La ecuación de Langmuir o isoterma de Langmuir o ecuación de adsorción de Langmuir relaciona la adsorción de moléculas en una superficie sólida con la presión de gas o concentración de un medio que se encuentre encima de la superficie sólida a una temperatura constante. La ecuación fue determinada por Irving Langmuir por concentraciones teóricas en 1916. Es una ecuación mucho más exacta para las isotermas de adsorcion del tipo 1.
Para ello postuló que: «Los gases, al ser adsorbidos por la superficie del sólido, forman únicamente una capa de espesor monomolecular». Además, visualizó que el proceso de adsorción consta de dos acciones opuestas, una de condensación de las moléculas de la fase de gas sobre la superficie, y una de evaporación de las situadas en la superficie hacia el gas. Cuando principia la adsorción, cada molécula que colisiona con la superficie puede condensarse en ella, pero al proseguir esta acción, cabe esperar que resulten adsorbidas aquellas moléculas que inciden en alguna parte de la superficie no cubierta todavía, pero además una molécula es capaz de liberarse por la agitación térmica escapándose hacia el gas. Cuando las velocidades de condensación y de liberación se hacen iguales entonces se establece el equilibrio.
La expresión de la ecuación es la siguiente:
donde:
- θ es la fracción de cobertura de la superficie,
- P es la presión del gas o su concentración, y
- α alpha es una constante, la constante de adsorción de Langmuir, que es mayor cuanto mayor sea la energía de ligadura de la adsorción y cuanto menor sea la temperatura.
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) = S - M
where : S - * , represents a vacant surface site
Note - in writing this equation we are making an inherent assumption that there are a fixed number of localised 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"
We may also 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 , as given below :
Rearrangement then gives the following expression for the surface coverage
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which is the usual form of expressing the Langmuir Isotherm.
Langmuir Isotherm - derivation from kinetic considerations
The equilibrium that may exist between gas adsorbed on a surface and molecules in the gas phase is a dynamic state, i.e. the equilibrium represents a state in which the rate of adsorption of molecules onto the surface is exactly counterbalanced by the rate of desorption of molecules back into the gas phase. It should therefore be possible to derive an isotherm for the adsorption process simply by considering and equating the rates for these two processes.
Expressions for the rate of adsorption and rate of desorption have been derived in Sections 2.3 & 2.6 respectively : specifically ,
Equating these two rates yields an equation of the form :
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[1]
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where the terms f(θ) & f '(θ) contain the pre-exponential surface coverage dependence of the rates of adsorption and desorption respectively and all other factors have been taken over to the right hand side to give a temperature-dependent "constant" characteristic of this particular adsorption process, C(T) .
We now need to make certain simplifying assumptions ... the first is one of the key assumptions of the Langmuir isotherm
Adsorption takes place only at specific localized sites on the surface and the saturation coverage corresponds to complete occupancy of these sites.
Let us initially further restrict our consideration to a simple case of reversible molecular adsorption. i.e.
S - + M (g) = S - M
where S - represents a vacant surface site and S - M the adsorption complex.
Under these circumstances it is reasonable to assume coverage dependencies for rates of the two processes of the form :
Adsorption, f (θ) = c (1-θ)
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i.e. proportional to the fraction of sites that are unoccupied.
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Desorption, f '(θ) = c'θ
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i.e. proportional to the fraction of sites which are occupied by adsorbed molecules.
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where θ is the fraction of sites occupied at equilibrium.
Note : these coverage dependencies are exactly what would be predicted by noting that the forward and reverse processes are elementary reaction steps , in which case it follows from standard chemical kinetic theory that
- the forward adsorption process will exhibit kinetics having a first order dependence on the concentration of vacant surface sites.
- the reverse desorption process will exhibit kinetics having a first order dependence on the concentration of adsorbed molecules.
Substitution into equation [1] then yields
where B(T) = (c'/c).C(T) . After rearrangement this gives the Langmuir Isotherm expression for the surface coverage
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where b ( = 1/B(T) ) is a function of temperature and contains an exponential term of the form
b = ...... exp [ ( Eades - Eaads ) / R T ] = ...... exp [ - ΔHads / R T ]
Consequently, b can only be regarded as a constant with respect to coverage if the enthalpy of adsorption is itself independent of coverage - this is the second major assumption of the Langmuir Isotherm.
Variation of Surface Coverage with Temperature & Pressure
Application of the assumptions of the Langmuir Isotherm leads to readily derivable expressions for the pressure dependence of the surface coverage (see Sections 3.2 and 3.3) - in the case of a simple, reversible molecular adsorption process the expression is
 | where b = b(T) |
This is illustrated in the graph below which shows the characteristic Langmuir variation of coverage with pressure for molecular adsorption.
Note that :
- θ → bP at low pressures
- θ → 1 at high pressures
At a given pressure the extent of adsorption is determined by the value of b : this in turn is dependent upon both the temperature (T) and the enthalpy (heat) of adsorption. Remember that the magnitude of the adsorption enthalpy (a negative quantity itself) reflects the strength of binding of the adsorbate to the substrate.
The value of b is increased by
- a reduction in the system temperature
- an increase in the strength of adsorption
Therefore the set of curves shown below illustrates the effect of either (i) increasing the magnitude of the adsorption enthalpy at a fixed temperature, or (ii) decreasing the temperature for a given adsorption system.
A given equilibrium surface coverage may be attainable at various combinations of pressure and temperature as highlighted below … note that as the temperature is lowered the pressure required to achieve a particular equilibrium surface coverage decreases.
- this is often used as justification for one of the main ideologies of surface chemistry ; specifically, that it is possible to study technologically-important (high pressure / high temperature) surface processes within the low pressure environment of typical surface analysis systems by working at low temperatures. It must be recognised however that, at such low temperatures, kinetic restrictions that are not present at higher temperatures may become important.
If you wish to see how the various factors relating to the adsorption and desorption of molecules influence the surface coverage then try out the Interactive Demonstration of the Langmuir Isotherm (note - this is based on the derivation given in Section 3.3 ).
Determination of Enthalpies of Adsorption
It has been shown in previous sections how the value of b is dependent upon the enthalpy of adsorption. It has also just been demonstrated how the value of b influences the pressure/temperature (P-T) dependence of the surface coverage.
The implication of this is that it must be possible to determine the enthalpy of adsorption for a particular adsorbate/substrate system by studying the P-T dependence of the surface coverage.
Various methods based upon this idea have been developed for the determination of adsorption enthalpies - one method is outlined below :
Step 1 : Involves determination of a number of adsorption isotherms (where a single isotherm is a coverage / pressure curve at a fixed temperature). |  |
Step 2 : It is then possible to read off a number of pairs of values of pressure and temperature which yield the same surface coverage |  |
Step 3 : The Clausius-Clapeyron equation 
may then be applied to this set of (P-T) data and a plot of ( ln P ) v's (1/T) should give a straight line, the slope of which yields the adsorption enthalpy.
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