Problem 194 - A Complete Resource Book in Chemistry for JEE Main

Question

Resistance of a conductivity cell filled with a solution of an electrolyte of concentration \(0.1 \mathrm{M}\) is \(100 \Omega\). The conductivity of this solution is \(1.29 \mathrm{~S} \mathrm{~m}^{-1}\). Resistance of the same cell when filled with \(0.2 \mathrm{M}\) of the same solution is \(520 \Omega\). The molar conductivity of \(0.02 \mathrm{M}\) solution of the electrolyte will be (a) \(124 \times 10^{-4} \mathrm{~S} \mathrm{~m}^{2} \mathrm{~mol}^{-1}\) (b) \(1240 \times 10^{-4} \mathrm{~S} \mathrm{~m}^{2} \mathrm{~mol}^{-1}\) (c) \(1.24 \times 10^{-4} \mathrm{~S} \mathrm{~m}^{2} \mathrm{~mol}^{-1}\) (d) \(12.4 \times 10^{-4} \mathrm{~S} \mathrm{~m}^{2} \mathrm{~mol}^{-1}\)

Step-by-Step Solution

Answer

The molar conductivity is (b) \(1240 \times 10^{-4} \text{ S m}^2 \text{ mol}^{-1}\).

Step 1: Determine the cell constant

The cell constant (denoted as \( G^* \)) can be calculated using the formula \( G^* = \, \text{Conductivity} \, \times \, \text{Resistance} \). Given that the conductivity is \( 1.29 \text{ S/m} \) and the resistance is \( 100 \, \Omega \), the cell constant is: \( G^* = 1.29 \times 100 = 129 \text{ m}^{-1} \).

Step 2: Calculate the new conductivity

The conductivity of the \( 0.2 \text{ M} \) solution can be calculated using the cell constant: \( \text{Conductivity} = \frac{G^*}{R} \), where \( R = 520 \, \Omega \). Thus, \( \text{Conductivity} = \frac{129}{520} = 0.2481 \text{ S/m} \).

Step 3: Calculate the molar conductivity at 0.02 M

Molar conductivity \( \Lambda_m \) can be calculated as \( \Lambda_m = \frac{\text{Conductivity}}{\text{Concentration}} \). For the \( 0.02 \text{ M} \) solution: \( \Lambda_m = \frac{0.2481}{0.02} = 12.405 \text{ S} \text{ m}^2 \text{ mol}^{-1} \).

Step 4: Convert to correct units

The molar conductivity is often expressed in \( \times 10^{-4} \text{ S m}^2 \text{ mol}^{-1} \). Therefore, \( 12.405 \text{ S} \text{ m}^2 \text{ mol}^{-1} \) can be written as \( 1240 \times 10^{-4} \text{ S} \text{ m}^2 \text{ mol}^{-1} \).

Key Concepts

Conductivity Cell

A conductivity cell is a device used to measure the electrical conductivity of an electrolyte solution. It typically consists of two or more electrodes, which are immersed in the solution. By applying a voltage across the electrodes, the movement of ions in the solution is detected and measured as an electrical current. This allows for the calculation of conductivity, which is crucial for understanding the properties of the electrolyte. The output from the cell helps determine how well electricity is transmitted through the electrolyte, and this property is largely influenced by the type of ions present and their concentration.

Electrolyte Concentration

Electrolyte concentration refers to the amount of an electrolyte (such as a salt, acid, or base) dissolved in a solvent, usually water, to form a solution. Concentration is typically expressed in molarity (M), which is the number of moles of solute per liter of solution.
In conductivity measurements, concentration plays a critical role as it impacts the mobility and number of ions in a solution, thus influencing conductivity. For example, a higher concentration means more ions are available to carry electric current, thus generally resulting in higher conductivity.

Cell Constant

The cell constant is a characteristic value of a conductivity cell and is defined as the product of conductivity and resistance. It is represented as \( G^* \) and has units of \( \text{m}^{-1} \).
The cell constant depends on the geometric arrangement of the electrodes in the cell and does not change with the solution's properties. It serves as a calibration factor that relates resistance measurements made with a specific cell to conductivity. Calculating the cell constant is an essential step in determining accurate conductivity values using a conductivity cell.

Resistance

Resistance is the opposition of a material to the flow of electric current through it, and is measured in ohms (\( \Omega \)). It is one of the critical parameters in a conductivity cell.
When an electrolyte solution fills a conductivity cell, the resistance varies with the type and concentration of the ions present. For example, with a higher concentration of an electrolyte, the resistance typically decreases as more ions are available to carry current.
To measure a solution's conductivity, resistance is often used in conjunction with the cell constant in the equation \( \text{Conductivity} = \frac{G^*}{R} \).

Conductivity

Conductivity (boted as \( \sigma \)) measures a solution's ability to conduct electricity, with the common unit being siemens per meter (\( \, \text{S} \cdot \text{m}^{-1} \)). It is directly influenced by the ions' mobility, concentration, and type.
The higher the concentration of ions, the higher the conductivity. However, the effect of concentration is not linear due to ion-ion interactions, particularly at higher concentrations. In a conductivity cell, conductivity is calculated using the cell constant and resistance obtained from measurements.

• For example, in the exercise, the initial conductivity is calculated as 1.29 S/m, using a cell constant derived from given resistance values.

Units Conversion

In scientific calculations, converting units is pivotal for standardization and comparison. When dealing with molar conductivity, it is often necessary to express the results in standardized units to align with scientific and industry practices.
Molar conductivity can be expressed in the format of \( \times 10^{-4} \, \text{S} \, \text{m}^2 \, \text{mol}^{-1} \), providing a uniform way to present findings. This is particularly useful when working with very large or small numbers, as it simplifies interpretation.

• In the example solution, a conversion to \( 1240 \times 10^{-4} \, \text{S} \, \text{m}^2 \, \text{mol}^{-1} \) allows for straightforward comparison with other data.