Let `Q and W` denote the amount of heat given to an ideal gas and the work done by it in an adiabatic process.

Options

A. `Q = 0 `
B. `W = 0`
C. `Q = W`
D. `Q!=W`

Answer

To analyze the situation regarding an ideal gas undergoing an adiabatic process, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**:
- In an adiabatic process, there is no heat transfer to or from the system. This means that the amount of heat \( Q \) given to the gas is zero.
- Therefore, we can conclude that:
\[ Q = 0 \]
- This corresponds to **Option 1**.

2. **Apply the First Law of Thermodynamics**:
- The first law of thermodynamics states:
\[ Q = \Delta U + W \]
- Where \( \Delta U \) is the change in internal energy and \( W \) is the work done by the gas.
- Since we established that \( Q = 0 \), we can rewrite the equation as:
\[ 0 = \Delta U + W \]
- Rearranging gives us:
\[ \Delta U = -W \]

3. **Analyze the Change in Internal Energy**:
- For an ideal gas, the internal energy \( U \) is a function of temperature. If the gas undergoes an adiabatic process, there will be a change in temperature, which implies a change in internal energy:
\[ \Delta U \neq 0 \]
- Since \( \Delta U \) is not zero, it follows that \( W \) must also not be zero because of the relationship \( \Delta U = -W \). Therefore:
\[ W \neq 0 \]
- This means **Option 2** is incorrect.

4. **Evaluate the Relationship Between \( Q \) and \( W \)**:
- We have established that \( Q = 0 \) and \( W \neq 0 \).
- Therefore, \( Q \) is not equal to \( W \):
\[ Q \neq W \]
- This means **Option 3** is also incorrect, and **Option 4** is correct.

5. **Conclusion**:
- The correct options are:
- **Option 1**: \( Q = 0 \)
- **Option 4**: \( Q \neq W \)

### Summary of Correct Options:
- **Option 1**: \( Q = 0 \) (Correct)
- **Option 4**: \( Q \neq W \) (Correct)