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) \(\mathrm{x} / \mathrm{m}=\mathrm{a} \cdot \mathrm{p}\) (c) \(\mathrm{x} / \mathrm{m}=1 / \mathrm{a} . \mathrm{p}\) (d) \(\mathrm{x} / \mathrm{m}=\mathrm{b} / \mathrm{a}\)

Step-by-Step Solution

Answer

(a) \( \frac{x}{m} = \frac{a}{b} \)

Step 1: Recall Langmuir Isotherm Equation

The Langmuir isotherm describes adsorption by relating the amount adsorbed to the pressure of the gas or concentration of solute. The general form of Langmuir's isotherm is \( \frac{x}{m} = \frac{ap}{1+bp} \), where \( x/m \) is the amount adsorbed per unit mass of adsorbent, \( p \) is the pressure, and \( a \) and \( b \) are constants.

Step 2: Consider High Pressure Scenario

Under high pressure, the term \( bp \) in the denominator becomes much larger than 1 (\( bp \gg 1 \)), therefore, the \( +1 \) can be neglected. The equation simplifies to: \( \frac{x}{m} = \frac{ap}{bp} = \frac{a}{b} \).

Step 3: Compare Options with Simplified Equation

Look at the options provided and match the simplified high-pressure Langmuir isotherm equation \( \frac{a}{b} \). The correct option from the given choices is (a) \( \frac{x}{m} = \frac{a}{b} \).

Key Concepts

Understanding Adsorption

Adsorption is a surface phenomenon where molecules, ions, or atoms from a gas, liquid, or dissolved solid adhere to a surface. It's different from absorption, which involves the entire volume of the material. Imagine it like a magnet; it's just the surface that attracts other particles. Adsorption occurs because of surface energy. When a surface is exposed, like the surface of charcoal, it seeks to reduce its energy by attaching these particles, forming a thin layer.

• Importance in Chemistry: Adsorption plays a crucial role in various chemical processes, such as catalysis and filtration, where it helps in concentrating molecules at the surface to increase reaction rates.
• Langmuir Isotherm: This is a model used to describe the adsorption process, specifically for gases on a solid surface. It assumes a monolayer adsorption, which means only one layer of molecules can attach to the surface.

Langmuir's equation is fundamental to understanding how particles interact with surfaces, guiding applications in environmental science and industry, including activated carbon filters and gas masks.

High Pressure Scenarios in Adsorption

In adsorption, pressure plays a key role. At high pressures, things change. Typically, Langmuir isotherm includes a term that accounts for the pressure of the gas or concentration of the solute. However, when pressure is very high, this influences the adsorption differently.

• Pressure Impact: Under high pressure, the term related to pressure in the Langmuir equation becomes dominant. This means the effect of the initial layer can be disregarded, simplifying the equation.
• Practical Implications: In situations involving high-pressure, such as in industrial gas storage and catalytic processes, understanding this simplified model is crucial. It allows for accurate predictions and better efficiency in processes.

Thus, high-pressure scenarios allow us to approximate interactions more easily, helping industries to better design and optimize systems.

Breaking Down Chemistry Equations

Chemistry equations might seem a bit tricky, but they help us understand how different elements and compounds interact. The Langmuir Isotherm equation, for example, uses symbols and mathematical terms to express these interactions. Let's break down its components.

• General Form: The Langmuir equation is given by \( \frac{x}{m} = \frac{ap}{1+bp} \). Here, \( x/m \) represents the amount adsorbed per unit weight of the adsorbent, \( p \) is the pressure, and \( a \) and \( b \) are constants reflective of the adsorption process.
• High Pressure Simplification: At high pressure, the term \( bp \) becomes so large that the +1 in the denominator becomes negligible, simplifying the equation to \( \frac{x}{m} = \frac{a}{b} \). This makes it easier to calculate the adsorption amount as pressure increases.

By understanding these equations, students can better appreciate the delicate balances and interactions taking place in chemical processes, especially those involving surface chemistry and adsorption.