NOTE ON THERn!IODYNAWIIC SURFACES BY J. E. TREVOR
Conditions of stability In his investigation of the energy surface for a one-component system,I Gibbs has given an ingenious and convincing demonstration of the theorem that, when $, 6' are the pressure and absolute temperature of a thermodynamic system in a state of dissipated energy determined by the volume v and the entropy r ] , and 6v,r ] 87 is another state, not a state of dissipated energy at p , 0, it is true that
+
+
6e
+ pSv
--
067
>
0.
Otherwise stated, in all variations of state imagined as occurring from a state of stable equilibrium at $, 8 to a state not in stable equilibrium at $,8, 6e> - ~ S V ear].
+
It is possible, however, to have a succession of univariant states or of invariant states that are all states of stable equilibrium at $, 0. I n such a case, no one state of the succession v, ill spontaneously pass into another, the equilibrium of each ::ate will be Lneutral'. If the change through the succesr:'m of stable states be reversibly effected, Gibbs's demonstratic:~yields 6e= -pSv 067.
+
For a state to be one of stable equilibrium at the 1, essure and temperature$, 8, it is therefore necessary that, in :i1 variations of the state,
2.
6e - -ppSv+
067.
On passing from any point v, r] on the
c
.ergy surface
e = 4 a , 7) for one-phase states, or for states of ivi given phases, or of three r gv, r ] -l- 8q on the surface, given phases, to an adjacent poir t I
Trans. Conn. Acad.
2,
387 , : 5 ' 7 2 ) .
J. E. Trevor
84
whence, because of the above criterion of stability, which may be written
it appears that the condition that the point v, 7, shall represent a state of equilibrium is
for all values of u,7. This condition expresses that no portion of the surface may lie below the tangent plane passing through the point a, 7. It expresses, in particular, that a succession of states in neutral equilibrium is represented by the points on a line common to the surface and the plane tangent thereto at the point u,7 ; and that, at a point representing a state of stable equilibrium that is not also a state of neutral equilibrium, the surface is convex downward in all directions. All portions of the surface that do not satisfy these conditions must represent thermodynamically instable states. When e = e(v, 7)
is the equation of the continuous succession of stable and instable one-phase states, a point representing a state of stable equilibrium must satisfy the condition
The necessary and sufficient conditions for this quadratic expression to be positive, i. e., for the surface to be convex downward in every direction at v,7, are H(e)
>0,
>
aze/av2
0,
where H(e) denotes 'the hessian of the function e(., 7).
A con-
Nole on Thermodynamic Surfaces
85
sequence of these ‘conditions of stability’ is the further inequality
A problem
It was further pointed out by Gibbs that the thermodynamic properties represented by the surface e = e ( v , 7)
are represented by the surfaces f=f(v,e> g = A P , r> h = h ( P , 6);
the functions f ; g, F, being defined by the equations /(a,
6) = e - 0v
g ( P , v> = e h ( p , 0) = e
+Pv
+p a
- 671;
wherefore we have the set of differential equations de = -Pdv
+ 0dv
df = -Pd v - vd0 dg= vd) f 0dv dh= VdP-vdB.
We may now seek to determine the curvatures of those portions of thef; g,h surfaces that represent one-phase stable states, i. e., that correspond to the convex portions of the primitive energy surface. Auxiliary equations In this search we shall require to use a series of equations, which shall now be assembled for reference. We have
whence, by successive elimination of dv and dv,
86
J. E. Trevor
whence follow the desired equations,
and from these, by division,
The second derivatives We now require to determine the signs of the second de. rivatives, and of the hessian, of each of the functions f; g, h at points representing stable one-phase states. For azf/av2, we have
by equation (a). For az7/ae2,we have
--1:(a7) -
ae 2,
