Problem 101 - A Complete Resource Book in Chemistry for JEE Main

Question

A plot of \(\log \mathrm{x} / \mathrm{m}\) versus log \(\mathrm{P}\) for the adsorption of a gas on a solid gives a straight line with slope equal to (a) \(1 / \mathrm{n}\) (b) \(\log \mathrm{K}\) (c) \(-\log \mathrm{K}\) (d) \(\mathrm{n}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the context

This problem refers to the Freundlich adsorption isotherm, an empirical relationship between the concentration of a gas and its adsorption on a solid surface. The representation involves plotting the logarithm of the adsorption against the logarithm of the pressure.

Step 2: Recall the Freundlich Isotherm Equation

The Freundlich isotherm is described by the equation: \(\frac{x}{m} = K P^{1/n}\), where \(x\) is the mass of the gas adsorbed, \(m\) is the mass of the adsorbent, \(K\) is a constant, \(P\) is the pressure, and \(n\) is a constant related to the adsorption intensity.

Step 3: Convert to Logarithmic Form

Taking the logarithm of both sides of the Freundlich equation, we get: \(\log \left(\frac{x}{m}\right) = \log K + \frac{1}{n} \log P\). This equation demonstrates that a plot of \(\log \left(\frac{x}{m}\right)\) versus \(\log P\) will yield a straight line.

Step 4: Identify the Slope

From the logarithmic form of the equation \(\log \left(\frac{x}{m}\right) = \log K + \frac{1}{n} \log P\), we can see that it is in the form of \(y = mx + c\), where \(y = \log \left(\frac{x}{m}\right)\), \(x = \log P\), \(m = \frac{1}{n}\) (the slope), and \(c = \log K\). Thus, the slope of the line is \(\frac{1}{n}\).

Key Concepts

Adsorption of Gases

Adsorption refers to the process by which gas molecules are attracted and held onto the surface of a solid. This is different from absorption, where the substance permeates into the material. When discussing the adsorption of gases, we are considering how these gas molecules settle onto the surface, often due to various forces such as van der Waals.
The efficiency of adsorption depends on several factors:

• The nature of the gas: Polar gases generally adsorb more strongly than nonpolar gases.
• The nature of the adsorbent: Materials with larger surface areas tend to adsorb more gas.
• Pressure and Temperature: Typically, adsorption increases with pressure and decreases with temperature.

Understanding adsorption at a fundamental level helps in analyzing how substances interact in various fields, such as catalysis and sensor design.

Logarithmic Transformations

Logarithmic transformations are mathematical tools that help simplify complex relationships. In the context of gas adsorption, we start with the Freundlich equation: \[ \frac{x}{m} = K P^{1/n} \]Converting this equation to a logarithmic form changes the multiplication into addition, making it easier to solve and interpret:\[ \log \left( \frac{x}{m} \right) = \log K + \frac{1}{n} \log P \]
This transformation is useful because it turns a power function into a linear equation:

• The relationship is more straightforward to analyze graphically.
• It simplifies the identification of important parameters like the constant \(K\) and the slope \(\frac{1}{n}\).

Overall, logarithmic transformations are essential for distilling complex data into more digestible visual formats.

Chemical Equilibrium

Chemical equilibrium is the state in which the rate of the forward reaction equals the rate of the backward reaction. For adsorption processes, this concept involves the equilibrium between the adsorbed molecules on a surface and those in the surrounding area.
Although the specifics of chemical equilibrium differ when comparing gases adsorbed on solids to those in solutions, the underlying principle remains. The system reaches a balance, and the concentration of adsorbed molecules no longer changes over time. This stage can be described by models like the Freundlich isotherm.
Maintaining equilibrium depends on various factors:

• External pressure and temperature.
• Surface properties of the adsorbent.
• The nature of the interacting substances.

Understanding equilibrium is crucial for optimizing industrial processes involving adsorption.

Adsorption Intensity

Adsorption intensity is a measure of the strength with which an adsorbate (e.g., a gas) is held onto an adsorbent surface. In the Freundlich adsorption isotherm, this concept is represented by the constant \(n\). The value of \(n\) provides insights into the adsorption characteristics:

• If \(n = 1\), adsorption is directly proportional to pressure.
• Values of \(n > 1\) indicate favorable adsorption, where it becomes easier to adsorb the gas as pressure increases.
• Values of \(n < 1\) suggest weaker adsorption intensity.

This parameter is essential for understanding how different substances interact with surfaces and aids in the design of efficient adsorption systems, such as those used in gas purification and environmental remediation.