Problem 91 - A Complete Resource Book in Chemistry for JEE Main

Question

Freundlich adsorption isotherm gives a straight line on plotting (a) \(\mathrm{x} / \mathrm{m}\) vs \(\mathrm{P}\) (b) \(\log \mathrm{x} / \mathrm{m}\) vs \(\mathrm{P}\) (c) \(\log \mathrm{x} / \mathrm{m}\) vs \(\log \mathrm{P}\) (d) \(\mathrm{x} / \mathrm{m}\) vs \(1 / \mathrm{P}\)

Step-by-Step Solution

Answer

(c) \( \log \mathrm{x} / \mathrm{m} \) vs \( \log \mathrm{P} \).

Step 1: Understanding the Freundlich Adsorption Isotherm

The Freundlich adsorption isotherm is an empirical equation that describes the adsorption of substances on surfaces. It is given by the formula: \( \frac{x}{m} = K P^{\frac{1}{n}} \), where \( x \) is the amount of adsorbate, \( m \) is the weight of the adsorbent, \( P \) is the pressure, \( K \) and \( n \) are constants.

Step 2: Transforming the Equation to a Linear Form

To find the condition for a straight line, we take the logarithm of both sides of the Freundlich equation. This gives us: \( \log \left( \frac{x}{m} \right) = \log K + \frac{1}{n} \log P \). This equation is in the form of a straight line, \( y = mx + c \), where \( y = \log \left( \frac{x}{m} \right) \), \( m = \frac{1}{n} \), \( x = \log P \), and \( c = \log K \).

Step 3: Identifying the Correct Plot for a Straight Line

Since we transformed the equation to a linear form \( \log \left( \frac{x}{m} \right) = \log K + \frac{1}{n} \log P \), we need to plot \( \log \left( \frac{x}{m} \right) \) against \( \log P \) to obtain a straight line.

Key Concepts

Adsorption Concepts

Adsorption is the process by which atoms, ions, or molecules from a gas, liquid, or dissolved solid adhere to a surface. In the context of the Freundlich adsorption isotherm, it deals with the adsorption of gases onto solid surfaces. Unlike absorption, where a substance is fully taken into another body's volume, adsorption occurs at the surface.

The Freundlich adsorption isotherm is an empirical model that helps us understand how molecules interact with surfaces. It's expressed with the equation \( \frac{x}{m} = K P^{\frac{1}{n}} \), where \( x \) is the amount of adsorbate absorbed on the adsorbent of mass \( m \), \( P \) is the pressure of the gas above the surface, and \( K \) and \( n \) are constants specific to the system.

Key points to consider in adsorption include:

• The nature of the adsorbate (substance being adsorbed) and adsorbent (material on which adsorption takes place).
• Amount of surface area available for adsorption.
• Pressure and temperature, as these conditions affect the state and behavior of the adsorbate.
• The physical and chemical environments of the adsorbate and adsorbent.

Understanding these factors helps in designing and analyzing systems that rely on adsorption processes.

Linearization of Equations

In many scientific analyses, converting nonlinear equations into linear form simplifies data interpretation and analysis. This process is known as linearization.

For example, the Freundlich adsorption isotherm's nonlinear equation \( \frac{x}{m} = K P^{\frac{1}{n}} \) can be challenging to work with directly. But by taking the logarithm of both sides, the equation becomes linear: \( \log \left( \frac{x}{m} \right) = \log K + \frac{1}{n} \log P \). This transformation aligns with the linear equation format \( y = mx + c \).

This form, with \( y = \log (x/m) \), \( x = \log P \), \( m = \frac{1}{n} \) as the slope, and \( c = \log K \) as the intercept, allows chemists to use simple linear regression techniques to analyze the data. Linearization is a powerful tool as it:

• Helps identify relationships between variables using straightforward graphing techniques.
• Makes it easier to extract physical parameters from the slope and intercept.
• Simplifies the calculation of constants like \( K \) and \( n \) in adsorption studies.

Mastering linearization is crucial for students and researchers alike as it opens the door to more advanced data manipulation and understanding.

Graphical Representation in Chemistry

Graphical methods are crucial in chemistry for visualizing and interpreting data. They provide tangible insight into relationships between variables and facilitate clearer understanding.

In the case of the Freundlich adsorption isotherm, plotting \( \log \left( \frac{x}{m} \right) \) against \( \log P \) offers a straightforward method to visualize the adsorption process. This plot results in a straight line due to the equation \( \log \left( \frac{x}{m} \right) = \log K + \frac{1}{n} \log P \), where the slope \( \frac{1}{n} \) and the intercept \( \log K \) hold critical information about the adsorption behavior.

Besides simplifying equation representation, graphical methods help:

• Confirm theoretical models against experimental data.
• Determine parameters: extracting the slope and intercept can reveal constants \( K \) and \( n \).
• Detect anomalies or deviations from expected behavior.

Graphical representation is not limited to adsorption but is a prevalent technique across various branches of chemistry. Understanding how to effectively use and interpret graphs is a foundational skill for successful analysis and scientific communication.