For a gas, molar specific heat in a process is greater then `C_V` . Which of the following relation is possible?

Options

A. `P^2 V^(1/3)=C`
B. `PV^2 = C`
C. `PV^3 = C`
D. ` P^2 V^2 = C`

Explanation

To solve the problem, we need to analyze the relationship between the molar specific heat in a process (C) and the molar specific heat at constant volume (Cv). We know that for a polytropic process, the molar specific heat can be expressed as:

\[ C = C_V + \frac{R}{1 - n} \]

Where:
- \( C \) is the molar specific heat in a process.
- \( C_V \) is the molar specific heat at constant volume.
- \( R \) is the universal gas constant.
- \( n \) is the polytropic index.

Given that \( C > C_V \), we can rearrange the equation:

\[ C - C_V > 0 \]

This implies:

\[ \frac{R}{1 - n} > 0 \]

For this inequality to hold true, the denominator \( (1 - n) \) must be positive, which means:

\[ 1 - n > 0 \]
\[ n < 1 \]

Thus, for the molar specific heat in a process to be greater than \( C_V \), the polytropic index \( n \) must be less than 1.

Next, we need to evaluate the provided options to see which one satisfies \( n < 1 \).

1. **Option 1:** \( P^2 V^{1/3} = \text{constant} \)
- This can be rewritten as \( PV^{1/6} = \text{constant} \) which implies \( n = \frac{1}{3} < 1 \) (Valid)

2. **Option 2:** \( P^2 V^2 = \text{constant} \)
- Here, \( n = 2 \) which is not less than 1 (Invalid)

3. **Option 3:** \( P^3 V^3 = \text{constant} \)
- Here, \( n = 3 \) which is also not less than 1 (Invalid)

4. **Option 4:** \( PV = \text{constant} \)
- This implies \( n = 1 \) which is not less than 1 (Invalid)

From the analysis, only **Option 1** satisfies the condition \( n < 1 \).

### Final Answer:
The possible relation is given by **Option 1**: \( P^2 V^{1/3} = \text{constant} \).