朗缪尔方程

${\displaystyle \theta ={\frac {\alpha \cdot P}{1+\alpha \cdot P}}}$

${\displaystyle \theta }$是表面覆盖范围分数，${\displaystyle P}$是气体压力或浓度，${\displaystyle \alpha }$为常数。

公式推导

${\displaystyle S^{*}+P\rightleftharpoons SP}$

${\displaystyle K={\frac {[SP]}{[S^{*}][P]}}}$

${\displaystyle \alpha ={\frac {\theta }{(1-\theta )p}}}$

${\displaystyle \theta =\alpha (1-\theta )p}$
${\displaystyle \theta =p\alpha -p\theta \alpha }$
${\displaystyle \theta +p\theta \alpha =p\alpha }$
${\displaystyle \theta (1+p\alpha )=p\alpha }$

${\displaystyle \theta ={\frac {\alpha \cdot p}{1+\alpha \cdot p}}}$

统计学推导

1. 假设有M个活性位点以供N个微粒结合。
2. 一个活性位点只能被一个微粒所占有。
3. 所有活性位点都是相互独立的。一个位点被占有的概率是与邻近位点的状态无关的。

N个微粒被吸附到M个位点之系统的配分函数（假设位点较微粒数量多）为：

${\displaystyle Q(N,M,T)={\frac {M!}{N!(M-N)!}}(q\lambda )^{N}}$

${\displaystyle q=q_{v}(T)^{3}}$${\displaystyle \lambda =e^{\beta \mu }}$.

${\displaystyle \Xi (\mu ,M,T)=\sum _{N=0}^{M}Q(N,M,T)=\sum _{N=0}^{M}{\binom {M}{N}}(q\lambda )^{N}=(1+q\lambda )^{M}}$

${\displaystyle \xi =1+q\lambda }$

${\displaystyle \langle N\rangle ={\frac {\partial {\ln {\Xi (\mu ,M,T)}}}{\partial {\beta \mu }}}=M{\frac {\partial {\ln {\xi (\mu ,M,T)}}}{\partial {\beta \mu }}}}$

${\displaystyle \langle s\rangle ={\frac {}{M}}={\frac {\partial {\ln {\xi (T)}}}{\partial {\beta \mu }}}=\lambda {\frac {\partial \ln {\xi (T)}}{\partial \lambda }}}$

${\displaystyle \langle s\rangle ={\frac {q\lambda }{1+q\lambda }}}$

方程拟合

${\displaystyle {\Gamma }=\Gamma _{max}{\frac {Kc}{1+Kc}}}$

${\displaystyle {\Gamma (c=K^{-1})}=\Gamma _{max}{\frac {KK^{-1}}{1+KK^{-1}}}={\frac {\Gamma _{max}}{2}}}$

${\displaystyle {\frac {1}{\Gamma }}={\frac {1}{\Gamma _{max}}}+{\frac {1}{\Gamma _{max}Kc}}}$

${\displaystyle \Gamma =\Gamma _{max}-{\frac {\Gamma }{Kc}}}$

${\displaystyle {\frac {\Gamma }{c}}=K\Gamma _{max}-K\Gamma }$

${\displaystyle {\frac {c}{\Gamma }}={\frac {c}{\Gamma _{max}}}+{\frac {1}{K\Gamma _{max}}}}$