A box of volume 2 V is divided into halves by a thin...

A box of volume $2 V$ is divided into halves by a thin partition. The left side contains a perfect gas at pressure $p_0$ and the right side is initially vacuum. A small hole of area $A$ is punched in the partition. What is the pressure $p_1$ in the left hand side as a function of time? Assume the temperature is constant on both sides. Express your answer in terms of the average velocity $v$.

Key Concepts

Average Molecular Velocity

The average molecular velocity is a measure of the typical speed of gas molecules at a given temperature. It plays a key role in determining the rate at which particles impinge on, and pass through, a small hole by directly affecting the flux. In effusion problems, this velocity is used to relate microscopic motion to the macroscopic flow of gas and the resulting pressure evolution.

Mass Conservation

Mass conservation (or particle number conservation) in a closed system implies that the decrease in the number of gas molecules in one region must equal the number flowing out. In the context of gas escaping through a small hole, this principle leads to a differential equation that describes the decay of density (and thus pressure) over time.

Exponential Decay Dynamics

When the rate of particle loss through a small aperture is proportional to the current number of particles, the resulting change in pressure follows an exponential decay law. This behavior is derived from the balance between the effusive flux and the remaining number of molecules in the compartment, assuming constant temperature.

Effusion

Effusion is the process by which gas molecules escape through a small opening into a low-pressure region or vacuum. In kinetic theory, the rate of effusion is determined by the number density of molecules and their average speed. This concept is crucial in understanding how gas escapes from a container through a small hole, leading to a time-dependent change in pressure.

Kinetic Theory of Gases

The kinetic theory of gases links the microscopic behavior of individual gas molecules—such as their speeds and collisions—to macroscopic properties like pressure and temperature. This theory provides the framework for analyzing how molecular motion results in a net flux of particles through a small hole, and how this flux relates to pressure changes within a container.

Transcript

Please give Ace some feedback