A thermally insulated container initially contains N molecules of an ideal monatomic gas at an absolute temperature of T. Molecules escape from the container through holes in the wall, and it can be shown that such process at temperature t, the average kinetic energy of escaping molecules is 2kt. How many molecules remain in the container when their temperature falls by half?

To tackle this problem, we need to analyze the behavior of the ideal monatomic gas as it loses molecules through the holes in the container. The key points to consider are the relationship between temperature, kinetic energy, and the number of molecules remaining in the container.

Understanding Kinetic Energy and Temperature

For an ideal monatomic gas, the average kinetic energy (KE) of the molecules is directly related to the temperature. The formula for the average kinetic energy of a single molecule is given by:

• KE = (3/2)kT

Here, k is the Boltzmann constant, and T is the absolute temperature. When the gas is at temperature T, the average kinetic energy of the molecules is:

• KE_initial = (3/2)kT

Behavior of Escaping Molecules

As the gas molecules escape through the holes, the remaining molecules in the container will experience a decrease in temperature. According to the problem, when the temperature of the gas falls to T/2, the average kinetic energy of the escaping molecules is given as:

• KE_escape = 2k(T/2) = kT

This indicates that the escaping molecules have a different average kinetic energy compared to the remaining molecules in the container. The remaining molecules will have an average kinetic energy of:

• KE_remaining = (3/2)k(T/2) = (3/4)kT

Calculating the Number of Remaining Molecules

Now, we need to determine how many molecules remain in the container after the temperature has dropped to T/2. The process of escaping molecules can be thought of as a dynamic equilibrium where the rate of molecules escaping is proportional to their kinetic energy.

Initially, we have N molecules at temperature T. As the temperature decreases, the average kinetic energy of the remaining molecules also decreases. The key point here is that the molecules that escape are those with higher kinetic energy, which corresponds to the higher temperature.

Since the average kinetic energy of the remaining molecules is now lower, we can infer that the number of molecules that remain in the container is proportional to the ratio of the average kinetic energies:

• KE_remaining / KE_initial = (3/4)kT / (3/2)kT = 1/2

This ratio indicates that half of the initial number of molecules remains in the container when the temperature falls to T/2.

Final Result

Therefore, the number of molecules remaining in the container when their temperature falls by half is:

• N_remaining = N * (1/2) = N/2

In summary, when the temperature of the gas in the thermally insulated container is reduced to half its initial value, half of the original molecules will remain in the container. This illustrates the relationship between temperature, kinetic energy, and the behavior of gas molecules in a confined space.