ON SOLUBILITY CURVES BY J.
E. TREVOR
Writing m,, for the mass of the first component in unit mass of the first phase of a two-component system in a two-phase state, the thermodynamic expression for the slope
of the family of solubility curves m,,(P,0) with p a r a m e t e r j is well known. I t is of interest to establish the corresponding expression for the slope dvz,,/d8 of the solubility curve m,1=
m11
= m11( 0)
of a two-component system in a three-phase state-a univariant state. When the system is in a two-phase state of thermodynamic equilibrium under the pressure j at the temperature 8, the differentials o f j , 8, m,, are connected by the equation
I n this equation, which is due to van der Waals, the letters
vi,rz,h, denote the specific volume, the specific entropy, and the specific thermodynamic potential of the i-th phase, and m y denotes the mass of the j - t h component in unit mass of the i-th phase. Now, the appearance of a third phase renders? a function of 8. If the equation remains otherwise unchanged, we shall have
J. E. Tyevov
IO0
or, replacing djjd0 by the ratio 6HiaV of the changes of entropy and volume of the system in an isothermal change of the state,
>. T h e object of this note is to show that the equation (I) is a consequence of the conditions of equilibrium of a two-component system in a three-phase state. If no constituents other than the two independent compofor the potential nents appear in the system, and if we write of the j - t h component in the i-th phase, the conditions of equilibrium are PI1 = PA19
PI1 = Pi1I
PI2 = P.22’
PI2
= P32.
Differentiating these equations within the region of their validity,
Let us recall the well-known equations that enable us to express the derivatives of the potentials pc by means of the specific extensities of the phases. T h e equation expressing that the thermodynamic potential H ( f i , 0, MI, M,) of a given phase of the phase system in homogeneous and of the first degree in the masses hl,, M, of the independently variable components of the phase is H = (MI t M d . /z
(P, 6, md.
Here 171, is the mass of the first component in unit mass of the phase. From this formulation we find, for the potentials, aH PI =
aM,
or, since m, = iM,/(Ml
+ M,),
ah am, Forming the derivatives of these potentials, noting that pr = A
- m, -.
and adding to each extensive quantity a subscript i to distinguish the phase, we get the equations in question :
Eliminating the derivatives of pzjbetween (2) and these equations, and combining the four resulting equations by eliminating between them the three quantities
we obtain the equation connecting the differentials of w,, and 8,
(3) where
J. E. Trevor
I02
(v3
- msl
h),.
. .
.
am91
D
=I
-
1
where
m,,
-q,
a,
x
a , - a3
+y
a,
-
- a,
- 71211
a,
- m,,
a, - a3
-I
+ m,, 0
-1
+
0
7
m3,
fit,,
0
0
m31
Replacing ai,x,y by their values, and collecting the coefficients of the ut and their derivatives, we find
_ -.
(4)
A == 6, - b,
6, - 6,
ZJl
v2
%
”11
?@21
m31
I
I
I
+u
+
a , - a2
- 62 61 - b3
*=
’
+x
-I
+ w~~~
a, - a3 -1-y
ZJ
0
- a2 a1 - a3
a1
61
(5)
I
-71
72
m,,m21 I
1
73
av
am,,
m21
0
-t-VI
32
v3
a-%
3%
3 2
7J3
?I2
73
’.
Wll
“23,
m,,
1
I
m21
1
m31
1
‘v1
I
1
1
1
vl I
SV
v2
v3 = - (m,, - mll) -
"21
m31
I
8%
1
Substituting these values in the equations (4),(5) for D, A, we find 6V D=-(??Z
21
- %l)
T h e equatioll (3) for d m , , / d ~ ,
hereupon becomes
This is the equation (I), which is thus seen to be, as asserted, a consequence of the conditions of equilibrium of a two-component system in a three-phase state. T h e equation will be expressed a little differently if we eliminate either 6%,, 6%,, or WK,, 892,, between equations of the set (6),(7), ( 8 ) ) (9)) instead of eliminating 6%,, am,. I n these successive cases we find
( IOb)
T h e formulations of (IO), (106) do not contain quantities characteristic of the third,,respectively of the second, phase of the system: It may here be remarked that equating the second members of the equations (IO), ( ~ o a )(106) , may serve as a check on the for example, work. Equating the second members of (IO), (~oa), we find i
whence
SJ4 -.
SV
-~
I1
.
I
I
I
the expressions substituted for the determinants of the third order being deduced, as above, from the equations (6), (7), (8),
(9). T h e equations (IO), IO^), and the equations ( r o a ) , IO^), can be compared in this same way. An interesting special case of (IO) is the equation for the slope of the solubility curve when only one of the phases has a variable composition. T o illustrate, suppose the first component of the system to be an involatile salt, the second component to be water, and the coexistent phases to b e : I . Saturated solutioti : 2 . Water vapor ; 3. Solid salt. I n this case nz,, = 0,and m3L= I , wherefore the equations (IO), ( ~ o a ) ob), , give the slope of the solubility curve in the forms
(114
I n any case the quantity 8H/8V is, of course, equal to the slope dpId0 of the boundary curve in the $,O-diagram. And it will not be overlooked that the function a2h!,/am:,is always positive for stable equilibrium. An obvious comment on the general equation ( I O ) is the following. From the three-phase system, in a state of equilibrium, suppose the third phase to be removed. For the remaining two-phase system, the equatiox of van der Waals cited at the beginning of this paper may be written
On comparing this equation with f orni dm,, -d0 ~
am,, fsH
ap
sv
(IO)
3%
-I 68-
we get
(IO)
in the
where the total derivatives relate to the three-phase state, and the partials to the two-phase state. Thus, the quantity %)
which for a one-phase state is independently variable, and for a two-phase state becomes a function 7% ( P , 0 ) , is altered on the appearance of a third phase to the implicit function of 6,
mn
(m0) 3
)
the form in $,0 remaining unchanged. Cowtell Umverszty December, '905.
