I could not understand the solution for the ques below. May be my concepts here are not so good. Plz explain me the reason........

Que. A box containing N molecules of a perfect gas at temperatue T1 and pressure P1. The number of molecules in the box is doubled keeping the Kinetic Energy (Total) constant. If new pressure is P2 and temperature is T2 then..

(1). P2=P1, T2=T1, (2) P2=P1 , T2=T1/2

(3). P2=2P1 , T2= T1 (4) P2=2P1, T2=T1/2

Ans given is (2)

Someone plz explain the reason?????

Thanks in advance..........

Let's break down the problem step by step to clarify the concepts involved. We have a box containing N molecules of a perfect gas at a certain temperature (T1) and pressure (P1). When the number of molecules is doubled while keeping the total kinetic energy constant, we need to analyze how this affects the pressure (P2) and temperature (T2) of the gas.

Understanding Kinetic Energy in Gases

The total kinetic energy of a gas is related to its temperature and the number of molecules. For an ideal gas, the total kinetic energy (KE) can be expressed as:

• KE = (3/2) * N * k * T

Here, k is the Boltzmann constant, N is the number of molecules, and T is the temperature in Kelvin. This equation shows that the total kinetic energy is directly proportional to both the number of molecules and the temperature.

Analyzing the Changes

Initially, we have:

• KE_initial = (3/2) * N * k * T1

When the number of molecules is doubled, we have:

• KE_final = (3/2) * (2N) * k * T2

Since the total kinetic energy is kept constant, we can set these two equations equal to each other:

• (3/2) * N * k * T1 = (3/2) * (2N) * k * T2

By simplifying this equation, we can cancel out common terms:

• N * T1 = 2N * T2

Dividing both sides by N (assuming N is not zero), we get:

• T1 = 2 * T2

This implies:

• T2 = T1 / 2

Relating Pressure and Temperature

Now, let's consider the relationship between pressure, volume, and temperature for an ideal gas, which is given by the ideal gas law:

• P * V = n * R * T

In our case, since we are doubling the number of molecules, we can say that:

• P1 * V = N * k * T1
• P2 * V = (2N) * k * T2

Substituting T2 = T1 / 2 into the second equation gives:

• P2 * V = (2N) * k * (T1 / 2)

This simplifies to:

• P2 * V = N * k * T1

From the first equation, we know that:

• P1 * V = N * k * T1

Thus, we can equate the two expressions:

• P2 * V = P1 * V

Since the volumes are the same, we conclude that:

• P2 = P1

Final Results

From our analysis, we have determined that:

• P2 = P1
• T2 = T1 / 2

This matches option (2) from your question. So, the correct answer is indeed:

• P2 = P1, T2 = T1 / 2

In summary, when the number of molecules is doubled while keeping the total kinetic energy constant, the pressure remains the same, but the temperature is halved. This is a great example of how gas laws interact with kinetic theory, and understanding these relationships is key to mastering gas behavior.