= Two-vessel effusive mixing
{title2=$\tau_s=4V/[A_h(\overline v_{s,L}+\overline v_{s,R})]$}
For a species $s$ in two well-mixed vessels of equal <volume (thermodynamics)> $V$ 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.
Back to article page
