Problem 8 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

For a linear plot of \(\log (x / m)\) versus \(\log p\) in a Freundlich adsorption isotherm, which of the following statements is correct? ( \(k\) and \(n\) are constants).(a) Only \(1 / n\) appears as the slope. (b) \(\log (1 / n)\) appears as the intercept. (c) Both \(k\) and \(1 / n\) appear in the slope term. (d) \(1 / n\) appears as the intercept.

Step-by-Step Solution

Answer

The correct statement is (a) Only \( 1/n \) appears as the slope.

Step 1: Understand the Freundlich Equation

The Freundlich adsorption isotherm is represented by the equation: \[ x/m = k \cdot p^{1/n} \] where \( x/m \) is the amount of adsorbate per unit mass of adsorbent, \( p \) is the pressure, and \( k \) and \( n \) are constants.

Step 2: Convert to Logarithmic Form

To fit the Freundlich isotherm to a linear plot, take the logarithm of both sides of the equation. The equation becomes: \[ \log (x/m) = \log (k \cdot p^{1/n}) \] Applying the properties of logarithms, this can be expanded to: \[ \log (x/m) = \log k + \frac{1}{n} \log p \]

Step 3: Identify the Linear Plot Components

For the linear plot of \( \log (x/m) \) versus \( \log p \), the equation \( \log (x/m) = \log k + \frac{1}{n} \log p \) takes the form of \( y = mx + c \), where \( y = \log (x/m) \), \( x = \log p \), the slope \( m = \frac{1}{n} \), and the intercept \( c = \log k \).

Step 4: Determine the Correct Statement

From the linear equation form, the slope is \( \frac{1}{n} \) and the intercept is \( \log k \). Therefore, the statement '(a) Only \( 1/n \) appears as the slope' is correct.

Key Concepts

Logarithmic Transformation

The Freundlich adsorption isotherm equation \[ x/m = k \cdot p^{1/n} \]serves as a model to quantify how gases and solutes adsorb onto surfaces at varied pressures. This relationship becomes more manageable by using a **Logarithmic Transformation**. By taking the logarithm on both sides of the equation, we transform it into a linear form:\[ \log(x/m) = \log k + \frac{1}{n} \log p \]This transformation is beneficial because logarithms simplify the multiplication in the isotherm into addition, revealing linear relationships. Such linearity makes it possible to identify key parameters like slope and intercept in a plot, thus aiding in the characterization of adsorption processes. By recognizing the critical constants, namely \( k \) and \( n \), through logarithmic transformation, students can better grasp complex adsorption concepts effectively.

Linear Plot

Once transformed to the logarithmic form, the Freundlich isotherm takes the linear expression \( y = mx + c \), perfectly fitting into our mathematical understanding of linear graphs.- **Linear Equation Representation**: - Here, \( y \) corresponds to \( \log(x/m) \) - \( x \) stands for \( \log p \). - The slope \( m \) equates to \( \frac{1}{n} \) - The intercept \( c \) is represented by \( \log k \).This linear relationship allows researchers to create a plot of \( \log(x/m) \) versus \( \log p \), showcasing a straight line. The slope of this line, \( \frac{1}{n} \), provides direct insights into adsorption intensity, while the intercept \( \log k \) indicates adsorption capacity. Altogether, this transformation and plotting offer an empirical mechanism to analyze adsorption metrics.Understanding the components of the plot aids in correctly identifying scientific data from graphical metrics, making mastery of these methods essential for exploring adsorption phenomena.

Adsorbate and Adsorbent

In the realm of adsorption isotherms, two key players are always present: the **Adsorbate** and the **Adsorbent**. Understanding these terms is crucial for applying the Freundlich adsorption isotherm effectively.- **Adsorbate**: - The adsorbate refers to the substance being accumulated on a surface. In many scenarios, this could be a gas or solution molecule. - The interaction between the adsorbate and the adsorbent determines the success of the adsorption process.- **Adsorbent**: - The adsorbent is the material that captures and holds onto the adsorbate. - This entity is typically a solid with a large surface area.The specific relationship between the adsorbate and adsorbent is described by \( x/m \) in the Freundlich isotherm, where \( x \) is the mass of adsorbate and \( m \) is the mass of adsorbent. Recognizing their respective roles helps one understand how adsorbents can be optimized for selective uptake of adsorbates under varying pressures and conditions.