Problem 140 - A Complete Resource Book in Chemistry for JEE Main

Question

The standard reduction potentials of \(\mathrm{Cu}^{2+} / \mathrm{Cu}\) and \(\mathrm{Cu}^{2+} /\) \(\mathrm{Cu}^{+}\)are \(0.337 \mathrm{~V}\) and \(0.153 \mathrm{~V}\) respectively. The standard electrode potential of \(\mathrm{Cu}^{+} / \mathrm{Cu}\) half cell is (a) \(0.184 \mathrm{~V}\) (b) \(0.827 \mathrm{~V}\) (c) \(0.521 \mathrm{~V}\) (d) \(0.490 \mathrm{~V}\)

Step-by-Step Solution

Answer

The standard electrode potential of \( \mathrm{Cu}^+ / \mathrm{Cu} \) is \( 0.184 \mathrm{~V} \).

Step 1: Write down the half-reaction equations

First, identify the given half-reactions and their respective standard reduction potentials. We have two half-reactions: 1. \( \mathrm{Cu}^{2+} + 2e^- \rightarrow \mathrm{Cu} \) with \( E^\circ = 0.337 \mathrm{~V} \)2. \( \mathrm{Cu}^{2+} + e^- \rightarrow \mathrm{Cu}^+ \) with \( E^\circ = 0.153 \mathrm{~V} \)

Step 2: Use Equation to Find Missing Potential

We want to find the standard reduction potential for \( \mathrm{Cu}^+ / \mathrm{Cu} \). The relation between the potentials is:\[ E^\circ(\mathrm{Cu}^{2+}/\mathrm{Cu}) = E^\circ(\mathrm{Cu}^{2+}/\mathrm{Cu}^+) + E^\circ(\mathrm{Cu}^+/\mathrm{Cu}) \]Thus,\[ 0.337 \mathrm{~V} = 0.153 \mathrm{~V} + E^\circ(\mathrm{Cu}^+/\mathrm{Cu}) \]

Step 3: Solve for the Unknown Potential

Rearrange the above equation to isolate \( E^\circ(\mathrm{Cu}^+/\mathrm{Cu}) \):\[ E^\circ(\mathrm{Cu}^+/\mathrm{Cu}) = 0.337 \mathrm{~V} - 0.153 \mathrm{~V} \]Perform the subtraction:\[ E^\circ(\mathrm{Cu}^+/\mathrm{Cu}) = 0.184 \mathrm{~V} \]

Step 4: Verify the Answer

Compare the calculated potential with the given options: The calculated \( E^\circ \) for \( \mathrm{Cu}^+/\mathrm{Cu} \) is \( 0.184 \mathrm{~V} \), which matches option (a).

Key Concepts

Standard Reduction Potentials

Standard reduction potentials are a vital part of understanding electrochemical cells. They help us determine how likely a chemical species is to gain electrons, or be reduced, in a chemical reaction. The standard reduction potential is measured in volts (V) and is conducted under standard conditions: 25°C, 1M concentration for each ion, and 1 atm pressure.

When examining reduction potentials, a positive value indicates a greater tendency to gain electrons and undergo reduction. Conversely, a more negative value implies a weaker tendency to be reduced. In electrochemical cells, these potentials allow us to predict the direction of electron flow, which is key in determining which reactions will occur spontaneously.

In our example:

• The reduction potential for the reaction enabled equation renderingecuation: \( \mathrm{Cu}^{2+} + 2e^- \rightarrow \mathrm{Cu} \) is 0.337 V,
• while for enabled equation renderingecuation: \( \mathrm{Cu}^{2+} + e^- \rightarrow \mathrm{Cu}^+ \) it is 0.153 V.

These values signify a higher tendency for enabled equation renderingecuation: \( \mathrm{Cu}^{2+} \) to reduce to copper metal enabled equation renderingecuation: \( \mathrm{Cu} \) than to the copper ion enabled equation renderingecuation: \( \mathrm{Cu}^+ \). This places a crucial role in determining how we calculate the missing potential for the half-reaction: enabled equation renderingecuation: \( \mathrm{Cu}^+ / \mathrm{Cu} \).

Electrochemistry

Electrochemistry is the branch of chemistry that deals with the relationship between electricity and chemical reactions. It's a captivating field that combines the concepts of chemistry and physics to explain how electrons move through chemical processes.

Central to electrochemistry are galvanic or voltaic cells, which convert chemical energy into electrical energy through spontaneous redox reactions. A redox reaction includes two half-reactions, one reduction, and one oxidation. The electrochemical potential differences between these half-reactions allow for the movement of electrons through an external circuit, generating current.

To understand electrochemical cells better, it's important to grasp:

• Electrode Potential: Dictates the energy level available for electron movement between chemicals.
• Cell Potential: The overall voltage or electromotive force between two electrodes in a cell.
• Standard Cell Potential: Calculated using standard reduction potentials from redox reactions provided within electrochemical tables.

By determining these factors, chemists can predict the feasibility of redox processes and harness energy effectively. Hence, the understanding of electrochemistry is critical when applying textbook solutions to solve such problems, as conditions such as concentration, temperature, and pressure might affect standard potentials.

Half-Reaction Equations

Half-reaction equations are an indispensable tool in understanding redox reactions in electrochemistry. Each half-reaction represents either an oxidation or a reduction process, depicting the transfer of electrons between chemical species.

1. **Reduction Half-Reaction** - This involves the gain of electrons where a substance decreases its oxidation state.2. **Oxidation Half-Reaction** - This process includes the loss of electrons resulting in an increase in the oxidation state of the substance.
In our exercise, we come across two reduction half-reactions:

• \( \mathrm{Cu}^{2+} + 2e^- \rightarrow \mathrm{Cu} \)
• \( \mathrm{Cu}^{2+} + e^- \rightarrow \mathrm{Cu}^+ \)

To solve for the unknown reduction potential of another half-reaction, enabled equation renderingecuation: \( \mathrm{Cu}^+ \rightarrow \mathrm{Cu} \), we use these known equations in combination.

Such scenarios highlight how equations are constructed and resolved in electrochemistry. Understanding half-reaction equations, thus, provides insight into the stoichiometry of reactions and helps in the calculation of reaction potentials, which is essential for determining a reaction's spontaneity.