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

Options

A. `q=0, DeltaT ne 0, W=0`
B. `q ne0, DeltaT=0, W=0`
C. `q=0, DeltaT=0, W=0`
D. `q=0, DeltaT lt 0, W ne 0`

Explanation

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

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**:
- In an adiabatic process, there is no heat transfer (Q = 0) between the system and the surroundings.

2. **Identify the Nature of Free Expansion**:
- Free expansion is a process where a gas expands into a vacuum without doing work (W = 0) and without heat exchange.

3. **Apply the First Law of Thermodynamics**:
- The First Law of Thermodynamics states:
\[ \Delta U = Q + W \]
- Since Q = 0 (no heat transfer) and W = 0 (no work done), we can substitute these values into the equation:
\[ \Delta U = 0 + 0 = 0 \]

4. **Conclusion on Internal Energy Change**:
- Since \(\Delta U = 0\), this means there is no change in the internal energy of the gas during the free expansion.

5. **Relate Internal Energy to Temperature**:
- For an ideal gas, the internal energy is a function of temperature. Therefore, if there is no change in internal energy, the temperature must also remain constant:
\[ \Delta T = 0 \]

6. **Summarize the Findings**:
- In summary, for the free expansion of an ideal gas under adiabatic conditions:
- Q = 0
- W = 0
- \(\Delta U = 0\)
- \(\Delta T = 0\)

7. **Choose the Correct Option**:
- From the options provided, the correct statement would be that Q = 0 and W = 0, leading to \(\Delta U = 0\) and \(\Delta T = 0\).

### Final Answer:
The correct option for the free expansion of an ideal gas under adiabatic conditions is that Q = 0, W = 0, and \(\Delta T = 0\).