Question

Consider a (classical) ideal gas of diatomic molecules occupying a volume $V$ at temperature $T$. The Hamiltonian describing one molecule is $H(\mathbf{r}_1, \mathbf{r}_2, \mathbf{p}_1, \mathbf{p}_2) = \frac{1}{2m}(p_1^2 + p_2^2) + \frac{K}{2}|\mathbf{r}_1 - \mathbf{r}_2|^2$, where $K$ is a (positive) constant, $(\mathbf{r}_1, \mathbf{r}_2)$ describe the positions of the two atoms, and $(\mathbf{p}_1, \mathbf{p}_2)$ the corresponding momenta. (a) Calculate the canonical partition function for a system containing $N$ molecules. [Hint: Use the center of mass and relative positions and the corresponding momenta; use the fact that the inter-atomic distance is much smaller than the size of the system]. (b) Calculate the free energy of the system and the equations of state (for $S$, $P$, and $\mu$). (c) Determine the energy of the system using the theorem of the equipartition of energy. Check the result by explicitly calculating the energy. (d) Determine the temperature dependence of $(\mathbf{r}_1 - \mathbf{r}_2)^2$.

          Consider a (classical) ideal gas of diatomic molecules occupying a volume $V$ at temperature $T$. The Hamiltonian describing one molecule is
$H(\mathbf{r}_1, \mathbf{r}_2, \mathbf{p}_1, \mathbf{p}_2) = \frac{1}{2m}(p_1^2 + p_2^2) + \frac{K}{2}|\mathbf{r}_1 - \mathbf{r}_2|^2$,
where $K$ is a (positive) constant, $(\mathbf{r}_1, \mathbf{r}_2)$ describe the positions of the two atoms, and $(\mathbf{p}_1, \mathbf{p}_2)$ the corresponding momenta.
(a) Calculate the canonical partition function for a system containing $N$ molecules. [Hint: Use the center of mass and relative positions and the corresponding momenta; use the fact that the inter-atomic distance is much smaller than the size of the system].
(b) Calculate the free energy of the system and the equations of state (for $S$, $P$, and $\mu$).
(c) Determine the energy of the system using the theorem of the equipartition of energy. Check the result by explicitly calculating the energy.
(d) Determine the temperature dependence of $(\mathbf{r}_1 - \mathbf{r}_2)^2$.

Consider a (classical) ideal gas of diatomic molecules occupying a volume V at temperature T. The Hamiltonian describing one molecule is
H(𝐫1, 𝐫2, 𝐩1, 𝐩2) = (1)/(2m)(p1^2 + p2^2) + (K)/(2)|𝐫1 - 𝐫2|^2,
where K is a (positive) constant, (𝐫1, 𝐫2) describe the positions of the two atoms, and (𝐩1, 𝐩2) the corresponding momenta.
(a) Calculate the canonical partition function for a system containing N molecules. [Hint: Use the center of mass and relative positions and the corresponding momenta; use the fact that the inter-atomic distance is much smaller than the size of the system].
(b) Calculate the free energy of the system and the equations of state (for S, P, and μ).
(c) Determine the energy of the system using the theorem of the equipartition of energy. Check the result by explicitly calculating the energy.
(d) Determine the temperature dependence of (𝐫1 - 𝐫2)^2.

Added by Katherine P.

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