To solve the problem of calculating the entropy change when an ideal gas expands and is heated, we can follow these steps:
### Step 1: Identify Initial Conditions
- Initial volume (V1) = 0.5 L
- Final volume (V2) = 1.0 L
- Initial temperature (T1) = 298 K (room temperature)
- Final temperature (T2) = 373 K
- Pressure (P) = 1 atm
### Step 2: Calculate the Number of Moles (n)
Using the ideal gas law, \( PV = nRT \):
\[
n = \frac{PV}{RT}
\]
Where:
- \( R = 0.082 \, \text{L atm K}^{-1} \text{mol}^{-1} \)
Substituting the values:
\[
n = \frac{(1 \, \text{atm}) \times (0.5 \, \text{L})}{(0.082 \, \text{L atm K}^{-1} \text{mol}^{-1}) \times (298 \, \text{K})}
\]
Calculating this gives:
\[
n = \frac{0.5}{24.4756} \approx 0.0204 \, \text{mol}
\]
### Step 3: Use the Entropy Change Formula
The entropy change (\( \Delta S \)) for an ideal gas can be calculated using the formula:
\[
\Delta S = nC_v \ln\left(\frac{T_2}{T_1}\right) + nR \ln\left(\frac{V_2}{V_1}\right)
\]
Where:
- \( C_v = 12.5 \, \text{J K}^{-1} \text{mol}^{-1} \)
### Step 4: Calculate Each Component
1. **Temperature Change Component**:
\[
nC_v \ln\left(\frac{T_2}{T_1}\right) = 0.0204 \times 12.5 \times \ln\left(\frac{373}{298}\right)
\]
First, calculate \( \frac{373}{298} \approx 1.25 \).
Then, using \( \ln(1.25) \approx 0.2231 \):
\[
nC_v \ln\left(\frac{T_2}{T_1}\right) \approx 0.0204 \times 12.5 \times 0.2231 \approx 0.0571 \, \text{J K}^{-1}
\]
2. **Volume Change Component**:
\[
nR \ln\left(\frac{V_2}{V_1}\right) = 0.0204 \times 0.082 \times \ln\left(\frac{1.0}{0.5}\right)
\]
Since \( \frac{1.0}{0.5} = 2 \), we have \( \ln(2) \approx 0.693 \):
\[
nR \ln\left(\frac{V_2}{V_1}\right) \approx 0.0204 \times 0.082 \times 0.693 \approx 0.0011 \, \text{J K}^{-1}
\]
### Step 5: Combine the Components
Now, sum both components to find the total entropy change:
\[
\Delta S \approx 0.0571 + 0.0011 \approx 0.0582 \, \text{J K}^{-1}
\]
### Step 6: Final Calculation
After rounding, we find:
\[
\Delta S \approx 0.058 \, \text{J K}^{-1}
\]
### Conclusion
The entropy change when the ideal gas expands and is heated under the given conditions is approximately \( 0.058 \, \text{J K}^{-1} \).
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