Problem 17 - A Complete Resource Book in Chemistry for JEE Main

Question

What is the equation form of Langmuir isotherm under high pressure? (a) \(\mathrm{x} / \mathrm{m}=\mathrm{a} / \mathrm{b}\) (b) \(\mathrm{x} / \mathrm{m}=\mathrm{a} . \mathrm{p}\) (c) \(\mathrm{x} / \mathrm{m}=1 / \mathrm{a}\).p (d) \(\mathrm{x} / \mathrm{m}=\mathrm{b} / \mathrm{a}\)

Step-by-Step Solution

Answer

The equation form under high pressure is \( \frac{x}{m} = \frac{a}{b} \). Option (a).

Step 1: Understanding Langmuir Isotherm

Langmuir isotherm represents the relationship between the gas adsorbed on a solid surface and the pressure of the gas above the surface. The general equation of the Langmuir isotherm is: \[ \frac{x}{m} = \frac{ap}{1 + bp} \]where \(x\) is the amount of gas adsorbed, \(m\) is the mass of the adsorbent, \(a\) is a constant related to adsorption capacity, \(b\) is a constant related to pressure, and \(p\) is the pressure.

Step 2: Applying the High Pressure Condition

At high pressure, the term \(bp\) becomes very large compared to 1, allowing us to make the approximation: \[ 1 + bp \approx bp \]Using this approximation, the Langmuir isotherm simplifies to: \[ \frac{x}{m} = \frac{ap}{bp} \]

Step 3: Simplifying the Equation

Simplify the expression \( \frac{ap}{bp} \) by cancelling \(p\) from the numerator and the denominator. This leads to:\[ \frac{x}{m} = \frac{a}{b} \] which matches the option \( (a) \).

Key Concepts

Understanding Adsorption

Adsorption is a process where molecules from a gas or liquid collect on the surface of a solid or a liquid. It is fundamental in understanding how pollutants or reactants interact at surfaces. This phenomenon is different from absorption, where the substance is distributed throughout the volume of a material. In adsorption:

• The adsorbed molecules form a layer on the adsorbent's surface.
• They do not penetrate into the bulk of the material.

Langmuir isotherm specifically describes adsorption quantitatively and helps determine how much gas will attach to a surface under various pressures. This is crucial for industries that rely on catalysts or need to purify gases.

Role of High Pressure in Langmuir Isotherm

In the context of the Langmuir isotherm, high pressure conditions have a significant impact on the adsorption process. At high pressures:

• The term involving pressure in the Langmuir equation \(bp\) becomes very large compared to 1.
• This simplification allows us to assume that \(1 + bp \approx bp\).

This assumption simplifies the isotherm equation to focus on the adsorption ratio alone, reducing it to \[ \frac{x}{m} = \frac{a}{b} \]. This relationship indicates that at high pressure, the system is dominated by the adsorption capacity and pressure constants, making calculation more straightforward.

Concept of Adsorbed Gas

Adsorbed gas refers to the gas molecules that attach to a surface through adsorption. When studying the adsorption of gases onto a solid surface:

• The gas molecules form a film that is one or more molecules thick.
• This attachment is temporary and can depend on factors like temperature, pressure, and the nature of the gas and adsorbent.

Understanding adsorbed gas is essential for processes like gas purification, where unwanted gases need to be separated from other gases. The Langmuir isotherm gives a theoretical framework to predict how gases will behave when adsorbed, which is vital for the design of efficient adsorption systems.