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# Computing U, H, F, G, S, and mu for nitrogen gas.

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For a mole of nitrogen (N_2) gas at room temperature and atmospheric pressure, compute the internal energy, the enthalpy, the Helmholtz free energy, the Gibbs free energy, the entropy, and the chemical potential. The rotational constant epsilon for N_2 is 0.00025 eV. The electronic ground state is not degenerate.

https://brainmass.com/math/real-analysis/computing-u-h-f-g-s-and-mu-for-nitrogen-gas-106351

#### Solution Preview

We need to evaluate the partition function for the rotational degrees of freedoms for one molecule in the high temperature limit, Z_{rot}.

We can write the total partition function as:

Z = Z_{rot}*Z_{trans}

where Z_{trans} is the partition function for the translational degrees of freedom. Because the thermodynamical quantities are functions of (derivatives of) logarithms of Z, they can be written as a sum of a purely translational part and a rotational part.
We can write any of the thermodynamic quantity X as:

X = X_{rot} + X_{trans}

The translational part is just:

U_{trans} = 3/2 NkT

S_{trans} = Sackur-Tetrode formula

F_{trans} = U_{trans} - T S_{trans}

H_{trans} = U_{trans} + PV = 5/2 NkT

G_{trans} = F_{trans} + PV = F_{trans} + NkT

The chemical potential mu follows from the fact that G = N mu (let me know if you don't know how this is ...

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