= Solution
{c}
In the <harmonic approximation>, each in-plane displacement coordinate contributes one quadratic <kinetic energy> term and one quadratic restoring <potential energy> term. The <equipartition theorem> assigns $k_BT/2$ to each. There are two such coordinates per <chemical atom>, hence
$$
U\simeq2N_ak_BT,\qquad
\boxed{\frac{C_v}{N_a}\longrightarrow2k_B.}
$$
Here the fixed-size <heat capacity> is at constant area in this two-dimensional model. On cooling, <phonon> excitation freezes out and the <heat capacity> decreases. For each nonzero <angular frequency>, the <quantum harmonic oscillator> contributes
$$
c_\omega=k_B\frac{x^2e^x}{(e^x-1)^2},\qquad x=\frac{\hbar\omega}{k_BT},
$$
which is below $k_B$ at finite $x$ and approaches zero as $T\to0$. Two linearly dispersing in-plane <acoustic phonon branches> give the familiar two-dimensional <Debye model> scaling $C_v\propto T^2$ at sufficiently low <temperature>. Out-of-plane modes are excluded by the specified model.
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