Problem 38 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Two students use same stock solution of \(\mathrm{ZnSO}_{4}\) and a solution of \(\mathrm{CuSO}_{4}\). The emf of one cell is \(0.03 \mathrm{~V}\) higher than the other. The conc. of \(\mathrm{CuSO}_{4}\) in the cell with higher emf value is \(0.5 \mathrm{M}\). Find out the conc. of \(\mathrm{CuSO}_{4}\) in the other cell \((2.203 \mathrm{RT} / \mathrm{F}=0.06\) ).

Step-by-Step Solution

Answer

The concentration of \(\mathrm{CuSO}_{4}\) in the other cell is \(0.05 \text{ M}\).

Step 1: Identify the Nernst Equation

The Nernst Equation helps us calculate the emf of a cell involving a concentration cell. For a zinc and copper cell, it can be expressed as: \[ E = E^\circ - \frac{0.0592}{n} \log \frac{[\text{low conc. Cu}^{2+}]}{[\text{high conc. Cu}^{2+}]} \] where \(n=2\) is the number of electrons exchanged.

Step 2: Set Up Equation for Both Cells

Assume the emf in the cell with higher concentration of \(\mathrm{CuSO}_{4}\) is \(E_1\) and the emf in the cell with lower concentration is \(E_2\). Given that \(E_2 = E_1 - 0.03 \, \text{V}\), we will use this comparison to find the relationship between the two concentrations.

Step 3: Write Nernst Equations for Both Cells

For the cell with higher emf, the equation is \[ E_1 = E^\circ - 0.06 \log \left( \frac{0.5}{[\text{low conc. Cu}^{2+}]_{1}} \right) \].For the other cell, it is \[ E_2 = E^\circ - 0.06 \log \left( \frac{[\text{low conc. Cu}^{2+}]_{2}}{0.5} \right) \].

Step 4: Substitute Difference in EMF Values

Since \(E_2 = E_1 - 0.03\), we can write: \[ E^\circ - 0.06 \log \left( \frac{[\text{low conc. Cu}^{2+}]_{2}}{0.5} \right) = E^\circ - 0.06 \log \left( \frac{0.5}{[\text{low conc. Cu}^{2+}]_{1}} \right) - 0.03 \].

Step 5: Solve for the Unknown Concentration

Simplify the equation: \[ 0.06 \log \left( \frac{[\text{low conc. Cu}^{2+}]_{2}}{0.5} \right) = 0.06 \log \left( \frac{0.5}{[\text{low conc. Cu}^{2+}]_{1}} \right) + 0.03 \]. Convert to exponential form to find \([\text{low conc. Cu}^{2+}]_{2}\).

Step 6: Calculate Unknown Concentration

Rearrange and use the properties of logarithms: \[ \log \left( \frac{[\text{low conc. Cu}^{2+}]_{2}}{0.5} \right) - \log \left( \frac{0.5}{[\text{low conc. Cu}^{2+}]_{1}} \right) = 0.5 \]. Solve this for \([\text{low conc. Cu}^{2+}]_{2}\): \( [\text{low conc. Cu}^{2+}]_{2} = 0.05 \text{ M} \).

Key Concepts

Electromotive Force (EMF)

Electromotive force (emf) is the voltage developed by any source of electrical energy, such as a battery or dynamo. In electrochemistry, it refers to the potential difference between the two electrodes of an electrochemical cell. This potential difference can drive an electric current, which can be used to perform work, such as moving charge through a circuit.
Understanding emf involves some fundamental concepts:

• Standard EMF (\( E^\circ \)): This is the emf of a cell when all components are in their standard states, typically at 1 molar concentration, 1 atm pressure, and 25°C.
• Cell Potential: Difference in electrochemical potential between two electrodes due to the redox reaction.
• Relationship with Gibbs Free Energy: The emf of a cell can also be linked to the Gibbs free energy change of a reaction, offering insights into the energy efficiency of the process.

In a concentration cell, which relies on the Nernst Equation, the emf can change based on ion concentrations. A higher concentration of ions can yield a higher emf, reflecting a stronger potential for electron flow.

Concentration Cell

A concentration cell is a type of electrochemical cell where the electrodes are identical, but the concentrations of the ionic solutions are different. These cells function due to the concentration gradient between solutions, driving the electrochemical reaction.
The crucial elements of a concentration cell include:

• Identical Electrodes: Both compartments of the cell use the same electrode material.
• Different Concentrations: The driving force of the cell's electricity comes purely from the difference in ion concentration in the two solutions.
• Nernst Equation Application: The Nernst Equation is used to calculate the difference in potential caused by concentration differences, rather than standard cell potentials.

Concentration cells continue to work until equilibrium is reached, meaning both sides reach the same concentration. At this point, the emf drops to zero, stopping the cell's operation.

Logarithm Properties

In chemical reactions and electrochemistry, logarithm properties can simplify complex equations, making them easier to solve and analyze. Logarithms help in manipulating equations like the Nernst Equation, used for calculating cell potentials.
Key properties useful in electrochemical contexts are:

• Product to Sum: \( ext{log}(a imes b) = ext{log}(a) + ext{log}(b) \).
• Quotient to Difference: \( ext{log}(\frac{a}{b}) = ext{log}(a) - ext{log}(b) \).
• Exponentiation: \( ext{log}(a^b) = b imes ext{log}(a) \).

These properties allow solving equations by reducing the number of terms and facilitating easier calculations. In the Nernst Equation, logarithms help convert voltage potential equations from exponential forms to logarithmic forms, simplifying the process of solving for unknown variables, such as ion concentration.