The internal energy of an ideal gas increases when it

Options

A. Expands
B. Is compressed
C. Is first expanded and then compressed
D. None of these

Answer

To determine when the internal energy of an ideal gas increases, we need to analyze the relationship between internal energy and temperature. The internal energy (U) of an ideal gas is directly proportional to its temperature (T).

### Step-by-Step Solution:

1. **Understanding Internal Energy**:
- The internal energy of an ideal gas is a function of its temperature. For an ideal gas, the internal energy can be expressed as:
\[ U = nC_vT \]
where \( n \) is the number of moles, \( C_v \) is the molar specific heat at constant volume, and \( T \) is the temperature.

2. **Effect of Temperature on Internal Energy**:
- Since internal energy is directly related to temperature, any increase in temperature will result in an increase in internal energy. Therefore, for the internal energy to increase, the temperature must increase.

3. **Analyzing the Given Options**:
- **Option 1: Expanded** - Expansion does not necessarily mean an increase in temperature. If the gas expands adiabatically (without heat exchange), its temperature may decrease, leading to no change or a decrease in internal energy.
- **Option 2: Compressed** - Compression can lead to an increase in temperature if work is done on the gas, but it is not guaranteed. If the gas is compressed isothermally, the temperature remains constant, resulting in no change in internal energy.
- **Option 3: First expand then compress** - Similar to the previous options, the overall change in internal energy depends on the temperature changes during these processes. If the temperature does not increase, the internal energy will not increase.
- **Option 4: None of these** - This option suggests that none of the previous options guarantee an increase in internal energy.

4. **Conclusion**:
- Since the internal energy of an ideal gas increases only when the temperature increases, and none of the provided options ensure a temperature increase, the correct answer is:
\[ \text{Option 4: None of these} \]