Problem 198 - A Complete Resource Book in Chemistry for JEE Main

Question

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

Step-by-Step Solution

Answer

Option (b) is correct: Only \( 1/n \) appears as the slope.

Step 1: Understanding the Freundlich adsorption isotherm equation

The Freundlich adsorption isotherm is expressed as \[ \text{x/m} = k \cdot p^{1/n} \] where \( \text{x/m} \) is the amount of solute adsorbed per unit mass of adsorbent, \( k \) is a constant related to the adsorption capacity, \( p \) is the pressure, and \( n \) is a constant related to the intensity of adsorption. Our task is to rewrite it in a linear form for plotting against \( \log(p) \).

Step 2: Taking the logarithm of the equation

To linearize the equation, take the logarithm on both sides: \[ \log(\text{x/m}) = \log(k \cdot p^{1/n}) \] This simplifies to: \[ \log(\text{x/m}) = \log(k) + \frac{1}{n} \cdot \log(p) \] Here, \( \log(k) \) is the intercept and \( \frac{1}{n} \) is the slope.

Step 3: Analyzing the linear form equation

The equation \[ \log(\text{x/m}) = \log(k) + \frac{1}{n} \cdot \log(p) \] shows that when \( \log(\text{x/m}) \) is plotted against \( \log(p) \), the slope of the line is \( \frac{1}{n} \) and the intercept is \( \log(k) \). Thus, \( \frac{1}{n} \) appears as the slope.

Step 4: Verifying the options

Now, verify the given options:- (a) Incorrect, as \( 1/n \) appears as the slope, not the intercept.- (b) Correct, as only \( 1/n \) appears as the slope.- (c) Incorrect, as \( \log(1/n) \) does not appear in the equation.- (d) Incorrect, \( k \) appears in the intercept as \( \log(k) \), not in the slope.

Key Concepts

Logarithmic plot interpretation

Understanding how to interpret a logarithmic plot is essential for analyzing adsorption isotherms, like the Freundlich isotherm. In this context, the plot of \( \log(\text{x/m}) \) against \( \log(p) \) effectively linearizes the relationship between the amount of a substance adsorbed and the pressure. By transforming the non-linear relationship into a linear one, you can easily determine characteristic constants of adsorption systems, such as the adsorption capacity and intensity.

In any logarithmic plot:

• A straight line indicates a power-law relationship between the variables involved.
• The slope of the line provides information about the exponent in the power law.
• The intercept helps reveal scaling constants, which are usually related to baseline levels or capacities.

This interpretation is particularly useful for verifying the behavior of adsorption according to theories like the Freundlich isotherm, providing a clear visualization of how efficiently an adsorbent interacts with a solute under changing pressures.

Linear form equation

To better analyze the Freundlich isotherm, it is crucial to transform it into a linear form equation. The original equation is \( \text{x/m} = k \cdot p^{1/n} \), which models adsorption as a non-linear, exponential relationship between pressure and the quantity of material adsorbed per unit mass. By taking the logarithm of both sides, we convert this into a more manageable form for analysis: \[ \log(\text{x/m}) = \log(k) + \frac{1}{n} \cdot \log(p) \]

This transformation converts the original problem into one involving straight-line equations, which:

• Facilitates easy computation and estimation of parameters involved.
• Enables the derivation of constants by direct observational data.
• Simplifies the interpretation, as any plotted linear graphs can be analyzed using basic tools rather than complex mathematical functions.

Understanding the linear form equation allows better predictions and comparisons across different adsorption scenarios and enhances the clarity of complex thermo-physical data.

Slope and intercept in adsorption isotherms

The slope and intercept in adsorption isotherms provide significant insights into the system's adsorption behavior. In the example of the Freundlich adsorption isotherm's linear form: \[ \log(\text{x/m}) = \log(k) + \frac{1}{n} \cdot \log(p) \] Understanding the slope: - The slope is \( \frac{1}{n} \) and is directly related to the adsorption efficiency in response to changes in pressure.- A steeper slope suggests that pressure changes have a significant influence, whereas a gentle slope indicates a moderate influence.

Understanding the intercept: - The intercept is \( \log(k) \), representing the adsorption capacity, or how much the adsorbent can hold when pressure is not a limiting factor.- It provides the baseline adsorption capacity, revealing how much material can be adsorbed at unit pressure.
Accurately determining these parameters helps in designing adsorption systems for various industrial applications or environmental management, promoting more tailored and efficient solutions.