What is the balanced half-reaction for the process that occurs at the cathode?

To determine the balanced half-reaction for the process occurring at the cathode in an acidic solution, we need to identify the species involved in the reduction reaction from the given electrochemical cell notation:

\[ \text{C(s) | V}^{2+}(\text{aq}), \text{V}^{3+}(\text{aq}) \| \text{VO}_2^{+}(\text{aq}), \text{VO}^{2+}(\text{aq}) \| \text{C(s)} \]

At the cathode, vanadium is transitioning from a higher to a lower oxidation state, indicating reduction. Thus, we need to determine the reduction half-reaction.

The relevant species at the cathode are \(\text{VO}_2^{+}\) and \(\text{VO}^{2+}\). The reduction half-reaction involves \(\text{VO}_2^{+}\) being reduced to \(\text{VO}^{2+}\).

The half-reaction in acidic solution can be written as:

\[ \text{VO}_2^{+} + 2H^+ + e^- \rightarrow \text{VO}^{2+} + H_2O \]

Now, let's balance this half-reaction. We'll ensure that the number of atoms and charges on both sides of the equation are balanced:

1. **Balance the atoms other than H and O:**
   - V is already balanced (1 V on each side).

2. **Balance oxygen atoms:**
   - There are 2 oxygen atoms on the left side (in \(\text{VO}_2^{+}\)).
   - To balance the oxygens, we need 1 water molecule (\(H_2O\)) on the right side.

3. **Balance hydrogen atoms:**
   - There are 2 hydrogen atoms on the right side (in \(H_2O\)).
   - To balance hydrogens, we need 2 \(H^+\) on the left side.

4. **Balance the charge:**
   - The left side has a total charge of \(+3\) (\(\text{VO}_2^{+}\) is \(+1\) and \(2H^+\) is \(+2\)).
   - The right side has a charge of \(+2\) (\(\text{VO}^{2+}\) is \(+2\) and \(H_2O\) is neutral).
   - To balance the charge, we need to add one electron (\(e^-\)) to the left side.

The balanced reduction half-reaction at the cathode is:

\[ \text{VO}_2^{+} + 2H^+ + e^- \rightarrow \text{VO}^{2+} + H_2O \]

Do you want more details or have any questions about this process? Here are some additional questions you might want to explore:

1. How do you calculate the overall cell potential for this electrochemical cell?
2. How do you combine the half-reactions to write the overall redox reaction?
3. What is the Nernst equation and how is it applied to this cell?
4. How do you determine which species gets oxidized and which gets reduced in a redox reaction?
5. What are the standard reduction potentials for \(\text{V}^{2+}/\text{V}^{3+}\) and \(\text{VO}_2^{+}/\text{VO}^{2+}\)?

**Tip:** Always ensure that the charges and atoms are balanced separately in both half-reactions before combining them into the overall redox reaction.

Explore the balanced half-reaction for the cathode process in electrochemistry. Understand how to determine reduction half-reactions in acidic solutions, using examples from vanadium electrochemistry. Learn about the Nernst equation and its application to electrochemical cells.

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