Let `(C_v) and (C_p)` denote the molar heat capacities of an ideal gas at constant volume and constant pressure respectively . Which of the following is a universal constant?

Options

A. `C_p/ C_v`
B. `C_p C_v`
C. `C_p - C_v`
D. `C_p + C_v`

Answer

To solve the question, we need to identify which of the given options represents a universal constant related to the molar heat capacities \(C_v\) and \(C_p\) of an ideal gas.

### Step-by-Step Solution:

1. **Understanding Molar Heat Capacities**:
- \(C_v\) is the molar heat capacity at constant volume.
- \(C_p\) is the molar heat capacity at constant pressure.

2. **Mayer's Relation**:
- There is a well-known relationship between \(C_p\) and \(C_v\) given by Mayer's equation:
\[ C_p - C_v = R \]
- Here, \(R\) is the universal gas constant.

3. **Identifying the Universal Constant**:
- From Mayer's equation, we can see that \(R\) is a constant value for all ideal gases, which makes it a universal constant.
- The value of \(R\) is approximately \(8.314 \, \text{J/(mol·K)}\) or \(2 \, \text{cal/(mol·K)}\).

4. **Analyzing Other Options**:
- While the ratio \(\frac{C_p}{C_v}\) (denoted as \(\gamma\)) is a ratio of heat capacities, it varies for different types of gases (e.g., \(\gamma\) is approximately \(1.67\) for monoatomic gases and \(1.4\) for diatomic gases). Thus, it is not a universal constant.

5. **Conclusion**:
- The only universal constant among the options given is \(R\), which is derived from the relationship between \(C_p\) and \(C_v\).

### Final Answer:
The universal constant is \(C_p - C_v = R\).