Question

Langmuir's adsorption equation which describes the amount of gas adsorbed on a solid surface is written as \(\frac{\mathrm{p}}{\mathrm{x} / \mathrm{m}}=\frac{1}{\mathrm{ab}}+\frac{\mathrm{p}}{\mathrm{b}}\) Here \(\mathrm{x} / \mathrm{m}\) is the extent of adsorption, a and \(\mathrm{b}\) are constants, and \(\mathrm{p}\) is the gas pressure. On the basis of the above equation indicate which of the following statements is correct? (1) At low value of \(\mathrm{p}, \mathrm{x} / \mathrm{m} \propto \mathrm{p}\). (2) At high value of \(\mathrm{p}, \mathrm{x} \longrightarrow \mathrm{b}\) (adsorption is independent of pressure). (3) Between the low and high value of \(\mathrm{p}\), the above equation becomes \(x / m=k p^{1 / n}\) which is the Freundlich equation (here \(\mathrm{k}\) and \(\mathrm{n}\) are constants) (a) 1 and 2 (b) 2 and 3 (c) \(\mathbb{l}\) and 3 (d) 1,2 and 3

Short answer

(a) 1 and 2

Step-by-step solution

Understand Langmuir's Adsorption Equation

The Langmuir adsorption equation is given by \( \frac{\mathrm{p}}{\mathrm{x} / \mathrm{m}}=\frac{1}{\mathrm{ab}}+\frac{\mathrm{p}}{\mathrm{b}} \). Here, \( \mathrm{x} / \mathrm{m} \) is the extent of adsorption, \( \mathrm{a} \) and \( \mathrm{b} \) are constants, and \( \mathrm{p} \) is the pressure of the gas. This equation relates the pressure and extent of adsorption.

Analyze the Equation at Low Pressure

At low values of \( \mathrm{p} \), the term \( \frac{1}{\mathrm{ab}} \) dominates because \( \frac{\mathrm{p}}{\mathrm{b}} \) is small. Therefore, the relationship approximately becomes \( \frac{\mathrm{p}}{\mathrm{x} / \mathrm{m}} \approx \frac{1}{\mathrm{ab}} \). Solving for \( \mathrm{x} / \mathrm{m} \), we get \( \mathrm{x} / \mathrm{m} \propto \mathrm{p} \), indicating proportionality between pressure and extent of adsorption.

Analyze the Equation at High Pressure

At high values of \( \mathrm{p} \), the term \( \frac{\mathrm{p}}{\mathrm{b}} \) becomes much larger than \( \frac{1}{\mathrm{ab}} \). Thus, the equation simplifies to \( \frac{\mathrm{p}}{\mathrm{x} / \mathrm{m}} \approx \frac{\mathrm{p}}{\mathrm{b}} \). Solving for \( \mathrm{x} / \mathrm{m} \), we find \( \mathrm{x} / \mathrm{m} \rightarrow \mathrm{b} \), meaning the adsorption approaches constant value \( \mathrm{b} \) and becomes independent of pressure.

Relation to Freundlich Equation

The Freundlich equation is \( \mathrm{x} / \mathrm{m} = k \mathrm{p}^{1/n} \). Langmuir's equation does not directly become the Freundlich equation between low and high \( \mathrm{p} \) values, as it inherently assumes a specific mechanism characterized by constant sites and no multilayer adsorption, unlike Freundlich's empirical nature. Hence, statement 3 is inconsistent with the Langmuir model and not applicable.

Key concepts

gas-solid adsorption

Gas-solid adsorption is a fascinating process where gas molecules accumulate on the surface of a solid material. This phenomenon is crucial in various fields, including catalysis, pollution control, and material science. The solid material, often referred to as the adsorbent, provides sites on its surface where gas molecules, also known as adsorbates, can attach or adhere.

• Adsorbents: These are usually porous solids like activated carbon, silica gel, or metal oxides, offering a large surface area for adsorption.
• Adsorbates: These are typically gas molecules that interact with the solid’s surface.
• Mechanism: The process can involve physical adsorption, which is reversible and usually relies on van der Waals forces, or chemical adsorption, which involves stronger, reactive interactions.

Understanding this process aids in determining how various materials can capture or release gases, impacting applications like air purification.

pressure dependence in adsorption

Pressure plays a significant role in determining the extent of adsorption in gas-solid systems. The relationship between pressure and adsorption can be represented and analyzed through different adsorption isotherms, such as Langmuir and Freundlich isotherms.

• Low Pressure: At low pressures, the adsorption tends to increase linearly with an increase in pressure. This occurs because initially the sites available on the solid surface are plenty, allowing more gas molecules to adhere as pressure increases.
• High Pressure: As the pressure continues to increase and most adsorption sites become occupied, we observe a plateau in adsorption. The process becomes saturated, meaning further increases in pressure do not significantly alter the extent of adsorption.
• Langmuir Isotherm: This model describes a situation where at low pressures, adsorption is proportional to pressure, whereas at high pressures, it becomes constant, showcasing saturation. This behavior is typical for surfaces with a finite number of binding sites available.

Analyzing pressure dependence helps in predicting and optimizing adsorption efficiency for different materials under varying conditions.

Freundlich adsorption isotherm

The Freundlich adsorption isotherm is an empirical model used to describe the adsorption process, particularly when it doesn't conform to the Langmuir model. Unlike Langmuir, the Freundlich isotherm applies well to adsorption on heterogeneous surfaces, indicating that different sites on the solid may host varying strengths of adsorption.

• Formula: It is given by the equation \( rac{x}{m} = k p^{1/n} \), where \( x/m \) is the amount of gas adsorbed per unit mass of solid, \( p \) is the gas pressure, and \( k \) and \( n \) are constants.
• Characteristics: The model does not reach a saturation point as can be seen with Langmuir’s model. This means it can better describe systems where multilayer adsorption occurs.
• Applications: It is especially useful for explaining adsorption in complex systems, such as those found in environmental pollution studies, where different substrates can adsorb various pollutants to different extents.

The versatility of the Freundlich adsorption isotherm makes it an indispensable tool in analyzing adsorption behavior across varied scientific fields.