The `E^(@)` at `25^(@)`C for the following reaction at the indicated concentrations is 1.50 V. Calculate the `DeltaG` in kJ`//`mol `25^(@)`C :

Options

A. `-140.94`
B. `-295`
C. `-212`
D. `-422.83`

Answer

To calculate the Gibbs free energy change (ΔG) for the given electrochemical reaction at 25°C, we can follow these steps:

### Step 1: Write down the reaction
The reaction is:
\[ \text{Cr (s)} + 3 \text{Ag}^+ (aq) \rightarrow 3 \text{Ag (s)} + \text{Cr}^{3+} (aq) \]

### Step 2: Identify the number of electrons transferred (n)
In this reaction, chromium (Cr) is oxidized from an oxidation state of 0 to +3, losing 3 electrons. Silver ions (Ag⁺) are reduced from +1 to 0, gaining 3 electrons. Therefore, the number of electrons transferred (n) is:
\[ n = 3 \]

### Step 3: Use the Nernst equation
The Nernst equation relates the standard cell potential (E°) to the cell potential (E) under non-standard conditions:
\[ E = E° - \frac{0.059}{n} \log \left( \frac{[\text{Cr}^{3+}]}{[\text{Ag}^+]^3} \right) \]

Given:
- \( E° = 1.50 \, \text{V} \)
- \( [\text{Cr}^{3+}] = 0.1 \, \text{M} \)
- \( [\text{Ag}^+] = 0.1 \, \text{M} \)

### Step 4: Substitute values into the Nernst equation
Substituting the known values into the Nernst equation:
\[ E = 1.50 - \frac{0.059}{3} \log \left( \frac{0.1}{(0.1)^3} \right) \]

### Step 5: Simplify the logarithm
The logarithm simplifies as follows:
\[ \log \left( \frac{0.1}{(0.1)^3} \right) = \log(0.1) - \log(0.1^3) = \log(0.1) - 3\log(0.1) = -1 + 3 = 2 \]

### Step 6: Calculate E
Now substituting back into the equation:
\[ E = 1.50 - \frac{0.059}{3} \times 2 \]
\[ E = 1.50 - \frac{0.059 \times 2}{3} \]
\[ E = 1.50 - 0.03933 \]
\[ E \approx 1.46067 \, \text{V} \]

### Step 7: Calculate ΔG using the formula
The Gibbs free energy change can be calculated using the formula:
\[ \Delta G = -nFE \]
Where:
- \( F \) (Faraday's constant) = 96500 C/mol

Substituting the values:
\[ \Delta G = -3 \times 96500 \times 1.46067 \]

### Step 8: Calculate ΔG
Calculating this gives:
\[ \Delta G = -3 \times 96500 \times 1.46067 \approx -422843.7 \, \text{J/mol} \]

### Step 9: Convert to kJ/mol
To convert from joules to kilojoules:
\[ \Delta G \approx -422.8437 \, \text{kJ/mol} \]

### Final Answer
Thus, the Gibbs free energy change (ΔG) is approximately:
\[ \Delta G \approx -422.84 \, \text{kJ/mol} \]