Question

(a) Derive Freundlich and Langmuir isotherm equations.

(b) An aqueous solution containing a valuable solute is colored by a small amount of an impurity. Before crystallization, the impurity is to be removed by adsorption on a decoloring carbon which adsorps only insignificant amount of the principal solute. Equilibrium data at constant temperature is as follows:

Kg C/Kg Solution	0	0.001	0.004	0.008	0.02	0.04
Equilibirum Color/Kg Solution	9.6	8.6	6.3	4.3	1.7	0.7

It is desired to reduce the color intensity to 10% of its original value (9.6)
(1) Find out Freundlich adsorption isotherm
(2) Determine the quantity of fresh carbon requirement per 100 kg of solution for a single stage operation.

Answer 1

(a)

Freundlich Isotherm

The Freundlich adsorption isotherm is mathematically expressed as:

where

x = mass of adsorbate

m = mass of adsorbent

c = Equilibrium concentration of adsorbate in solution.

It is also written as:

Langmuir Isotherm

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 K_a represents equilibrium constant for forward reaction and K_d represents equilibrium constant for backward direction.

According to Kinetic theory,

Rate of forward reaction = K_a [A] [B]

Rate of backward reaction = K_d [AB]

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

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 ‘ θ ’ is introduced. Let θ 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 – θ).

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

imilarly, 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.

K_a P (1 – θ) = K_d θ

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

K_aP – K_aP θ = K_d θ

K_aP = K_aP θ + K_d θ

K_aP = (K_d + K_aP) θ

Divide numerator and denominator on RHS by K_d, we get

Now put

in above equation we get

Langmuir Adsorption Equation

This is known as Langmuir Adsorption Equation.

(b)

x/m	c	ln(x/m)	ln C
0	9.6		2.261763
0.001	8.6	-6.90776	2.151762
0.004	6.3	-5.52146	1.84055
0.008	4.3	-4.82831	1.458615
0.02	1.7	-3.91202	0.530628
0.04	0.7	-3.21888	-0.35667

now plot ln x/m vs ln c

(1)So from graph

y=-1.315 x-3.397

So

1/n=-1.315

n=-0.76

lnK=-3.397

K=0.00374

So Freundlich adsorption isotherm

X/m=0.00374 C^1/-0.76=0.00374 C^-1.315