Problem 204 - A Complete Resource Book in Chemistry for JEE Main

Question

Resistance of \(0.2 \mathrm{M}\) solution of an electrolyte is \(50 \Omega\) The specific conductance of the solution is \(1.4 \mathrm{~S} \mathrm{~m}^{-1}\). The resistance of \(0.5 \mathrm{M}\) solution of the same electrolyte is \(280 \Omega\) The molar conductivity of \(0.5 \mathrm{M}\) solution of the electrolyte in \(\mathrm{S} \mathrm{mt}^{2} \mathrm{~mol}^{-1}\) is: (a) \(5 \times 10^{3}\) (b) \(5 \times 10^{2}\) (c) \(5 \times 10^{-4}\) (d) \(5 \times 10^{-3}\)

Step-by-Step Solution

Answer

The molar conductivity is \(5 \times 10^3 \, \mathrm{S} \, \mathrm{cm}^2 \, \mathrm{mol}^{-1}\), so option (a) is correct.

Step 1: Understanding Given Information

We are given the resistance of a \(0.2 \mathrm{M}\) solution of an electrolyte as \(50 \, \Omega\) and the specific conductance (conductivity) of this solution as \(1.4 \, \mathrm{S} \, \mathrm{m}^{-1}\). In addition, we have the resistance of a \(0.5 \mathrm{M}\) solution of the same electrolyte as \(280 \, \Omega\).

Step 2: Find Cell Constant

The cell constant \(G^*\) can be calculated using the relationship between resistance \(R\), specific conductance \(K\), and the cell constant: \[G^* = K \times R\] For the \(0.2 \mathrm{M}\) solution:\[G^* = 1.4 \, \mathrm{S} \, \mathrm{m}^{-1} \times 50 \, \Omega = 70 \, \mathrm{m}^{-1}\]

Step 3: Calculate Conductivity of 0.5 M Solution

Using the cell constant, we can find the specific conductance \(K\) of the \(0.5 \mathrm{M}\) solution: \[K = \frac{G^*}{R} = \frac{70 \, \mathrm{m}^{-1}}{280 \, \Omega} = 0.25 \, \mathrm{S} \, \mathrm{m}^{-1}\]

Step 4: Calculate Molar Conductivity

Molar conductivity \(\Lambda_m\) can be calculated using the formula: \[\Lambda_m = \frac{K}{c}\] where \(c\) is the concentration in \(\mathrm{mol} \, \mathrm{m}^{-3}\). For the \(0.5 \mathrm{M}\) solution:\[\Lambda_m = \frac{0.25 \, \mathrm{S} \, \mathrm{m}^{-1}}{0.5 \, \mathrm{mol} \, \mathrm{m}^{-3}} = 0.5 \, \mathrm{S} \, \mathrm{m}^2 \, \mathrm{mol}^{-1}\]

Step 5: Convert Units for Final Answer

Convert the molar conductivity from \(\mathrm{S} \, \mathrm{m}^2 \, \mathrm{mol}^{-1}\) to \(\mathrm{S} \, \mathrm{cm}^2 \, \mathrm{mol}^{-1}\): Just multiply by \(10^4\) to convert from \(\mathrm{m}^2\) to \(\mathrm{cm}^2\):\[\Lambda_m = 0.5 \, \mathrm{S} \, \mathrm{m}^2 \, \mathrm{mol}^{-1} \times 10^4 = 5 \times 10^3 \, \mathrm{S} \, \mathrm{cm}^2 \, \mathrm{mol}^{-1}\]

Step 6: Conclusion: Select the Correct Option

The molar conductivity of the \(0.5 \mathrm{M}\) solution of the electrolyte is \(5 \times 10^3 \, \mathrm{S} \, \mathrm{cm}^2 \, \mathrm{mol}^{-1}\). Therefore, the correct answer is option (a) \(5 \times 10^3\).

Key Concepts

Specific Conductance

Specific conductance, often referred to as conductivity (denoted by \( K \)), is a measure of a material's ability to conduct an electric current. It's a crucial property in the field of chemistry, especially when dealing with electrolyte solutions.
For electrolyte solutions, specific conductance is impacted by both the nature of the ions present and their concentration. Key aspects to remember about specific conductance include:

• Unit of Measurement: The standard unit for specific conductance is Siemens per meter (\( \mathrm{S} \mathrm{m}^{-1} \)).
• Relationship with Resistance: The specific conductance of a solution is inversely related to its resistance. If the resistance is high, the specific conductance will be low, and vice versa.

Specific conductance can tell us a lot about how well a liquid can carry an electric charge, which in turn is important for reactions that rely on ion movement. Knowing this helps in understanding how changes in the solution, like concentration variations, affect its conductive abilities.

Cell Constant

The cell constant (denoted as \( G^* \)) is a proportionality factor that relates the measured conductance of a solution to its specific conductance. It is specific to the particular measuring setup and can vary between different setups.Understanding the cell constant involves several points:

• Calculation: It is calculated using the formula \( G^* = K \times R \), where \( R \) is the resistance and \( K \) is the specific conductance of the solution.
• Dimensions: The units of the cell constant are typically per meter (\( \mathrm{m}^{-1} \)).

The cell constant is vital for determining accurate conductance measurements of a solution by compensating for the geometry of the electrochemical cell in use. By applying the cell constant, researchers can relate conductance readings back to the intrinsic properties of the solution itself, paving the way for meaningful analytical studies.

Electrolyte Solution Resistance

Resistance in electrolyte solutions is a measure of how much the path of an electric current is hindered. While metals have relatively low resistance, electrolyte solutions can have a wide range depending on various factors like concentration and temperature. Routine points about resistance:

• Resistance Unit: The resistance is measured in ohms (\( \Omega \)).
• Inverse Relation: As resistance increases, specific conductance decreases. This is because fewer ions are capable of carrying electric current.

When studying resistance in electrolyte solutions, it is important to consider both the solution's physical properties and other influencing factors. This helps in understanding how easily current flows through a solution, which is critical in electrochemical cell applications.

Concentration in Chemistry

Concentration in chemistry refers to the amount of solute present in a given quantity of solvent or solution. It is a fundamental concept, especially in analytical chemistry and reactions involving electrolytes.Key elements of concentration:

• Measurements: Concentration is often measured in molarity, which is the number of moles of solute per liter of solution (\( \mathrm{mol} \, \mathrm{L}^{-1} \)).
• Concentration's Impact: The concentration of ions affects a solution's specific conductance and its ability to conduct electrical current. A more concentrated solution generally means more ions and consequently higher specific conductance.

Understanding concentration is crucial since it affects reaction rates, equilibria, and overall electrical properties of solutions. Mastery of this concept aids in predicting how changes in concentration can alter the chemical and physical behavior of a solution.