Question

A gas consisting of N noninteracting diatomic molecules is confined within a volume V. The kinetic energy of each molecule contains terms associated with both translation and rotation, so that the corresponding (single particle) Hamiltonian is $H_1(r, \theta, \phi, p, p_\theta, p_\phi) = \frac{p \cdot p}{2m} + \frac{p_\theta^2}{2I} + \frac{p_\phi^2}{2I \sin^2 \theta}$, where $m$ is the mass of the molecule and $I$ its moment of inertia. (a) Calculate the partition function $Z(T, V, N)$ in the canonical ensemble and the free energy of the system, $F(T, V, N)$. (b) Calculate the coefficients $c_V = C_V/N$ and $c_p = C_p/N$, and the adiabatic coefficient $\gamma$ for the diatomic gas. Compare the results with the corresponding quantities for the monoatomic ideal gas.

          A gas consisting of N noninteracting diatomic molecules is confined within a volume V. The kinetic energy of each molecule contains terms associated with both translation and rotation, so that the corresponding (single particle) Hamiltonian is
$H_1(r, \theta, \phi, p, p_\theta, p_\phi) = \frac{p \cdot p}{2m} + \frac{p_\theta^2}{2I} + \frac{p_\phi^2}{2I \sin^2 \theta}$,
where $m$ is the mass of the molecule and $I$ its moment of inertia.
(a) Calculate the partition function $Z(T, V, N)$ in the canonical ensemble and the free energy of the system, $F(T, V, N)$.
(b) Calculate the coefficients $c_V = C_V/N$ and $c_p = C_p/N$, and the adiabatic coefficient $\gamma$ for the diatomic gas. Compare the results with the corresponding quantities for the monoatomic ideal gas.

A gas consisting of N noninteracting diatomic molecules is confined within a volume V. The kinetic energy of each molecule contains terms associated with both translation and rotation, so that the corresponding (single particle) Hamiltonian is
H1(r, θ, ϕ, p, pθ, pϕ) = (p · p)/(2m) + (pθ^2)/(2I) + (pϕ^2)/(2I sin^2 θ),
where m is the mass of the molecule and I its moment of inertia.
(a) Calculate the partition function Z(T, V, N) in the canonical ensemble and the free energy of the system, F(T, V, N).
(b) Calculate the coefficients cV = CV/N and cp = Cp/N, and the adiabatic coefficient γ for the diatomic gas. Compare the results with the corresponding quantities for the monoatomic ideal gas.

Added by Carlos Y.

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