Physical Chemistry Third Edition

2.4 Calculation of Amounts of Heat and Energy Changes 61

The Heat Capacity at Constant Volume

For a closed simple system at equilibriumnis fixed and we can chooseTandVas

independent variables:

dU

(

∂U

∂T

)

V,n

dT+

(

∂U

∂V

)

T,n

dV (closed simple system) (2.4-1)

From the first law of thermodynamics an infinitesimal amount of heat is given by

dqdU−dwdU+P(transmitted)dV (closed simple system) (2.4-2)

IfVis constant,dV0 anddw0:

dqdU

(

∂U

∂T

)

V,n

dT (Vconstant, simple system) (2.4-3)

Comparison of this equation with Eq. (2.2-1) shows that

CV

(

∂U

∂T

)

V,n

(simple system) (2.4-4)

whereCVis the heat capacity at constant volume. Although the heat capacity in general

is not a derivative of any function, the heat capacity at constant volume of a simple

closed system is equal to the partial derivative ofUwith respect toT.

The Ideal Gas as an Example System

It is found experimentally that all gases at sufficiently low pressure (dilute gases) obey

the ideal gas equation of state to an adequate approximation:

PV≈nRT (dilute gas) (2.4-5)

Dilute gases also have an internal energy that is independent of the volume or the

pressure:

U≈U(T,n) (dilute gas) (2.4-6)

We now define anideal gasas one that obeys Eqs. (2.4-5) and (2.4-6) exactly for all

temperatures and pressures. We will show in Chapter 4 that Eq. (2.4-6) is a consequence

of Eq. (2.4-5).

To an excellent approximation, the heat capacities at constant volume of monatomic

gases are given from Eq. (2.3-10) by

CV

(

∂U

∂T

)

V,n

≈

3

2

nR (dilute monatomic gas) (2.4-7)

wherenis the amount of the gas in moles andRis the ideal gas constant. If the

vibrational and electronic motions of molecules can be ignored, Eqs. (2.3-11) and

(2.3-12) give

CV≈

5

2

nR (diatomic or linear polyatomic dilute gas) (2.4-8)