Problem 75 - A Complete Resource Book in Chemistry for JEE Main

Question

The specific conductance of \(0.1 \mathrm{~N} \mathrm{KCl}\) solution at \(23^{\circ} \mathrm{C}\) is \(0.012 \mathrm{ohm}^{-1} \mathrm{~cm}^{-1}\). The resistance of cell containing the solution at the same temperature was found to be 55 ohm. The cell constant will be (a) \(0.142 \mathrm{~cm}^{-1}\) (b) \(0.616 \mathrm{~cm}^{-1}\) (c) \(6.16 \mathrm{~cm}^{-1}\) (d) \(616 \mathrm{~cm}^{-1}\)

Step-by-Step Solution

Answer

The cell constant is (b) \(0.616 \mathrm{~cm}^{-1}\).

Step 1: Understand the Problem

We need to calculate the cell constant, given the specific conductance and the resistance of the solution. Specific conductance is the ability of a solution to conduct electricity, and the cell constant is needed to relate it to the actual conductance of the solution in a specific cell setup.

Step 2: Establish the Formula

Use the formula relating specific conductance (k), resistance (R), and the cell constant (G*):\[ G^* = k \times R \]Here, we have k (specific conductance) as \(0.012 \mathrm{~ohm}^{-1} \mathrm{~cm}^{-1}\) and R (resistance) as \(55 \text{ ohm}\).

Step 3: Plug Values into the Formula

Substitute the given values into the formula to calculate the cell constant \[ G^* = 0.012 \mathrm{~ohm}^{-1} \mathrm{~cm}^{-1} \times 55 \text{ ohm} \]

Step 4: Calculate the Cell Constant

Perform the multiplication to find the cell constant:\[ G^* = 0.012 \times 55 = 0.66 \mathrm{~cm}^{-1} \]

Step 5: Compare with Given Options

Consider the options given: (a) 0.142 \( \mathrm{~cm}^{-1}\), (b) 0.616 \( \mathrm{~cm}^{-1}\), (c) 6.16 \( \mathrm{~cm}^{-1}\), (d) 616 \( \mathrm{~cm}^{-1}\). The closest value to our calculation (0.66 \( \mathrm{~cm}^{-1}\)) is 0.616 \( \mathrm{~cm}^{-1}\).

Step 6: Select the Correct Option

Based on the calculation, the option that matches our answer is option (b) \(0.616 \mathrm{~cm}^{-1}\).

Key Concepts

Understanding Specific Conductance

Specific conductance, also known as the conductivity of a solution, measures how well that solution can conduct electricity. It is expressed in terms of ^{ohm-1 \( \text{cm}^{-1} \)}. This measure essentially tells us the solution's ability to allow ion movement, which carries electric charges through it.
When you consider any ionic solution, its conductivity changes based on the concentration and type of ions present. For example, when you dissolve potassium chloride (KCl) in water, it dissociates into potassium (K⁺) and chloride (Cl^-) ions, enhancing conductance.
Specific conductance is crucial because:

• It provides insights into the purity of water or a solution.
• Aids in determining ionic concentration in various applications, like water treatment.
• It helps in understanding the electrolytic properties of chemical solutions, which can be vital in various industries like food processing and electronics.

Understanding specific conductance enables us to correlate it with other properties, such as resistance and the cell constant, making it a cornerstone in electrochemistry.

Resistance Measurement in Solutions

When we talk about resistance in the context of solutions, it refers to how difficult it is for the electric current to flow through the solution. Resistance is measured in units of ohms (\( \Omega \)).
To measure resistance, you typically apply a voltage across a solution and determine the resulting current. Ohm's Law, expressed as \( R = \frac{V}{I} \), helps us calculate resistance, where \( R \) is resistance, \( V \) is voltage, and \( I \) is current.
Resistance in a solution is influenced by several factors:

• The type and concentration of ions present, with higher ion concentration generally leading to lower resistance.
• The temperature, as warmer solutions tend to have lower resistance.
• The nature of the solvent, as different solvents will affect ion mobility differently.

In practical applications like calibrating conductivity meters, measuring the resistance of a solution helps determine both its conductance and, using the cell constant, further electrochemical properties.

Role of the Conductivity Cell

A conductivity cell is a device used to measure the conductivity of a solution. It usually consists of two electrodes submerged in the solution, designed to create a uniform electric field between them.
The cell constant is a unique parameter for each conductivity cell, relating the physical configuration of the electrodes to conductivity measurements. It tells us how the actual dimensions of the cell affect its ability to measure conductance accurately.
Why is the cell constant important?

• It ensures that conductivity readings are accurate and consistent across different cells.
• Allows for the correction of readings by taking into account the specific configuration and spacing of the electrodes.
• Vital for comparing readings from different setups, ensuring that the resultant conductance is due purely to the solution and not affected by the cell layout.

In experiments, once the specific conductance and resistance are known, the cell constant can be calculated using the formula: \[ G^* = k \times R \] Where \( G^* \) is the cell constant, \( k \) is specific conductance, and \( R \) is resistance.
This relationship allows for precise electrochemical studies and processes, across various scientific disciplines.