A box contains `N` molecules of a perfect gas at temperature `T_(1)` and temperature `P_(1)`. The number of molecule in the box is double keeping the total kinetic energy of the gas same as before. If the new pressure is `P_(2)` and temperature `T_(2), then

Options

A. `P_(2) = P_(1), T_(2) = T_(1)`
B. `P_(2) = P_(1), T_(2) = (T_(1))/(2)`
C. `P_(2) = 2P_(1), T_(2) = T_(1)`
D. `P_(2) = 2P_(1), T_(2) = (T_(1))/(2)`

Explanation

To solve the problem step by step, we will analyze the relationship between the initial and final states of the gas in terms of the number of molecules, pressure, and temperature.

### Step 1: Understanding the Initial and Final States
Initially, we have:
- Number of molecules, \( N_1 = N \)
- Temperature, \( T_1 \)
- Pressure, \( P_1 \)

Finally, the number of molecules is doubled:
- Number of molecules, \( N_2 = 2N \)
- Temperature, \( T_2 \)
- Pressure, \( P_2 \)

### Step 2: Kinetic Energy Relation
The total kinetic energy of a gas is given by the formula:
\[ E = \frac{3}{2} N k T \]
where \( k \) is the Boltzmann constant.

For the initial state:
\[ E_1 = \frac{3}{2} N k T_1 \]

For the final state:
\[ E_2 = \frac{3}{2} (2N) k T_2 \]

According to the problem, the total kinetic energy remains the same:
\[ E_1 = E_2 \]
Thus, we have:
\[ \frac{3}{2} N k T_1 = \frac{3}{2} (2N) k T_2 \]

### Step 3: Simplifying the Kinetic Energy Equation
We can cancel out common terms from both sides:
\[ N k T_1 = 2N k T_2 \]

Dividing both sides by \( N k \) (assuming \( N \neq 0 \) and \( k \neq 0 \)):
\[ T_1 = 2 T_2 \]

### Step 4: Relating Pressure and Temperature
Using the ideal gas law, we have:
\[ PV = nRT \]
In terms of molecules, this can be expressed as:
\[ PV = N k T \]

For the initial state:
\[ P_1 V = N k T_1 \]

For the final state:
\[ P_2 V = (2N) k T_2 \]

### Step 5: Setting Up the Equation
Since the volume \( V \) is constant, we can set the two equations equal to each other:
\[ P_1 V = N k T_1 \quad \text{and} \quad P_2 V = (2N) k T_2 \]

This gives us:
\[ P_1 = \frac{N k T_1}{V} \]
\[ P_2 = \frac{(2N) k T_2}{V} \]

### Step 6: Relating Pressures
From the equations, we can relate \( P_1 \) and \( P_2 \):
\[ P_1 = \frac{N k T_1}{V} \quad \text{and} \quad P_2 = \frac{(2N) k T_2}{V} \]

Substituting \( T_1 = 2 T_2 \) into the equation for \( P_2 \):
\[ P_2 = \frac{(2N) k (T_1/2)}{V} \]
\[ P_2 = \frac{N k T_1}{V} = P_1 \]

### Conclusion
Thus, we find that:
\[ P_1 = P_2 \]

### Final Answer
The relationship between the pressures and temperatures is:
\[ P_1 = P_2 \quad \text{and} \quad T_1 = 2 T_2 \]