At time t=0, a thin walled vessel of volume V, kept at...

At time $t=0$, a thin walled vessel of volume $V$, kept at constant temperature, contains $N_0$ ideal gas molecules which begin to leak out through a small hole of area $A$. Assuming negligible pressure outside the vessel, calculate the number of molecules leaving through the hole per unit time and the number remaining at time $t$. Express your answer in terms of $N_0, A, V$, and the average molecular velocity, $\bar{v}$.

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Key Concepts

Kinetic Theory of Gases

This theory explains that gas molecules move in random, isotropic directions and possess a range of speeds, typically described statistically by the Maxwell-Boltzmann distribution. It provides the foundation to relate macroscopic quantities such as temperature and pressure to the microscopic behavior of individual molecules, including their average speed and energy.

Effusion

Effusion is the process in which gas molecules escape from a container through a small hole. In kinetic theory, the rate at which molecules impinge on a surface is proportional to the number density of the gas and its average molecular speed, with the flux given by one fourth of the product of density and average speed. This concept explains how molecules leak in systems where the external pressure is negligible.

Exponential Decay in Leaking Systems

When the rate of loss of gas molecules is proportional to the number of molecules present, the system exhibits exponential decay. The differential equation dN/dt = -(constant)·N, where the constant involves the area of the hole, the average speed, and the volume, leads to a solution where the number of molecules decreases exponentially with time. This behavior is typical in processes where effusion determines the rate of loss.

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