Question

A plot of \(\log \left(\frac{x}{M}\right)\) against \(\log P\) for the adsorption of a gas on a solid gives a straight line with slope equal to: (a) \(\frac{1}{n}\) (b) \(n\) (c) \(\log \mathrm{K}\) (d) \(\mathrm{K}\)

Short answer

The slope of the straight line plot is equal to (a) \( \frac{1}{n} \) as it corresponds to the exponent in the Freundlich adsorption isotherm.

Step-by-step solution

Identify the Adsorption Isotherm

The relationship described resembles the Freundlich adsorption isotherm, which has the form \( P = K \left(\frac{x}{M}\right)^{1/n} \) where \( P \) is the pressure, \( x/M \) is the amount of gas adsorbed per unit mass of the adsorbent, \( n \) is the heterogeneity factor, and \( K \) is the Freundlich constant.

Apply Logarithms to the Isotherm

By taking the logarithm of both sides of the Freundlich isotherm, we get: \( \log P = \log K + \frac{1}{n}\log \left(\frac{x}{M}\right) \).

Compare with the Given Linear Form

In the given linear plot of \( \log \left(\frac{x}{M}\right) \) against \( \log P \),it is implied that \( \log P \) is the y-axis and \( \log \left(\frac{x}{M}\right) \) is the x-axis. The slope of this line is the coefficient of the x-term, which in our logarithmic form of the isotherm is \( \frac{1}{n} \) following standard \( y = mx + c \) form where \( m \) is the slope.

Key concepts

Adsorption of Gases

Adsorption is a surface phenomenon where molecules of a gas or liquid accumulate on the surface of a solid or a liquid. When it comes to gases, this process is fundamental in various industrial applications—ranging from gas masks that filter harmful substances from air to the storage and separation of gases.

Several factors affect the adsorption of gases, such as the nature of the gas, the surface area of the adsorbent, and the temperature. Typically, increasing the surface area of the adsorbent and reducing the temperature enhances the extent of gas adsorption, as more molecules will find a spot to adhere to on the adsorbent's surface.

In physical chemistry, especially for Joint Entrance Examination (JEE) aspirants, understanding adsorption is vital. It is studied using various adsorption isotherms, one of which is the Freundlich adsorption isotherm. This model was derived empirically and helps explain how the pressure of a gas relates to the amount adsorbed by non-ideal heterogeneous surfaces.

Chemical Kinetics

Chemical kinetics is the study of the speed or rate at which chemical reactions occur and the factors that affect these rates. While adsorption itself is not a chemical reaction, it still represents an interaction at the molecular level that is influenced by kinetic principles. The rate at which a gas is adsorbed onto a surface can be indicative of the efficiency of the adsorbent material.

Within the context of physical chemistry and JEE preparation, understanding the kinetics behind adsorption is crucial. Students should be aware that temperature, as well as the properties of the reactive species, can drastically impact the rate of adsorption. Furthermore, the application of concepts from chemical kinetics can be extended to analyze the Freundlich adsorption isotherm, providing insights into the dynamic aspects of adsorption processes.

Physical Chemistry for JEE

For students preparing for competitive exams like the JEE, physical chemistry is a key component. The discipline merges the principles of physics and chemistry to explain how chemical systems behave. Topics like thermodynamics, electrochemistry, and chemical kinetics are pillars of physical chemistry and are rigorously tested in the JEE.

An understanding of adsorption isotherms, particularly the Freundlich isotherm, fits into this broader context. The JEE aspirants are expected to know not just the mathematical representation of the isotherm, but also its physical significance—how it describes the variation in adsorption with pressure. Students often encounter problems, like the textbook example provided, which demand both a conceptual understanding as well as an ability to navigate through the mathematical relationships to interpret graphical data and solve problems.