Problem 4 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Adsorption of a gas follows Freundlich adsorption isotherm. \(x\) is the mass of the gas adsorbed on mass \(\mathrm{m}\) of the adsorbent. The plot of \(\frac{x}{m}\) versus \(\log p\) is shown in the given graph. \(\frac{x}{m}\) is proportional to :(a) \(p^{2 / 3}\) (b) \(p^{3 / 2}\) (c) \(p^{3}\) (d) \(p^{2}\)

Step-by-Step Solution

Answer

Option (b): \( p^{3/2} \).

Step 1: Understanding Freundlich Adsorption Isotherm

The Freundlich adsorption isotherm is expressed by the equation \( \frac{x}{m} = Kp^{1/n} \), where \( \frac{x}{m} \) is the amount of gas adsorbed per unit mass of adsorbent, \( K \) is a constant, and \( n \) is a measure of adsorption intensity or capacity.

Step 2: Taking Logarithm of Both Sides

To linearize the Freundlich equation for analysis, take the logarithm of both sides. This gives the equation \( \log \left( \frac{x}{m} \right) = \log K + \frac{1}{n} \log p \). This equation is of the form \( y = mx + c \), where \( m = \frac{1}{n} \).

Step 3: Relating Graph to Equation

The graph of \( \frac{x}{m} \) versus \( \log p \) is a straight line, suggesting a relationship consistent with the log-transformed Freundlich equation. The slope of this line is equal to \( \frac{1}{n} \).

Step 4: Analyzing the Slope

Given the slope of the line on the graph (interpreted from options), match it to the exponent required for each option by inverting \( \frac{1}{n} \) to find \( n \). Each option suggests a specific value of \( n \): - For option (a): \( \frac{1}{n} = \frac{2}{3} \Rightarrow n=\frac{3}{2} \)- For option (b): \( \frac{1}{n} = \frac{3}{2} \Rightarrow n=\frac{2}{3} \)- For option (c): \( \frac{1}{n} = 3 \Rightarrow n=\frac{1}{3} \)- For option (d): \( \frac{1}{n} = 2 \Rightarrow n=\frac{1}{2} \)

Step 5: Selecting the Correct Option

From the given options, \( \frac{1}{n} = \frac{3}{2} \) corresponds to the slope observed. This matches with option (b). Therefore, \( \frac{x}{m}\) is proportional to \( p^{3/2} \).

Key Concepts

Adsorption

Adsorption is a fascinating process that involves molecules sticking to a surface. Imagine a sponge soaking up water; that's similar to how adsorption works on a molecular level. Molecules from a gas or liquid phase cling to the surface of a solid material. This surface-based process is different from absorption, where a substance permeates the whole volume. Adsorption is influenced by several factors, such as:

• Surface area of the adsorbent
• Pressure of the gas or concentration of the solute
• Temperature

Various types of adsorption exist, including physical adsorption (physisorption) and chemical adsorption (chemisorption). Physisorption is where weak Van der Waals forces hold the molecules, while chemisorption involves strong chemical bonds.

Isotherms

Isotherms are essential concepts in understanding adsorption phenomena. They describe how a gas or liquid interacts with a solid surface at a constant temperature. Basically, isotherms provide a relationship between the pressure (or concentration) of the adsorbate and the amount adsorbed on the surface. Popular adsorption isotherm models include:

• Langmuir Isotherm: This model assumes a monolayer adsorption on a surface with finite sites.
• Freundlich Isotherm: This model suggests a multi-layer adsorption and is suitable for heterogeneous surfaces.

Freundlich isotherms, in particular, describe how an increase in gas pressure results in a greater amount of gas adsorbing. The Freundlich equation mathematically expresses this relationship, making it easier to translate experimental data into a comprehensible form.

Gas Adsorption

When we talk about gas adsorption, we're looking at how gases adhere to solid surfaces. This is significant in numerous applications, including air purification and catalysis. During gas adsorption, gas molecules come into contact with a solid surface and stick to it due to various forces. For effective gas adsorption:

• Higher surface area of the adsorbent is desirable.
• Lower temperatures typically enhance adsorption capacity.
• Pressure often plays a critical role; higher pressure can increase the rate of adsorption.

Gas adsorption can be utilized in industries to remove impurities or for storing gases under specific conditions. It’s a vital process that aids in both environmental and industrial applications.

Adsorbent

An adsorbent is a material that facilitates the adsorption process by providing a surface for molecules to attach. Common adsorbents include activated carbon, silica gel, and zeolites. These materials are chosen for their high surface area and specific physical and chemical properties. Key properties of effective adsorbents include:

• High porosity, allowing for more sites to trap molecules.
• Tunable surface properties to target specific molecules.
• Chemical stability, so the adsorbent doesn't degrade easily.

Adsorbents have diverse uses, from purifying water to capturing pollutants from the air. By designing adsorbents with specific characteristics, scientists can target nearly any gas or liquid molecules for capture, making adsorption a versatile solution for handling various chemistry problems.

Chemistry Problems

Solving chemistry problems involving adsorption can enhance understanding of how substances interact at surfaces. It requires knowledge of the principles that govern surface interactions and the mathematical models that describe these processes. When addressing such problems:

• Identify the type of isotherm that best fits the data.
• Understand and apply the relevant equations, such as those of Freundlich or Langmuir isotherms.
• Analyze graphical data to determine relationships, such as slopes or intercepts, which inform about the adsorption process.

Understanding the chemistry behind adsorption helps in designing better materials and processes across various scientific fields. Whether in catalysis, material science, or environmental science, mastering these concepts can tackle real-world issues effectively.