An ideal gas undergoes a process in which co-efficient of volume expansion of gas `gamma`, varies with absolute temperature by the relation `gamma=(2)/(T)`. Let C is molar heat capacity in this process and `C_(p).C_(V)` are molar heat capacities at constant pressure and volume respectively. Then

Options

A. `C=3C_(V)+2R`
B. `C=2C_(V)+2R`
C. `C=C_(V)+2R`
D. `C=C_(V)-R`

Answer

To solve the problem, we need to find the molar heat capacity \( C \) of an ideal gas undergoing a specific process where the coefficient of volume expansion \( \gamma \) varies with absolute temperature \( T \) as given by the relation \( \gamma = \frac{2}{T} \).

### Step-by-Step Solution:

1. **Understanding the Coefficient of Volume Expansion**:
The coefficient of volume expansion \( \gamma \) is defined as:
\[ \gamma = \frac{1}{V} \left( \frac{dV}{dT} \right) \]
Given that \( \gamma = \frac{2}{T} \), we can equate this to the definition:
\[ \frac{dV}{dT} = \gamma V = \frac{2V}{T} \]

2. **Rearranging the Equation**:
We can rearrange the equation to express \( dV \):
\[ dV = \frac{2V}{T} dT \]

3. **Applying the First Law of Thermodynamics**:
According to the first law of thermodynamics:
\[ dq = du + dw \]
Where:
- \( dq \) is the heat added to the system,
- \( du \) is the change in internal energy,
- \( dw \) is the work done by the system.

We can express these in terms of molar heat capacities:
\[ dq = nC dT, \quad du = nC_V dT, \quad dw = P dV \]

4. **Substituting for Work Done**:
The work done \( dw \) can be expressed as:
\[ dw = P dV = P \left( \frac{2V}{T} dT \right) \]
Using the ideal gas law \( PV = nRT \), we can substitute \( P \):
\[ P = \frac{nRT}{V} \]
Therefore:
\[ dw = \frac{nRT}{V} \cdot \frac{2V}{T} dT = 2nR dT \]

5. **Combining the Equations**:
Now substituting \( du \) and \( dw \) back into the first law:
\[ nC dT = nC_V dT + 2nR dT \]
Dividing through by \( n dT \) (assuming \( dT \neq 0 \)):
\[ C = C_V + 2R \]

6. **Final Result**:
Thus, the molar heat capacity \( C \) in this process is given by:
\[ C = C_V + 2R \]

### Conclusion:
The correct answer is \( C = C_V + 2R \), which corresponds to option number 3 in the provided choices.