Problem 7 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

If the standard electrode potential for a cell is \(2 \mathrm{~V}\) at \(300 \mathrm{~K}\), the equilibrium constant \((\mathrm{K})\) for the reaction \(\mathrm{Zn}(\mathrm{s})+\mathrm{Cu}^{2+}(\mathrm{aq}) \rightleftharpoons \mathrm{Zn}^{2+}(\mathrm{aq})+\mathrm{Cu}(\mathrm{s})\) at \(300 \mathrm{~K}\) is approximately \(\left(\mathrm{R}=8 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}, \mathrm{~F}=96000 \mathrm{C} \mathrm{mol}^{-1}\right)\) [Main Jan. 9, 2019 (II)] (a) \(\mathrm{e}^{-80}\) (b) \(\mathrm{e}^{-160}\) (c) \(\mathrm{e}^{320}\) (d) \(\mathrm{e}^{160}\)

Step-by-Step Solution

Answer

The equilibrium constant is approximately \(e^{160}\), option (d).

Step 1: Identify the Equation

We need to use the Nernst equation and the relationship between the standard cell potential \(E^0\), the gas constant \(R\), the temperature \(T\), Faraday's constant \(F\), and the equilibrium constant \(K\). The relevant equation is: \[E^0 = \frac{RT}{nF} \ln K\] where \(n\) is the number of electrons transferred in the balanced equation for the cell reaction.

Step 2: Determine Reaction Details

For the reaction \(\mathrm{Zn}(\mathrm{s})+\mathrm{Cu}^{2+}(\mathrm{aq}) \rightleftharpoons \mathrm{Zn}^{2+}(\mathrm{aq})+\mathrm{Cu}(\mathrm{s})\), the number of moles of electrons transferred, \(n\), is 2 because zinc is oxidized from \(\mathrm{Zn}\) to \(\mathrm{Zn}^{2+}\) which corresponds to a transfer of two electrons.

Step 3: Substitute Known Values

Substitute the known values into the equation: \(E^0 = 2 \mathrm{~V}\), \(R = 8 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\), \(T = 300 \mathrm{~K}\), \(F = 96000 \mathrm{C} \mathrm{mol}^{-1}\), and \(n = 2\). This gives:\[2 = \frac{8 \times 300}{2 \times 96000} \ln K\]

Step 4: Simplify and Solve for \(\ln K\)

Simplify the equation: \[2 = \frac{8 \times 300}{2 \times 96000} \ln K = \frac{2400}{192000} \ln K = \frac{1}{80} \ln K\]Multiply both sides by 80 to isolate \(\ln K\): \[160 = \ln K\]

Step 5: Find \(K\) from \(\ln K\)

Since \(\ln K = 160\), convert this to exponential form to find \(K\): \[K = e^{160}\]

Step 6: Select the Correct Answer

From the options given, the answer that matches \(K = e^{160}\) is option (d) \(e^{160}\).

Key Concepts

Nernst equation

The Nernst equation is a fundamental part of electrochemistry as it helps predict the potential of an electrochemical cell under non-standard conditions. In its basic form, the equation looks like this:\[E = E^0 - \frac{RT}{nF} \ln Q\]This equation relates several key aspects:

• \(E\) is the cell potential under non-standard conditions.
• \(E^0\) symbolizes the standard electrode potential, a measure of the electromotive force of a cell when all components are at standard conditions.
• \(R\) is the universal gas constant, \(8.314 \text{ J K}^{-1} \text{ mol}^{-1}\).
• \(T\) stands for temperature in Kelvin.
• \(n\) is the moles of electrons exchanged in the redox reaction.
• \(F\) is Faraday's constant, \(96485 \text{ C mol}^{-1}\).
• \(Q\) is the reaction quotient, representing the ratio of product and reactant concentrations.

The Nernst equation is vital to understanding how changes in concentration and temperature affect cell potential. It is frequently used to calculate equilibrium constants and predict the direction of redox reactions in a cell.

Standard electrode potential

The standard electrode potential, denoted as \(E^0\), is an important concept in electrochemistry. This value reflects the potential difference between an electrode and a standard hydrogen electrode under specified conditions.It is measured under the following conditions:

• All solutes have a concentration of \(1 \text{ mol L}^{-1}\).
• Gases are at a pressure of \(1 \text{ atm}\).
• Temperature is usually at \(298 \text{ K}\) (25°C).

The standard electrode potential tells us how likely a compound is to be reduced. Higher values indicate stronger oxidizing agents, while lower values suggest stronger reducing agents. For cell reactions, the cell's overall standard electrode potential \(E^0_{cell}\) is derived by subtracting the standard electrode potential of the anode from that of the cathode:\[E^0_{cell} = E^0_{cathode} - E^0_{anode}\]Understanding \(E^0\) helps in assessing the feasibility of electrochemical reactions and aids in the design of galvanic cells, batteries, and other electrochemical devices.

Equilibrium constant calculation

In electrochemical reactions, the equilibrium constant \(K\) can be calculated using the standard electrode potential \(E^0\), utilizing the relationship with the Nernst equation. These steps help bridge a clear connection between electrochemical cells and their equilibrium states.The key equation derived from the Nernst equation is:\[E^0 = \frac{RT}{nF} \ln K\]Here’s how to find the equilibrium constant from this:

• Insert known values for \(E^0\), \(R\), \(T\), \(n\), and \(F\) into the equation.
• Rearrange to solve for \(\ln K\).
• Convert \(\ln K\) to \(K\) using the exponential function: \(K = e^{\ln K}\).

This calculation method illustrates how a redox reaction reaches equilibrium, connecting the concept of electrochemical potential to standard chemistry equilibria. It is significant in understanding the extent to which a reaction will proceed or how spontaneous it is.