The translational K.E. of the molecules of a gas at absolute temperature (T) can be doubled

Options

A. by increasing T to 4T
B. by increasing T to `sqrt2T`
C. by decreasing T to `T/2`
D. by increasing T to 2T

Explanation

To solve the problem of how to double the translational kinetic energy of the molecules of a gas at absolute temperature \( T \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Kinetic Energy**:
The translational kinetic energy (TKE) of a gas molecule is given by the formula:
\[ KE = \frac{3}{2} k_B T \]
where \( k_B \) is the Boltzmann constant and \( T \) is the absolute temperature.

2. **Identify the Current Kinetic Energy**:
At temperature \( T \), the translational kinetic energy is:
\[ KE = \frac{3}{2} k_B T \]

3. **Determine the New Kinetic Energy**:
To double the kinetic energy, we want:
\[ KE' = 2 \times KE = 2 \times \frac{3}{2} k_B T = 3 k_B T \]

4. **Relate the New Kinetic Energy to Temperature**:
The new kinetic energy can also be expressed in terms of a new temperature \( T' \):
\[ KE' = \frac{3}{2} k_B T' \]
Setting this equal to the doubled kinetic energy:
\[ \frac{3}{2} k_B T' = 3 k_B T \]

5. **Solve for the New Temperature**:
Dividing both sides by \( \frac{3}{2} k_B \):
\[ T' = 2T \]

6. **Conclusion**:
To double the translational kinetic energy of the gas molecules, the temperature must be increased to \( 2T \).

### Final Answer:
The translational kinetic energy of the molecules of a gas at absolute temperature \( T \) can be doubled by increasing the temperature to \( 2T \).

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