## Langmuir isotherm and Freundlich isotherm, Adsorption Phenomenon (Part2) and MCQs for GPAT, NIPER, Pharmacist and Drug Inspector exam

Many attempts have been made to develop a mathematical expression that relates the amount of the adsorbate per unit weight of adsorbent to a function of the solute concentration remaining in the solution at a fixed temperature. The most frequently used equations to fit the experimentally observed isotherm are Langmuir, Freundlich, and Brunner, Emmett and Teller (BET).

**Langmuir isotherm: **Langmuir tried to explain adsorption in terms of dynamic equilibrium between the rates of adsorption and desorption. He derived an equation based on the following facts:

- Every active site of the adsorbent acts in the same way
- Adsorbate is adsorbed on the surface of the solid (adsorbent) to form a monolayer
- The rate of adsorption is proportional to the concentration of the adsorbate [A] and the number of unoccupied sites available (1 – α)

Rate of adsorption of solute = k_{a} [A] (1 – α)

where ka is the adsorption rate constant.

- The adsorbed substances tend to escape from the surface and therefore the rate of desorption is proportional to the number of occupied sites (α)

Rate of desorption of solute = k_{d} α

where kd is the desorption rate constant.

- At equilibrium, the rates of adsorption and desorption are equal

k_{a} [A] (1 – α) = k_{d} α

or

α = k_{a} [A] / k_{a} [A] + k_{d}

If a monolayer of the solute covers the surface of the adsorbent, the amount of solute, adsorbed per unit weight of adsorbent, is directly proportional to the fraction D of the surface occupied with the solute:

q_{e} = k’ α

where k’ is a constant. Substituting above equations and dividing the resulting equation by kd yields the following:

q_{e} = (k_{a} k’/ k_{d}) [A] / 1 + (k_{a}/k_{d}) [A] = k_{1}A/1+k_{2}A

where, k_{1 = }k_{a} k’/ k_{d }and k_{2} = k_{a}/k_{d}

In a double reciprocal form, the equation is rearranged as follows:

1/ q_{e }= k_{1}/ k_{2} + 1/ k_{1} (1/[A])

Hence the plot of 1/qe versus 1/[A] gives a slope of 1/K1 and an intercept of K1/K2.

**Freundlich isotherm: **There are two special cases of the Langmuir isotherm.

For very low concentrations (i.e. K2[A] << 1) the specific adsorption is proportional to the concentration of the adsorbate:

q_{e} = k_{1}/[A]

For very high concentrations (i.e. K2[A] >> 1) the specific adsorption is independent of the concentration of the adsorbate:

q_{e }= k_{1}/k_{2}

For intermediate concentration, the Freundlich equation for adsorption at a given temperature is

q_{e }= k[A]^{n}

where k and n are constants and the value of n ranges from 0 to 1.

When n = 1, the Freundlich equation is identical to the very low concentration case of the Langmuir isotherm.

When n = 0, the Freundlich equation is identical to the very high concentration case of the Langmuir isotherm.

The logarithmic form of Freundlich equation is

log qe = log k + n log [A]

Plotting qe against log [A] gives a straight line with a slope n and intercept log k

**Multiple choice questions (MCQs)**

**1.BET stands for**

a)Brunner, Emmett and Teller

b)Burner, Emmett and Teller

c)Brunner, Emet and Teller

d)Brunner, Emmett and Tailor

**2.Langmuir tried to explain adsorption in terms of _____ between the rates of adsorption and desorption. **

a)Dynamic equilibrium

b)Static equilibrium

c)Both of these

d)None of these

**3.Langmuir derived an equation based on which of the following facts?**

a)Every active site of the adsorbent acts in the same way

b)Adsorbate is adsorbed on the surface of the solid (adsorbent) to form a monolayer

c)The rate of adsorption is proportional to the concentration of the adsorbate [A] and the number of unoccupied sites available (1 – α)

d)All of these

**4.In adsorption at liquid interface, when the added molecules move on their own according to the interface, this process is called**

a)Negative adsorption

b)Positive absorption

c)Positive desorption

d)Positive adsorption

**5.The Freundlich equation is identical to the very low concentration case of the Langmuir isotherm when**

a)n = 1

b)n = 0

c)n>1

d)n=2

**6.The Freundlich equation is identical to the very high concentration case of the Langmuir isotherm when**

a)n = 1

b)n = 0

c)n>1

d)n=2

**7.According to Freundlich adsorption isotherm, which of the following is correct?**

a)x/m α p^{1/n}

b)x/m α p^{1}

c)x/m α pº

d)all are correct at different ranges of pressure

**8.According to Langmuir isotherm the adsorbed substances tend to escape from the surface and therefore the rate of desorption is proportional to the number of occupied sites (α).**

a)True

b)False

**9.A plot of log x/m versus log p for the adsorption of a gas on a solid gives a straight line with slope equal to**

a)N

b)1/n

c)Log k

d)–log k

**10.According to Langmuir isotherm at equilibrium, the rates of adsorption and desorption are equal.**

a)True

b)False

**11.For very high concentrations (i.e. K2[A] << 1) the specific adsorption is proportional to the concentration of the adsorbate.**

a)True

b)False

**12.For very high concentrations (i.e. K2[A] >> 1) the specific adsorption is ____ of the concentration of the adsorbate.**

a)Dependent

b)Independent

c)More

d)Less

**13.For very low concentrations (i.e. K2[A] << 1) the specific adsorption is _____ to the concentration of the adsorbate.**

a)Proportional

b)Inversely proportional

c)Independent

d)Higher

**14.For intermediate concentration, the Freundlich equation for adsorption at a given temperature is**

a)q_{e} = k_{1}/[A]

b)q_{e }= k[A]^{n}

c)q_{e }= k_{1}/k_{2}

d)log qe = log k + n log [A]

**15.The logarithmic form of Freundlich equation is**

a)q_{e} = k_{1}/[A]

b)q_{e }= k[A]^{n}

c)q_{e }= k_{1}/k_{2}

d)log qe = log k + n log [A]

**Solutions:**

- a)Brunner, Emmett and Teller
- a)Dynamic equilibrium
- d)All of these
- d)Positive adsorption
- a)n = 1
- b)n = 0
- d)all are correct at different ranges of pressure
- a)True
- b)1/n
- a)True
- b)False
- b)Independent
- a)Proportional
- b)q
_{e }= k[A]^{n} - d)log qe = log k + n log [A]

**References: **

1. GAURAV KUMAR JAIN – THEORY & PRACTICE OF PHYSICAL PHARMACY, 1st edition 2012 Elsevier, page no. 128-136.

2. Martins Physical Pharmacy, 6th edition 2011, page no. 669-696.

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