Problem 178 - A Complete Resource Book in Chemistry for JEE Main

Question

Conductivity (Seimens S) is directly proportional to area of the vessel and the concentration of the solution in it and is inversely proportional to the length of the vessel, then constant of proportionality is expressed in [2002] (a) \(\mathrm{S} \mathrm{m} \mathrm{mol}^{-1}\) (b) \(\mathrm{S}^{2} \mathrm{~m}^{2} \mathrm{~mol}^{-2}\) (c) \(\mathrm{S} \mathrm{m}^{2} \mathrm{~mol}^{-1}\) (d) \(\mathrm{S}^{2} \mathrm{~m}^{2} \mathrm{~mol}\)

Step-by-Step Solution

Answer

The correct answer is (a) \( \mathrm{S} \mathrm{m} \mathrm{mol}^{-1} \).

Step 1: Understand the Relationship

Given, conductivity \( C \) is directly proportional to the area \( A \) of the vessel and the concentration \( c \) of the solution, and inversely proportional to length \( L \) of the vessel. This implies \( C \propto \frac{A \times c}{L} \).

Step 2: Write the Proportionality Equation

To convert the proportionality into an equation, introduce a constant of proportionality \( k \). Thus, \( C = k \frac{A \times c}{L} \).

Step 3: Analyze the Units of Conductivity

Conductivity \( C \) is measured in Siemens \( \mathrm{S} \). The area \( A \) has units \( \mathrm{m}^2 \), concentration \( c \) has units \( \mathrm{mol} \), and length \( L \) has units \( \mathrm{m} \). Substitute these into the equation to find the units of \( k \).

Step 4: Dimension Analysis for Constant \( k \)

Rearranging \( k \) gives \( k = C \frac{L}{A \times c} \). Therefore, the units of \( k \) are \( \mathrm{S} \cdot \frac{\mathrm{m}}{\mathrm{m}^2 \cdot \mathrm{mol}} = \mathrm{S} \cdot \mathrm{m}^{-1} \cdot \mathrm{mol}^{-1} \times \mathrm{m}^2 = \mathrm{S} \cdot \mathrm{m} \cdot \mathrm{mol}^{-1} \).

Step 5: Match with Given Options

Compare the calculated unit \( \mathrm{S} \cdot \mathrm{m} \cdot \mathrm{mol}^{-1} \) with the given options. It matches with option (a) \( \mathrm{S} \mathrm{m} \mathrm{mol}^{-1} \).

Key Concepts

Physical Chemistry

Physical chemistry is an important branch of chemistry that bridges the gap between physics and chemistry. It focuses on understanding the physical properties of molecules, the forces that act upon them, and the interactions that govern chemical reactions. In physical chemistry, concepts like energy, velocity, entropy, and molecular structure are of significant interest.

One key aspect of physical chemistry is its use of mathematical equations and models to describe chemical phenomena. This includes the study of thermodynamics, kinetics, quantum mechanics, and statistical mechanics. By understanding these concepts, scientists can predict how chemicals will behave in various environments and under different conditions.

In the context of conductivity, physical chemistry helps us understand how ions and molecules move within a solution to conduct electricity. This movement is influenced by the concentration of ions, the size of the vessel, and other physical properties of the solution. Analyzing such factors can provide insight into the efficiency of materials in conducting electricity and their potential applications in various fields.

Dimensional Analysis

Dimensional analysis is a powerful tool used in physical chemistry to simplify complex equations and verify the correctness of mathematical expressions. It involves examining the dimensions of physical quantities involved in a problem to ensure consistent and accurate calculations.

In the given exercise, dimensional analysis is used to determine the units of the constant of proportionality, denoted as \( k \). By analyzing the dimensions, we can see how quantities such as conductivity, area, concentration, and length interact with each other. This ensures that the resultant unit of \( k \) matches the expected outcome, which in this case is expressed as \( \text{S}\cdot \text{m}\cdot \text{mol}^{-1} \).

• Conductivity \( C \) is measured in Siemens \( \text{S} \).
• The area \( A \) is measured in squared meters \( \text{m}^2 \).
• Concentration \( c \) is measured in moles \( \text{mol} \).
• Length \( L \) is measured in meters \( \text{m} \).

To find the dimensional formula of \( k \), you rearrange the equation and substitute the dimensions, thereby checking that all terms are dimensionally consistent.

Proportionality Constants

Proportionality constants play a critical role in mathematical equations in physical chemistry. They relate different physical quantities and facilitate the conversion of proportional relationships into equations. These constants hold specific values that depend on the conditions of the system being studied.

In the exercise, the constant \( k \) connects the conductivity of a solution to its physical properties. The relationship is represented by \( C = k \frac{A \times c}{L} \), where \( C \) is the conductivity, \( A \) is the area, \( c \) is the concentration, and \( L \) is the length of the vessel. By determining the proportionality constant, you can predict how a change in one of these variables affects the conductivity.

Understanding the role of proportionality constants helps students grasp how equations like Ohm's law or Van der Waals equation function. These equations predict measurable quantities based on initial conditions, helping scientists and engineers design experiments and develop new technologies effectively.