Near room temperature, the classical Helmholtz free energy of \(N\) diatomic gas molecules in a container of volume \(V\) is given by

\[A(N, V, T)=-N k T \ln \left(\frac{V}{N \lambda^{3}}\left(\frac{2 I k T}{\hbar^{2}}\right)\right)-N k T\]

where \(\lambda=h / \sqrt{2 \pi m k T}\) is the thermal de Broglie wavelength, \(m=m_{1}+m_{2}\) is the mass of the molecule, \(I=\mu a^{2}\) is the moment of inertia, \(\mu=m_{1} m_{2} /\left(m_{1}+m_{2}\right)\) is the reduced mass, and \(a\) is the bond length. The molecules are attracted to the walls of the container by van der Waals forces with binding energy \(\varepsilon\). If they are otherwise free to move and rotate about one axis, the surface Helmholtz free energy of \(N_{S}\) molecules on surface area \(A\) is given by

\[A_{\mathrm{S}}\left(N_{s}, A, T\right)=-N_{s} \varepsilon-N_{s} k T \ln \left(\frac{A}{N_{s} \lambda^{2}} \sqrt{\frac{2 \pi I k T}{\hbar^{2}}}\right)-N_{s} k T\].

Calculate the equilibrium surface number density \(N_{S} / A\). Estimate the number of nitrogen and oxygen molecules in the bulk and on the surface of a spherical 1 liter vacuum chamber at room temperature and \(1 \mathrm{~atm}\) for surface binding energy \(\varepsilon / k \approx 1000 \mathrm{~K}\). As the vacuum chamber is pumped down, the molecules trapped on the walls only very slowly unbind. Show that this interferes with maintaining an ultrahigh vacuum below about \(10^{-7}\) pascal if the vacuum chamber is simply pumped down starting from \(1 \mathrm{~atm}\). What experimental method might one use to achieve low pressures by reducing the number of molecules trapped on the walls?