# Langmuir Adsorption Isotherm

## Postulates or Assumptions of Langmuir Adsorption Isotherm

The main postulates or assumptions of Langmuir Adsorption Isotherm are as follows-1. Adsorption of adsorbate molecules takes place only on fixed number of adsorption sites that are available on the surface of solid adsorbent.

2. Adsorption is a process of

*sticky collision*. This means when a gas molecule hits a vacant site on the adsorbent surface, it will get adsorbed; however, if it hits an occupied site, an elastic collision occurs and the gas molecule gets rebound. This implies that adsorption takes place only on a vacant site, where each such site can hold only one gas molecule.

3. All active sites on the adsorbent surface are energetically equivalent, i.e., they involve constant heat of adsorption.

4. The surface of the solid adsorbent is assumed to be completely flat and uniform on microscopic dimensions.

5. Under conditions of low pressure and moderately high temperature, a monomolecular layer of adsorbate molecules is formed on the adsorbent surface.

6. There are no interactions between the gas molecules that are getting adsorbent on the adsorbent surface; adsorption of gas molecules takes place independent of the occupation of the neighboring sites. These gaseous molecules are thus assumed tobehave ideally.

7. The adsorbed layer on the solid adsorbent is uni-molecular in thickness.

8. Dynamic equilibrium exists between adsorbed gaseous molecules and the free gaseous molecules. Adsorption takes place on vacant sites and desorption takes place from occupied sites, till a state of equilibrium is attained.

At equilibrium, Rate of adsorption = rate of desorption.

## Derivation of Langmuir Adsorption Isotherm

Langmuir derived an adsorption isotherm on the basis of kinetic theory of gases. He considered that the gas molecules strike a solid surface and then adsorbed. Some of these molecules are evaporated or desorbed fairly rapidly. Thus eventually a dynamic equilibrium is set up between the two opposing processes.The rate of adsorption of the gas on the adsorbent surface will be proportional to the rate at which gas molecules strike the surface.

According to the kinetic theory of gas, the rate of striking of gas molecules is proportional to the pressure of the gas (P) at constant temperature. If θ is the fraction of the surface covered by adsorbed molecules at any instance, then the fraction of uncovered surface is (1- θ). Therefore-

Rate of adsorption ∝ (1- θ)P

or, Rate of adsorption = K

_{1}(1- θ)P

and

Rate of desorption ∝ θ

or, Rate of desorption = K

_{2}θ

where K

_{1}and K

_{2}are rate constant

at equilibrium-

Rate of adsorption = Rate of desorption

or, K

_{1}(1- θ)P = K

_{2}θ

or, θ = K

_{1}P/(K

_{1}P + K

_{2})

The mass w of the gas adsorbed per unit mass of adsorbent is directely proportional to fraction of surface covered(θ).

That is- w = K

_{3}θ

where K

_{3}is proportionality constant.

Thus,

w = K

_{1}K

_{3}P/(K

_{1}P + K

_{2})

or, w = (K

_{1}K

_{3}/K

_{2})P/(K

_{1}/ K

_{2})P +1

or, w = AP/(BP + 1)

where A and B are constant.

This relation is Langmuir Adsorption Isotherm.

and is written as-

P/w = 1/A + (B/A)P

If pressure (P) is very low-

BP/A may be ignored and the above equation may be written as-

w = AP

If pressure (P) is very high-

1/A may be ignored and the above equation may be written as-

w = A/B

Hence at low pressure, the amount of gas adsorbed is directely proportional to pressure(P) while at high pressure the mass adsorbed (w) reached a constant value (A/B) when the surface is completely covered with one molecule thick layer of gas. At this stage, adsorption is independent of pressure. Thus-

at low pressure-

w = KP'

and high pressure-

w = KP

^{o}

and at intermediate pressure-

w = KP

^{n}

where n = 0 and 1

## Limitations of Langmuir Adsorption Equation

1. The Langmuir Equation is valid under low pressure only. The adsorbed gas has to behave ideally in the vapor phase. This condition can be fulfilled at low pressure conditions only.2. Langmuir Equation assumes that adsorption is monolayer. But, monolayer formation is possible only under low pressure condition. Under high pressure condition the assumption breaks down as gas molecules attract more and more molecules towards each other.

3. Another assumption was that all the sites on the solid surface are equal in size and shape and have equal affinity for adsorbate molecules i.e. the surface of solid if homogeneous. But we all know that in real solid surfaces are heterogeneous.

4. Langmuir Equation assumed that molecules do not interact with each other. This is impossible as weak force of attraction exists even between molecules of same type.

5. The adsorbed molecules have to be localized i.e. decrease in randomness is zero (ΔS = 0). This is not possible because on adsorption liquefaction of gases taking place, which results into decrease in randomness but the value is not zero.