Consider \(1 \mathrm{~kg}\) of pure diatomic \(\mathrm{N}_{2}\) in thermodynamic equilibrium. The fundamental vibrational frequency of \(\mathrm{N}_{2}\) is \(v=7.06 \times 10^{13} / \mathrm{s}\), the molecular weight \(\mathscr{M}_{\mathrm{N}_{2}}=28\), Planck's constant \(h=6.625 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}\), and the Boltzmann constant is \(k=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K}\).

a. Calculate and plot on graph paper the number of \(\mathrm{N}_{2}\) molecules in each of the first three vibrational energy levels, \(\varepsilon_{o}, \varepsilon_{1}\), and \(\varepsilon_{2}\) as a function of temperature from \(T=300\) to \(3500 \mathrm{~K}\), using \(400 \mathrm{~K}\) increments.

b. Calculate and plot on graph paper the sensible enthalpy (including translation, rotation, and vibration) in joules per kilogram as a function of temperature from \(T=300\) to \(3500 \mathrm{~K}\).

c. Calculate and plot on graph paper the specific heat at constant pressure as a function of temperature from \(T=300\) to \(3500 \mathrm{~K}\).