LANGMUIR ADSORPTION ISOTHERM

Langmuir proposed the relationship between W and P based on kinetic theory of gases.

ASSUMPTION :-

1) Langmuir assumed that adsorption is monolayer in nature.

2) Langmuir assumed that the rate of adsorption of gas over adsorbent is equal to the rate of desorption of gases.

                                According to langmuir, the rate of adsorption depends on the available surface of adsorbent and the pressure of the gas.

rate of adsorption, 

∝ available surface 

∝ pressure of gas

 

θ - represents the total fraction of surface covered 

1-θ - represents the fraction of uncovered surface. 

rate of adsorption   ∝ (1−θ) 

                                   ∝ P

                                   ∝ P (1−θ)

rate of adsorption = k1 P (1−θ)..............(1)

rate of desorption ∝ fraction of covered area, 

                         ∝ θ

rate of desorption = k₂θ........................ (2)

According to langmuir, rate of adsorption and desorption are equal. 

            P (1−θ) = k₂θ

         k₁p − k₁p θ = k₂ θ

     k₁p = k₂ θ + k₁p θ

     k₁p = θ [ k₂ + k₁p]

          θ =   k₁ P         .........................(3)

             (k₂ + k₁p)

             

Here, 

k₁ = rate constant for adsorption. 

k₂ = rate constant for desorption. 

                        The amount of gas adsorbed depends on θ

                     W ∝ θ

                     W = k₃ θ

                    W = k₃    [  k₁ P  ]

                                 (k₂ + k₁p)

W =  k₃ k₁ P

        k₂+ k₁P

Dividing k₂,

       k₃k₁  P

          k₂

w=                     

    k₂   +  k₁   P

    k₂        k₂

      W=    k₃ k₁    P

                k₃

                                 P

           1+ k₁/k₂

                   

Here,

          k₃  k₁    = A,   k₁   = B, 

           k₂                  k₂

A, and B are constants.

            W=    AP      ............................ (4)

                    1+ BP

Inverting the above equation,

   1      =    1+ BP   

  W              AP

 P        =   1+ BP  

 W               A

      P  =   1   + ( B/A) P ................... (5)

      W      A

The equation (4) and (5) are called langmuir adsorption isotherm.

The above equation gives a straight line. The plot of P/W versus P, will give a straight line with slop B/A and the intercap is 1/A.

The above the graph is also known as langmuir adsorption isotherm.

SPECIAL CASES :

1) AT LOW PRESSURE :

Under low pressure the term (B/A) P' can be neglected.

        P/W = 1/A (or) W= AP

                    W ∝ P

Under low pressure, the amount of gas adsorbed is directly proportional to the pressure of the gas.

2) AT HIGH PRESSURE :

                                          Under high pressure the term 1/A can be neglected

P/W = (B/A) P

1/W = B/A

W = A/B

W = constant

∴ under the pressure, the amount of gas adsorbed is constant.

3) AT INTERMEDIATE PRESSURE :

At low P,  W ∝ P (or)    W = kp'

At high P,  W ∝ constant (or) W = kp⁰

At intermediate P,     W = kpⁿ

Here, the value of ' n ' lies between 0 and 1.