## Friday, November 13, 2009

Langmuir derived an equation based on some theoretical considerations. The assumptions of Langmuir’s theory are,
1. Valencies at the surface of the adsorbent are not fully satisfied.
2. The adsorbed gas layer on the solid surface is only one molecule thick.
3. The surface of the solid is homogeneous, so the adsorbed layer is uniform all over the adsorbent.
5. The adsorbed gas molecules do not move around the surface.
6. The process of adsorption is a dynamic process, which consists of two opposite processes.

Condensation Process
It involves the condensation of the molecules of the gas on the surface of the solid.
Evaporation Process
It involves evaporation of the molecules of the adsorbate from the surface of the adsorbent.

Derivation of Langmuir Isotherm
According to Langmiur’s assumptions, when the gas molecules strike a solid surface, some of the molecules are adsorbed where some are desorbed. Thereby dynamic equilibrium is established between adsorption and desorption.
Consider an adsorbing surface of area 1 square cm exposed to a gas. Then

q - Fraction of the surface area covered by gas molecules
1- q - Fraction of the uncovered area (vacant area)
r - Pressure of the gas

The rate of adsorption = Ka (1-q)p – (1)
The rate of evaporation desorption of gas molecules = Kd q - (2)
Where K1 & K2 are proportionality constant for a given system.
At equilibrium :
Rate of adsorption = Rate of desorption
Ka(1-q)r = Kdq
Kar = Kdq+ Karq
Kar = q (Kd+ Kar)
q = Kar/Kd+Kar - (3)
Dividing the Eq. (1) by ‘ Kd’ it becomes
q = Ka/Kd) r/ 1+ (Ka/Kd) r
q = K r / 1+ K r
Where, Ka/Kd = b = equilibrium constant, called the adsorption coefficient.
But the amount of gas adsorbed per gram of the adsorbent (x)/m is proportional to the fraction (q) of the surface covered.
On comparing Eq (2) & (3) it becomes
x/k = k’kr /1+kr - (4)
where K’ = the new constant.
Equation (4) gives the relation between the amount of gas adsorbed to the pressure of the gas at constant T is known as Langmuir adsorption isotherm.
The above Eq. (4) may be rewritten as
1+kr = k’kr / x
1/k’k + kr / k’k = r/x - (5)
Equation (5) represent a straight line (i.e. y = c + mx) If the graph is plotted between r/x Vs r, we should get a straight line with slope k/k’k and the Intercept 1/k’k
This equatin is found valid in all cases.
Plot of r/x Vs r

Case (1) : At low pressure
If the pressure r is very low k/k’k , the r term is negligible.
i.e. 1/k’k >> k/k’k r
Hence Eq. (5) becomes
1/k’k = r/x (or) x = rk’k - (6)
x µ r
i.e. amount of adsorption per unit weight of adsorbent is directly proportional to the r at low pressure.
Case (ii) at high pressure.
If the pressure r is high 1/k’k term is negligilble,
i.e. k/k’k
Hence Eq (5) becomes
k/k’k r = r/x or x = k’ (constant)
x = k’p° - (7)
i.e. extend of adsorption is independent of pressure of the gas, because the surface becomes completely covered at high pressure.
Case (ii) At normal pressure
If the pressure r is normal (intermediate) the eq. (7) becomes
X = k’pn - (8)
Where n lies between 0 to 1
Equation (8) is called Frendliech’s adsorption isotherm.
Merits – Langmuir's adsorption isotherm holds good at low pressure.
Demerits - It fails at high pressure.