= Langmuir adsorption model
{c}
{title2=$\theta=bP/(1+bP)$}
{wiki}
= Langmuir isotherm
{c}
{synonym}
Consider an <ideal gas> in <chemical equilibrium> with homogeneous, identical <adsorption sites> at fixed <temperature>. Each intact <molecule> occupies one site; only one <monolayer> is allowed. Occupation of distinct sites is <independent>: neighboring bound <molecules> do not change binding or release rates. The simplest <Langmuir adsorption model> has one adsorbing <chemical species>, no splitting of <molecules>, and no competitors. <Irving Langmuir>'s 1918 paper treats simple <adsorption> alongside more complicated cases; its scope exceeds this particular model.<The Adsorption of Gases on Plane Surfaces of Glass, Mica and Platinum by Irving Langmuir>[Langmuir, The Adsorption of Gases on Plane Surfaces of Glass, Mica and Platinum, 1918, theoretical part: Case I and subsequent cases]{ref}
The derivation is a balance between arrivals finding vacancies and departures from occupied <adsorption sites>. Let $k_{\rm a}P$ be the arrival-and-binding rate per vacant site and $k_{\rm d}$ the departure rate per occupied site. Thus $k_{\rm a}$ has units of inverse <pressure> per unit <time>, whereas $k_{\rm d}$ has units of inverse <time>. The mean <surface coverage> obeys
$$
\frac{d\theta}{dt}=k_{\rm a}P(1-\theta)-k_{\rm d}\theta.
$$
At <chemical equilibrium>, setting the two terms equal gives
$$
k_{\rm a}P(1-\theta)=k_{\rm d}\theta,
\qquad b=\frac{k_{\rm a}}{k_{\rm d}},
\qquad \boxed{\theta=\frac{bP}{1+bP}}.
$$
The product $bP$ is <dimensionless>. Here $P$ is the adsorbing <gas>'s <partial pressure>, expressed in modern bar, with $1\,\mathrm{bar}=10^5\,\mathrm{Pa}$, and $b$ is expressed in $\mathrm{bar}^{-1}$. The <pascal (unit)> is the <SI> unit of <pressure>. This dimensional $b$ is not itself a dimensionless thermodynamic <equilibrium constant>. A larger $b$ gives greater <surface coverage> at the same $P$ and <temperature>, but does not specify how quickly <chemical equilibrium> is reached: multiplying both rate coefficients by the same factor leaves $b$ unchanged.
Choose $b=0.5$, $2$, and $8\,\mathrm{bar}^{-1}$ as illustrative parameters at one fixed <temperature>, without assigning a particular material. At $P=0.5\,\mathrm{bar}$, substitution gives, respectively,
$$
\theta=\frac{0.25}{1.25}=\frac15,\qquad
\theta=\frac{1}{2},\qquad
\theta=\frac45.
$$
Solving $\theta=1/2$ gives $P_{1/2}=1/b$: the three half-occupation <pressures> are $2$, $0.5$, and $0.125\,\mathrm{bar}$. For $b=2\,\mathrm{bar}^{-1}$, reaching $\theta=0.9$ instead requires $bP=9$, or $P=4.5\,\mathrm{bar}$. If the complete-<monolayer> capacity is stipulated to be $q_{\rm m}=10\,\mathrm{mmol\,g^{-1}}$, then $q=q_{\rm m}\theta$ predicts $q=5\,\mathrm{mmol\,g^{-1}}$ at $P=0.5\,\mathrm{bar}$. This last number is an original hypothetical example, not a measured uptake.
\Image[/langmuir-adsorption-isotherms.png]
{title=<Langmuir isotherms> for three chosen values of $b$}
{id=image-langmuir-adsorption-isotherms}
{description=Original model curves $\theta=bP/(1+bP)$ at fixed <temperature>, with $P$ in modern bar and $b=0.5$, $2$, and $8\,\mathrm{bar}^{-1}$. Open circles mark the calculated half-occupation points, not experimental observations. The panels show the same curves on linear and logarithmic <pressure> axes; the horizontal dashed line marks complete-<monolayer> capacity. No curve is historical data or a fit to a particular material.}
The shape follows directly from the finite number of <adsorption sites>. For $bP\ll1$, $\theta\approx bP$: nearly every arriving <molecule> finds a vacancy. For $bP\gg1$, $1-\theta=1/(1+bP)\approx1/(bP)$: vacancies become scarce. More formally,
$$
\frac{d\theta}{dP}=\frac{b}{(1+bP)^2}>0,
\qquad
\frac{d^2\theta}{dP^2}=-\frac{2b^2}{(1+bP)^3}<0.
$$
Thus <surface coverage> rises with diminishing slope and approaches one only as $P\to\infty$. This saturation is a site-capacity limit, not a claim that the <gas> has reached its condensation <pressure>. A second layer, different site binding strengths, interactions between bound <molecules>, or a changing surface invalidate parts of the setup. A good fit alone does not establish identical <adsorption sites>: a restricted <pressure> range can hide such differences. The <Langmuir adsorption model> is useful because it makes the capacity and the <pressure> scale of filling separate, testable quantities.
Back to article page
