derivation of the langmuir and bet isotherms isotherm - Langmuiradsorption isothermderivationPDF derivation Unraveling the Mysteries: The Derivation of the Langmuir and BET Isotherms

BETtheory PDF In the realm of surface chemistry and materials science, understanding how gases or solutes interact with solid surfaces is paramount. This interaction, known as adsorption, is often quantified by isotherms, which describe the equilibrium relationship between the amount of adsorbate and its concentration at a constant temperature. Two of the most foundational and widely applied models in this field are the Langmuir isotherm and the BET (Brunauer, Emmett and Teller) isothermDerivation of the BET and Langmuir Isotherms. This article delves into the derivation of these crucial models, exploring their underlying assumptions, mathematical frameworks, and the invaluable insights they offer into surface area analysis and material characterization.

The Langmuir Isotherm: A Monolayer Marvel

The Langmuir isotherm, pioneered by Irving Langmuir, provides a fundamental model for monolayer adsorptionTheBET(Brunauer, Emmet and Teller)isothermallows to extend ideas and hypotheses that are adopted to obtain theLangmuir isothermto describe the adsorption .... It is based on a set of key assumptions regarding the adsorption process:

1.Kinetic Approach to Multilayer Sorption: Equations of ... Adsorption occurs at specific, localized sites on the solid surfaceLangmuir vs BET Isotherm Explained | PDF | Adsorption. These sites are assumed to be energetically equivalent.

2.2019年6月25日—BETTheory extends theLangmuirtheory from monolayer adsorption to ...isothermsin the range of 0.05 < pp/pp0. < 0.30. Page 11 ... Adsorption is limited to a single layer (monolayer) of adsorbate molecules on the surfaceIt outlines the theory's extensions from Langmuir theory andprovides various equations and isotherm typesfor monolayer and multilayer formations under .... Once a site is occupied, it cannot accommodate another molecule.Brunauer-Emmett-Teller (BET) surface area analysis

3. Adsorbed molecules do not interact with each other. The presence of an adsorbed molecule does not influence the adsorption of neighboring molecules.

4. The rate of adsorption is equal to the rate of desorption at equilibrium.

The derivation of the Langmuir isotherm can be approached from a kinetic or thermodynamic perspective.Brunauer-Emmett-Teller (BET) surface area analysis A common kinetic approach considers the dynamic equilibrium between adsorption and desorptionThe equation that mathematically representsLangmuir isothermis θ = (K * P) / (1 + K * P). ... V_m * P_0). ... like activated carbon or zeolites.. Let $\theta$ represent the fraction of the surface covered by adsorbed molecules.

The rate of adsorption is proportional to the concentration of the adsorbate ($P$, for gas phase, or $C$, for liquid phase) and the fraction of unoccupied surface sites $(1-\theta)$:

Rate of Adsorption $\propto P(1-\theta)$

The rate of desorption, on the other hand, is proportional to the fraction of occupied surface sites ($\theta$):

Rate of Desorption $\propto \theta$

At equilibrium, the rate of adsorption equals the rate of desorption.Langmuir vs BET Isotherm Explained | PDF | Adsorption Introducing proportionality constants, $k_{ads}$ and $k_{des}$:

$k_{ads} P (1-\theta) = k_{des} \theta$

Rearranging this equation to solve for $\theta$, we get:

$k_{ads} P = k_{des} \theta + k_{ads} P \theta$

$k_{ads} P = \theta (k_{des} + k_{ads} P)$

$\theta = \frac{k_{ads} P}{k_{des} + k_{ads} P}$

Dividing both the numerator and denominator by $k_{des}$, and defining an equilibrium constant $K = \frac{k_{ads}}{k_{des}}$:

$\theta = \frac{K P}{1 + K P}$

This is the most common form of the Langmuir isotherm equation. Often, $\theta$ is expressed in terms of the volume of gas adsorbed ($V$) and the maximum monolayer capacity ($V_m$), such that $\theta = V/V_m$. Thus, the Langmuir equation can also be written as:

$\frac{V}{V_m} = \frac{K P}{1 + K P}$

Or, in its linear form, often used for graphical analysis:

$\frac{P}{V} = \frac{1}{K V_m} + \frac{P}{V_m}$

This linear form allows for the determination of $V_m$ (the maximum adsorption capacity) and $K$ (related to the adsorption energy) from the intercept and slope of a plot of $P/V$ versus $P$作者：R Afonso·2016·被引用次数：46—TheLangmuirequation is one of the most successful adsorptionisothermequations, being widely used to fit Type I adsorptionisotherms. In this article we show .... Langmuir's model is particularly effective for Type I isotherms, which represent low pressures and monolayer formation.

The BET Isotherm: Embracing Multilayers

While the Langmuir isotherm is invaluable for describing monolayer adsorption, it fails to account for the possibility of multilayer adsorption, where subsequent layers of adsorbate molecules can form on top of the first layer. This limitation led Brunauer, Emmett, and Teller (BET) to develop a more comprehensive theory in 1938.A Statistical Derivation of Langmuir's Adsorption Isotherm The BET theory extends the principles of Langmuir's reasoning to describe the formation of multiple adsorbed layers.

The BET isotherm derivation builds upon the Langmuir model by considering the equilibrium between gas molecules and the solid surface, allowing for condensation and evaporation between layers. The core assumptions of the BET theory are:

* Adsorption occurs on a homogeneous surface.

* Adsorption is multilayer.

* The Langmuir theory is applicable to each layer separately3.1BETadsorptionisotherm. 3.2DerivationofBETadsorptionisotherm. 3.3 Physical significance of constant C. 3.4 Explanation of different types of .... This means that the heat of adsorption for the first layer is constant (heat of adsorption, $\Delta H_{ads}$), while the heat of adsorption for the second and subsequent layers is equal to the heat of liquefaction ($\Delta H_{vap}$) of the adsorbate gas.

* The molecules in the upper layers are assumed to be in equilibrium with the gas phase, as in condensationDerivation of the Langmuir isotherm​​For molecules in contact with a solid surface at a fixed temperature, the Langmuir Isotherm, developed by Irving Langmuir ....

The BET equation, derived from these assumptions, relates the volume of gas adsorbed ($V$) at a given relative pressure ($P/P_0$, where $P_0$ is the saturation vapor pressure of the adsorbate at the given temperature) to the monolayer capacity ($V_m$) and a constant ($C$) related to the adsorption energy:

$\frac{P/P_0}{V(1 - P/P_0)} = \frac{1}{V_m C} + \frac{C-1}{V_m C} \left(\frac{P}{P_0}\right)$

The constant $C$ is given by:

$C = \exp\left(\frac{\Delta H_{ads} - \Delta H_{vap}}{RT}\right)$

where $R$ is the ideal gas constant and $T$ is the absolute temperature.

The BET equation is typically linearized