The following sets of values for `C_(v)` and `C_(p)` of an ideal gas have been reported by different students. The units are cal `"mole"^(-1)K^(-1)`. Which of these sets is most reliable?

Options

A. `C_(v) = 3, C_(p) = 5`
B. `C_(v) = 4, C_(p) = 6`
C. `C_(v) =3, C_(p) = 2`
D. `C_(v) = 3, C_(p) = 4.2`

Answer

To determine which set of values for \( C_p \) and \( C_v \) of an ideal gas is most reliable, we can use the relationship between \( C_p \), \( C_v \), and the universal gas constant \( R \). The relationship is given by:

\[ C_p - C_v = R \]

### Step-by-Step Solution:

1. **Identify the value of R**:
The universal gas constant \( R \) is given as \( 8.314 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1} \). We need to convert this value into calories since the given values for \( C_p \) and \( C_v \) are in calories.

2. **Convert R to calories**:
We know that \( 1 \, \text{cal} = 4.186 \, \text{J} \). Therefore, to convert \( R \) from joules to calories, we divide by \( 4.186 \):

\[ R = \frac{8.314 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1}}{4.186 \, \text{J/cal}} \approx 1.98 \, \text{cal} \, \text{mol}^{-1} \, \text{K}^{-1} \approx 2 \, \text{cal} \, \text{mol}^{-1} \, \text{K}^{-1} \]

3. **Apply the relationship \( C_p - C_v = R \)**:
Now that we have \( R \approx 2 \, \text{cal} \, \text{mol}^{-1} \, \text{K}^{-1} \), we can check each set of values for \( C_p \) and \( C_v \) to see if they satisfy the equation \( C_p - C_v = 2 \).

4. **Evaluate each set**:
- **Set 1**: \( C_p = 5 \), \( C_v = 3 \)
\[ C_p - C_v = 5 - 3 = 2 \quad (\text{Valid}) \]
- **Set 2**: \( C_p = 6 \), \( C_v = 4 \)
\[ C_p - C_v = 6 - 4 = 2 \quad (\text{Valid}) \]
- **Set 3**: \( C_p = 2 \), \( C_v = 3 \)
\[ C_p - C_v = 2 - 3 = -1 \quad (\text{Invalid}) \]
- **Set 4**: \( C_p = 4.2 \), \( C_v = 3 \)
\[ C_p - C_v = 4.2 - 3 = 1.2 \quad (\text{Invalid}) \]

5. **Conclusion**:
The most reliable sets are Set 1 and Set 2, as they both satisfy the equation \( C_p - C_v = 2 \).