PHYSICAL CHEMISTRY IN BRIEF

CHAP. 3: FUNDAMENTALS OF THERMODYNAMICS [CONTENTS] 97

VolumeV 1 is often chosen as large as to allow for a system in its initial state to behave as
an ideal gas. In the limitV 1 →∞at constant amount of substance, equation (3.79) rearranges
to

S(T, V) =S◦(T 1 , Vst) +

∫T

T 1

CV◦(T)

T

dT+nRln

V

Vst

+

∫V

∞

[(

∂p

∂T

)

V

−

nR

V

]

dV , (3.80)

whereVst=nRT /pst, andpstis a standard pressure.
If the volume stays unchanged during a thermodynamic process, equation (3.68) simplifies
to

S(T, V) =S(T 1 , V) +

∫T

T 1

CV(T, V)

T

dT , [V]. (3.81)

For an isothermal process, equation (3.79) becomes

S(T, V) =S(T, V 1 ) +

∫V

V 1

(

∂p

∂T

)

V

dV , [T]. (3.82)

3.5.4.2 Temperature and pressure dependence for a homogeneous system

By integrating the total differential (3.50) with respect to the general prescription (3.30) we
obtain

S(T, p) =S(T 1 , p 1 ) +

∫T

T 1

Cp(T, p 1 )

T

dT−

∫p

p 1

(

∂V

∂T

)

p

dp. (3.83)

Ifp 1 = 0, the system in its initial state behaves as an ideal gas and equation (3.83) becomes

S(T, p) =S◦(T 1 , pst) +

∫T

T 1

Cp◦(T)

T

dT−nRln

p

pst

+

∫p

0



nR

p

−

(

∂V

∂T

)

p



dp , (3.84)

wherepstis a chosen standard pressure.
If the pressure stays unchanged during a thermodynamic process, equation (3.83) simplifies
to

S(T, p) =S(T 1 , p) +

∫T

T 1

Cp(T, p)

T

dT , [p]. (3.85)

For an isothermal process, equation (3.83) becomes

S(T, p) =S(T, p 1 )−

∫p

p 1

(

∂V

∂T

)

p

dp , [T]. (3.86)