Physical Chemistry Third Edition

160 4 The Thermodynamics of Real Systems

Equation (4.2-10) is one of a class of equations calledMaxwell relations. A common

use of these relations is to replace a partial derivative that is hard to measure with

one that can more easily be measured. For example, it would be difficult to measure

(∂P/∂S)V,n, but much easier to measure (∂T /∂V)S,n.

The Maxwell relations are named for
James Clerk Maxwell, 1831–1879,
a great British physicist who made
fundamental contributions to
electromagnetic theory, gas kinetic
theory, and thermodynamics.

EXAMPLE 4.1

From the relation in Eq. (4.2-10), find an expression for (∂P/∂S)V,nfor an ideal gas with

constant heat capacity.

Solution

Equation (2.4-21) gives for a reversible adiabatic process in an ideal gas with constant heat

capacity

TT 1

(

V 1

V

)nR/CV

where we omit the subscripts on the final values ofTandV. Since a reversible adiabatic

process corresponds to constant entropy, differentiation of this formula with respect toV

corresponds to constantS:

(

∂P

∂S

)

V,nn

−

(

∂T

∂V

)

S,n

−T 1 (V 1 )nR/CV

(

nR

CV

)

V−(nR/CV)−^1



nR T 1

CVV

(

V 1

V

)

VnR/CV

To complete the solution, we replaceT 1 V 1 nR/CVbyTVnR/CV:

(

∂P

∂S

)

V,n

−

nRT

CVV

−

RT

CV, mV

Exercise 4.1

a.Find the value of (∂P/∂S)V,nfor 1.000 mol of helium at 1.000 atm (101325 Pa) and 298.15 K.

Assume that helium is ideal withCV 3 nR/2.

b.Find the value of (∂P/∂S)V,nfor 2.000 mol of helium at 1.000 atm (101325 Pa) and 298.15 K.

Explain the dependence on the amount of substance.

c.Find the value of (∂P/∂S)V,nfor 1.000 mol of helium at 2.000 atm (202650 Pa) and 298.15 K.

Explain the dependence on the pressure.

We now write the differentialdHfor a closed system from the definition ofH:

dHdU+PdV+VdPTdS−PdV+PdV+VdP (4.2-11)

TdS+VdP (closed system)