Physical Chemistry Third Edition

1090 26 Equilibrium Statistical Mechanics. II. Statistical Thermodynamics

The Translational Energy of a Dilute Gas

The translational contribution to the energy is the same for all dilute gases:

UtrNkBT^2

(

∂ln(ztr)

∂T

)

V

NkBT^2

(

∂

∂T

ln

(

2 πmkBT

h^2

) 3 / 2

V

)

V



3

2

NkBT^2

d

dT

ln(T)

Utr

3

2

NkBT (26.2-3)

The Electronic Energy of a Dilute Gas

The electronic contribution to the energy is obtained by manual summation:

UelNkBT^2

(

dln(zel)

dT

)

UelNkBT^2

d

dT

[

ln(g 0 e−ε^0 /kBT+g 1 e−ε^1 /kBT+ ···

]

NkBT

1

zel

[

g 0

ε 0

kBT

e−ε^0 /kBT+g 1

ε 1

kBT

e−ε^1 /kBT+ ···

]



N

zel

[g 0 ε 0 e−ε^0 /kBT+g 1 ε 1 e−ε^1 /kBT+ ···] (26.2-4)

In most atoms and molecules, the ground electronic level is nondegenerate and the first

excited level is sufficiently high in energy that at room temperature the second and

further terms in the sum can be neglected. In this case the electronic partition function

is very nearly equal tog 0 e−ε^0 /kBTand

Uel≈Nε 0

(

molecules with high

first excited level

)

(26.2-5a)

If the ground-state electronic energyε 0 is chosen to equal zero, then

Uel≈ 0

⎛

⎝

molecules with high

first excited level

andε 0  0

⎞

⎠ (26.2-5b)

Some substances such as NO have an excited level that is near the ground level, so that

Eq. (26.2-5) does not apply, as will be illustrated in the following example.

EXAMPLE26.3

Find the electronic contribution to the thermodynamic energy of 1.000 mol of gaseous NO at

1.000 atm and 298.15 K. The ground electronic level is a^2 Π 1 / 2 term. The first excited level

is a^2 Π 3 / 2 term with an energy 2.380× 10 −^21 J above the ground level. Both levels have a

degeneracy equal to 2.