A container has N molecules at absolute temperature T. If the number of molecules is doubled but kinetic energy in box remains the same as before, the absolute temperature of the gas is

Options

A. T
B. `(T)/(2)`
C. `3T`
D. `4T`

Answer

To solve the problem, we need to analyze the relationship between the number of molecules, kinetic energy, and temperature in a gas.

### Step-by-Step Solution:

1. **Understand the relationship of kinetic energy and temperature**:
The average kinetic energy (KE) of a single molecule in a gas is given by the formula:
\[ KE = \frac{f}{2} k_B T \]
where \( f \) is the degrees of freedom, \( k_B \) is the Boltzmann constant, and \( T \) is the absolute temperature.

2. **Calculate total kinetic energy for N molecules**:
For \( N \) molecules, the total kinetic energy can be expressed as:
\[ KE_{total} = N \cdot \frac{f}{2} k_B T \]

3. **Consider the new scenario**:
If the number of molecules is doubled (i.e., \( N \) becomes \( 2N \)), but the total kinetic energy remains the same, we can denote the new temperature as \( T' \). The total kinetic energy for \( 2N \) molecules at the new temperature will be:
\[ KE'_{total} = 2N \cdot \frac{f}{2} k_B T' \]

4. **Set the total kinetic energies equal**:
Since the total kinetic energy remains the same, we can set the two expressions for kinetic energy equal to each other:
\[ N \cdot \frac{f}{2} k_B T = 2N \cdot \frac{f}{2} k_B T' \]

5. **Simplify the equation**:
We can cancel out common terms from both sides of the equation:
\[ T = 2 T' \]

6. **Solve for the new temperature**:
Rearranging the equation gives us:
\[ T' = \frac{T}{2} \]

### Conclusion:
The new absolute temperature \( T' \) of the gas when the number of molecules is doubled while keeping the kinetic energy constant is:
\[ T' = \frac{T}{2} \]

Thus, the answer is \( T/2 \).