Physical Chemistry Third Edition

4.2 Fundamental Relations for Closed Simple Systems 161

Therefore

(

∂H

∂S

)

P,n

T (4.2-12)

and

(

∂H

∂P

)

S,n

V (4.2-13)

Using the Euler reciprocity relation, we obtain a second Maxwell relation:

(

∂T

∂P

)

S,n



(

∂V

∂S

)

P,n

(a Maxwell relation) (4.2-14)

EXAMPLE 4.2

Find an expression for (∂V /∂S)P,nfor an ideal gas with constant heat capacity.

Solution

For a reversible adiabatic process,

T 2

T 1



(

P 2

P 1

)nR/(CV+nR)

Drop the subscript 2:

TT 1

(

P

P 1

)nR/(CV+nR)

(

∂V

∂S

)

P,n



(

∂T

∂P

)

S,n



nR

CV+nR

T 1

PnR/ 1 (CV+nR)

PnR/(CV+nR)−^1



nR

CV+nR

T 1

PnR/ 1 (CV+nR)

PnR/(CV+nR)

1

P



nR

CV+nR

T

P



R

CV, m+R

T

P

Exercise 4.2

a.Evaluate (∂V /∂S)P,nfor 1.000 mol of helium (assumed ideal) at 1.000 atm and 298.15 K.

TakeCV, m 3 R/2.

b.Evaluate (∂V /∂S)P,nfor 2.000 mol of helium at 1.000 atm and 298.15 K. Explain the depen-

dence on the amount of substance.