The **main difference between Langmuir and BET isotherm** is that Langmuir isotherm assumes monolayer adsorption with no interaction between adsorbed molecules, while BET isotherm extends the model to include multilayer adsorption and considers interactions between adsorbed molecules.

Langmuir and BET Isotherms are two essential models used to describe gas adsorption on surfaces. Moreover, these isotherms provide valuable insights into the adsorption behavior at different surface coverage levels.

### Key Areas Covered

**1. What is Langmuir Isotherm**

* – Definition, Features, Calculation*

**2. What is BET Isotherm**

* – Definition, Features, Calculation*

**3. Similarities Between Langmuir and BET Isotherm**

* – Outline of Common Features*

**4. Difference Between Langmuir and BET Isotherm**

* – Comparison of Key Differences*

### Key Terms

*Langmuir Isotherm, BET Isotherm*

## What is Langmuir Isotherm

The Langmuir isotherm is a fundamental model that describes the adsorption behavior of gases on solid surfaces. It is named after Irving Langmuir, an American chemist who proposed the model in 1918. The Langmuir isotherm assumes that adsorption occurs on a surface with a limited number of equivalent adsorption sites. It suggests that the adsorption process reaches equilibrium once a monolayer of adsorbate molecules fully covers the surface. Moreover, the model assumes that there is no interaction between the adsorbed molecules and that each adsorption site can accommodate only one molecule. This implies that the adsorption process does not lead to the formation of multiple layers.

### Mathematical Expression of Langmuir Isotherm

The following equation mathematically represents the Langmuir isotherm:

**θ = (K * P) / (1 + K * P)**

θ is the fractional surface coverage or the ratio of the number of adsorbed molecules to the total number of adsorption sites. K is the Langmuir constant, which is related to the affinity of the adsorbate for the adsorbent surface, and P is the pressure or concentration of the gas in the gas phase.

The Langmuir equation describes the relationship between the fractional surface coverage (θ) and the pressure or concentration of the gas (P). It demonstrates that the fractional surface coverage increases with increasing pressure or concentration until it reaches a maximum value, known as saturation coverage.

### Uses of Langmuir Isotherm

The Langmuir isotherm is particularly useful for analyzing the adsorption behavior of gases on solid surfaces with limited adsorption capacity. It is commonly applied in catalysis, where it helps understand the interaction between the gas molecules and the catalyst surface. The Langmuir isotherm provides information about the maximum surface coverage and the affinity of the adsorbate for the adsorbent surface, which is valuable for optimizing catalyst performance.

Furthermore, the Langmuir isotherm can be utilized to determine important parameters such as the Langmuir constant (K) and the maximum adsorption capacity (θ_max). These parameters can be obtained by fitting experimental data to the Langmuir equation using techniques such as linear regression or nonlinear curve fitting. The Langmuir isotherm is also utilized to calculate other quantities of interest, such as the adsorption energy or heat of adsorption.

However, the Langmuir isotherm has certain limitations. One major assumption of the Langmuir model is that there is no interaction between the adsorbed molecules. In reality, adsorbed molecules can interact with each other, leading to deviations from the ideal Langmuir behavior. The Langmuir isotherm also assumes a homogeneous surface with equivalent adsorption sites, which may not be the case for many real-world systems. Additionally, the Langmuir isotherm is limited to describing monolayer adsorption and does not account for multilayer adsorption.

## What is BET Isotherm

The BET (Brunauer-Emmett-Teller) isotherm is a widely used model for analyzing gas adsorption on solid surfaces, particularly on porous materials. It was developed by Stephen Brunauer, Paul Hugh Emmett, and Edward Teller in 1938 as an extension of the Langmuir isotherm. The BET isotherm provides insights into the multilayer adsorption behavior and allows for the calculation of important parameters such as the specific surface area and the adsorption energy.

The BET isotherm extends the Langmuir model by considering the formation of multilayer adsorption on the solid surface. It assumes that the adsorbed gas molecules in each layer interact with each other but not with the gas molecules in other layers. The model postulates that the adsorption energy decreases as additional gas molecules are adsorbed. This leads to the formation of multiple layers until equilibrium is reached.

### Mathematical Expression of BET Isotherm

The mathematical representation of the BET isotherm is based on the assumption that a series of Langmuir monolayer adsorptions can represent multilayer adsorption. The equation is given as follows:

**P / (V * (P_0 – P)) = (1 / (C * V_m)) + (C – 1) / (C * V_m * P_0)**

where P is the pressure of the gas, V is the amount adsorbed, P_0 is the saturation pressure, C is the constant related to the interaction between gas molecules in adjacent layers, and V_m is the monolayer adsorption capacity.

The BET equation allows for the calculation of the monolayer adsorption capacity (V_m) and the constant C. These parameters are crucial for determining the specific surface area of the material. The specific surface area is a measure of the total surface area per unit mass of the adsorbent material and is an important parameter in various fields such as materials science, catalysis, and gas separation.

The BET isotherm can be derived by plotting the experimental data of P / (V * (P_0 – P)) against P / (P_0 – P). The resulting plot should exhibit a linear relationship for a valid application of the BET model. The slope and intercept of the linear portion of the plot can be used to calculate the parameters V_m and C, respectively.

The BET isotherm finds broad applications in the characterization of porous materials, such as activated carbon, zeolites, and metal-organic frameworks. It is used to determine the specific surface area, pore size distribution, and adsorption capacity of these materials. The BET model is valuable in evaluating the performance of adsorbents in gas storage, gas separation, and purification processes.

## Similarities Between Langmuir and BET Isotherm

- Both Langmuir and BET isotherms describe the adsorption behavior of gases on solid surfaces.
- Both isotherms assume that the adsorption process reaches equilibrium; here, the rate of adsorption is equal to the rate of desorption.
- Moreover, both isotherms consider the formation of adsorption layers on the surface of the solid.

## Difference Between Langmuir and BET Isotherm

### Definition

The Langmuir isotherm is a model that assumes single-layer adsorption on a surface, with no interaction between the adsorbed gas molecules. On the other hand, the BET isotherm is an extension of the Langmuir model that includes multilayer adsorption, where multiple layers of gas molecules can adsorb on the surface with interactions within each layer.

### Surface Coverage

The Langmuir isotherm provides information about the maximum surface coverage or saturation level. It assumes that the adsorption process reaches equilibrium once adsorbate molecules fully cover the surface. In contrast, the BET isotherm considers the formation of multiple layers, allowing for the analysis of adsorption on both the monolayer and multilayer levels. It provides insights into the adsorption behavior at different surface coverage levels.

### Mathematical Formulation

Mathematically, the Langmuir isotherm uses a simple equation relating fractional surface coverage to gas pressure/concentration. However, the BET isotherm involves a more complex model incorporating adsorption energies and multilayer formation. The equation that mathematically represents Langmuir isotherm is θ = (K * P) / (1 + K * P). Meanwhile, the equation that mathematically represents BET isotherm is P / (V * (P_0 – P)) = (1 / (C * V_m)) + (C – 1) / (C * V_m * P_0).

### Applications

The Langmuir isotherm is suitable for analyzing gases adsorbed on surfaces with limited capacity, like solid catalysts. The BET isotherm is used for surfaces with the potential for multilayer formation, such as gas adsorption on porous materials like activated carbon or zeolites.

## Conclusion

In brief, Langmuir and BET isotherms describe the adsorption behavior of gases on solid surfaces. Langmuir isotherm assumes monolayer adsorption with no interaction between adsorbed molecules. In contrast, BET isotherm extends the model to include multilayer adsorption and considers interactions between adsorbed molecules. Thus, this is the main difference between Langmuir and Bet isotherm.

##### Reference:

1. “Langmuir Isotherm – An Overview.” Science Direct.

##### Image Courtesy:

1. “Langmuir Adsorption Schema” By Ichwarsnur – Own work (CC BY-SA 4.0) via Commons Wikimedia

2. “BET Multilayer Adsorption” By Life of Riley – Own work (CC BY-SA 3.0) via Commons Wikimedia

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