Problem 133 - A Complete Resource Book in Chemistry for JEE Main

Question

For the electrochemical cell, \(\mathrm{M}\left|\mathrm{M}^{+} \| \mathrm{X}^{-}\right| \mathrm{X}\) \(E^{\circ} \mathrm{M}^{+} / \mathrm{M}=0.44 \mathrm{~V}\) and \(E^{\circ} \mathrm{X} / \mathrm{X}^{-}=0.33 \mathrm{~V}\) From these data, one can deduce that (a) \(\mathrm{M}+\mathrm{X} \longrightarrow \mathrm{M}^{\prime}+\mathrm{X}^{-}\)is the spontaneous reaction (b) \(\mathrm{M}^{+}+\mathrm{X}^{-} \longrightarrow \mathrm{M}+\mathrm{X}\) is spontaneous reaction (c) \(E_{\text {cell }}=0.77 \mathrm{~V}\) (d) \(E_{\text {cell }}=-0.77 \mathrm{~V}\)

Step-by-Step Solution

Answer

Answer: (b) is spontaneous, calculated cell potential is \(-0.11 \, \text{V}\).

Step 1: Write Half Reactions

Identify the half-reactions for the given electrochemical cell: - Oxidation half-reaction: \( \mathrm{M} \rightarrow \mathrm{M}^{+} + e^{-} \) - Reduction half-reaction: \( \mathrm{X} + e^{-} \rightarrow \mathrm{X}^{-} \)

Step 2: Determine Standard Potential of the Electrochemical Cell

Calculate the standard cell potential \( E^{\circ}_{\text{cell}} \) using the formula:\[ E^{\circ}_{\text{cell}} = E^{\circ}_{\text{reduction}} - E^{\circ}_{\text{oxidation}} \]Substitute the given values: \[ E^{\circ}_{\text{cell}} = 0.33\, \text{V} - 0.44\, \text{V} = -0.11\, \text{V} \]

Step 3: Interpret Standard Cell Potential

Since \( E^{\circ}_{\text{cell}} \) is negative, the reaction is not spontaneous in the direction given by the reduction of \( \mathrm{X} \) and the oxidation of \( \mathrm{M} \).Hence, the reverse reaction (from \( \mathrm{M}^{+} + \mathrm{X}^{-} \longrightarrow \mathrm{M} + \mathrm{X} \)) is expected to be spontaneous since it would have a positive cell potential.

Step 4: Conclusion

The reverse reaction \( \mathrm{M}^{+} + \mathrm{X}^{-} \longrightarrow \mathrm{M} + \mathrm{X} \) is spontaneous, and the cell potential in the forward direction calculated was \(-0.11\, \text{V}\). Thus, neither \( E_{\text{cell}} = 0.77\, \text{V} \) nor \( E_{\text{cell}} = -0.77\, \text{V} \) are correct representations of the calculated cell potential.

Key Concepts

Standard Cell Potential

In electrochemical cells, the standard cell potential is crucial as it indicates the cell's overall ability to produce an electric voltage through chemical reactions. It is denoted by \( E^{\circ}_{\text{cell}} \) and is measured in volts. This potential is derived from the difference in the standard reduction potentials of two half-reactions occurring in the cell.

To calculate the standard cell potential, you use the formula:

• \( E^{\circ}_{\text{cell}} = E^{\circ}_{\text{reduction}} - E^{\circ}_{\text{oxidation}} \)

Substituting values from our example exercise, we find:

• Reduction potential of \( \mathrm{X/X}^- \) is \( 0.33 \text{ V} \)
• Oxidation potential of \( \mathrm{M^+/M} \) is \( 0.44 \text{ V} \)

Plugging these into the formula gives a standard cell potential of \(-0.11 \text{ V}\).

A positive \( E^{\circ}_{\text{cell}} \) signals that the reaction is spontaneous whereas a negative value, like in our case, indicates non-spontaneity in the assumed direction of the reaction.

Half-reactions

Half-reactions are the building blocks of electrochemical cells. They represent the individual oxidation and reduction processes that occur in the electrochemical reaction.

In our example:

• The oxidation half-reaction is: \( \mathrm{M} \rightarrow \mathrm{M}^{+} + e^{-} \).
• The reduction half-reaction is: \( \mathrm{X} + e^{-} \rightarrow \mathrm{X}^{-} \).

In the oxidation half-reaction, \( \mathrm{M} \) loses an electron to become \( \mathrm{M}^{+} \). Conversely, in the reduction half-reaction, \( \mathrm{X} \) gains an electron to transform into \( \mathrm{X}^{-} \).

Adding these half-reactions gives the full cell reaction. They are essential for calculating the cell potential and for understanding whether the reaction will occur spontaneously as intended. Each half-reaction has its own standard potential, which when combined, helps in determining the overall cell potential.

Spontaneity of Reactions

The spontaneity of a chemical reaction in electrochemistry is primarily determined by the sign of the cell potential \( E_{\text{cell}} \). Spontaneous reactions in an electrochemical cell occur when the cell potential is positive.

In the case of our example exercise, the \( E^{\circ}_{\text{cell}} \) calculated was \(-0.11 \text{ V}\), indicating that the reaction direction initially evaluated is not spontaneous.

However, if we consider the reverse reaction, where \( \mathrm{M}^{+} + \mathrm{X}^{-} \rightarrow \mathrm{M} + \mathrm{X} \), the cell potential of this reverse process would be positive, making it spontaneous.

Therefore, cell potentials guide us in analyzing whether an electrochemical cell will work spontaneously. They help us decide if the direction of electron flow as hypothesized leads to a viable system or not. Understanding these concepts aids in practical applications of electrochemistry, from batteries to electrolysis processes.