Closed-vessel thermal transpiration ($\Delta p=2p_{0}q(q-1)/(q+1)$)

Two equal sealed vessels of fixed volume, initially at $T_0,p_0$, exchange an ideal gas through a thin effusive opening. Hold one at $T_1$ and the other at $T_0$, with both interiors locally thermalized. Zero net particle transfer gives $p_h/\sqrt{T_1}=p_c/\sqrt{T_0}$. Writing $q=\sqrt{T_1/T_0}$, conservation of total particle number gives $p_h=2p_0{q}^2/(1+q)$ and $p_c=2p_{0}q/(1+q)$. Thus $\Delta p=2p_{0}q(q-1)/(1+q)$. The alternative $p_c=p_0$, $\Delta p=p_0(q-1)$ requires a cold reservoir whose pressure is externally held fixed; it is not the closed equal-volume problem.