Physical Chemistry Third Edition

170 4 The Thermodynamics of Real Systems

Exercise 4.7

a.Find an expression for (∂H/∂P)T,nfor a gas obeying the truncated pressure virial equation

of state:

PVmRT+A 2 P

whereA 2 is a function ofT. It has been shown that the second pressure virial coefficientA 2

is equal toB 2 , the second virial coefficient.

b.Evaluate (∂H/∂P)T,nfor 1.000 mol of argon at 1.000 atm and 298.15 K. Data onB 2 and

dB 2 /dTare found in Example 4.3.

We can now obtain a useful relation betweenCPandCVfor systems other than ideal

gases. Equation (2.5-11) is

CPCV+

[(

∂U

∂V

)

T,n

+P

](

∂V

∂T

)

P,n

(4.3-7)

TheCVterm represents the energy change due to an increase in temperature that

would occur if the volume were constant. The (∂U/∂V) (internal pressure) term repre-

sents the energy absorbed in raising the potential energy of intermolecular attraction,

and theP(∂V /∂T) term represents work done against the pressure exerted by the

surroundings.

We now use the thermodynamic equation of state to write

CPCV+

[

T

(

∂P

∂T

)

V,n

+P−P

](

∂V

∂T

)

P,n

CV+T

(

∂P

∂T

)

V,n

(

∂V

∂T

)

P,n

(4.3-8)

We apply the cycle rule, Eq. (B-15) of Appendix B, in the form:

(

∂P

∂T

)

V,n

(

∂T

∂V

)

P,n

(

∂V

∂P

)

T,n

− 1 (4.3-9)

which gives

CPCV−T

(

∂P

∂V

)

T,n

[(

∂V

∂T

)

P,n

] 2

(4.3-10)

We obtain

CPCV−T

(

∂P

∂V

)

T,n

[(

∂V

∂T

)

P,n

] 2

CV+

TV α^2

κT

(4.3-11)

whereαis the coefficient of thermal expansion andκTis the isothermal compressibility.

Sinceα(which is occasionally negative) is squared and sinceκTis always positive,CP

is never smaller thanCV.