Problem 6 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

If \(x\) gram of gas is adsorbed by \(m\) gram of adsorbent at pressure \(P\), the plot of \(\log \frac{x}{m}\) versus \(\log P\) is linear. The slope of the plot is: (n and \(k\) are constants and \(\mathrm{n}>1\) )(a) \(\log \mathrm{k}\) (b) \(\frac{1}{n}\) (c) \(2 \mathrm{k}\) (d) \(\mathrm{n}\)

Step-by-Step Solution

Answer

The slope is \(\frac{1}{n}\).

Step 1: Understanding the problem

We are given that the plot of \( \log \frac{x}{m} \) versus \( \log P \) is linear. This implies a relationship between \( \frac{x}{m} \) and \( P \) that follows a power law. To find the slope, we'll first determine the equation representing the line.

Step 2: Applying the Freundlich Isotherm

The problem describes a process known as adsorption, often modeled by the Freundlich isotherm: \( \frac{x}{m} = kP^{1/n} \). Here, \( k \) and \( n \) are constants, and \( n > 1 \) is given.

Step 3: Taking the logarithm of the Freundlich equation

We take the logarithm of both sides of the Freundlich equation to transform it into a linear form: \[ \log \frac{x}{m} = \log (kP^{1/n}) \] This can be expanded to: \[ \log \frac{x}{m} = \log k + \frac{1}{n} \log P \]

Step 4: Identifying the slope in the equation

The equation \( \log \frac{x}{m} = \log k + \frac{1}{n} \log P \) is in the form of \( y = c + mx \), which is a linear equation where \( m \) is the slope. By comparing, we identify that \( m = \frac{1}{n} \).

Key Concepts

Adsorption

Adsorption is a fascinating natural process where a solid (often called the adsorbent) captures molecules of a gas or a liquid on its surface. Imagine tiny particles sticking to a surface, held there by attractive force. This is different from absorption, where substances are taken up into the interior of a material. In the world of chemistry and physics, adsorption is crucial because it's essentially how substances cling onto surfaces, which can be helpful in various applications, like filtering pollutants. There are different types of adsorption, including physical and chemical. In this context, the focus is mostly on the equilibrium state, where the rate of adsorption equals the rate of desorption, providing a stable system for analysis. The amount of gas adsorbed depends on several factors:

• The nature and surface properties of the adsorbent.
• The pressure of the gas above the surface.
• The temperature at which the process occurs.

These factors are pivotal in describing the adsorption isotherms, which quantitatively describe how molecules adsorb onto surfaces at constant temperature.

Linear Plot

A linear plot is a graphical representation where you get a straight line, illustrating a direct proportional relationship between two variables. In mathematics, this means that as one variable increases, the other does, at a steady rate, so you can trace a straight path between them. In the context of adsorption, we use a linear plot to interpret the Freundlich Isotherm. By plotting \( \log \frac{x}{m} \) against \( \log P \), researchers can determine critical parameters about how adsorption behaves under different pressures. This method offers several benefits:

• It efficiently displays the relationship between variables with clarity.
• Identifies the behavior of adsorption as pressure changes.
• Helps in determining constants that describe the adsorption characteristics.

Using logs transforms a power law (curved line) into a linear graph, which is simpler to interpret and helps in understanding underlying relationships.

Power Law Relationship

The power law relationship is an important concept in understanding how various quantities relate to each other in a real-world context. It's a mathematical relationship where one quantity varies as a power of another. In its general form, it can be expressed as: \[ y = ax^b \]Where:

• \( y \) and \( x \) are the quantities of interest.
• \( a \) is a constant that scales the equation.
• \( b \) is the exponent that shows how one variable reacts to changes in another.

In adsorption, the Freundlich Isotherm shows a power law relationship between the amount of gas adsorbed (\( \frac{x}{m} \)) and pressure (\( P \)). This is given by the equation:\[ \frac{x}{m} = kP^{1/n} \]Taking logs converts this to a linear relationship, so we can easily determine the slope (\( \frac{1}{n} \)) and intercept (\( \log k \)) from the graph. The power law highlights how sensitive \( \frac{x}{m} \) is to changes in \( P \), with \( n \) coming into play as it modulates the influence of pressure on the adsorption process. This understanding is vital, particularly in optimizing processes where adsorption is key, like in gas purification.