ON THE F U N D A M E N T A L E Q U A T I O N S O F T H E MULTIPLE P O I N T BY PAUL SAUREL
+
Consider a system of n z phases formed by means of n independent components and denote by vi, vZthe entropy and the volume of the unit of mass of the i-th phase, and by mzj the mass of the j - t h component which enters into the unit of mass of the i-th phase. If we denote by dIIIJdT, the slope of the pressure-temperature curve of the i-th univariant system that can be formed from the invariant system, that is to say, of the univariant system that can be formed from the invariant system by suppressing the i-th phase, then at the multiple point the following equations' hold :
In a previous note2 we have shown that these equations of Riecke can be written in the form 268
Riecke. Gottinger Nachrichten, p. 223 (1890). Zeit. phys. Chem. 6, (Isgo). Jour. Phys. Chem. 5, 170 (1901).
*
PuuZ Saurel
262 n + 2
1 1 1 2
in which SVi, SH, denote the changes in the volume and the entropy of the i-th univariant system, corresponding to a certain reversible change of that system at the multiple point. I t was shown that the reversible changes are such that
2
n + 2
ni-2
2
SVi = 0,
SH, = 0 ,
(4)
+
and it should have been pointed out that the n 2 reversible changes used in defining 6V, and SH, constitute, when taken in succession, a reversible cycle of the invariant system. Our result can accordingly be stated as follows : Consider at the temperature and under the pressure of the multiple point a reversible cycle of the invariant system, which 2 reversible changes, each of which incan be divided into n volves the phases of one of the n 2 univariaiit systems. During the z-th of these reversible changes, the volume and the entropy of the invariant system receive increments 6V2, 6H,. These increments satisfy not only equations 4, but also equations 3. Riecke's equations can be put into another form, which is even simpler than that just given. Denote by M, the m a s of the i-th phase, by ' 9 I j the mass of the j-th compoment, arid by H, V, the entropy and the volume of the invariant system. Then the following equations hold :
+
+
n -1-
2
2 M , v ~ V, =
/1
i=
I
t=1
n+z
2
M m , = '%,
I.=
I , 2,
..., n.
(5)
2=1
Consider at the temperature and under the pressure of the multiple point a reversible change which leaves the volume unaltered. T h e preceding equations give US
Equations
of the MuZt$Ze Point
263
From these equations 6, we obtain at once
T h e coefficient of 6H in this equation is the same as the coefficient of dII,/dTiin equation I. It follows from this without difficulty that equation I can be written in the form
z = 1
Consider now at the temperature and under the pressure of the multiple point a reversible change which leaves the entropy nnaltered. Equations 5 now give us n
+
n + z
2
i=
1 = 1
I
nt-2
2 m,6M, z=
= 0,
I
From these we obtain at once
j=
I , 2,
..
e ,
n.
264
Equations of the Mu Zt$Ze Point
mi- I , n mi+,,,
- (-1)i+16V
T h e coefficient of 6V in this equation is the same as the coefficient of dT,/dITi in equation 2 . I t follows that equation 2 can be written in the form
t= I
T h e fundamental equations of the multiple point can accordingly be put into the form of equations 8 and 11. It should be remembered that in the first of these equations 6M, denotes the increment of Mi, due to a reversible change in which the volume is kept constant, while in the second 6 M i denotes the increment of Mi due to a reversible change in which the entropy is kept constant. New Yovk, ApviZ ra, ~90.2.
