In class, we assumed that all monatomic gases have 3 degrees of freedom (\(f = 3\)). In this question, we explore the possibility that a monatomic gas might have additional degrees of freedom due to the electrons orbiting the nucleus, or the rotation of the nucleus. To answer this question, you will need to use the equipartition theorem and understand how quantized energy levels affect the application of the equipartition theorem.
(a) An atom of helium can store energy by bumping its electron from its lowest orbital energy level to a higher orbital energy level. Moving an electron from the lowest state to the first excited state would store an energy of 24.6 eV (24.6 electron-volts). Give a quantitative explanation (i.e. by comparing quantities) that shows we can ignore this energy storage mode when calculating the heat capacity of helium gas at ordinary temperatures.
(b) The helium-4 nucleus can be modelled as a solid spherical object with mass \(m\), radius \(r\), and moment of inertia \(I=(2/5)mr^2\). If the nucleus starts to rotate, it would have rotational kinetic energy \(K_{\text{rotation}}=L^2/(2I)\), where \(L\) is the angular momentum. Usually the helium-4 nucleus has \(L = 0\), however, it can be excited to a non-zero angular momentum state with \(L \approx \hbar\), or \(2\hbar\), or \(3\hbar\), etc. (\(L\) is quantized). Give a quantitative explanation that shows we can ignore this energy storage mode when calculating the heat capacity of helium gas at ordinary temperatures.