= Solution
Chemical equilibrium in $n+\nu_e\leftrightarrow p+e^-$ gives
$$
\mu_n+\mu_{\nu_e}=\mu_p+\mu_e.
$$
The negligible lepton <chemical potentials> therefore imply $\mu_n\simeq\mu_p$. Nonrelativistic Maxwell--Boltzmann number densities satisfy
$$
n_i\simeq g_i\left(\frac{m_ik_BT}{2\pi\hbar^2}\right)^{3/2}
\exp\left(\frac{\mu_i-m_ic^2}{k_BT}\right).
$$
Neutrons and protons have equal spin degeneracy and nearly equal masses, so their prefactor ratio is approximately one. Hence
$$
\boxed{\frac{n_n}{n_p}\simeq
\exp\left[-\frac{(m_n-m_p)c^2}{k_BT}\right]
=e^{-Q/(k_BT)}.}
$$
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