ON 31AIXn’ELL’STHEOREX
BY P h U L SIUREL
T h e object of the present note is to give a demonstration of the theorem that the slim of the areas included between the actual and the theoretical isotherms is equal to zero, provided the areas above the actual isotherm are reckoned positive and those below negative. Originally, the theorem referred to tlie isotherms of a substance containing a single component, but it has been extended by Rliimcke’ to the isotherms of a substance containing two or more components, and a demonstration of the generalized theorem has recently been given by Duhem. Consider a heterogeneous inas5 XI, consisting of 71 components of inasses 91cr, 91t2, . . . , 9!tp, and existing in the form of Y honiogeiieoiis phases. If we denote the energ>-,entropy, volume, temperature, pressure, and inasses of the components, of the i-th phase, by E,, H,, TrZ,t,,p , , Nz2,. . . , M,,, (so that AIv denotes the inass of they-th component in the i-th phase), we shall have the relation’ d E , L t,dH
-
+
@ , d V ,- k , , d h l , ,Ix 2 d l I L o
*
e
-
X&lI
j j .
(I)
In order that tlie various phases should coexist it is necessar5 3 that t = t = ,.. = f , = t
p =p
x
...
=$I,=#
x , , = x = . . . = - x,,= x, x,,= A,, . . . = A,, = x, 21
... x, = A:
1
1
=
e . .
= A,,,= x,.
Blurncke. Zeit. phys. Cheni. 6, 153 (1890). Gihhs. On the Equilibrium of Heterogeneous Suhstaiices. p. I 16. Gihbs. Ibid. p. 119.
T a k i n g the sum of the E
91T1=
7'
equations of the form
(I),
and writing
= ZE,2t H = ',H ~, l7= ST',, 9 l T * = X h I z 2 , . . . , 911i>t= CMZ,,,
we shall have, for the whole mass, dE
= tdH
-# d v
+ X,&31LI + h,d9lL* + . -
e
$- XJ8&?Ii?t.
(2)
More'over, as the energy is a homogeneous function of the first degree of the entropy, voliime, and inasses of the components, we have E = i" - $1- $- X,?lTl + XeyKI . . + A,'2ITJ,. ( 3 )
If we write
equations
(2)
+
and (3) become
h I d ~ Ed11 = M (tdq-jdv (177 - PV X,C,
+
+ E
=7 t7
-$-J
+ X,dC, - X , ~ C ,-
X,C,
- *
+ h,c, + X,c, -
-
- + h dc,,)
X,,C, ) d l I ,
* T
+ X,,c,,
and consequently de
tdq
-$&I
T
X,dc,- X-dc, -
*
* 7
h dc,,.
(4)
T h i s equation connects the changes in the average energy, entropy, volume and concentrations diiring any actual change in the system. O n the other hand, if we denote the energy, entropy, volume, temperature, pressure, :tiid masses of the components, of the unit of mass of the it11 phase, by E,, q,, ZJ,? t,, PI, m,,, mZ2, . . . VI^,^, the reasoning which served to establish equation (I) will also serve to establish the equation dez = f d ~ ,
-t pL&Z,,
+ p a . d ~ -~ , ,- - + p2]dm,,,. *
(j)
E, is thus a function of q r , z ;, 7)zZ1, w z 2 ,. . ., HZ,,,; and we shall assume that the functions obtained in this manner for the various phases are not distinct functions but are parts of one and the same uniform analytical function. If, then, the system uiidergoes a transformation hy which it passes from one homogeneous
phase to a second homogeneous phase, vie may calculate the change in the energy either by equation ( 3 ) or hj- equation (5). If, in particular, \ye snppose the transformation to take place at constant temperature and with constant concentrations, we shall have froin equation (3),
1
12
E, - E ,
t(T1-77)) -
#&I,
- 1
and from equation (5) E,
-E, =f (T2- 71))
-
[2#zdL - 1
X comparison of these equations yields
and since p and 21 are the pressure and volunie along the actual isotherni, and P I and ZJ, are the pressure and voluine along the theoretical isotherm, the theorem is demonstrated. Bovdenux,Jazunvy 23, rS99
