
254 The Principles of Thermodynamics
With P
I
(h) not even being interpretable as the pressure in the interface, the lhs can
not be given the meaning of a chemical potential for this region. In fact, it is hard
to make any sensible use of this equation. Remarkably, van der Waals manages to
derive the following relation (again for planar interfaces)
u
I
(h) −Ts
I
(h)+P
e
v
I
(h)=
μ
+
c
2
2
1
ρ
I
(h)
d
ρ
I
(h)
dh
2
−
d
2
ρ
I
(h)
dh
2
(12.56)
Where P
e
is the external pressure. The reversal in the sign of the second derivative
term will be seen to be crucial. The crucial observation of vdW is that neither with P
nor with P
e
is this expression equal to the common chemical potential for ...