Calculate the maximum work that can be obtained from the decimolar Daniell cell at `25^(@)C`.
Given `E^(c-)._((Zn^(2+)|Zn))=-0.76V` and `E^(c-)._((Cu^(2+)|Cu))=0.34V`

Options

A. `193.0k J`
B. `212.3kJ`
C. `81.06kJ`
D. `40.53kJ`

Explanation

To calculate the maximum work that can be obtained from the decimolar Daniell cell at 25°C, we will follow these steps:

### Step 1: Write the half-reactions and identify the anode and cathode.
- **Anode (oxidation)**:
\[ \text{Zn} \rightarrow \text{Zn}^{2+} + 2e^- \]
- **Cathode (reduction)**:
\[ \text{Cu}^{2+} + 2e^- \rightarrow \text{Cu} \]

### Step 2: Write the overall cell reaction.
The overall reaction can be written as:
\[ \text{Zn} + \text{Cu}^{2+} \rightarrow \text{Zn}^{2+} + \text{Cu} \]

### Step 3: Calculate the standard cell potential \(E^\circ_{\text{cell}}\).
Using the standard reduction potentials provided:
- \(E^\circ_{\text{Zn}^{2+}/\text{Zn}} = -0.76 \, \text{V}\)
- \(E^\circ_{\text{Cu}^{2+}/\text{Cu}} = +0.34 \, \text{V}\)

The standard cell potential is calculated as:
\[ E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}} = 0.34 \, \text{V} - (-0.76 \, \text{V}) = 0.34 \, \text{V} + 0.76 \, \text{V} = 1.10 \, \text{V} \]

### Step 4: Calculate the cell potential under non-standard conditions.
Given that both concentrations of \(\text{Zn}^{2+}\) and \(\text{Cu}^{2+}\) are decimolar (0.1 M), we can use the Nernst equation:
\[ E = E^\circ_{\text{cell}} - \frac{0.0591}{n} \log \left( \frac{[\text{products}]}{[\text{reactants}]} \right) \]

Here, \(n = 2\) (the number of electrons transferred), and the concentrations are:
- \([\text{Zn}^{2+}] = 0.1 \, \text{M}\)
- \([\text{Cu}^{2+}] = 0.1 \, \text{M}\)

Substituting these values into the Nernst equation:
\[ E = 1.10 \, \text{V} - \frac{0.0591}{2} \log \left( \frac{0.1}{0.1} \right) \]
Since \(\log(1) = 0\):
\[ E = 1.10 \, \text{V} - 0 = 1.10 \, \text{V} \]

### Step 5: Calculate the maximum work \(W\).
The maximum work done is given by:
\[ W = -\Delta G = nFE \]
Where:
- \(F\) (Faraday's constant) = 96500 C/mol
- \(n = 2\)
- \(E = 1.10 \, \text{V}\)

Substituting the values:
\[ W = 2 \times 96500 \, \text{C/mol} \times 1.10 \, \text{V} \]
\[ W = 2 \times 96500 \times 1.10 = 212300 \, \text{J} = 212.3 \, \text{kJ} \]

### Final Answer:
The maximum work that can be obtained from the decimolar Daniell cell at \(25^\circ C\) is **212.3 kJ**.

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