For this consideration makes it possible to apply four of the equations (E) to (H) to changes in which both independent
Thermodyizamic FundaineiafaZ Fuiactions
575
variables of the differentiated function are maintained constant, and thus to obtain the eqnations
T h e quantities ( EZ- E J , (Ez- E,)),?) (F, - PI),, e ) (G, - GI)),0 ) have here been replaced by their values Q,,, W,,, W12,Q 1 2 . In these equations, the quantities
(F,- F1>v,e (Gz -- GI)$,1 (HA- HI)P,O are, indeed, identically zero ; but their indicated derivatives are not zero- they are either - (7, - 7,) or (a,- ur).
-+
111. Individual phases
T h e equation for the differential of the energy of an individual phase of a system of phases is
J. E. Trevov
576
. . . ) = -pdv-- 70'0 + C p d ~ z d G ( p ,7, . . .) = vdp + 0 d +~ Z',/*d?Jz d H ( p , 0, .* ) ~ dp vd0 + z p d n ~ de ( v, -ql p, . . . ) = -jab + 0d7 - Cnzdp df (v, 0, pl . . . ) = - p d v - - 7d0 - Cmdp d F ( v , 0,
773,
112,
YIL,
d H P , 7, h
* * *>
=
VdP T
-
>
(6)
CmdP
successively in all the eight possible choices of sets of independent variables, taken three at'a time from the pairs $1
2)s
0, 7 ,
ZP,
X.?%,
the members of no pair being taken together. T h e energy E of the phase being a homogeneous function of the first degree of 71, 7,wzI . . ., we have, by Etiler's theorem of homogeneous functions, the integral relation
From this equation follows at once that our functions E . . . h are respectively equal to the expressions set opposite them below :
Themzoa'yizn~zicFzindanzental Fzinctlbm
E
+ 017 - .E,uL~z
= ---Pv
+ .E,u*.~ 017 + X , U L ~ Z
e = -pv
F = -Pv G= H-
CpLnz
577
+ 017
f =-pv
1
g=
017
h=
0.
Dropping from consideration the function h) i. e., dropping the equation
our task is to express each of the functions E . . . g in terms of each of the others and its independent variables and derivatives. We find, from the definition equations (5) and the derivatives that appear in equations (6), the following :
E=F$0v
=H--v+Orl = e -Yfiu?n /i'
/ j
F=E-07
__ - H --)V
=f
+ e,
= 8'
-- p z ,
=g-ppv-e,
= f +e,-pv
580
J. E. Tyevov All these equations have one or another of the forms
ZI
which may be written, more compactly,
ZI
By means of these transformations, our equations become the following ; which are arranged in sets of six, each set representing all but one of the fundamental ' functions E . . . g in terms of that one and its derivatives and independent variables. (
TheP-modynnmzc Fuizdameiztal Fuizctioizs
581
J. E. TYevor
-(a+
1)g.
I n each of these seven sets, the equations are arranged in order with reference to the variables with respect to which the differentiations are successively indicated. T h e latter three of each set are sums of the former three ; which is easily seen to follow from the relations
584
Thermodynamic FulzdamentaZ Fuizctions
i
E+ f=e+F
E + g = e + G IF g = f + G
+
fH+ e = G L f (E) ( H + J = F , (0 I H t g -G, (7) which follow from the definition equations (5) and the relation
Efjv-
6'7 - 2pm-
0.
T h e relation that applies in each case is indicated by the appended Greek letter. Discarding the three redundant equations of each set, our definitive results for individual phases are the twenty-one equations (E') to (g').These are in every way analogous to the corresponding general results (E) to (H) derived for closed systems. T h e ap.plication of them, as in the previous case, to finite reversible thermodynamic changes is too obvious to call for consideration in detail. Cornell University, Apvil, zgoo.
