FrontierScience / 1e2f4cbe-4a38-4dd3-9378-14df40f7599f / A thin-walled insulated container of mass `M` (ignoring the thickness of the container) floats in space (considered as a vacuum), initially at rest, filled with `N₀` molecules of n…

A thin-walled insulated container of mass `M` (ignoring the thickness of the container) floats in space (considered as a vacuum), initially at rest, filled with `N₀` molecules of nitrogen gas (`N₂`), at an equilibrium temperature of `T₀`, and with a mass per molecule of nitrogen of `m`. In this case, we assume that `M≫N₀m`. A small hole appears in the container, which is so small that it is much smaller than the mean free length of the gas (that is, `√(A)<<λ` where `A` is the area of the hole and `λ` is the mean free path length). The mean free path length refers to the average distance a gas molecule travels before colliding with another during the random motion. The rate of escape of the gas is so slow that we can treat the gas in the container as an equilibrium state of uniform temperature at every instant. The Boltzmann constant is `k_(B)` and all temperatures are absolute.

When the gas in the container is at equilibrium temperature `T`, what is the average amount of energy (`E_(avg)`) carried by each molecule of nitrogen that escapes from the small breach? Express your answer in terms of the fundamental constants stated above.

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.

A thin-walled insulated container of mass `\(M\)` (ignoring the thickness of the container) floats in space (considered as a vacuum), initially at rest, filled with `\(N_0\)` molecules of nitrogen gas (`\( N_2 \)`), at an equilibrium temperature of `\(T_0\)`, and with a mass per molecule of nitrogen of `\(m\)`. In this case, we assume that `\(M\gg N_0m\)`. A small hole appears in the container, which is so small that it is much smaller than the mean free length of the gas (that is, `\( \sqrt{A} << \lambda \)` where `\( A \)` is the area of the hole and `\( \lambda \)` is the mean free path length). The mean free path length refers to the average distance a gas molecule travels before colliding with another during the random motion. The rate of escape of the gas is so slow that we can treat the gas in the container as an equilibrium state of uniform temperature at every instant. The Boltzmann constant is `\(k_B\)` and all temperatures are absolute.

When the gas in the container is at equilibrium temperature `\(T\)`, what is the average amount of energy (`\( E_{avg} \)`) carried by each molecule of nitrogen that escapes from the small breach? Express your answer in terms of the fundamental constants stated above.

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.