THE F U N D A M E N T A L EQUATION O F A MULTIPLE POINT
BY PAUL SAUREL
At the triple point formed by the iiitersection of the three curves which represent in the pressure-temperature plane the equilibrium of three one-coniponent univariant systems, there exists a well known relation between the slopes of the three curves. T h e object of the present note is to show that at any multiple point an analogous relation exists between the slopes of the intersecting curves. Consider the multiple point which represents in the pressure-temperature plane an invariant system of n p phases formed by means of n independent components. If one of the phases of the invariant system be removed there remains a univariant system. T o each of the n 2 univariant systems which can be formed in this may, there corresponds a curve in the pressure-temperature plane, Let us consider the univariant system which has been obtained by removing the first phase of the invariant system and let us denote the slope of the corresponding
+
+
curve by
__ ,
T h e slope is given by an equation which is due
dI',
to Gibbs,' and which can be written in either of the forms :
Cf. Jour. 1 On the Equilibriutn of Heterogeneous Substances, p, I 54. Phys. Chem. 5, j o (1901).
1 . .
172 1
...
737
e . .
... ...
..
.)
e . .
.. .
2'8 !
Vn
+
21
. . .,
I n these equations aiand T~ denote the volume and the entropy of the unit of mass of the i-th phase, and m, denotes the mass of the j-tli component, which is present in the unit of mass of the i-th phase. If to the above equations we add the equations which correspond to the other univariant'systems (the signs of the equations with even indices having previously been changed), we obtain :
In theseequations
d n -
1 denotes
d T,
the slopeat the multiple point of
the curve which corresponds to the univariant system obtained from the invariant system by removing the i-th phase. Moreover, the values of at, v,, my are the equilibrium values which correspond to the pressure and the temperature of the multiple point. If we recollect that from the definition of the concentrations 792, we have
2
m,, = I ,
i=
.. ., + 2 ,
I, 2,
12
I=I
it follows without difficulty that the determinants which form the right-hand members of the last pair of equations are identically equal to zero. TTe thixs obtain the following equations, which are due to Riecke : 1
dn,
1 -dTT
'
dn* ~
d?',
u, '
7%
. . .,
??z;r&
2'2 !
TlTpl!
* '
%?L
e ,
I
1 1=0,
1
I 1-
1
(1)
.
.
-
dnn 2 dTn+z I
UTZ+
,
"mn-z,~!
Gottinger Sachrichten, p.
223
.*.,
(1890).
mn+z,x
I
Zeit. phys. Chem. 6, 268 (1890).
If the invariant system is formed by means of a single cornponent, we have ml1= 1?Z21 = PZ SI - 1,
and equations
I
and
2
become
Equation 3, which has been called the fundaniental equation of the triple point, mas established almost simultaneously by Riecke', Katanson' and Duhem.3 From it can be derived equations which had previously been used by Ies a reversible change in which the mass of the i-th phase is increased by 8M2. Some of these increments will of course be negative. T h e condition that, during the change under consideration, tlie total mass of each component remains the same, is expressed by the equations :
2=2
From these 72 homogeneous equations we can obtain at once the ratios of the 12 I increments. These ratios are giveii by the equations :
+
SX,]
Consider now the coefficient of d E, in equation ~~
d *I
I.
The
equations just written show that this coefficient is proportional to v,6;11, Z$M, + . . . z'Tz+ * 6hlw+ 2 . (6)
+
+
If the 6nI's be so chosen that each of the fractions in equation 5 is equal to unity, the coefficient under consideration becomes equal to the expression 6. But this last expression is the increase which the volume of the system undergoes during the reversible change that we have been considering. We shall denote this change in volume bp 6YZ. In the same way it may be shown that, for a suitable choice of the increments 6h1, the coefficient of
drI d T'
-pi is
equal to the
change in the volume of the i-th univariant system during the corresponding reversible change at the multiple point. Denoting this change in volume by 6Vi, we may write equation I in the form
Similarly, it can be shown that equation 2 may be written in the form
I n these equations, 6Y7and 6H,are the changes in the volume and the entropy of the i-th univariant system during a reversible change at the multiple point. This reversible change is such that the change in the mass of the phase k is equal to that minor of the determinant I , which corresponds to the product
drI dTZ
--L
vk,or, what is the same thing, to that minor of
dT, the determinant z which corresponds to the product __
drIZ
vk*
Since the changes under consideration are reversible, \ye may write : 6H, == -2 Q T '
I 76
Paul Saurel
in which T denotes the temperature of the multiple point and Qi denotes the heat absorbed by the 2-th univariant system during the corresponding change at the multiple point. Equation I: may thus be written in the second form :
Equations 7, 8 and 9 are the generalizations of equations 3 and 4 ; any one of them might be called the fundamental equatiod of the multiple point. T h e demonstration that me have just given follows directly from Riecke's equations. T h e following demonstration, while essentially the same as the preceding, may perhaps seem simpler. For each of the ynirariant systems that we have considered there holds an equation of the form dn,- 6K
dT,
SV, *
in which AH, and 6T', are the simultaneous changes of entropy and volume during a reversible change at constant temperature and under constant pressure. T h e changes in the masses are here subject only to the conditions 5 for the system I and to the analogous conditions for the other systems. T h e above relation enables us to write, for the multiple point, t2
n+2
i= I
z= I
11
n i z
Let us suppose that each of the fractions in equations 5 is equal to XI,and that each of the fractions in the equations for the i-th univariant system is equal to h,. T h e n it can be seen without much difficulty that
I77
I
I
If now we take
x =x 1
2
=
... = = x n + 2 = x ,
and if we recollect that =$,zj=
I,
i=
I, 2,
.. ., + 2 , 72
I
it follows that the determinants in the preceding equations are identically equal to zero. We thus obtain as before,
I n our first demonstration, h has the value unity. Aiew Yoi%,Janzsaty 26, rgor.
