Problem 110 - A Complete Resource Book in Chemistry for JEE Main

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 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 \(\mathrm{x} / \mathrm{m}=\mathrm{kp}^{1 / \mathrm{n}}\), which is the Freundlich equation (here \(\mathrm{k}\) and \(\mathrm{n}\) are constants) (a) 1 and 2 (b) 2 and 3 (c) 1 and 3 (d) 1,2 and 3

Step-by-Step Solution

Answer

Option (a) 1 and 2 are correct.

Step 1: Understanding the Langmuir Equation

The Langmuir adsorption equation is given by: \( \frac{p}{x/m} = \frac{1}{ab} + \frac{p}{b} \). In this equation, \( x/m \) represents the extent of adsorption, and \( p \) is the gas pressure. Constants \( a \) and \( b \) relate to specific properties of the surface and the adsorbing gas.

Step 2: Analyzing Low Pressure Conditions

At low pressure \( p \rightarrow 0 \), the equation simplifies as \( \frac{p}{x/m} \approx \frac{1}{ab} \), leading to \( x/m \approx ab \cdot p \). This shows that \( x/m \propto p \) at low pressures. Hence, statement (1) is correct.

Step 3: Analyzing High Pressure Conditions

At high pressure \( p \rightarrow \infty \), the term \( \frac{1}{ab} \) in comparison to \( \frac{p}{b} \) becomes negligible, leading to \( \frac{p}{x/m} \approx \frac{p}{b} \). Solving for \( x/m \), we get \( x/m \approx b \), indicating that adsorption becomes independent of \( p \) and \( x/m \) approaches a maximum of \( b \). Therefore, statement (2) is correct.

Step 4: Assessing Intermediate Pressure Conditions

The intermediary situation doesn't exactly match the Freundlich equation \( x/m = kp^{1/n} \). The Langmuir equation does not transition into a form represented by a power law of \( p \). Thus, statement (3) is incorrect.

Step 5: Conclusion of Correct Statements

From analyzing the statements above, only statements (1) and (2) are correct. Therefore, the correct answer to the problem is option (a) which represents the correct statements.

Key Concepts

Extent of Adsorption

The term "extent of adsorption" refers to how much a gas can be absorbed on a solid surface. It is denoted by the variable \( \frac{x}{m} \) in the Langmuir adsorption equation. This expression encompasses two critical components: \( x \), which represents the mass of gas adsorbed, and \( m \), the mass of the adsorbent. Therefore, \( \frac{x}{m} \) offers a quantifiable measure of adsorption capacity per unit mass of the adsorbent.

At low gas pressures, the Langmuir equation simplifies to indicate that the extent of adsorption is directly proportional to the gas pressure \( p \). This proportionality occurs because, at low pressures, there are plenty of available sites for the gas molecules to adhere, which results in a direct increase in adsorption with an increase in pressure. Understanding this concept is essential for evaluating how effectively different materials can capture and hold gases.

Adsorption Pressure Dependence

Adsorption pressure dependence illustrates how the behavior of gas adsorption changes with variations in pressure. The Langmuir adsorption equation is particularly insightful in analyzing this aspect. At low pressures, because most adsorption sites are empty, the extent of adsorption is directly proportional to the pressure of the gas. This means that as pressure increases, more gas molecules "stick" to the surface of the adsorbent.

Conversely, at high pressures, nearly all adsorption sites get occupied. Hence, further increases in pressure do not significantly boost adsorption. This leads to a plateau in the adsorption graph, indicating that the extent of adsorption becomes almost independent of pressure. This dual relationship showcases why the Langmuir model is used to explain the surface adsorption properties under varying pressure conditions.

Surface Adsorption Properties

The surface adsorption properties are integral to the adsorption process. They illuminate how different solid surfaces interact with gases under various conditions. In the Langmuir equation, the constants \( a \) and \( b \) give insight into these properties. These constants are specific to the adsorbent and the adsorbing gas, relating to factors like surface area and surface energy.

The properties determine the extent and efficiency of adsorption and help predict how effective an adsorbent will be in different industrial or laboratory settings. By carefully studying these properties, scientists can select suitable materials for specific adsorption-based applications, like gas purification or separation processes. Thus, understanding these surface characteristics is pivotal in deploying the Langmuir model effectively.

Freundlich Adsorption Isotherm

The Freundlich adsorption isotherm is an empirical model describing how adsorbates interact with heterogeneous surfaces. Unlike the Langmuir isotherm, which considers adsorption on a uniform surface, the Freundlich model accommodates non-uniform adsorbent surfaces with different affinities for the adsorbate.

Represented by the equation \( x/m = kp^{1/n} \), the Freundlich isotherm shows that adsorption increases with pressure and is also dependent on the concentration intensity parameters \( k \) and \( n \). These parameters are indicative of the adsorption capacity and intensity, respectively. While the Langmuir equation assumes a uniform adsorption process, the Freundlich isotherm better describes scenarios where the surface has varied energies of adsorption sites.

Understanding both models allows for more comprehensive analysis and prediction of adsorption behaviors across different materials and conditions. This serves well in practical applications like pollutant control and catalyst design.