26 CHAPTER 3. FULL THERMODYNAMIC THEORY
independent variables, and the objective of this section is to express changes in the three
dependent variables , ", and s in terms of changes in T and p.
The rst law of thermodynamics is usually given with s and as the independent
variables,
1
d" = T ds �� p d
= T ds +
p
2 d: (3.1a)
By a standard Legendre transformation of " to the Gibbs free energy g per unit mass,
g = " �� Ts + p 1
, we obtain the rst law with T and p as the independent variables,
dg = ��s dT +
1
dp: (3.1b)
We furthermore introduce the following three thermodynamics coecients: the isobaric
heat capacity per unit mass cp, the isobaric thermal expansion coecient p, and the
isothermal compressibility T , as
cp = T
@s
@T
p
; (3.2a)
p = ��
1
@
@T
p
; (3.2b)
T =
1
@
@p
T
: (3.2c)
The subscript p or T refers to evaluating the partial derivative while keeping p, respectively,
T constant, e.g. (@g=@T)p = ��s, Eq. (3.1b). Moreover, as a standard step towards getting
rid of explicit references to the entropy, we derive from Eqs. (3.1b) and (3.2b) the following
Maxwell relation based on Schwarz' theorem, which states that the order of dierentiation
is irrelevant,
@s
@p
T
=
(
@
@p
��
@g
@T
p
)
T
(
= ��
@
@T
@g
@p
T
)
p
"
= ��
@
@T
1
#
p
= ��
1
p: (3.3)
Using Eqs. (3.2) and (3.3) we express ds and d in terms of dT and dp
Tds = cp dT ��
pT
dp; (3.4a)
1
d = T dp �� p dT: (3.4b)
Substituting ds in Eq. (3.1a) by Eq. (3.4a), d" is written in terms of dT, dp, and d,
d" = cp dT �� pT dp +
p
d; (3.5a)