`Delta_rG^@` for the cell with the cell reaction : `Zn_((s))+ Ag_2O_((s)) + H_2O_((l)) to Zn_((aq))^(2+) + 2Ag_((s)) + 2OH_((aq))^(-)`
`[E_(Ag_2O // Ag)^(@)=0.344 V, E_(Zn^(2+)//Zn)^@=-0.76 V]`

Options

A. `2.13xx10^5 "J mol"^(-1)`
B. `-2.13xx10^5 "J mol"^(-1)`
C. `1.06xx10^5 "J mol"^(-1)`
D. `-1.06xx10^5 "J mol"^(-1)`

Explanation

To calculate the standard Gibbs free energy change (Δ_rG^@) for the given cell reaction, we will follow these steps:

### Step 1: Write down the cell reaction
The cell reaction is given as:
\[ \text{Zn}_{(s)} + \text{Ag}_2\text{O}_{(s)} + \text{H}_2\text{O}_{(l)} \rightarrow \text{Zn}^{2+}_{(aq)} + 2\text{Ag}_{(s)} + 2\text{OH}^-_{(aq)} \]

### Step 2: Identify the half-reactions
From the overall cell reaction, we can identify the oxidation and reduction half-reactions:
- Oxidation (Anode):
\[ \text{Zn}_{(s)} \rightarrow \text{Zn}^{2+}_{(aq)} + 2e^- \]
- Reduction (Cathode):
\[ \text{Ag}_2\text{O}_{(s)} + 2e^- \rightarrow 2\text{Ag}_{(s)} + 2\text{OH}^-_{(aq)} \]

### Step 3: Determine the standard reduction potentials
The standard reduction potentials are given as:
- For Ag2O/Ag: \( E^\circ_{\text{Ag}_2\text{O}/\text{Ag}} = 0.344 \, \text{V} \)
- For Zn2+/Zn: \( E^\circ_{\text{Zn}^{2+}/\text{Zn}} = -0.76 \, \text{V} \)

### Step 4: Calculate the standard cell potential (E°cell)
The standard cell potential can be calculated using the formula:
\[ E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}} \]
Substituting the values:
\[ E^\circ_{\text{cell}} = 0.344 \, \text{V} - (-0.76 \, \text{V}) \]
\[ E^\circ_{\text{cell}} = 0.344 \, \text{V} + 0.76 \, \text{V} = 1.104 \, \text{V} \]

### Step 5: Calculate the Gibbs free energy change (Δ_rG^@)
The Gibbs free energy change can be calculated using the formula:
\[ \Delta_rG^@ = -nFE^\circ_{\text{cell}} \]
Where:
- \( n = 2 \) (number of moles of electrons transferred)
- \( F = 96500 \, \text{C/mol} \) (Faraday's constant)

Substituting the values:
\[ \Delta_rG^@ = -2 \times 96500 \, \text{C/mol} \times 1.104 \, \text{V} \]
\[ \Delta_rG^@ = -2 \times 96500 \times 1.104 \]
\[ \Delta_rG^@ = -213,130.4 \, \text{J/mol} \]
\[ \Delta_rG^@ \approx -213.1 \, \text{kJ/mol} \]

### Conclusion
The value of \( \Delta_rG^@ \) for the cell reaction is approximately \( -213.1 \, \text{kJ/mol} \), which corresponds to option B.

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