# 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: