= Solution
The given binding <energy> $\epsilon$ belongs to the formation of two carbon-dioxide <molecules>. Relative to the reference used in (a), the internal <energy> offset of one carbon-dioxide <molecule> is therefore $-\epsilon/2$, not $-\epsilon$. Its mass is $m_c=m_{\rm CO}+m_{\rm O_2}/2$ and its <thermal de Broglie wavelength> is $\lambda_c=h/\sqrt{2\pi m_ck_BT}$, where the subscript $c$ denotes $\mathrm{CO_2}$.
The single-molecule <canonical partition function> is $(V/\lambda_c^3)e^{\beta\epsilon/2}$. For $N_c$ indistinguishable noninteracting <molecules>,
$$
\boxed{Z_{\rm CO_2}=\frac{(V/\lambda_c^3)^{N_c}}{N_c!}\exp\left(\frac{\beta\epsilon N_c}{2}\right).}
$$
The corresponding <Helmholtz free energy>, using <Stirling's approximation>, is
$$
\boxed{F_{\rm CO_2}=N_ck_BT\left[\ln\left(\frac{N_c\lambda_c^3}{V}\right)-1\right]-\frac{\epsilon N_c}{2}.}
$$
The negative sign in the last term expresses the stabilizing binding <energy> and ensures that low <temperature> favors association.
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