1. The gravitational potential energies available to a molecule near the surface of the earth are \(\epsilon \left(h\right)=mgh\). Each height, \(h\), corresponds to a unique energy, so we can infer that the degeneracy of \(\epsilon \left(h\right)\) is unity. Derive the probability density function for the distribution of molecules in the earth’s atmosphere. (See Problem 19 in Chapter 3.)

2. The value of the molecular partition function approximates the number of quantum states that are available to the molecule and whose energy is less than \(kT\). How many such quantum states are available to a molecule of molecular weight \(40\) that is confined in a volume of \({10}^{-6}\ {\mathrm{m}}^3\) at \(300\) K?