Contribution to the heat capacity of vibrational energy levels.

[lxazy]

Homework Statement

The carbon-dioxide has nondegenerate vibrational energies εr=（h/2π）ω(r+1);r=0,1,2..., where ω=(1.26*10^14)s^-1. What is the contribution of these vibrational modes to the molar heat capacity of carbon-dioxide gas at T=400K?

Homework Equations

Z=∑exp(-（h/2π）ω(r+1)/k_BT)

F=-k_BTlnZ

Cv=-T(d^2F/dT^2) keeping the volume constant

The Attempt at a Solution

I checked in some books, but did not find any relevant examples about this type of problems.

I found some formulas and process for solving this kind of question, but for diatomic molecules. Could someone please tell me whether these equations are the ones needed in this problem, if not, please let me know the correct equations and method. THANK YOU!

Z=∑exp(-（h/2π）ω(r+1)/k_BT)

F=-k_BTlnZ

Cv=-T(d^2F/dT^2) keeping the volume constant

 

[Astronuc]

What is the energy (heat capacity) in a mole?

How many molecules in a mole? How much energy is in the lowest energy state of one molecule? Then the next? . . . .

Or who would one expect the energy to be partitioned at 400 K?

 

[kloptok]

You should be able to calculate Z explicitly, it is a geometric series, right?

With the equations you've listed it's basically just mathe-magic from there, I think.

What is it you've found for diatomic molecules?

 

[lxazy]

You should be able to calculate Z explicitly, it is a geometric series, right?

With the equations you've listed it's basically just mathe-magic from there, I think.

What is it you've found for diatomic molecules?

Thanks for your reply! For diatomic molecules I would have used these steps and formulas to calculate the heat capacity. However, it was carbon-dioxide in this problem, which was a polyatomic molecule, and I did not find anything relevant from the statistical physics books. That is why I tried to seek for help on what procedure and formulas and techniques that should be used to solve this kind of problem.

 

[paranormal]

Hello,

The equations you have listed are indeed relevant for solving this problem. The first equation, Z, is the partition function which takes into account the possible energy levels of the molecule. The second equation, F, is the Helmholtz free energy which is related to the partition function. And the third equation, Cv, is the heat capacity at constant volume.

To solve this problem, you will need to calculate the partition function, Z, for the given vibrational energies of carbon dioxide. This can be done by plugging in the values for εr, ω, and T into the first equation. Once you have the value for Z, you can use it to calculate the Helmholtz free energy, F, using the second equation. Finally, you can use the third equation to calculate the heat capacity, Cv, at the given temperature of 400K.

I hope this helps. Let me know if you have any further questions or need clarification. Good luck!