Physical Chemistry Third Edition

4.3 Additional Useful Thermodynamic Identities 167

4.3 Additional Useful Thermodynamic Identities

We have asserted without proof that (∂U/∂V)T,nvanishes for an ideal gas. We can now

prove this assertion and can obtain a formula that allows evaluation of this derivative

for nonideal gases and liquids. We convert Eq. (4.2-3) to a derivative equation by

nonrigorously “dividing” bydV, converting the quotients to partial derivatives, and

specifying thatTandnare held fixed. The process is mathematically indefensible, but

gives the correct derivative relation:

(

∂U

∂P

)

T,n

T

(

∂S

∂V

)

T,n

−P

(

∂V

∂V

)

T,n

T

(

∂S

∂V

)

T,n

−P (4.3-1)

We apply the Maxwell relation of Eq. (4.2-18) to the first term to obtain

(

∂U

∂V

)

T,n

T

(

∂P

∂T

)

V,n

−P

(the thermodynamic

equation of state)

(4.3-2)

The relation shown in Eq. (4.3-2) is called thethermodynamic equation of state. For

an ideal gas,

T

(

∂P

∂T

)

V,n

T

nR

V

P (4.3-3)

so that

(

∂U

∂V

)

T,n

P−P 0 (4.3-4)

It is now necessary only to specify thatPVnRTto define an ideal gas.

EXAMPLE 4.7

Show thatPis proportional toTin an ideal gas ifVandnare constant, using Eqs. (4.3-1)

and (4.3-4).

Solution

For an ideal gas

T

(

∂P

∂T

)

V,n

P

(

∂P

∂T

)

V,n



P

T

At constantVandn,

1

P

dP

1

T

dT

ln (P)ln (T)+ln (constant)

P

T

constantat constantVandn