- #1
Gravitino
- 13
- 0
[SOLVED] Equipartition Theorem
1. Kerson Huang, P 7.2:
Consider a classical system of N noninteracting diatomic molecules in a box of volume V at temperature T. The Hamiltonian for a single molecule is taken to be
H=\frac{1}{2m}(\vec{p_1}^2+\vec{p_2}^2) +\frac{K}{2}(|\vec{r_1}-\vec{r_2}|^2).
Obtain the internal energy and show that it is consistent with equipartition theorem.
2. Homework Equations and attempt
I used the formulae for partition function
Q_N = \frac{1}{N!}Q_1
where
Q_1 = \int d \omega e^{-\beta H}
Further, I moved to the center of mass frame, and wrote the Hamiltonan in this form:
H= \frac{P_{cm}^2}{4m} + \frac{p^2}{4m} +\frac{Kr^2}{2}
where now \vec{r}=\vec{r_1}-\vec{r_2} and the mass of two atoms the same, while we all know what center of mass frame is (it is not too imprtant for my question to show all the way of derivations, I guess).
After all, using E =-\frac{\partial}{\partial \beta} \ln Q_N
I found that E=\frac{9}{2}Nk_BT.
3. Question
According to equipartition theorem and using my last modified Hamiltonian looks I have 9 degrees of freedom and everything looks like being consistent. However, as far as I know, the diatomic molecule has 6 degrees of freedom (in case it is not rigid molecule). If I think, like that, then I am missing the factor (instead of 6 I have 9). Can you help me, where I am wrong?
1. Kerson Huang, P 7.2:
Consider a classical system of N noninteracting diatomic molecules in a box of volume V at temperature T. The Hamiltonian for a single molecule is taken to be
H=\frac{1}{2m}(\vec{p_1}^2+\vec{p_2}^2) +\frac{K}{2}(|\vec{r_1}-\vec{r_2}|^2).
Obtain the internal energy and show that it is consistent with equipartition theorem.
2. Homework Equations and attempt
I used the formulae for partition function
Q_N = \frac{1}{N!}Q_1
where
Q_1 = \int d \omega e^{-\beta H}
Further, I moved to the center of mass frame, and wrote the Hamiltonan in this form:
H= \frac{P_{cm}^2}{4m} + \frac{p^2}{4m} +\frac{Kr^2}{2}
where now \vec{r}=\vec{r_1}-\vec{r_2} and the mass of two atoms the same, while we all know what center of mass frame is (it is not too imprtant for my question to show all the way of derivations, I guess).
After all, using E =-\frac{\partial}{\partial \beta} \ln Q_N
I found that E=\frac{9}{2}Nk_BT.
3. Question
According to equipartition theorem and using my last modified Hamiltonian looks I have 9 degrees of freedom and everything looks like being consistent. However, as far as I know, the diatomic molecule has 6 degrees of freedom (in case it is not rigid molecule). If I think, like that, then I am missing the factor (instead of 6 I have 9). Can you help me, where I am wrong?
Last edited:
