124 The Principles of Thermodynamics
The important lesson is that the requirements of extensivity of S,U and V completely
fixes the chemical potential
μ
, and furthermore it is fixedtobethesameasthemolar
Gibbs potential (note that for one mole of ideal gas RT = PV
m
)! This is not an ac-
cidental result for the ideal gas only. The problem below will show it to be true for
the van der Waals case also. Actually it is true for any system as will be shown now.
Let S
m
,V
m
,U
m
be the molar quantities at T,P. Then the first law for one mole of the
substance reads:
TdS
m
= dU
m
+ PdV
m
(6.12)
whereas for variable molar systems we have
TdS= dU + PdV −
μ
dn (6.13)
Consistency between eqn(6.12), eqn.(6.13), and S = nS
m
,V = nV
m
,U = nU
m
imme-
diately gives
μ
= U
m
−TS
m
+ PV
m
, i.e the chemical ...