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

Step-by-Step Solution

Answer

The equation is \( \frac{x}{m} = \frac{a}{b} \), matching option (a).

Step 1: Recall the Langmuir Isotherm Equation

The Langmuir isotherm describes the adsorption of molecules onto a solid surface and is expressed as \( \frac{x}{m} = \frac{ap}{1 + bp} \), where \( x/m \) is the amount adsorbed per unit mass, \( p \) is the pressure, and \( a \) and \( b \) are constants related to the system.

Step 2: Simplify the Equation Under High Pressure Conditions

At high pressure, the term \( bp \) in the denominator becomes much larger than 1. Therefore, the equation simplifies to \( \frac{x}{m} = \frac{ap}{bp} = \frac{a}{b} \), as the \( bp \) terms cancel each other out.

Step 3: Identify the Correct Option

Look at the given options and match them with the simplified form \( \frac{x}{m} = \frac{a}{b} \). Option (a) \( \frac{x}{m} = \frac{a}{b} \) corresponds to the equation we derived.

Key Concepts

Adsorption

Adsorption is a process where molecules from a gas or liquid adhere to the surface of a solid material. It's a surface phenomenon,
distinct from absorption, where a substance is taken up within the volume of another. In adsorption:

• Molecules form a film on the surface of the adsorbent.
• The extent of adsorption depends on factors like pressure and temperature.

Adsorption is significant in various industries, such as filtration systems, catalysis, and even air purification. Because it involves interaction with a surface, the nature and area of the adsorbent material greatly affect the process. The Langmuir isotherm describes this adsorption process under the assumption that:

• There is a finite number of adsorption sites on the surface.
• Once a site is occupied, no further adsorption can happen at that site.
• All sites are equally available and possess equal energy levels.

Understanding adsorption is critical for predicting how substances behave in various conditions, which is why Langmuir's theory is a foundational concept here.

High Pressure

When considering adsorption at high pressures, the behavior of molecules changes due to increased collision frequency. Under high pressure:

• Molecules possess higher kinetic energy.
• The collision rates with the surface increase.
• Certain simplifications can be made in adsorption equations.

In the context of the Langmuir isotherm, the factor to consider is the dominance of the pressure in the isotherm equation. The original form, \( \frac{x}{m} = \frac{ap}{1 + bp} \), becomes simplified by canceling the \( bp \) term since it becomes significantly large, reducing the equation to \( \frac{x}{m} = \frac{a}{b} \).
This simplification highlights the direct relationship between pressure and adsorption capacity at elevated pressures, assuming the surface coverage reaches a maximized state. Thus, learning how adsorption behaves at high pressure is essential for applications involving adsorption systems in pressurized environments.

Chemical Equilibrium

Chemical equilibrium relates to the balance that is achieved when a chemical reaction reaches a state where the concentrations of reactants and products remain constant over time. In the context of the Langmuir isotherm:

• The equilibrium of the adsorption process is crucial.
• It determines how much a substrate surface can adsorb at a given pressure.

The Langmuir isotherm offers a simplified description of adsorption equilibrium, assuming constant temperature and pressure conditions. It implies that:

• Adsorption sites no longer change in occupancy after equilibrium is reached.
• The rate of adsorption equals the rate of desorption, leading to a stable state.

Understanding chemical equilibrium in adsorption helps predict how efficiently a material can retain adsorbed molecules, which is vital for optimizing industrial processes such as gas storage, petrochemical transformations, and environmental decontamination. Mastery of these concepts allows for improved designs and processes involving adsorption equilibria.