Physical Chemistry Third Edition

168 4 The Thermodynamics of Real Systems

EXAMPLE 4.8

For a gas obeying the truncated virial equation of state,

PVm

RT

 1 +

B 2

Vm

show that

(

∂U

∂V

)

T,n

RT^2

(dB 2 /dT)

Vm^2

Solution

P

RT

Vm

+

RT B 2

Vm^2

From Eq. (4.3-2)

(

∂U

∂V

)

T,n

T

(

∂P

∂T

)

V,n

−P

If bothVandnare held fixed, thenVmis held fixed;

T

(

∂P

∂T

)

V,n



RT

Vm

+

RT

Vm^2

(

B 2 +T

dB 2

dT

)

(

∂U

∂V

)

T,n



RT

Vm

+

RT

Vm^2

(

B 2 +T

dB 2

dT

)

−

RT

Vm

−

RT B 2

Vm^2



RT^2

Vm^2

dB 2

dT

Exercise 4.5

Find the value of (∂U/∂V)T,nfor 1.000 mol of argon at 1.000 atm and 298.15 K, using the

truncated virial equation of state.

The partial derivative (∂U/∂V)T,nis an important measure of the deviation of a

system from ideal gas behavior. It has the same units as pressure and is known as the

internal pressure.If the internal pressure is positive, the energy increases as the volume

increases and work is done against the attractive intermolecular forces. The internal

pressure is a measure of the net cohesive forces of the system and can have a large

positive value for liquids.

To derive a useful equation for the internal pressure, we begin with Eq. (4.3-2)

and apply the cycle rule to the partial derivative on the right-hand side of the

equation:

(

∂U

∂V

)

T,n

−T

(

∂P

∂V

)

T,n

(

∂V

∂T

)

P,n

−P