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

Answer

To determine which set of values for \( C_v \) and \( C_p \) of an ideal gas is most reliable, we can use the relationships between these specific heats for an ideal gas.

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- \( C_v \) is the specific heat at constant volume.
- \( C_p \) is the specific heat at constant pressure.
- For an ideal gas, the relationship between \( C_p \) and \( C_v \) is given by:
\[ C_p = C_v + R \]
where \( R \) is the ideal gas constant.

2. **Values for Monatomic and Diatomic Gases**:
- For a monatomic ideal gas, the values are:
\[ C_v = \frac{3}{2} R \quad \text{and} \quad C_p = \frac{5}{2} R \]
- For a diatomic ideal gas, the values are:
\[ C_v = \frac{5}{2} R \quad \text{and} \quad C_p = \frac{7}{2} R \]

3. **Converting to Caloric Units**:
- The value of \( R \) in calories is approximately \( 2 \, \text{cal} \, \text{mole}^{-1} \, \text{K}^{-1} \).
- Therefore, for a diatomic gas:
\[ C_v = \frac{5}{2} \times 2 = 5 \, \text{cal} \, \text{mole}^{-1} \, \text{K}^{-1} \]
\[ C_p = \frac{7}{2} \times 2 = 7 \, \text{cal} \, \text{mole}^{-1} \, \text{K}^{-1} \]

4. **Evaluating the Options**:
- We need to check the reported values against these theoretical values. The most reliable set will have \( C_v \) and \( C_p \) that closely match the calculated values for diatomic gases.
- The ideal values for \( C_v \) and \( C_p \) for diatomic gases are \( 5 \) and \( 7 \) respectively.

5. **Conclusion**:
- The set of values that matches \( C_v = 5 \) and \( C_p = 7 \) is the most reliable.

### Final Answer:
The most reliable set of values for \( C_v \) and \( C_p \) is:
- \( C_v = 5 \, \text{cal} \, \text{mole}^{-1} \, \text{K}^{-1} \)
- \( C_p = 7 \, \text{cal} \, \text{mole}^{-1} \, \text{K}^{-1} \)