The average translational kinetic energy of the molecules of a gas will be doubled if at constant

Options

A. Volume, its pressure is doubled
B. temperature, its pressure is doubled
C. pressure, its volume is halved
D. temperature, its volume is doubled

Explanation

To determine under which condition the average translational kinetic energy of the molecules of a gas will be doubled, we can analyze the relationship between pressure, volume, temperature, and kinetic energy.

### Step-by-step Solution:

1. **Understanding the Relationship**:
The average translational kinetic energy (U) of the gas molecules is given by the formula:
\[ U = \frac{3}{2} nRT \]
where \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the absolute temperature in Kelvin.

2. **Identifying the Effect of Doubling Kinetic Energy**:
To double the average kinetic energy, we need to double the temperature (T), since \( U \) is directly proportional to \( T \). Thus, if \( T \) is doubled, \( U \) will also be doubled.

3. **Analyzing the Options**:
- **Option 1**: Keeping the volume constant and doubling the pressure.
- According to the ideal gas law \( PV = nRT \), if the volume (V) is constant and pressure (P) is doubled, the temperature (T) must also double. Therefore, this option will double the average kinetic energy.
- **Option 2**: Keeping the temperature constant and doubling the pressure.
- If the temperature is constant, the kinetic energy will not change regardless of pressure changes. This option is incorrect.
- **Option 3**: Keeping the pressure constant and halving the volume.
- If the pressure is constant and the volume is halved, the temperature will decrease, leading to a decrease in kinetic energy. This option is incorrect.
- **Option 4**: Keeping the temperature constant and halving the volume.
- Similar to option 2, if the temperature is constant, the kinetic energy will not change. This option is also incorrect.

4. **Conclusion**:
The only option that results in the average translational kinetic energy being doubled is **Option 1**: Keeping the volume constant and doubling the pressure.

### Final Answer:
The average translational kinetic energy of the molecules of a gas will be doubled if, at constant volume, the pressure is doubled.

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