To find the maximum work done by the cell under standard conditions, we can follow these steps:
### Step 1: Identify the Cell Representation
The cell representation is given as:
\[ M(s) | M^{2+}(aq) || Ag^{+}(aq) | Ag(s) \]
### Step 2: Write Down the Given Standard Reduction Potentials
We have the following standard reduction potentials:
- For the metal M:
\[ E^\circ_{M/M^{2+}} = +2.37 \, \text{V} \]
- For the silver-silver chloride electrode:
\[ E^\circ_{Ag/Ag^{+}} = +0.80 \, \text{V} \]
### Step 3: Determine the Half-Cell Reactions
The half-cell reactions are:
- Oxidation at the anode (Metal M):
\[ M(s) \rightarrow M^{2+}(aq) + 2e^- \]
- Reduction at the cathode (Silver):
\[ Ag^{+}(aq) + e^- \rightarrow Ag(s) \]
### Step 4: Calculate the Number of Electrons Transferred (n)
From the half-cell reactions, we see that:
- 2 electrons are transferred in the oxidation of M.
- 1 electron is involved in the reduction of Ag.
To balance the electrons, we will need to multiply the silver half-reaction by 2:
\[ 2Ag^{+}(aq) + 2e^- \rightarrow 2Ag(s) \]
Thus, \( n = 2 \).
### Step 5: Calculate the Standard Cell Potential (E°cell)
The standard cell potential is calculated as:
\[ E^\circ_{cell} = E^\circ_{cathode} - E^\circ_{anode} \]
Where:
- \( E^\circ_{cathode} = E^\circ_{Ag/Ag^{+}} = +0.80 \, \text{V} \)
- \( E^\circ_{anode} = E^\circ_{M/M^{2+}} = +2.37 \, \text{V} \)
Thus,
\[ E^\circ_{cell} = 0.80 \, \text{V} - 2.37 \, \text{V} = -1.57 \, \text{V} \]
### Step 6: Calculate the Maximum Work Done (ΔG)
The maximum work done by the cell can be calculated using the formula:
\[ \Delta G = -nFE^\circ_{cell} \]
Where:
- \( F = 96500 \, \text{C/mol} \) (Faraday's constant)
- \( n = 2 \)
- \( E^\circ_{cell} = -1.57 \, \text{V} \)
Substituting the values:
\[ \Delta G = -2 \times 96500 \, \text{C/mol} \times (-1.57 \, \text{V}) \]
\[ \Delta G = 2 \times 96500 \times 1.57 \]
\[ \Delta G = 303,610 \, \text{J} \]
### Step 7: Convert to Kilojoules
To convert from joules to kilojoules:
\[ \Delta G = 303.61 \, \text{kJ} \]
### Final Answer
The maximum work done by the cell under standard conditions is approximately:
\[ \Delta G \approx 303.61 \, \text{kJ} \]
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