If the `Pb^(2+)` ion concentration is maintained at `1.0 M` , what is the `[Cu^(2+)]` concentration when the cell potential drops to zero? `E^(o) ""_(cell)= 0.473V`, and cell reaction is `Pb (s)abs(Pb^(2+ )(1.0M))abs(Cu^(2+)(1.0xx10^(-4)M)) Cu(s)`

Options

A. `1xx10^(-16)M`
B. `1xx10^(16)M`
C. `1xx10^(-14)M`
D. `1xx10^(14)M`

Answer

To solve the problem, we need to find the concentration of \([Cu^{2+}]\) when the cell potential drops to zero. We will use the Nernst equation for this calculation.

### Step-by-Step Solution:

1. **Identify the Cell Reaction**:
The cell reaction is given as:
\[ Pb(s) + Cu^{2+}(aq) \rightarrow Pb^{2+}(aq) + Cu(s) \]
Here, lead (Pb) is oxidized to lead ions (\(Pb^{2+}\)), and copper ions (\(Cu^{2+}\)) are reduced to copper metal (Cu).

2. **Determine the Standard Cell Potential**:
The standard cell potential \(E^\circ_{cell}\) is given as \(0.473 V\).

3. **Use the Nernst Equation**:
The Nernst equation is given by:
\[ E_{cell} = E^\circ_{cell} - \frac{0.0591}{n} \log \frac{[Pb^{2+}]}{[Cu^{2+}]} \]
where \(n\) is the number of moles of electrons transferred in the balanced equation. In this case, \(n = 2\) because 2 electrons are transferred.

4. **Set the Cell Potential to Zero**:
Since we want to find the concentration when the cell potential drops to zero, we set \(E_{cell} = 0\):
\[ 0 = 0.473 - \frac{0.0591}{2} \log \frac{[1.0]}{[Cu^{2+}]} \]

5. **Rearranging the Equation**:
Rearranging the equation gives:
\[ \frac{0.0591}{2} \log \frac{[1.0]}{[Cu^{2+}]} = 0.473 \]
\[ \log \frac{[1.0]}{[Cu^{2+}]} = \frac{0.473 \times 2}{0.0591} \]

6. **Calculating the Logarithm**:
Calculate the right-hand side:
\[ \log \frac{[1.0]}{[Cu^{2+}]} = \frac{0.946}{0.0591} \approx 16.01 \]

7. **Exponentiating to Find \([Cu^{2+}]\)**:
From the logarithmic equation:
\[ \frac{1.0}{[Cu^{2+}]} = 10^{16.01} \]
Therefore:
\[ [Cu^{2+}] = \frac{1.0}{10^{16.01}} \approx 10^{-16.01} \text{ M} \]

8. **Final Result**:
Thus, the concentration of \([Cu^{2+}]\) when the cell potential drops to zero is approximately:
\[ [Cu^{2+}] \approx 10^{-16} \text{ M} \]