The molar entropy content of `1 ` mole of oxygen `(O_(2))` gas at `300 K` and `1 atm` is `250 J mol e^(-1)K^(-1)`. Calculate `DeltaG` when 1 mole of oxygen is expanded reversibility and isothermally from `300K`, 1 atm to double its volume ( Take `R=8.314J mol e^(-1)K^(-1),log e=2.303)`

Options

A. `1.728 KJ mol e^(-1)K^(-1)`
B. 0
C. `-1.728 KJ mol e^(-1)K^(-1)`
D. `0.75KJ mol e^(-1)K^(-1)`

Explanation

To calculate the change in Gibbs free energy (ΔG) when 1 mole of oxygen gas (O₂) is expanded reversibly and isothermally from 300 K and 1 atm to double its volume, we can follow these steps:

### Step 1: Understand the Given Information
- Molar entropy of O₂, S = 250 J/(mol·K)
- Temperature, T = 300 K
- Initial volume, V₁ = x (assume)
- Final volume, V₂ = 2x (since it is doubled)
- Ideal gas constant, R = 8.314 J/(mol·K)

### Step 2: Use the Gibbs Free Energy Equation
For an isothermal process, the change in Gibbs free energy can be expressed as:
\[ \Delta G = \Delta H - T \Delta S \]

### Step 3: Determine ΔH and ΔS
Since the process is isothermal and there is no change in temperature:
- The change in internal energy (ΔU) is 0 for an ideal gas during isothermal expansion.
- Therefore, ΔH = 0.

Thus, the equation simplifies to:
\[ \Delta G = -T \Delta S \]

### Step 4: Calculate ΔS (Change in Entropy)
For an isothermal reversible expansion, the change in entropy (ΔS) can be calculated using the formula:
\[ \Delta S = nR \ln\left(\frac{V_2}{V_1}\right) \]

Substituting the values:
- n = 1 mole
- R = 8.314 J/(mol·K)
- \( \frac{V_2}{V_1} = \frac{2x}{x} = 2 \)

Thus,
\[ \Delta S = 1 \cdot 8.314 \cdot \ln(2) \]

Using the relationship \( \ln(2) \approx 0.693 \):
\[ \Delta S = 8.314 \cdot 0.693 \]
\[ \Delta S \approx 5.764 \, \text{J/K} \]

### Step 5: Substitute ΔS into the ΔG Equation
Now substituting ΔH and ΔS back into the ΔG equation:
\[ \Delta G = -T \Delta S \]
\[ \Delta G = -300 \cdot 5.764 \]
\[ \Delta G \approx -1729.2 \, \text{J} \]

### Step 6: Convert to Kilojoules
To convert ΔG from joules to kilojoules:
\[ \Delta G \approx -1.729 \, \text{kJ} \]

### Final Answer
Thus, the change in Gibbs free energy (ΔG) when 1 mole of oxygen gas is expanded isothermally and reversibly to double its volume is approximately:
\[ \Delta G \approx -1.729 \, \text{kJ} \]