The graphical relationship between the amount of the substance being adsorbed per unit concentration of the adsorbate and the applied pressure is called adsorption isotherm. Various adsorption isotherms are employed for explaining the extent of adsorption such as Freundlich adsorption isotherm and Langmuir adsorption isotherm. The Freundlich adsorption isotherm is associated with multilayer adsorption on the surface whereas, the Langmuir adsorption isotherm is associated with monolayer adsorption.

Freundlich’s adsorption isotherm is empirical while Langmuir’s adsorption isotherm is theoretical. At a given temperature, the adsorption of a gas on the surface of a liquid or solid increases with the increase in pressure.

## Freundlich vs Langmuir Adsorption Isotherms

Freundlich adsorption isotherm | Langmuir adsorption isotherm |

The surface of solid adsorbent is heterogeneous for Freundlich adsorption isotherm | The surface of solid adsorbent is homogeneous for Langmuir adsorption isotherm |

It is associated with multi-molecular layer adsorption | It is associated with mono-molecular layer adsorption |

No equilibrium can be established for FAI | Dynamic equilibrium is established in LAI |

Interaction between adsorbed molecules can occur because it is multilayered | There is no interaction between adsorbed molecules |

The surface of adsorbent is not uniform | The surface of adsorbent is uniform |

No chemical forces are not involved | Chemical forces are involved |

Freundlich's equation is purely empirical | Langmuir's equation has theoretical basis |

It does not explain chemisorption | It explains the mechanism of chemisorption |

Freundlich adsorption isotherm does not give quantitative explanations | Langmuir adsorption isotherm gives quantitative explanations |

It gives comparatively less satisfactory explanation of adsorption of gases on solids | It gives more satisfactory explanation of adsorption of gases on solids |

Only adsorption is involved in Freundlich adsorption isotherm | Both adsorption and desorption are involved in Langmuir adsorption isotherm |

## Freundlich adsorption isotherm

In 1909, Herbert Freundlich gave an empirical relationship to explain the variation of the amount of gas adsorbed per unit mass of the adsorbent (solid) with pressure at a constant temperature.

where,

- w = mass of gas adsorbed
- m = mass of adsorbent
- P = pressure
- k = constant
- n = number of moles

According to the above equation, a curved isotherm is obtained when plotting the mass of the gas adsorbed per unit mass of adsorbent (w/m) against the equilibrium pressure.

The graph between log w/m against the log P at higher pressure shows a slight curvature, which makes this isotherm, invalid for higher pressures.

The above graph shows that the Freundlich isotherm is not applicable at higher pressure. By taking the log on both sides of the Freundlich equation, we get;

log w / m = log k + 1 / n log P

This equation is for a straight line. A plot of log w/m against log P should be a straight line with slope 1/n and its intercept shall be log k. However, it is observed that a straight line is only obtained at low pressure. It gives a slight curvature at higher pressure, especially with low temperature which is why Freundlich adsorption isotherm is not applicable to the adsorption of gases on solids at higher pressures.

### Uses of FAI

- Freundlich adsorption isotherm is used to determine the extent of the adsorption of gases on solids at low pressures.
- It is also used for the adsorption of solute in a solution.

### Limitations of FAI

- Freundlich’s equation is purely empirical. It does not have any theoretical basis.
- The constants in the equations ‘k’ and ‘n’ vary at different temperatures.
- It is valid only for a specific range of pressure.

## Langmuir adsorption isotherm

Failure of Freundlich adsorption isotherm at higher pressure indicates that his concept is not entirely correct. In 1916, Langmuir introduced a new concept that includes the effect of chemical (intermolecular) forces. He derived a mathematical equation based on the kinetic molecular theory of gases for the adsorption phenomenon, which is called the Langmuir adsorption isotherm.

θ = bP / (1 + bP)

where,

- θ is the fraction of the surface covered by adsorbed species
- b is adsorption coefficient
- P is the applied pressure

### Assumptions for LAI

Following assumptions are made necessary for the Langmuir adsorption isotherm to work:

- The solid surface should be homogenous.
- It should have a fixed number of adsorption sites.
- Each site must adsorb only one molecule (mono-layered adsorption).
- The adsorption of molecules is confined to a monomolecular layer.
- The adsorbed gas behaves ideally in the vapor phase.
- There is no interaction between the adsorbed molecules.
- The rate of adsorption and desorption becomes equal at their dynamic equilibrium.

### Derivation of LAI

According to Langmuir, the gas molecules strike the surface of a solid. Some of the striking molecules get adsorbed and others start to evaporate or desorb. Therefore, a dynamic equilibrium is established between the number of adsorbed and desorbed molecules.

Suppose theta (θ) is the fraction of the total surface molecules, adsorbed on the surface. The total surface (1) is covered by molecules and the fraction of the naked area is (1-θ).

The rate of desorption (R_{d}) is proportional to the covered surface that is theta (θ). Thus;

R_{d} = k_{d} θ (i)

where k_{d} is the rate constant for desorption

The rate of adsorption R_{a} is proportional to the naked surface (1-θ) and the pressure of the gas.

R_{a} = k_{a} (1-θ) P (ii)

where k_{a} is the rate constant for adsorption

The rate of desorption and rate of adsorption become equal at equilibrium, so, at equilibrium from (i) and (ii)

k_{d} θ = k_{a} (1-θ) P (iii)

Or,

k_{d}θ = k_{a}P – K_{a}θP

k_{d}θ + k_{a}θP = k_{a}P

θ (k_{d}+k_{a}P)= k_{a}P

θ = k_{a} P / (K_{d} + k_{a}P) (iv)

By dividing the numerator and denominator with k_{d, }we get;

θ

θ

As b = k_{a} /k_{d},

θ = bP / (1 + bP)

The extent of adsorption is given by ‘b’ (k_{a}/k_{d}) also known as the adsorption coefficient.

The amount of the gas that is adsorbed per gram of the adsorbent is (x) which is proportional to theta (θ), so;

x ∝ k b P / (1 + bP)

Or,

x = k’ b P / (1 + bP)

where k’ is the new constant. The above equation gives a relationship between the gas adsorbed upon applying pressure at a constant temperature. It is also known as Langmuir adsorption isotherm.

The above equation can be rearranged as:

P/x = 1/k’ + P/k’’

where k’ = k’/b = constant.

The above equation can be plotted as a straight-line equation. P/x can be plotted against P, the slope is 1/k’’ and the intercept is 1/k’.

The above graph is the verification of Langmuir isotherm for adsorption of nitrogen molecules on mica at the temperature of 90K.

### Uses of LAI

- Langmuir adsorption isotherm gives a quantitative explanation of adsorption.
- Unlike Freundlich adsorption isotherm, it explains the mechanism of chemisorption.
- It gives a more satisfactory quantification of adsorption as compared to Freundlich’s adsorption isotherm when explaining the adsorption of gases on solid surfaces.

### Limitations of LAI

- Langmuir adsorption isotherm is only applicable at low pressure and fails at high pressures.

P/x = 1/k’ + P/k’’

At low pressure, P/k’’ may be ignored and the isotherm becomes x = k’ P. At high pressures, 1/k’ may be ignored and the isotherm becomes x = k’’.

- This theory does not explain all the experimentally observed adsorption isotherms.
- It assumes that the surface of a solid is capable of adsorbing a single layer molecule thickness, but in actual practice, it is a much thicker layer.
- The adsorption saturation value is generally independent of temperature, but the experiment shows that the saturation value decreases with an increase in temperature.

## Key Takeaway(s)

**Additional articles:**

## Concepts Berg

**What is Langmuir adsorption isotherm?**

Langmuir adsorption isotherm is a straight line equation that gives us an idea of how gas molecules are adsorbed on a solid surface. It explains molecules adsorbed on the surface of a solid in a monolayer structure.

P/x = 1/k’ + P/k’’

Langmuir equation has a theoretical background and it is based on the kinetic molecular theory of gases.

**What is Freundlich adsorption isotherm?**

Freundlich adsorption isotherm is the graphical representation that describes the variations of the amount of gas adsorbed per unit mass of the adsorbent with pressure at a constant temperature. The empirical equation is:

w / m = k P^{1/n}

- w = mass of gas adsorbed
- m = mass of adsorbent
- P = pressure
- k = constant
- n = no. of moles

**How is the Langmuir isotherm different from the Freundlich isotherm in terms of validity?**

Langmuir’s isotherm is more satisfactory compared to Freundlich’s to explain the adsorption of gases on solids. It explains the mechanism of chemisorption while Freundlich isotherm does not.

**Reference books**

- Surface and Colloid Chemistry: (Principles and Applications) by
**K.S Birdi**(Berkeley, University of California & Unilever, Copenhagen, Denmark) **Essential of Physical Chemistry:**2nd edition By**B.S Bahl**(Gurdaspur, India) and**Arun Bahl**(RSC, UK) and**G.D. Tuli**(Delhi University, India)**Principle of Physical Chemistry**by**Haq Nawaz Bhatti**(University of Agriculture, Faisalabad, Pakistan)

**Reference links**

- Adsorption isotherms (toppr.com)
- Modeling and Interpretation of Adsorption Isotherms (hindawi.com)
- Adsorption isotherms (byjus.com)