Question
Sun November 27, 2011 By: Divya Khare

# Please can i get a complete detailed explanation of Langmuir Adsorption Isotherm with its derivation?

Mon November 28, 2011

In 1916, Irving Langmuir proposed an Adsorption Isotherm which explained the variation of Adsorption with pressure. Based on his theory, he derived Langmuir Equation which depicted a relationship between the number of active sites of the surface undergoing adsorption and pressure.

Assumptions of Langmuir Isotherm

Langmuir proposed his theory by making following assumptions.

1. Fixed number of vacant or adsorption sites are available on the surface of solid.
2. All the vacant sites are of equal size and shape on the surface of adsorbent.
3. Each site can hold maximum of one gaseous molecule and a constant amount of heat energy is released during this process.
4. Dynamic equilibrium exists between adsorbed gaseous molecules and the free gaseous molecules.

Where A (g) is unadsorbed gaseous molecule, B(s) is unoccupied metal surface and AB is Adsorbed gaseous molecule.

5. Adsorption is monolayer or unilayer.

### Derivations of the Langmuir Adsorption Equation

#### Calculation of Equilibrium Constant

Langmuir proposed that dynamic equilibrium exists between adsorbed gaseous molecules and the free gaseous molecules. Using the equilibrium equation, equilibrium constant can be calculated.

Where Ka represents equilibrium constant for forward reaction and Kd represents equilibrium constant for backward direction.

According to Kinetic theory,

Rate of forward reaction = Ka [A] [B]

Rate of backward reaction = Kd [AB]

At equilibrium, Rate of forward reaction is equal to Rate of backward reaction

The above equation represents the equilibrium constant for distribution of adsorbate between the surface and the gas phase.

#### Derivation

Langmuir Equation which depicts a relationship between the number of active sites of the surface undergoing adsorption (i.e. extent of adsorption) and pressure.

To derive Langmuir Equation and new parameter Â‘ ? Â’ is introduced. Let ? the number of sites of the surface which are covered with gaseous molecules. Therefore, the fraction of surface which are unoccupied by gaseous molecules will be (1 Â– ?).

Now, Rate of forward direction depends upon two factors: Number of sited available on the surface of adsorbent, (1 Â– ?) and Pressure, P. Therefore rate of forward reaction is directly proportional to both mentioned factors.

Similarly, Rate of backward reaction or Rate of Desorption depends upon number of sites occupied by the gaseous molecules on the surface of adsorbent.

At equilibrium, rate of adsorption is equal to rate of desorption.

Ka P (1 Â– ?) = Kd ?

We can solve the above equation to write it in terms of ?.

KaP Â– KaP ? = Kd ?

KaP = KaP ? + Kd ?

KaP = (Kd + KaP) ?

Divide numerator and denominator on RHS by Kd, we get

Now put

in above equation we get