= Solution
Hydrostatic balance and the ideal-gas <adiabatic temperature gradient> imply
$$
\left|\frac{dT}{dz}\right|_{\rm ad}
=\frac{g}{c_p}.
$$
The radiative region is stable while $K\rho/T^3<g/c_p$. Equality at the <radiative-convective boundary>, together with $\rho=\mu P/(\mathcal RT)$, gives
$$
\boxed{P_{\rm rc}
=\frac{g\mathcal R}{K\mu c_p}T_{\rm rc}^4
=\frac{g\nabla_{\rm ad}}{K}T_{\rm rc}^4}.
$$
For radiative diffusion carrying intrinsic flux $F_{\rm int}=\sigma T_{\rm int}^4$, $K=3\kappa F_{\rm int}/(16\sigma)$, so
$$
\boxed{P_{\rm rc}
=\frac{16g\nabla_{\rm ad}}{3\kappa}
\left(\frac{T_{\rm rc}}{T_{\rm int}}\right)^4}.
$$
For an irradiated hot Jupiter with $g\simeq10\,{\rm m\,s^{-2}}$, $\nabla_{\rm ad}\simeq2/7$, $\kappa\simeq10^{-2}\,{\rm m^2\,kg^{-1}}$, $T_{\rm rc}\simeq1500\,{\rm K}$, and $T_{\rm int}\simeq150$--$200\,{\rm K}$, this gives roughly $P_{\rm rc}\sim50$--$200\,{\rm bar}$.
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