Problem 7 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Adsorption of a gas on a surface follows Freundlich adsorption isotherm. Plot of \(\log \frac{x}{m}\) versus \(\log p\) gives a straight line with slope equal to \(0.5\), then :( \(\frac{x}{m}\) is the mass of the gas adsorbed per gram of adsorbent)(a) Adsorption is independent of pressure. (b) Adsorption is proportional to the pressure. (c) Adsorption is proportional to the square root of pressure. (d) Adsorption is proportional to the square of pressure.

Step-by-Step Solution

Answer

Adsorption is proportional to the square root of pressure (option c).

Step 1: Understand the Problem

Understand that the problem is about the Freundlich adsorption isotherm, which indicates a logarithmic relationship between the amount of gas adsorbed on a surface and the pressure.

Step 2: Freundlich Isotherm Formula

Recognize the Freundlich adsorption isotherm is expressed as: \( \log \frac{x}{m} = \log K + \frac{1}{n} \log p \). Here, \( K \) is the adsorption capacity, \( \frac{1}{n} \) is the slope, \( \frac{x}{m} \) is the mass of gas adsorbed per unit mass, and \( p \) is the pressure.

Step 3: Analyzing the Given Data

Notice that you are told the slope of the \( \log \frac{x}{m} \) versus \( \log p \) graph is \( 0.5 \), which means \( \frac{1}{n} = 0.5 \). Thus, \( n = 2 \).

Step 4: Determine the Relationship

Since \( n = 2 \), from the formula \( \frac{1}{n} \log p \), it follows that \( \log \frac{x}{m} = \log K + 0.5 \log p \), indicating \( \frac{x}{m} \) is proportional to \( p^{0.5} \) or \( \sqrt{p} \).

Step 5: Select the Correct Option

From the options given, option (c) correctly represents the relationship: Adsorption is proportional to the square root of pressure.

Key Concepts

Adsorption Isotherm

Adsorption isotherms are fundamental in understanding how gases interact with solid surfaces. When a gas comes into contact with a solid, molecules from the gas phase can "stick" or adsorb onto the solid's surface. This process is not random but follows specific patterns or isotherms that describe how the amount of adsorbed gas relates to parameters like pressure at a constant temperature. The Freundlich adsorption isotherm, in particular, is used to describe systems where the adsorption sites are not uniform in terms of energy. It provides a logarithmic relationship between the gas adsorbed and pressure, making it versatile for different surfaces and gases.

Gas Adsorption

Gas adsorption occurs when gas molecules adhere to a solid surface. It is a surface phenomenon, distinct from absorption, which involves the entire volume of the solid. This process depends heavily on the physical and chemical properties of both the gas and the surface. Several factors can influence gas adsorption:

• The nature of the adsorbent, such as its surface area and pore structure.
• The properties of the gas, including molecular size and polarity.
• Environmental factors like temperature and pressure.

These interactions are crucial for various applications, including catalysis, pollution control, and gas storage.

Adsorption Proportionality

Adsorption proportionality refers to the mathematical relationship between the amount of gas adsorbed (\( rac{x}{m} \)) and the influencing factors like pressure. In the context of the Freundlich isotherm, the amount adsorbed is related to the pressure raised to a power, as represented by \( rac{1}{n} \). For instance, when \( rac{1}{n} = 0.5 \), the relationship becomes \( rac{x}{m} = K \, p^{0.5} \). This indicates that adsorption is proportional to the square root of pressure. Understanding this proportionality helps in predicting how changes in pressure will affect gas adsorption.

Pressure and Adsorption Relationship

The relationship between pressure and adsorption is critical in determining the effectiveness of a surface in adsorbing gas molecules. As pressure increases, more gas molecules are available to potentially interact with the surface, generally leading to an increase in adsorption until a saturation point is reached. In the Freundlich adsorption isotherm, this relationship is described by a power law rather than being linear. The specific form of the relationship \( rac{x}{m} = K \, p^{1/n} \) explains that the amount of gas adsorbed is proportional to the pressure raised to the power \( 1/n \). In scenarios where \( n = 2 \), this results in the adsorption being proportional to the square root of the pressure. Such insights are essential for applications that require precise control of adsorption processes.