An ideal gas with adiabatic exponent (`gamma=1.5`) undergoes a process in which work done by the gas is same as increase in internal energy of the gas. Here R is gas constant. The molar heat capacity C of gas for the process is:

Options

A. C=4R
B. C=0
C. C=2R
D. C=R

Explanation

To solve the problem, we need to find the molar heat capacity \( C \) of an ideal gas undergoing a specific process where the work done by the gas is equal to the increase in its internal energy. We know the adiabatic exponent \( \gamma = 1.5 \).

### Step-by-Step Solution:

1. **Understand the relationship between work done and internal energy**:
Given that the work done \( W \) by the gas is equal to the increase in internal energy \( \Delta U \), we can write:
\[ W = \Delta U \]

2. **Apply the First Law of Thermodynamics**:
The First Law of Thermodynamics states:
\[ \Delta Q = W + \Delta U \]
Since \( W = \Delta U \), we can substitute this into the equation:
\[ \Delta Q = \Delta U + \Delta U = 2\Delta U \]

3. **Express \( \Delta Q \) in terms of heat capacity**:
The heat added to the system can also be expressed in terms of the molar heat capacity \( C \):
\[ \Delta Q = nC \Delta T \]
where \( n \) is the number of moles and \( \Delta T \) is the change in temperature.

4. **Express \( \Delta U \) in terms of heat capacity**:
The change in internal energy can be expressed as:
\[ \Delta U = nC_V \Delta T \]
where \( C_V \) is the molar heat capacity at constant volume.

5. **Substitute \( \Delta U \) into the equation for \( \Delta Q \)**:
From the previous steps, we have:
\[ \Delta Q = 2\Delta U = 2(nC_V \Delta T) \]
Thus,
\[ nC \Delta T = 2(nC_V \Delta T) \]

6. **Cancel out common terms**:
Since \( n \) and \( \Delta T \) are common on both sides (assuming \( n \neq 0 \) and \( \Delta T \neq 0 \)), we can simplify:
\[ C = 2C_V \]

7. **Relate \( C_V \) to \( R \) and \( \gamma \)**:
We know that:
\[ C_V = \frac{R}{\gamma - 1} \]
Substituting \( \gamma = 1.5 \):
\[ C_V = \frac{R}{1.5 - 1} = \frac{R}{0.5} = 2R \]

8. **Substitute \( C_V \) back into the equation for \( C \)**:
Now substituting \( C_V \) into the equation \( C = 2C_V \):
\[ C = 2(2R) = 4R \]

### Final Answer:
The molar heat capacity \( C \) of the gas for the process is:
\[ C = 4R \]