Concise Physical Chemistry

c03 JWBS043-Rogers September 13, 2010 11:24 Printer Name: Yet to Come

48 THE THERMODYNAMICS OF SIMPLE SYSTEMS

3.10 THE HEAT CAPACITY OF AN IDEAL GAS

An ideal gas consists of point particles that cannot vibrate or rotate. It can have

only kinetic energyUkin=^12 mv^2. For a large collection of particles, the total kinetic

energy is

Ukin=^12 Nmu ̄^2

whereNis the number of atoms or molecules andu ̄is their average speed. We also

know from the kinetic theory of gases that

pV=RT=^13 NAmu ̄^2 =^23

( 1

2 NAmu ̄

2 )= 2

3 Ukin

This (slightly circular) reasoning leads to the simple statement

Ukin=^32 RT

for one mole of an ideal gas. No direction is favored over any other in Cartesian

3-space, so we can split the kinetic energy components into three equal parts of

1

2 RTper degree of freedom along any arbitraryx,y,zspace coordinates. This division

is general. On a molar basis, we expect to find^12 RTof energy per mole per degree

of freedom or^12 kBTper particle (molecule or atom) per translational, rotational,

vibrational or, rarely, electronic degree of freedom. (Recall thatkBis the gas constant

per particle.)

If the gas is ideal, we obtain amolar heat capacity

CV=

(

∂U

∂T

)

V

=

∂

( 3

2 RT

)

∂T

=^32 R

SinceR∼=2 cal K−^1 mol−^1 , the heat capacity of an ideal gas is about 3 cal K−^1 mol−^1 =

12.5 J K−^1 mol−^1. Table 3.1 shows that this is true for the monatomic gases helium He

and mercury vapor Hg but that it is not true for more complicated molecular species.

TABLE 3.1 Heat Capacities andγfor Selected Gases.

Gas CV(J K−^1 mol−^1 ) Cp(J K−^1 mol−^1 ) γ(unitless)

He 12.5 20.8 1.67

Hg 12.5 20.8 1.67

H 2 20.5 28.9 1.41

NH 3 27.5 36.1 1.31

Diethyl ether 57.5 66.5 1.16