For molecules in contact with a solid surface at a fixed temperature, the Langmuir Isotherm, developed by Irving Langmuir in 1916, describes the partitioning between gas phase and adsorbed species as a function of applied pressure. |
The adsorption process between gas phase molecules, A, vacant surface sites, S, and occupied surface sites, SA, can be represented by the equation,
assuming that there are a fixed number of surface sites present on the surface.
An equilibrium constant, K, can be written:
q = Fraction of surface sites occupied (0 <q< 1)
Note that
Thus it is possible to define the equilibrium constant, b:
Rearranging gives the expression for surface coverage:
The rate of adsorption will be proportional to the pressure of the gas and the number of vacant sites for adsorption. If the total number of sites on the surface is N, then the rate of change of the surface coverage due to adsorption is:
The rate of change of the coverage due to the adsorbate leaving the surface (desorption) is proportional to the number of adsorbed species:
In these equations, k_{a} and k_{d} are the rate constants for adsorption and desorption repectively, and p is the pressure of the adsorbate gas. At equilibrium, the coverage is independent of time and thus the adorption and desorption rates are equal. The solution to this condition gives us a relation for q:
where b = k_{a}/k_{d}.
Dependence of b on external parameters:
b is only a constant if the enthalpy of adsorption is independent of coverage.
As with all chemical equilibria, the position of equilibrium (determined by the value of b) will depend upon a number of factors:
where b_{3} > b_{2} > b_{1}
Consider a surface:
Definition:
q_{0}, q_{1}, ..., q_{n} = Surface area (cm^{-2}) covered by 0, 1, ..., n layers of adsorbed molecules. |
At Equilibrium:
q_{0} must remain constant. |
. Rate of Evaporation Rate of Condensation . . = from First Layer onto Bare Surface
Similarly, at equilibrium q_{1} must remain constant.
. Rate of Condensation Rate of Condensation . . on the Bare Surface on the 1st Layer + = + Rate of Evaporation Rate of Evaporation from the second layer from the second layer |
. . . | k_{1}Pq_{0} + k_{-2}q_{2} = k_{2}Pq_{1} + k_{-1}q_{1} |
Substituting into (I) gives
k_{-2}q_{2} = k_{2}Pq_{1}
Extending this argument to other layers,
Definitions:
Total surface area of the catalyst,
Total volume of gas adsorbed on surface
. . . |
where v_{m} is the volume of gas adsorbed when the entire surface is covered with a complete monolayer.
From (I),
If we assume that the properties of the 1st, 2nd, ... layers are equivalent, then,
Similarly,
q_{3} =xq_{2} =x^{2}q_{1} |
Generally,
q_{i} =xq_{i-1} =x^{i-1}q_{1} =x^{i-1}yq_{0} =cx^{i}q_{0} {c=x/y} |
Substituting into (V),
Also,
. . . |
must be infinite. This means that at P_{0}, x must equal 1.
. . . | g = P_{0} | (From definition of x) |
. . . | x = P/P_{0} |
Substituting into VII, we arrive at the recognized form of the BET isotherm,
This can be rearranged to give,