PHYSICAL CHEMISTRY IN BRIEF

CHAP. 3: FUNDAMENTALS OF THERMODYNAMICS [CONTENTS] 85

3.4.4 Total differential of entropy as a function ofT,V andT,p

The total differential of entropy as a function of temperature and volume,S=f(T, V), is

dS=

(

∂S

∂T

)

V

dT+

(

∂S

∂V

)

T

dV. (3.46)

At a fixed volume it follows from (3.6) and (3.17)

(

∂S

∂T

)

V

=

1

T

(

̄dQ

dT

)

isochoric

=

1

T

(

∂U

∂T

)

V

=

CV

T

. (3.47)

For the differentiation of entropy with respect to volume, Maxwell relation (3.44) applies.
Equation (3.46) can be rearranged to

dS=

CV

T

dT+

(

∂p

∂T

)

V

dV. (3.48)

In the same way we obtain for entropy as a function of temperature and pressure

(

∂S

∂T

)

p

=

1

T

(

̄dQ

dT

)

isobaric

=

1

T

(

∂H

∂T

)

p

=

Cp

T

(3.49)

and using Maxwell relation (3.45) we get

dS=

Cp

T

dT−

(

∂V

∂T

)

p

dp. (3.50)

3.4.5 Conversion from natural variables to variablesT,V orT,p.

Internal energy may be converted from the function of natural variables to the function of
variablesT,V using equation (3.33), into which we substitute relation (3.48) for dS.Hence we
have

dU=CVdT+

[

T

(

∂p

∂T

)

V

−p

]

dV. (3.51)