If `E_(Cu^(2+)|Cu)^(@) = 0.34V` and `E_(Cu^(2+)|Cu^(+))^(@)= 0.15 V` then the value for disproportion for `Cu^(+)` is :

Options

A. `(-0.19V)`
B. `(-0.38V)`
C. `(0.94)`
D. `(0.38V)`

Explanation

To find the value for the disproportionation potential of \( Cu^+ \), we can follow these steps:

### Step 1: Understand the Reactions
We have two half-reactions involving copper:
1. The reduction of \( Cu^{2+} \) to \( Cu \):
\[ Cu^{2+} + 2e^- \rightarrow Cu \quad (E^\circ = 0.34 \, V) \]
2. The reduction of \( Cu^{2+} \) to \( Cu^+ \):
\[ Cu^{2+} + e^- \rightarrow Cu^+ \quad (E^\circ = 0.15 \, V) \]

The disproportionation reaction for \( Cu^+ \) can be represented as:
\[ 2Cu^+ \rightarrow Cu^{2+} + Cu \]

### Step 2: Write the Gibbs Free Energy Change Equations
The Gibbs free energy change (\( \Delta G \)) for each half-reaction can be calculated using the formula:
\[ \Delta G = -nFE^\circ \]
where \( n \) is the number of moles of electrons transferred and \( F \) is Faraday's constant.

For the first reaction (reduction of \( Cu^{2+} \)):
\[ \Delta G_1 = -2F(0.34) \]

For the second reaction (reduction of \( Cu^{2+} \)):
\[ \Delta G_2 = -F(0.15) \]

### Step 3: Set Up the Disproportionation Reaction
For the disproportionation reaction, we can express its Gibbs free energy change as:
\[ \Delta G_{disproportionation} = \Delta G_1 - 2\Delta G_2 \]

Substituting the values from above:
\[ \Delta G_{disproportionation} = (-2F(0.34)) - (-F(0.15)) \]
\[ = -2F(0.34) + F(0.15) \]
\[ = -2F(0.34) + F(0.15) \]

### Step 4: Calculate the Disproportionation Potential
Now, we can express the overall Gibbs free energy change in terms of the standard potential for the disproportionation reaction:
\[ \Delta G_{disproportionation} = -nFE^\circ_{disproportionation} \]
For the disproportionation of \( Cu^+ \), \( n = 1 \):
\[ -2F(0.34) + F(0.15) = -FE^\circ_{disproportionation} \]

Dividing through by \( -F \):
\[ E^\circ_{disproportionation} = 2(0.34) - 0.15 \]
\[ = 0.68 - 0.15 \]
\[ = 0.53 \, V \]

### Step 5: Final Calculation
Thus, the value for the disproportionation potential of \( Cu^+ \) is:
\[ E^\circ_{disproportionation} = 0.53 \, V \]

### Summary
The value for the disproportionation potential of \( Cu^+ \) is \( 0.53 \, V \).