= Solution
Write $\beta=(k_{\rm B}T)^{-1}$ and $x=\beta\delta$. The one-site <partition function> includes both degenerate lower states,
$$
q=e^{\beta\varepsilon}(2+e^{-x}).
$$
The occupied <adsorption sites> are distinguishable positions, so the <canonical partition function> is a product, \b[without an additional $N!$ divisor]:
$$
\boxed{Z_N=q^N=e^{\beta N\varepsilon}(2+e^{-\beta\delta})^N.}
$$
Differentiating $\ln Z_N$ gives the <internal energy>,
$$
\boxed{U=-\left(\frac{\partial\ln Z_N}{\partial\beta}\right)_N
=-N\varepsilon+\frac{N\delta e^{-\beta\delta}}{2+e^{-\beta\delta}}
=-N\varepsilon+\frac{N\delta}{1+2e^{\beta\delta}}.}
$$
The final factor is the population of the perpendicular orientation. The shared binding offset $-\varepsilon$ shifts all <energy levels> equally and does not determine their relative populations.
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