Problem 135 - A Complete Resource Book in Chemistry for JEE Main

Question

The emf of the following three galvanic cells (1) \(\mathrm{Zn}\left|\mathrm{Zn}^{2+}(1 \mathrm{M}) \| \mathrm{Cu}^{2+}(1 \mathrm{M})\right| \mathrm{Cu}\) (2) \(\mathrm{Zn}\left|\mathrm{Zn}^{2+}(0.1 \mathrm{M}) \| \mathrm{Cu}^{2+}(1 \mathrm{M})\right| \mathrm{Cu}\) (3) \(\mathrm{Zn}\left|\mathrm{Zn}^{2+}(1 \mathrm{M}) \| \mathrm{Cu}^{2+}(0.1 \mathrm{M})\right| \mathrm{Cu}\) are represented by \(E_{1}, E_{2}\) and \(E_{y}\). Which of the following statement is true? (a) \(E_{2}>E_{1}>E_{3}\) (b) \(E_{3}>E_{2}>E_{1}\) (c) \(E_{1}>E_{2}>E_{3}\) (d) \(E_{3}>E_{1}>E_{2}\)

Step-by-Step Solution

Answer

Option (a) \(E_{2} > E_{1} > E_{3}\) is true.

Step 1: Understand Nernst Equation

To determine the emf of each cell, we use the Nernst equation: \( E = E^0 - \frac{RT}{nF} \ln Q \). Here, \( E^0 \) is the standard cell potential, \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, \( n \) is the number of electrons transferred in the redox reaction, \( F \) is the Faraday constant, and \( Q \) is the reaction quotient.

Step 2: Calculate Reaction Quotient for Each Cell

For the reaction \( \mathrm{Zn} + \mathrm{Cu}^{2+} \rightarrow \mathrm{Zn}^{2+} + \mathrm{Cu} \), the reaction quotient \( Q \) is \( \frac{[\mathrm{Zn}^{2+}]}{[\mathrm{Cu}^{2+}]} \). Therefore, \( Q_1 = \frac{1}{1} = 1 \), \( Q_2 = \frac{0.1}{1} = 0.1 \), and \( Q_3 = \frac{1}{0.1} = 10 \).

Step 3: Standard Cell Potential

The standard cell potential \( E^0 \) for the reaction is the difference in standard reduction potentials of the cathode and anode: \( E^0 = E^0_{\text{Cu}^{2+}/\text{Cu}} - E^0_{\text{Zn}^{2+}/\text{Zn}} = 0.34 - (-0.76) = 1.10 \text{ V} \).

Step 4: Apply Nernst Equation to Calculate Emfs

For cell 1, since \( Q_1 = 1 \), \( E_1 = 1.10 \) V (no change from standard). For cell 2, \( E_2 = 1.10 - \frac{RT}{nF} \ln 0.1 \), which increases \( E \). For cell 3, \( E_3 = 1.10 - \frac{RT}{nF} \ln 10 \), which decreases \( E \).

Step 5: Compare Emfs

Since \( \ln 0.1 \) is negative and \( \ln 10 \) is positive, \( E_2 > E_1 \) and \( E_3 < E_1 \). Therefore, the order is \( E_2 > E_1 > E_3 \).

Key Concepts

Nernst equation

The Nernst equation is a fundamental formula in electrochemistry that relates the electromotive force (emf) of a galvanic cell to the concentrations of the reactants and products. It provides insight into how voltage changes with varying concentrations.

The equation is expressed as:

\[E = E^0 - \frac{RT}{nF} \ln Q\]where:

• \(E^0\) is the standard cell potential, the voltage under standard conditions (1 M concentration, 1 atm pressure, 25°C).
• \(R\) is the universal gas constant (8.314 J/mol·K).
• \(T\) is the temperature in Kelvin.
• \(n\) is the number of moles of electrons transferred in the redox reaction.
• \(F\) is the Faraday constant (96485 C/mol).
• \(Q\) is the reaction quotient, indicating the relative concentration of products to reactants.

The Nernst equation allows us to calculate the cell potential for non-standard conditions. For instance, the standard cell potentials are adjusted based on concentrations, thus impacting the overall voltage.

Galvanic cells

Galvanic cells, or voltaic cells, are devices that convert chemical energy into electrical energy through a spontaneous redox reaction. They consist of two half-cells, each containing an electrode and an electrolyte.

Key components include:

• **Anode:** where oxidation occurs, losing electrons.
• **Cathode:** where reduction occurs, gaining electrons.
• A salt bridge: allows the flow of ions to maintain charge neutrality.
• A wire: connects the electrodes, allowing electron flow from anode to cathode.

In the provided exercise, a galvanic cell is set up using zinc and copper. These cells have different concentrations of ion solutions which affect emf based on the reaction taking place. The flow of electrons from zinc to copper generates electricity, as the errors in measurements and conditions can also affect performance. Understanding galvanic cells is crucial for learning about battery functions and industrial applications.

Standard cell potential

Standard cell potential, denoted as \(E^0\), is the measure of voltage that a galvanic cell can produce under standard conditions: 1 M solutions, 1 atm pressure, and 25°C (298 K).

This potential is calculated using the standard reduction potentials of the electrodes:

• The standard reduction potential is the tendency of a substance to gain electrons.
• Positive values indicate a greater tendency to be reduced (gain electrons).
• The formula for calculating \(E^0\) is:\[E^0 = E^0_{cathode} - E^0_{anode}\]

For zinc and copper, as in the exercise, the standard cell potential is:
\(E^0 = 0.34 \, V - (-0.76 \, V) = 1.10 \, V\).
This validates the capability of the cell to do work. Having a positive value, as in most spontaneous reactions, shows that the galvanic cell can effectively generate electricity.

Reaction quotient

The reaction quotient, denoted as \(Q\), is a measure used in chemistry to understand the direction of a chemical reaction that is not at equilibrium. It is comparable to the equilibrium constant but is calculated at any given set of conditions.

In a galvanic cell, the reaction quotient is defined as the ratio of the concentrations of products to reactants, raised to the power of their coefficients in the balanced chemical equation:

• For the reaction \(\mathrm{Zn} + \mathrm{Cu}^{2+} \rightarrow \mathrm{Zn}^{2+} + \mathrm{Cu}\), the quotient \(Q\) is expressed as:
• \[Q = \frac{[\mathrm{Zn}^{2+}]}{[\mathrm{Cu}^{2+}]}\]

Changes in the concentration of ions directly affect \(Q\), which in turn influences the cell potential through the Nernst equation. An increase in \(Q\) could mean the cell is less effective at producing voltage, while a decrease suggests a stronger driving force for the reaction. Understanding \(Q\) helps in predicting how changes in conditions alter the cell's efficiency.