Physical Chemistry Third Edition

26.2 Working Equations for the Thermodynamic Functions of a Dilute Gas 1089

b.How will the rotational partition functions of the two

molecules compare at equal volumes and equal

temperatures?

c.How will the vibrational partition functions of the two

molecules compare at 300 K?

d.How will the energies of 1.000 mol of each substance

compare at 1.000 bar (100,000 Pa) and 300 K?

e.How will the entropies of 1.000 mol of

each substance compare at 1.000 bar and

300 K?

26.2 a.Find the molar entropy of helium at 323.15 K and
1.000 bar (100,000 Pa).
b.Using the value from part a, estimate the value ofΩfor
1.000 mol of helium at 323.15 K and 1.000 bar
(100,000 Pa). Find the ratio of this value to the
value of Example 26.1 and comment on this
ratio.

26.3 Using the thermodynamic relation

μ

(

∂G

∂N

)

T,P

derive the formula for the chemical potential of a dilute

gas. You must express the molecular partition function in

terms ofTandP.

26.4 Calculate the Helmholtz energy of 1.000 mol of Cl 2 gas at

298.15 K and 1.000 bar pressure.

26.5 Calculate the molar entropy of water vapor at 100◦C and

1.000 bar. The bond distances are equal to 95.8 pm and the

bond angle is equal to 104.45◦. The vibrational frequencies

are 4.7817× 1013 s−^1 , 1.0947× 1014 s−^1 , and

1.1260× 1014 s−^1.

26.6 Calculate the value ofμ−ε 0 for water vapor at 100◦C and

1.000 bar, whereε 0 is the ground-state energy of the

molecule.

26.7 Calculate the molar heat capacity at constant pressure for

carbon dioxide at 298.15 K.

26.2 Working Equations for the Thermodynamic

Functions of a Dilute Gas

In the previous section we obtained general formulas for the thermodynamic variables

of a dilute gas in terms of the logarithm of the molecular partition function. In this

section we write working formulas for the different contributions to the thermodynamic

functions. If the partition function is factored as in Eq. (25.4-6), the logarithm is a sum

of terms:

ln(z)ln(ztr)+ln(zrot)+ln(zvib)+ln(zelec) (atoms) (26.2-1a)

ln(z)ln(ztr)+ln(zrot)+ln(zvib)+ln(zel) (molecules) (26.2-1b)

The Internal Energy of a Dilute Gas

Since the logarithm of the partition function is a sum of terms, the internal energy of a

dilute gas is a sum of contributions. For a monatomic substance

UUtr+Uel (26.2-2a)

For a molecular substance

UUtr+Uel+Urot+Uvib (26.2-2b)