Entropy is a state function and its value depends on two or three variable temperature (T), Pressure(P) and volume (V). Entropy change for an ideal gas having number of moles(n) can be determined by the following equation.
`DeltaS=2.303 "nC"_(v)"log"((T_(2))/(T_(1))) + 2.303 "nR log" ((V_(2))/(V_(1)))`
`DeltaS=2.303 "nC"_(P) "log"((T_(2))/(T_(1))) + 2.303 "nR log" ((P_(1))/(P_(2)))`
Since free energy change for a process or a chemical equation is a deciding factor of spontaneity , which can be obtained by using entropy change `(DeltaS)` according to the expression, `DeltaG = DeltaH -TDeltaS` at a temperature `T`
What would be the entropy change involved in thermodynamic expansion of `2` moles of a gas from a volume of `5` L to a volume of `50` L at `25^(@)` C [Given `R=8.3 J//"mole"-K]`

Options

A. `38.23` J/K mol
B. `26.76` J/K
C. `20` J/k
D. `28.23` J/K

Explanation

To find the entropy change involved in the thermodynamic expansion of 2 moles of a gas from a volume of 5 L to a volume of 50 L at 25°C, we can use the formula for entropy change for an ideal gas during an isothermal expansion:

\[ \Delta S = n \cdot 2.303 \cdot R \cdot \log\left(\frac{V_2}{V_1}\right) \]

### Step-by-Step Solution:

1. **Identify the Variables**:
- Number of moles, \( n = 2 \) moles
- Initial volume, \( V_1 = 5 \) L
- Final volume, \( V_2 = 50 \) L
- Gas constant, \( R = 8.3 \, \text{J/(mole·K)} \)
- Temperature, \( T = 25^\circ C = 298 \, \text{K} \) (not needed for this calculation since the process is isothermal)

2. **Calculate the Volume Ratio**:
\[ \frac{V_2}{V_1} = \frac{50 \, \text{L}}{5 \, \text{L}} = 10 \]

3. **Calculate the Logarithm**:
\[ \log\left(\frac{V_2}{V_1}\right) = \log(10) = 1 \]

4. **Substitute Values into the Entropy Change Formula**:
\[ \Delta S = 2 \cdot 2.303 \cdot 8.3 \cdot 1 \]

5. **Perform the Calculation**:
\[ \Delta S = 2 \cdot 2.303 \cdot 8.3 \]
\[ = 2 \cdot 19.1159 \approx 38.23 \, \text{J/K} \]

### Final Answer:
The entropy change involved in the thermodynamic expansion is approximately \( \Delta S \approx 38.23 \, \text{J/K} \).