# Langmuir Adsorption Isotherm

In 1916 Langmuir proposed his theory which said that adsorption of a gas on the surface of a solid to be made up of elementary sites each of which could absorb one gas molecule.

It is assumed that all the adsorption sites are equivalent and the ability of the gas molecule to get bound to any one site is independent of whether the neighboring sites are occupied or not.

It is further assumed that a dynamic equilibrium exists between the adsorbed molecule and the free molecule.

If A is the gas molecule and M is the surface site then,

Where “K_{a}” and “K_{d}” are the rate constants for adsorption and desorption, respectively.

The rate of adsorption is proportional to the pressure of A, viz, P_{A} and number of vacant sites on the surface, viz, N(1-θ) where N is the total number of sites and θis the fraction of surface sites occupied by the gas molecules, i.e.

θ = Number of adsorptions sites occupied / Number of adsorptions sites are available

Thus the rate of adsorption = k_{a}p_{A}N(1- θ) ……. (1)

The rate of desorption is proportional to the number or adsorbed molecules, Nθ .

Thus the rate of desorption = k_{d}Nθ ……. (2)

Since at equilibrium, the rate of adsorption is equal to the Rate of desorption, we can write from equation (1) and (2)

K_{a}p_{A}N(1-θ) = k_{d}Nθ ……. (3)

Or, Kp_{A}(1-θ) = θ ……. (4)

where, K = k_{a} / K_{d}

Equation (4), may thus be written as

(1-θ) / θ = 1 / Kp_{A} ……. (5)

Or, (1 / θ) – 1 = 1 / Kp_{A} ……. (6)

(1 / θ) = (1 / Kp_{A} ) + 1 = (1 + Kp_{A}) / Kp_{A} ……. (7)

Hence, *θ*** = Kp_{A}/ (1 + Kp_{A} )** ……. (8)

Equation (8) is called the “** Langmuir adsorption isotherm**”.

The following five assumptions are involved in derivation of the Langmuir adsorption isotherm:

1. 1. The absorbed gas behaves ideally in the vapour phase.

2. 2. Only a monolayer is formed by the adsorbed gas.

3. 3. The surface of the solid is homogeneous so that each binding site has the same affinity for the gas molecules.

4. 4. There is no lateral interaction between the adsorbate molecules.

5. 5. The adsorbed gas molecules are localized, i.e. they do not move around on the surface.

The first assumption holds at low pressure, the second assumption breaks down when the pressure of the gas is increased. The third assumption is not strictly true because the real surfaces are quite heterogeneous so that affinity for gas molecule is different at different sites. Crystal imperfections and cracks lead to the creation of different sites on the surface. The fourth and fifth assumptions, too, are not strictly valid.