Langmuir Adsorption Isotherm refers to the equilibrium between the adsorbent and the adsorbate system. In this method, adsorbate adsorption is always limited to one molecular layer. It happens at or before one reaches the relative pressure of unity. This equation is used in multiple systems with moderately low coverage. Most importantly, it is used to describe the behaviour of the binary absorption system. According to the Langmuir Adsorption Isotherm, the surface is homogeneous with the assumption that no lateral interaction takes place between the adjacent absorbed molecules. Even when a single molecule occupies a single surface site, this remains the same.

Assumptions of Langmuir Adsorption Isotherms

The Langmuir Adsorption Isotherm was derived using the kinetic energy of gases along with the following assumptions:

1. The adsorption includes a monolayer at the surface.
2. No interaction takes place between the molecules on different sites.
3. Each site can hold only a single adsorbed molecule.
4. The heat of adsorption is not dependent on the number of sites. It is equal for everyone.

Equation Derivations

Langmuir Adsorption Isotherm predicts linear adsorption at a maximum surface coverage and low adsorption densities. It happens at higher solute metal concentrations.

This isotherm takes the following form in the process:

KaCe (1 – θ) = ​Kd θ 

Here,

Ka refers to the respective rate constant required for adsorption,

Kd refers to the respective rate constant required for desorption. 

Θ stands for the fraction of the surface covered by an adsorbed molecule,

Calculation of Langmuir Adsorption Isotherm 

Langmuir Adsorption Isotherm is calculated as follows:

Ce/Qe = Ce/Qm + 1/(Qm*KL)

Here,

Ce refers to the equilibrium concentration of the said adsorbate,

Qe refers to the adsorption capacity that gets adsorbed at equilibrium,

Qm refers to the maximum adsorption capacity,

KL refers to the Langmuir Adsorption Constant.

Langmuir Constant

The Langmuir Constant is commonly referred to as K. It indicates the level of interaction between the surface and the adsorbate. If the value of this constant is larger, it indicates a strong interaction between the adsorbent and the adsorbate. On the other hand, K having a smaller value indicates a weaker interaction between the surface and the adsorbate.

Position of the Equilibrium

The position of the equilibrium depends on the factors mentioned below:

1. The relative stability of the gas phase species and the adsorbed molecules.
2. The temperature of the said system (both the surface and the gas).
3. The pressure of the gas is calculated above the surface. 

Derivation of Equilibrium Considerations

The Langmuir Isotherm is derived by treating the adsorption process as any other equilibrium process. However, the exceptional fact about this case is that the equilibrium happens between the gas phase molecules and the species adsorbed on the surface that also includes the vacant surface sites. 

This equilibrium is always in a dynamic phase. It means that the equilibrium represents a state in which the rate of desorption of molecules counterbalances the rate of adsorption of molecules. According to the model of Langmuir Adsorption Isotherm, both desorption and adsorption are easily reversible processes. 

Moreover, this model is also responsible for explaining the effects of pressure on the surroundings. That is why the adsorbent is assumed to be an ideal solid surface. It is composed of distinct sites that are capable of holding the adsorbate together. This binding is treated as a perfect chemical reaction that occurs between the gaseous molecules and a sorption site that is usually empty. 

The most important thing to be kept in mind here is that the Langmuir Adsorption Isotherm is applicable only for monolayer adsorption. Moreover, this process takes place on a homogeneous surface when there is zero interaction between the already adsorbed species. Although this process was earlier suitable for describing only the chemisorption process, now it is followed in many systems, most of which have moderately low coverage. 

Limitations of Langmuir Adsorption Isotherms

• The adsorbed gas is required to behave ideally in the vapour phase. Moreover, this condition can only be fulfilled under low pressure.

   

• Langmuir Adsorption Isotherms assume that adsorption is a monolayer. However, this formation is also possible only under low pressure. This assumption breaks down completely under high pressure. That is because gas molecules attract each other.

   

• The assumption held in this equation is that all sites present on the solid surface are equal in shape and size. It is also assumed that they have an equal affinity for the molecules that are adsorbed. In other words, the surface of a solid is considered homogeneous. However, it is heterogeneous.

   

• This equation also assumes that molecules do not interact with each other. However, that is not the case because all molecules have at least a weak force of attraction present among them.

   

• It further assumes that the adsorbed molecules must be localised. However, that is not true, as the adsorption liquefaction of gases results in a decrease in randomness. However, the value that remains is not zero.

Conclusion

Langmuir Adsorption Isotherms are based on a simple assumption. This theory holds that there is a homogeneous distribution of the reactive groups over the particulate’s surface. Moreover, no lateral interaction happens between these groups. So, it is possible to obtain only semi-empirical parameters. 

This model explains the process of adsorption with another key assumption: that is the adsorbate always behaves like an ideal gas during isothermal conditions. An important fact about Langmuir Adsorption Isotherm is that it always assumes monolayer adsorption. The above study material notes on Langmuir Adsorption Isotherms will help you understand the entire equilibrium process in detail.