Two-vessel effusive mixing (${\tau }_s=4V/[A_h({\overline{v}}_{s,L}+{\overline{v}}_{s,R})]$)

For a species $s$ in two well-mixed vessels of equal volume $V$ joined by a thin opening of area $A_h$, the effusion law gives ${\dot{n}}_{s,L}=-(A_h/4V)({\overline{v}}_{s,L}{n}_{s,L}-{\overline{v}}_{s,R}{n}_{s,R})$. Each species' total inventory is conserved. With fixed temperatures and total density sum $n_{s,L}+n_{s,R}=n_{s,\Sigma }$, the left density relaxes exponentially with time constant ${\tau }_s=4V/[A_h({\overline{v}}_{s,L}+{\overline{v}}_{s,R})]$ to $n_{s,L}^{\infty }=n_{s,\Sigma }\sqrt{T_R}/(\sqrt{T_L}+\sqrt{T_R})$.

At equal temperatures, ${\tau }_s=2V/(A_h{\overline{v}}_s)\propto \sqrt{m_s}$, and initially separated species mix at different rates. At unequal temperatures, the final density ratio is $n_{s,L}/n_{s,R}=\sqrt{T_R/T_L}$ for every species, while the partial-pressure ratio is $p_{s,L}/p_{s,R}=\sqrt{T_L/T_R}$. The mass-independent final ratios do not imply mass-independent transient rates. Maintaining different temperatures requires external heat exchange even after net particle transfer vanishes.