Thermal Physics Problem: CO2 Gas Heat Capacity at Different Temperatures

[kottur]

1. Homework Statement

Consider a CO2 molecule, which is linear and has vibrational modes with frequency
corresponding to 2565 cm-1 (an asymmetric stretch), 1480 cm-1 (a symmetric stretch)
526 cm-1 (bends). Sketch a curve showing how the constant volume heat capacity of CO2
gas varies with temperature and mark the values of plateaus. (Recall: the spacing between
rotational levels is smaller than the spacing between the vibrational levels).

3. The Attempt at a Solution

I think Z=$\frac{1}{1-e^{-h\varpi/\tau}}$ but after that I have problems with finding U so that I can find the heat capacity with: C$_{v}$=$\left(\frac{dU}{d\tau}\right)_{v}$

 

[anathema]

Thank you for your post. Your initial equation for partition function, Z, is correct. To find the internal energy, U, you can use the relation U=\frac{3}{2}NkT+\frac{1}{2}N(h\nu_1+n\nu_2+...), where N is the number of molecules, k is the Boltzmann constant, T is the temperature, and h is the Planck's constant. The second term in this equation accounts for the energy of the vibrational modes. You can then differentiate U with respect to temperature to find the heat capacity at constant volume, C_v. Remember that for a linear molecule, the vibrational modes are degenerate, so you will need to account for this in your calculations.

I hope this helps. Good luck with your calculations.
Scientist