- #1
Euclid
- 213
- 0
This is a GRE question.
A classical model of a diatomic molecule is a springy dumbbell, where the dumbbell is free to rotate about axes perpendicular to the spring. In the limit of high temperature. what is the specific heat per mole at constant volume?
The answer is 7R/2. This site
http://www.grephysics.net/v2006loader.php?serial=4&prob=15&yload=1
recommends to use a brute force approach to solve the problem. I believe the problem is much easier than suggested here, but since I have no experience with statistical mechanics, I would like help solving it this way.
1) How does one write down the partition function for this model? Taking into account only vibrational states, is it given by Z=\sum \limits_{n=0}^{\infty} \exp(-\beta (n+1/2)h\omega)?
If it is, the energy <E> should be given by
<E> = -\frac{1}{Z} \frac{\partial Z}{\partial \beta} = \frac{h \omega}{2} + \frac{1}{Z}\sum \limits_n^\infty n \exp(-\beta (n+1/2)h\omega)
The specific heat should then be given by
c=\frac{\partial <E>}{\partial T} =\frac{h\omega}{kT^2}(<n E> - <E><n>),
where I have used shorthand that is hopefully clear. Evaluating the limit T--> infty seems unpleasant.
2) Now, what if I want to solve the problem completely, by taking into account the quantization of rotational states? How do I write down the partition function in this case?
How do I take into account translational energy?
A classical model of a diatomic molecule is a springy dumbbell, where the dumbbell is free to rotate about axes perpendicular to the spring. In the limit of high temperature. what is the specific heat per mole at constant volume?
The answer is 7R/2. This site
http://www.grephysics.net/v2006loader.php?serial=4&prob=15&yload=1
recommends to use a brute force approach to solve the problem. I believe the problem is much easier than suggested here, but since I have no experience with statistical mechanics, I would like help solving it this way.
1) How does one write down the partition function for this model? Taking into account only vibrational states, is it given by Z=\sum \limits_{n=0}^{\infty} \exp(-\beta (n+1/2)h\omega)?
If it is, the energy <E> should be given by
<E> = -\frac{1}{Z} \frac{\partial Z}{\partial \beta} = \frac{h \omega}{2} + \frac{1}{Z}\sum \limits_n^\infty n \exp(-\beta (n+1/2)h\omega)
The specific heat should then be given by
c=\frac{\partial <E>}{\partial T} =\frac{h\omega}{kT^2}(<n E> - <E><n>),
where I have used shorthand that is hopefully clear. Evaluating the limit T--> infty seems unpleasant.
2) Now, what if I want to solve the problem completely, by taking into account the quantization of rotational states? How do I write down the partition function in this case?
How do I take into account translational energy?
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