A DERIVATION O F T H E P H A S E R U L E BY J. E. TREVOR
Although several forms of Gibbs’s derivation of the phase rule have been presented in recent years, I venture in the present paper to offer another, in the hope that its simplicity will entitle it to consideration. I . The case of yz constituents Let us first consider a phase-system having ?Z independently variable “con~ponents”,each one being a component of each of the phases of the system. At equilibrium, the energy of each of the Y coexistent phases is a function of the volume, the entropy, and the component-masses, of the phase,
E,(va, vt1 mz,,mz2
*,
(i”
WG,).
I, 2 ,
. ..] r ) .
By differentiation of these functions, we find expressions for the ci intensities ” that correspond to these variables (the extensities ”1, i. e., for the pressure, the temperature, and the chemical potentials of the components,
i
I (I)
I
p
_
_
= 4L(% 8,s c,,,
_ -
e = +zz(vzl v,, _ _ cL1=+13(nt1 v,,
...
( p>c = +++q
cs,,
c,,,
c,,,
C,I,
Cm,
_
_
(V%,
v,,
c,,,
* ‘ *
* . ‘ I
c,,,
, ,
C*JI
(i=
I,
21
‘ e * ,
r)
CWJ I CW>,
. * ‘ I
c,,,).
Here for the specific volumes, specific entropies, and specific component-masses or mass concentrations ”, of the phases we introduce the notation ((
T h e first derivatives of the functions Ei may be regarded as functions of these variables, since the functions Ei are honiogeneous and of the first degree in their variables.
J. E. Trevor
I 86
Differentiation of the functions Ei yields primarily, of course, ( e 2)r equations for the phase intensities pi, Bi, pLij; but between these functions obtain the ( e 2)r conditions of equilibrium,
+
+
p,
I
{
p,
=
8, = 8,
=
=
PI, = p2* =
...
PIS = p.,,, =
=p ... = 8, = e = p,
e . .
. . . - Prr == p l _.
= PCL, = pu,,,
'
which serve for the elimination of the phase intensities, whereby we obtain the ( G 2). equations (I). T h e only further independent relations between the variables of (I) are the r equations
+
I.
We introduce the ~ Z concentrations Y cii, and therefore also these equations ( 2 ) ) instead of restricting ourselves to functions of the (?z - .)I ratios of the component-masses, in order that our set of variables shall include the phase-concentrations of all the coniponents. Relating to our system of Y coexistent phases, we have above the (a
+
2 ) Y S y
equations (I) and (2)) between (n
+
2)Y+
(E
3-2 )
variables. Of these variables, therefore, the number that remains independent is the difference of these two numbers, or n+2-YY,
It appears, that is to say, that of the variables
P,
8, P,,
_ _
pn,
*>
Pn,
', cz,, (i= I , 2, . .* , 7 ) the number remaining independently variable, the variance u of the system, is v*, 72, c,,,
c,,,
* *
A Derzhatioiz of’ the Phase v
-=71
Rti le
187
-+ 2 - Y.
This is the phase rule for the case in question. I t niay be, however, that certain of the components are absent from certain of the phases. This fact will be expressed by p equations m, == 0.
The effect of these equations is to came p of the concentrations c , ~to disappear from the set of variables, and p of the equations P? = 4 z ( 2 + 2 ) 9
of the set of equations (I), to disappear from among the equations between them. Further, whenever the number of the components of a phase reduces to unity, an additional concentration disappeirs from the set of variables, and an additional equation,
vanishes. T h e number of disappearing equations being thus equal, in either case, to the number of disappearing variables, the number of variables remaining independent is still, as before, n+z-r.
T h e above result, but with reference to a smaller set of varidbles, is obtained by means of the above line of reasoning if we start from any one of the sets of auxiliary fundamental functions, w!~,
e,
m,,,nz,,,
Gz(#, 72, m,,, me*, H A P , 8, m,,, m,,,
. . .)
*
--
Pn,,,) mz,,> - , ?n,,,>. - 9
+
The case of n h constituents T h e case considered in the foregoing is not wholly general. I t may be that the number of substances whose masses vary i n the phases of the system exceeds the number of independently variable components of the system as a whole. Denoting the phase-masses of the former substances, the “constituents ”, by 2.
A,,,
A,,
*
-
e ,
k(,+ A),
(i’
I, 2,
...,
Y)
J. E. Trevor
I 88
8
_ _ = *z*(%,
%,
- _
c,,, Cz,, * ' * ,
C*(*Wz))
and here again the phase-intensities are replaced by the equilibrium values of the intensities with reference to the system as a whole, by means of the equations representing the conditions of equilibrium. T h e only further relations between the variables of (3) are the r eqtiations (4)
and h equations between some or all of the Xj's. Relating to our system of r coexistent phases, we have above
+ i z + ) + + k? + h + + + k? + 2 y
( E
equations bet ween (E
2)y
(E
2)
variables. Of these variables, therefore, the number that remains independent is the difference of these two numbers, or n+2-yr.
I t appears, that is to day, that of the variables,
A Devivatioiz p , 0,
_ -
A,,
v*, %,
of ihe Phase Rule
-, '* *,
A,,
A,+?,
*.
c,,, c-,
189
Cqrr+k),
'i(
I , 2,
..., r )
the number remaining independently variable, the variance the system is u=
n
+2
v
of
- Y.
I t may be, however, that certain of the constituents are absent from certain of the phases. T h i s will be expressed by q equations A, = 0.
T h e effect of these equations is to cause q of the concentrations C,. to disappear from the set of variables, and q of the equations h, = k(J +
2)>
of the set of equations (3) to disappear from among the equations between them. Further, whenever the number of the constituents of a phase reduces to unity, an additional concentration disappears from the set of variables, and an additional equation
vanishes.
T h e variance thus remains in any case V=
as before.
72
+
2
- Y,
T h e phase rule, that is to say, is true generally.
Here, as for the simpler case treated under the first caption, the same result, but with reference to a smaller set of variables, is obtained by the same reasoning if we set out from any one of the sets of auxiliary fundamental functions, F,(V,,
8,
A . ~ , A,,,
Am GE(PI H , ( p , 6, A,,, A,,, 172,
~
~
1
,
* * *
* * * *
7
- 9
-
Aq=+h,
1
Aqqz7~))
Az(,z+h)).
T h e treatment under this second caption, T h e Case of n h Constituents, may be regarded as the general derivation of the phase rule. T h a t under the first is a particular case of it, and may be looked upon as merely introductory.
+
Cornel2 Universiiy
