Derive an expression for the equilibrium constant \(K(T)\) for the reaction \(\mathrm{N}_{2}+\mathrm{O}_{2} ightleftarrows 2 \mathrm{NO}\) in terms of the ground state energy change \(\Delta \varepsilon_{0}=2 \varepsilon_{\mathrm{NO}}-\varepsilon_{\mathrm{N}_{2}}-\varepsilon_{\mathrm{O}_{2}}\) and the vibrational and rotational partition functions of the diatomic molecules, using results from Section 6.5. Give predictions for the ranges of temperatures where the rotational modes are classically excited but the vibration modes are suppressed and for higher temperatures where both the rotational and vibrational models are classically excited.