For a diatomic gas near room temperature, the internal partition function is simply the rotational partition function multiplied by the degeneracy Ze of the electronic ground state.

(a) Show that the entropy in this case is

S = Nk [ln (VZeZrot/NvQ) + 7/2.

Calculate the entropy of a mole of oxygen (Ze = 3) at room temperature and atmospheric pressure, and compare to the measured value.

(b) Calculate the chemical potential of oxygen in earth's atmosphere near sea level, at room temperature. Express the answer in electron-volts.

We can calculate the entropy from the partition function as follows. The partition function for an ideal gas is:

Z = Z1^(N)/N!

where Z1 is the partition function for a single molecule. Z1 factorizes:

Z1 = Z_{trans}*Z_{rot}*Z_{vib}*Z_{elec}*etc.

The translational part of the partition function is:

Z_{trans} = V*(2 pi m kT/h^2)^(3/2)

Since a partition function must be dimensionless the factor

(2 pi m kT/h^2)^(3/2)

must have the dimensions of an inverse volume. You can easily check this. This is the volume at which quantum effects become important. We define the quantity:

Vq = (2 pi m kT/h^2)^(-3/2)

to simplify expressions, but note that Vq depends on the temperature.

We ignore the vibrational part of the partition function and the electronic part is given by the degeneracy of the ground state, which ...

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... and atmospheric pressure, compute the internal energy, the enthalpy, the Helmholtz free energy, the Gibbs free energy, the entropy, and the chemical potential. ...