ON I N D I F F E R E N T POINTS BY PAUL SAUREL
If a bivariant or tnultivariant system is in equilibrium at a given temperature and under a given pressure, the state of equilibrium of the system is, in general, stable ; the masses of the phases as well as their concentrations are determinate. However, at exceptional points, which Duhem has called indifferent points, the state of equilibrium of a bivariant or multivariant system is indifferent. At such a point the system can undergo, without change of temperature or pressure, a reversible change which charges the masses of the phases while leaving their concentrations unchanged. Let us denote by n the number of independent components and by Y the number of phases of the system. Let denote the total mass of the j - t h component and let 7nj denote the mean concentration of this component throughout the system, that is to say, let m.=-----. mj I
12
2 j =I
Finally, let mii denote the mass of the j-th component which is present in the unit of mass of the i-th phase. Then, as we have shown,' the necessary and sufficient conditions for an indifferent point are that each determinant of the order Y that can be formed froin the matrix m,,
m,,, m,,,
112.2,
m21, m y 2 1
*
' 'I
mn
' ',
mva
1
(1) M n
3
min,
nzan,
should be equal to zero.
-
Jour. Phys. Chem. 5 , 48 (1901).
* *
mnr
314
Pau I Sazr ye I
I t may perhaps help in making the meaning of this result clear if we state it in geometrical terms. But for this purpose it will be necessary first to recall a few elementary propositions in analytical geometry. Consider a straight line. T h e position of any point on the line is determined by its distance from any fixed point of the line or more symmetrically by its distances from two fixed points of the line. Take two fixed points at a unit's distance apart and let xI, x 2 denote the distances from the two fixed points to any given point on the line. Then we have not only x,
but also
+ x, =
(2)
I,
+
A,x, A2x2= 0 , (3) in which '4, and AZare constants. Equation 3 may be called the equation of a point. If a second point whose coordinates are y,, jigis to coincide with the point x,, xz,we must have not only equation 3, but also the equation A#,
+ A2Y2 =
0.
(4)
From these two equations it follows that
is the condition that the two points x and y coincide. Consider next a plane. T h e position of any point in the plane is determined by its distances from two fixed intersecting lines or more symmetrically by its distances from three fixed lines forming a triangle. Take three fixed lines forming an xz, x3 equilateral triangle whose altitude is equal to I, and let xI, denote the distances from these three lines to any given point in the plane. Then it can be easily shown that x1+x,+x8=I.
(6)
It can also be shown that the equation of any line in the plane is of the form A,x, AZXS Asx, = 0, (7)
+
+
in which AT,AD,A3 are constants. If we wish to express the fact that the three points x , Y, and z lie in the same straight line, we must write the equations
is the condition that the three points x , y and z lie on the saiiie straight line. A point in the plane is determined by the intersection of two straight lines. Thus the position of a point is deterinined by the equations A,x, B,x,
+ A2x1+ A3x3= + B,x2 + B3xS=
0,
(10)
0,
in which the A’s and B’s are constants. From these we find that
If we express the fact that a second point y coincides with the point x we shall find that y,,y,,,y , satisfy equations I O and 11. From this it follows at once that
Accordingly when two points x and y coincide, the following conditions are satisfied : XI,
Yr I = 0 ,
1x1,
XZ,
1’2
I x 3 .
Y1 ’ = 0 ,
x2, Y2
l=o. (13) Y3 1 ”;, Y3 Consider finally an ordinary space of three dimensions. T h e position of any point in space is determined by its distances from three fixed planes which meet in a point or more sym-
’
I
~
0.
Pau I Saure I
316
metrically by its distances from four fixed planes forming a tetrahedron. Take four fixed planes forming a regular tetrahedron whose altitude is equal to I , and let x I ,xz, x xqdenote the dis?’ tames from these planes to any given point in space. Then it can easily be shown that
+ x, + x, +
x1
xp=
(14)
1.
I t can also be shown that the equation of any plane in space is of the form
+
+
+
A,x, A,x, A3x3 A,x, = 0 , (15) in which AI, AI, A3, A4 are constants. If we wish to express the fact that four points x, y , z and w lie in the same plane we must write the four equations
+ A,x, + A,x3 + A4x4= + A,Y, + + A,Y, = + A,z, + A3z3 + = A,w, -I A p 2 + A p 3 + A,w, =
A,x, AlY,
&Y3
42,
42,
From these it follows that
1 1;; I
Yll
21,
y,,
22,
w1 w2
x3,
Y3+ ‘%:
w3
X4?
Y43
w4
0,
0 9
(16)
0, 0.
j
1
=o
(17)
is the condition that four points lie in the same plane. A straight line is determined by two planes. Thus the equations of a straight line are A,x, B,x,
+ A,x, + A3x3+ A4x, = + B p , + B3x3+ B,x4 =
0, 0,
(18)
in which the A’s and B’s are constants. If we wish to express the fact that three points x, y and z are on the same line, we must write equations 18 and the four analogous equations obtained by replacing x by y and by z. These equations may be written in two groups of three each. From the first group we find that
On Indzferent Points
-
1
I
Ylt
21
XZ,
Y2,
22
x4,
Y4,
24
31 7
xz,
Y2,
Equations 2 0 are thus the conditions that three points x,y,and z lie on the same line. A point is determined by three planes. Thus the equations of a point are Alxl BlX, ClX,
+ A,x2 + A3x3+ A4x4= + + + = + c2x, + + = B2x2
0,
B&3
B4x4
0 1
C&3
CF4
0.
(21)
From these equations we find that x I ,x2,x3,x, are proportional to certain determinants which it is unnecessary to write. If we wish to express the fact that a pointy coincides with point x , we shall find that yr,y,,y3,y, are proportional to the same determinants. It follows that
Accordingly when two points x and y coincide the following conditions are satisfied :
Paul Sauvel
318 Xl Y
1
X21
y1
Ys
I
=o, 1 ~
1
y1 = o ,
x3, Y3 xs, Y2 ~=o, Y3 I
Y1 x4, y 4 = O , I Y2 Y3 = o , 1 =o, I Y4 I xu Y4 1
1
~
X2, x4,
I
(23)
We may extend this reasoning to space of more than three dimensions. Denote by xI, xz,. ..., x,, the n symmetrical coordinates of a point in a flat space of n - I dimensions. If two points x and y coincide it can be shown that every determinant of the order two that can be formed from the matrix Xl,
Y1
.. Yz..
X*I
. .
Xn, y n
is equal to zero. If three points x , y,and z lie on the same straight line it can be shown that every determinant of the order three that can be formed from the matrix
is equal to zero. If four points x, y , z and w lie on the same plane it can be shown that every determinant of the order four that can be formed from the matrix
is equal to zero. Finally it can be shown that the necessary and sufficient condition that Y points x, y , z, w , . . should lie in the same flat space of Y - 2 dimensions is that every determinant of order Y that can be formed from the matrik
XI,
3'1,
.x,,
y2'
z,,
w,,
.. '
z,,
w,,
*' *
(27) Xit, Y,t, z n , wn, * * * be equal to zero. If we observe that the concentrations mi and mty satisfy the conditions n
J = I
n
J = I
it follows at once from the preceding discussion that we can state condition I as follows : In order that a bivariant or multivariant system of Ypliases be at a n indifferent point it is necessary and sufficient that the representative point of the system and the representative points of the Y phases lie in the same flat space of Y - 2 dimensions. In particular, for a two-phase system the representative points must coincide, for a three-phase system they must lie in the same straight line, for a four-phase system they must lie in the same plane. T h e indifferent points of a bivariant or multivariant system form a series which is analogous to the series of states of equilibrium of a univariant system. At the indifferent point which corresponds to a given temperature there are a determinate equilibrium pressure and determinate equilibrium concentrations. In general, in passing from an indifferent point to an adjacent indifferent point, the equilibrium concentrations of the phases change. In general, then, in passing from an indifferent point to an adjacent indifferent point, it will be necessary to change the mean concentrations m3. However, the mean conc'entrations mJ can be kept constant if the equilibrium concentrations of the indifferent point vary with the temperature in such a manner that the point, line, plane or Y - 2 dimensional flat which they determine always contains the point which the mean concentrations m3 determine. Thus, for a two-phase sys-
tein, the concentrations of the two phases must remain equal. This will surely be the case if one of the phases is a phase of fixed composition. For a three-phase system, it is necessary that the line determined by the concentrations of the three phases always pass through the point determined by the mean concentrations m3. This will surely be the case if two of the phases are phases of fixed composition. For a four-phase system it is necessary that the plane determined by the concentrations of the four phases always pass through the point determined by the concentrations wj, This will surely be the case if three of the phases are phases of fixed composition. T h e simultaneous values of the temperatures and pressures of the indifferent points of a given system of phases can be represented by a curve, the slope of which is given by Clapeyron's equation,'
dn:---Q dT - T8V
*
In this equation 'I' and II denote the temperature and the pressure, Q denotes the heat absorbed by the system during a reversible change a t the indiffekent point, and 6V denotes the accompanying increase in volume., In general, as we pass along this curve we must change the mean concentrations m3. I n the special case, when the mean concentrations can be kept constant, the system behaves like a univariant system. I n conclusion it may be observed that in the case of systems consisting of three or more phases the indifferent point corresponding to a given temperature can occur in systems whose mean concentrations are different. For a three-phase system the point q can take any one of an infinite number of positions on a limited portion of a straight line. For a four-phase system the point in,, can take any one of an infinite number of positions on a limited portion of a plane. iVezv York, Muy3, 1902. Jour. Phys. Chem. 5 , 54 (1901).
