统计热力学与速率理论/渗透

气体从容器中渗透

考虑一个充满一定压力的气体的容器。容器与另一个不含气体的容器（即真空）之间用一个带有小孔的隔板隔开，小孔的表面积为$A_{0}$。第一个容器中的气体将以一定速率通过小孔进入第二个容器。这种现象称为渗透，气体通过小孔的速度称为渗透速率。

渗透速率可以定义为小孔表面积$A_{0}$与粒子碰撞容器壁的速度$Z_{w}$的乘积。

速率 =${\frac {dN}{dt}}=Z_{w}A_{0}$

其中$Z_{w}$由下式给出

$Z_{w}={\frac {p}{\left(2\pi mk_{B}T\right)^{1/2}}}$

压力对时间依赖性的推导

下面推导的渗透速率是相对于气体从充满的容器中逸出的速率。

速率 =$-{\frac {dN}{dt}}=-Z_{w}A_{0}={\frac {-pA_{0}}{\left(2\pi mk_{B}T\right)^{1/2}}}$

$N={\frac {pV}{k_{B}T}}$

${\frac {d}{dt}}{\frac {pV}{k_{B}T}}={\frac {-pA_{0}}{\left(2\pi mk_{B}T\right)^{1/2}}}$

${\frac {V}{k_{B}T}}{\frac {dp}{dt}}={\frac {-pA_{0}}{\left(2\pi mk_{B}T\right)^{1/2}}}$

${\frac {dp}{dt}}={\frac {-A_{0}p(k_{B}T)^{1/2}}{V\left(2\pi m\right)^{1/2}}}$

${\frac {dp}{dt}}={\frac {-A_{0}p(k_{B}T)^{1/2}}{V\left(2\pi m\right)^{1/2}}}$

通过积分求解微分方程。

${\frac {1}{p}}dp={\frac {-A_{0}}{V}}{\left({\frac {k_{B}T}{2\pi m}}\right)}^{1/2}dt$

$\int _{p_{0}}^{p}\!{\frac {1}{p}}dp=-\int _{0}^{t}\!{\frac {A_{0}}{V}}{\left({\frac {k_{B}T}{2\pi m}}\right)}^{1/2}dt$

$\int _{p_{0}}^{p}\!{\frac {1}{p}}dp=-{\frac {A_{0}}{V}}{\left({\frac {k_{B}T}{2\pi m}}\right)}^{1/2}\int _{0}^{t}\!dt$

$\left[\ln \left(p\right)\right]_{p_{0}}^{p}=-{\frac {A_{0}}{V}}{\left({\frac {k_{B}T}{2\pi m}}\right)}^{1/2}\left[t\right]_{0}^{t}$

$\ln \left(p\right)-\ln \left(p_{0}\right)=-{\frac {A_{0}}{V}}{\left({\frac {k_{B}T}{2\pi m}}\right)}^{1/2}\left(t-0\right)$

$\ln \left({\frac {p}{p_{0}}}\right)=-{\frac {A_{0}}{V}}{\left({\frac {k_{B}T}{2\pi m}}\right)}^{1/2}t$

$p=p_{0}\exp \left[-{\frac {A_{0}}{V}}\left({\frac {k_{B}T}{2\pi m}}\right)^{1/2}t\right]$