In an adibatic process, no transfer of heat takes place between system and surroundings. Choose the correct option for free expansion of an ideal gas under adiabatic conditions from the following :

Options

A. `q = 0, Delta T !=0, W = 0`
B. `q!=0, Delta T = 0, W = 0`
C. `q = 0, Delta T = 0, W = 0`
D. `q =0, Delta T lt 0, W != 0`

Explanation

To solve the problem regarding the free expansion of an ideal gas under adiabatic conditions, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Adiabatic Process**:
- In an adiabatic process, there is no heat exchange between the system and its surroundings. Therefore, we can state:
\[ Q = 0 \]

2. **Identifying Free Expansion**:
- Free expansion refers to the expansion of a gas into a vacuum. In this case, the gas does not perform any work against external pressure. Hence, we can conclude:
\[ W = 0 \]

3. **Applying the First Law of Thermodynamics**:
- The First Law of Thermodynamics states:
\[ \Delta U = Q + W \]
- Substituting the values we have:
\[ \Delta U = 0 + 0 = 0 \]
- This indicates that the change in internal energy (\(\Delta U\)) is zero.

4. **Relating Change in Internal Energy to Temperature**:
- The change in internal energy for an ideal gas can also be expressed as:
\[ \Delta U = n C_v \Delta T \]
- Where \(n\) is the number of moles, \(C_v\) is the molar heat capacity at constant volume, and \(\Delta T\) is the change in temperature.
- Since we established that \(\Delta U = 0\), we can set the equation to:
\[ n C_v \Delta T = 0 \]

5. **Analyzing the Equation**:
- The number of moles \(n\) and the heat capacity \(C_v\) are both positive quantities and cannot be zero. Therefore, for the equation to hold true, we must have:
\[ \Delta T = 0 \]

6. **Conclusion**:
- From our analysis, we have determined that:
- \(Q = 0\)
- \(W = 0\)
- \(\Delta U = 0\)
- \(\Delta T = 0\)

7. **Choosing the Correct Option**:
- Now, we can evaluate the options provided in the question. The correct option should reflect that:
- \(Q = 0\)
- \(\Delta T = 0\)
- \(W = 0\)
- Based on our findings, the correct option is:
- **Option C**: \(Q = 0\), \(\Delta T = 0\), \(W = 0\)