Problem 81 - A Complete Resource Book in Chemistry for JEE Main

Question

The emf of a Daniell cell at \(298 \mathrm{~K}\) is \(E_{\text {. }}\) \(\mathrm{Zn}\left|\mathrm{ZnSO}_{4} \| \mathrm{CuSO}_{4}\right| \mathrm{Cu}\) \((0.01 \mathrm{M}) \quad(1.0 \mathrm{M})\) when the concentration of \(\mathrm{ZnSO}_{4}\) is \(1.0 \mathrm{M}\) and that of \(\mathrm{CuSO}_{4}\) is \(0.01 \mathrm{M}\), the emf changed to \(E_{2^{\prime}}\) What is the relationship between \(E_{1}\) and \(E_{2}\) ? (a) \(E_{1}=E_{2}\) (b) \(E_{2} \neq E_{1}\) (c) \(E_{1}>E_{2}\) (d) \(E_{1}

Step-by-Step Solution

Answer

\( E_1 > E_2 \)

Step 1: Understanding the Nernst Equation

The Nernst equation is used to calculate the electromotive force (EMF) of a cell under non-standard conditions. It is given by \[ E = E^0 - \frac{RT}{nF} \ln{Q} \]where \(E\) is the EMF, \(E^0\) is the standard EMF, \(R\) is the gas constant, \(T\) is the temperature in Kelvin, \(n\) is the number of moles of electrons transferred, \(F\) is the Faraday constant, and \(Q\) is the reaction quotient.

Step 2: Setting Up the Reaction Quotient (Q)

For the Daniell cell, the overall cell reaction is \[ \text{Zn} + \text{Cu}^{2+} \rightarrow \text{Zn}^{2+} + \text{Cu} \].The reaction quotient \(Q\), is given by \[ Q = \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]} \].Here, \([\text{Zn}^{2+}]\) and \([\text{Cu}^{2+}]\) represent the concentrations of zinc and copper ions respectively.

Step 3: Calculate Q for Both Scenarios

For the first case, \\( [\text{Zn}^{2+}] = 0.01 \, \text{M} \) and \( [\text{Cu}^{2+}] = 1.0 \, \text{M} \), so \\[ Q_1 = \frac{0.01}{1.0} = 0.01 \].\For the second scenario, when \( [\text{Zn}^{2+}] = 1.0 \, \text{M} \) and \( [\text{Cu}^{2+}] = 0.01 \, \text{M} \), so \[ Q_2 = \frac{1.0}{0.01} = 100 \].

Step 4: Analyzing the Effects of Q on EMF

From the Nernst equation, we know that the larger the value of \(Q\), the more negtive the term \( -\frac{RT}{nF} \ln{Q} \) becomes, thus decreasing the EMF \(E\). In this case, since \( Q_1 < Q_2 \), it implies that \(E_1 > E_2\).

Step 5: Conclusion

After substituting the values of \(Q_1\) and \(Q_2\) into the Nernst equation, we can see that increasing the concentration of zinc ions while decreasing that of copper ions results in a decreased EMF. Hence, the correct relationship is \( E_1 > E_2 \).

Key Concepts

Nernst Equation

The Nernst Equation is a vital formula in electrochemistry that allows us to determine the electromotive force (EMF) of an electrochemical cell under non-standard conditions. Traditionally, the EMF or cell potential is measured under standard conditions. However, many reactions do not occur in such ideal environments.
The Nernst Equation is expressed as follows:
\[ E = E^0 - \frac{RT}{nF} \ln{Q} \]
Here, **\(E\)** represents the EMF under non-standard conditions, while **\(E^0\)** is the standard EMF. The constants **\(R\)**, the universal gas constant, and **\(T\)**, the temperature in Kelvin, play crucial roles. Additionally, **\(n\)** stands for the number of moles of electrons transferred into the electrochemical reaction and **\(F\)** is Faraday's constant, which is crucial for calculated charge transfers in molecules. Lastly, **\(Q\)**, the reaction quotient, gives a snapshot of the cell's conditions.

Reaction Quotient

The Reaction Quotient, commonly denoted as **\(Q\)**, is an important concept in chemistry, particularly electrochemistry. It helps us understand the position of a reaction at any given point in its progression towards equilibrium. Unlike the equilibrium constant, which applies only when a reaction has reached equilibrium, the reaction quotient can describe a reaction at any stage.
For a simple overall reaction such as:
\[ \text{Zn} + \text{Cu}^{2+} \rightarrow \text{Zn}^{2+} + \text{Cu} \]
The Reaction Quotient **\(Q\)** is calculated by the expression:
\[ Q = \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]} \]
Where **\([\text{Zn}^{2+}]\)** and **\([\text{Cu}^{2+}]\)** are the concentrations of zinc and copper ions. By knowing \(Q\), we can determine how the concentrations of the reactants or products affect the reaction's progression and influence the cell potential.

Electromotive Force

Electromotive Force, or EMF, could be likened to the "push" behind the movement of electrons in an electrochemical cell. It’s the maximum potential difference between two electrodes of a cell when no current flows and is a measure of the energy provided by a cell or battery per coulomb of charge passing through it.
This force is essential in various applications, from batteries to chemical production. It is represented as \(E\) and calculated using the Nernst Equation. When a cell operates under non-standard conditions, its EMF can fluctuate, influenced by factors like concentration.
The EMF of a cell:

• Provides insights into the cell's efficiency
• Affects the current and voltage produced by the cell
• Is influenced by the reaction quotient \(Q\)

Understanding EMF is essential for designing effective electrochemical cells.

Daniell Cell

The Daniell Cell is a classic example of a galvanic cell, a type of electrochemical cell that generates electrical energy from spontaneous chemical reactions. Named after its inventor, John Daniell, this cell paved the way for modern batteries by demonstrating how chemical reactions can be harnessed to produce electricity.
In a typical Daniell Cell setup, the zinc and copper electrodes are dipped into solutions of their corresponding sulfate salts. This creates two half-cells:

• Anode: where zinc undergoes oxidation (loses electrons)
• Cathode: where copper undergoes reduction (gains electrons)

These reactions result in the flow of electrons from the zinc to the copper electrode, generating an EMF. The concepts behind the Daniell Cell are fundamental to understanding how modern batteries operate.

Standard Conditions

Standard conditions in electrochemistry refer to a set of baseline conditions for measuring cell potentials and other chemical properties. These conditions are crucial because they provide a uniform basis for comparing different cells and reactions.
In standard conditions, the temperature is generally set at **298 K** (or 25°C), the pressure at **1 atm**, and all concentrations at **1 M** for solutions. Under these circumstances, the standard electromotive force (\(E^0\)) of a cell is determined, which forms the basis for calculations involving the Nernst Equation.
Standard conditions are essential because:

• They allow for consistent comparison of different electrochemical cells.
• They provide a reference point for calculating changes and shifts under real-world conditions using the Nernst Equation.

Keeping the standard conditions in mind helps us comprehend the differences when reactions happen in varied environments.