Student handout: Heat capacity of N2

Contemporary Challenges 2021
Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”

For N2 gas molecules in a 10 cm cubic box, the rules of QM dictate the discrete allowed values for:

  1. Translational K.E. in one dimension: \(\left\{ 1\times 10^{-40}\text{ J}, 4\times 10^{-40}\text{ J}, 9\times 10^{-40}\text{ J}, \ldots \right\}\)
  2. Rotational K.E.: \(\left\{ 0\text{ J}, 0.8\times 10^{-22}\text{ J}, 0.8\times 10^{-22}\text{ J}, 0.8\times 10^{-22}\text{ J}, 2.5\times 10^{-22}\text{ J}, \ldots \right\}\)
  3. Vibrational energy: \(\left\{ 2.3\times 10^{-20}\text{ J}, 6.9\times 10^{-20}\text{ J}, 11.5\times 10^{-20}\text{ J}, \ldots \right\}\)
Sketch a graph of \(\frac{dU}{dT}\) of \(10^{22}\) molecules of N2 gas in the temperature range of 70 K (the temperature at which \(N_2\) becomes liquid at 1 atm of pressure) to 5000 K (at which temperature N2 breaks apart).

(use a logarithmic temperature axis)

Keywords
equipartition quantum energy levels
Learning Outcomes