5.23: Isotherms are Plots of Surface Coverage as a Function of Gas Pressure at Constant Temperature
- Page ID
- 547560
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Whenever a gas is in contact with a solid there will be an equilibrium established between the molecules in the gas phase and the corresponding adsorbed species (molecules or atoms) which are bound to the surface of the solid. As with all chemical equilibria, the position of equilibrium will depend upon a number of factors:
- The relative stabilities of the adsorbed and gas phase species involved
- The temperature of the system (both the gas and surface, although these are normally the same)
- The pressure of the gas above the surface
In general, factors (2) and (3) exert opposite effects on the concentration of adsorbed species - that is to say that the surface coverage may be increased by raising the gas pressure but will be reduced if the surface temperature is raised.
The Langmuir isotherm was developed by Irving Langmuir in 1916 to describe the dependence of the surface coverage of an adsorbed gas on the pressure of the gas above the surface at a fixed temperature. There are many other types of isotherm (Temkin, Freundlich ...) which differ in one or more of the assumptions made in deriving the expression for the surface coverage; in particular, on how they treat the surface coverage dependence of the enthalpy of adsorption. Whilst the Langmuir isotherm is one of the simplest, it still provides a useful insight into the pressure dependence of the extent of surface adsorption.
Note: Surface Coverage & the Langmuir Isotherm
When considering adsorption isotherms it is conventional to adopt a definition of surface coverage (\(θ\)) which is defined as the fraction of the maximum (saturation) surface coverage \(N_{max}\) of a particular adsorbate on a given surface:
\[\theta = \frac{N}{N_{max}},\]
This means the maximum \(\theta\) is 1, (\(θ_{max} = 1\)).
This way of defining the surface coverage differs from that usually adopted in surface science where the more common practice is to equate \(θ\) with the ratio of adsorbate species to surface substrate atoms (which leads to saturation coverages which are almost invariably less than unity).
The Langmuir Isotherm - Derivation from Equilibrium Considerations
We may derive the Langmuir isotherm by treating the adsorption process as we would any other equilibrium process - except in this case the equilibrium is between the gas phase molecules (\(M\)), together with vacant surface sites, and the species adsorbed on the surface. Thus, for a non-dissociative (molecular) adsorption process, we consider the adsorption to be represented by the following chemical equation :
\[S - * + M_{(g)} \rightleftharpoons S - M \label{eq1} \]
where :
- \(S - *\) represents a vacant surface site
Assumption 1
In writing Equation \(\ref{eq1}\) we are making an inherent assumption that there are a fixed number of localized surface sites present on the surface. This is the first major assumption of the Langmuir isotherm.
We may now define an equilibrium constant (\(K\)) in terms of the concentrations of "reactants" and "products"
\[ K = \dfrac{[S-M]}{[S-*][M]}\label{2} \]
The units are not the same as when considering homogenous equilibria in a volume. [S-*] and [S-M] are formally concentrations in per unit area of surface (e.g. m-2) rather than concentrations per unit volume.
Note that
- [ S - M ] is proportional to the surface coverage of adsorbed molecules, i.e. proportional to \(θ\)
- [ S - * ] is proportional to the number of vacant sites, i.e. proportional to \(1-θ\)
- [ M ] is proportional to the pressure of gas, \(P\)
Hence, it is also possible to define another equilibrium constant, b , appropriate for coverage as:
\[ b =\dfrac{\theta}{(1- \theta)P}\label{3} \]
Rearrangement then gives the following expression for the surface coverage
\[ \theta =\dfrac{b P}{1 + bP}\label{4} \]
which is the usual form of expressing the Langmuir Isotherm. As with all chemical reactions, the equilibrium constant, \(b\), is both temperature-dependent and related to the Gibbs free energy and hence to the enthalpy change for the process.
Assumption 2
\(b\) is only a constant (independent of \(\theta\)) if the enthalpy of adsorption is independent of coverage. This is the second major assumption of the Langmuir Isotherm.
A plot of \(\theta\) vs. \(bP\) shows that as the pressure increases, \(\theta\) approaches 1, meaning that nearly the entire surface is coated with a monolayer of adsorbed gas (Figure \(\PageIndex{1}\)).
Equation \(\ref{4}\) can be rearranged to the form
\[\dfrac{1}{\theta} = 1 + \dfrac{1}{bP} \label{5} \]
showing that the inverse of the fraction of occupied surface sites is a linear function of the inverse of the pressure. If we plot experimental data for the adsorption of diatomic oxygen and carbon monoxide onto a silica surface, we can see that the Langmuir adsorption isotherm describes the data well (figure \(\PageIndex{2}\)).
The Langmuir Isotherm from a kinetics
The assumptions are the same as for the equilibrium argument:
- Adsorption takes place only at specific localized sites on the surface and the saturation coverage corresponds to complete occupancy of these sites.
- The rate of adsorption and desorption is independent of the surface coverage.
This is the case of a reversible molecular adsorption, i.e.
\[S- * + \ce{M_{(g)}} \overset{k_a}{\underset{k_d}{\leftrightharpoons}} \ce{S-M} \label{33Eq2} \]
where
- \(S-*\) represents a vacant surface site and
- \(\ce{S-M}\) the adsorption complex.
- \(k_a\) is the rate constant for adsorption
- \(k_d\) is the rate constant for desorption
At equilibrium the rate of adsorption equals the rate of desorption \(R_a = R_d\) or equivalently [S-M] is constant :
\[\frac{d[S-M]}{dt}\Big|_e = 0 = k_a[S-*]_e[M]_e - k_d[S-M]_e,\label{kineq_cond}\]
where subscript e indicates we are at equilibrium. We are interested in the concentration adsorbed to the surface, [S-M]e. Solving for that yields:
\[ [S-M]_e = \frac{k_a}{k_d}[S-*]_e[M]_e. \label{[S-M]}\]
We would like to get this in a form equivalent to equation \(\ref{4}\) in terms of \(\theta\). To do this we recognize that the maximum possible coverage is the number of binding sites per unit area on a completely clean surface, which we can represent as \([S-*]_o\). Since each S-M created removes an S-*,
\[[S-*] = [S-*]_o - [S-M].\nonumber\]
Substituting this in to equation \(\ref{[S-M]}\) and dividing through by [S-*]o:
\[ \frac{[S-M]_e}{[S-*]_o} = \frac{k_a}{k_d}\left(1 -\frac{[S-M]_e}{[S-*]_o}\right)[M]_e. \label{[S-M]2}\]
The ratio \(\frac{[S-M]_e}{[S-*]_o}\) is \(\theta\). Thus, making the substitution and solving for \(\theta\) we get:
\[ \theta = \frac{k_a}{k_d}\left(1 -\theta\right)[M]_e\implies \theta + \frac{k_a}{k_d}[M]_e\theta = \frac{k_a}{k_d}[M]_e \implies \theta= \frac{\frac{k_a}{k_d}[M]_e}{1 + \frac{k_a}{k_d}[M]_e}.\label{theta_kin}\]
Comparing this equation with equation \(\ref{4}\) we recognize that [M]e is proportional to the pressure (P) and the ratio of the adsorption and desorption rates (the equilibrium constant for adsorption, Kads) is proportional to b. If the gas above the solid is at low enough pressure to behave ideally you can show that P = RT[M]e and b = (ka/kd)/(RT).
Energetics
We know that to a first approximation rate constants follow the Arrhenius model for their temperature dependence:
\[k = A exp \left(\frac{-E_a}{RT}\right).\nonumber\]
Thus, the ratio of \(k_a/k_d\) in equation \(\ref{theta_kin}\) can be written as:
\[\frac{k_a}{k_d} = \frac{A_{ads} exp \left(\frac{-E_{ads}}{RT}\right)}{A_{des} exp \left(\frac{-E_{des}}{RT}\right)} = \frac{A_{ads}}{A_{des}} exp \left(\frac{E_{des}-E_{ads}}{RT}\right) = \frac{A_{ads}}{A_{des}} exp \left(\frac{-\Delta H_{ads}}{RT}\right), \label{ka/kd}\]
where subscript 'des' refers to the desorption process and 'ads' refers to the adsorption process. Thus the equilibrium constant between adsorption and desorption will increase as the enthalpy change on adsorption gets more favorable (more negative). As you would expect the probability of sticking increases the more strongly the molecule binds to the surface.
Contributors and Attributions
-
Roger Nix (Queen Mary, University of London)
-
Jonathan Gutow (UW Oshkosh)