An insulated box of volume 2 V is divided into two equal...

An insulated box of volume 2 V is divided into two equal parts by a thin, heat-conducting partition. One side contains a gas of hard-sphere molecules at atmospheric pressure and $T=293 \mathrm{~K}$. (a) Show that the number of molecules striking the partition per unit area and unit time is $n \bar{v} / 4$. (b) A small round hole of radius $r$ is opened in the partition, small enough so that thermal equilibrium between the two sides is maintained via heat conduction through the partition. Calculate the pressure and temperature as functions of time in both halves of the box. (c) Suppose the partition is a non-conductor of heat. Discuss briefly and qualitatively any deviations from the time-dependence of temperature and pressure found in part (b).

An insulated box of volume 2 V is divided into two equal parts by a thin, heat-conducting partition. One side contains a gas of hard-sphere molecules at atmospheric pressure and $T=293 \mathrm{~K}$.
(a) Show that the number of molecules striking the partition per unit area and unit time is $n \bar{v} / 4$.
(b) A small round hole of radius $r$ is opened in the partition, small enough so that thermal equilibrium between the two sides is maintained via heat conduction through the partition. Calculate the pressure and temperature as functions of time in both halves of the box.
(c) Suppose the partition is a non-conductor of heat. Discuss briefly and qualitatively any deviations from the time-dependence of temperature and pressure found in part (b).

Key Concepts

Kinetic Theory of Gases

Kinetic theory links the microscopic motion of gas molecules with macroscopic properties such as pressure and temperature. It assumes that gas molecules move in random directions and interact only through elastic collisions, allowing for the derivation of expressions for pressure, average speed, and the rate of collisions with surfaces.

Molecular Flux

Molecular flux describes the number of molecules striking a unit area per unit time. It is derived from the statistical analysis of molecular speeds (using the Maxwell-Boltzmann distribution) and plays a crucial role in connecting microscopic dynamics to macroscopic effects like pressure and mass transport, especially in processes such as effusion.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution gives the probability distribution of speeds for the molecules in an ideal gas. This statistical framework is fundamental for calculating average speeds, energies, and collision rates, thereby underpinning many key results in kinetic theory including the derivation of molecular flux.

Effusion

Effusion is the process by which gas molecules escape through a small opening into a region of lower pressure or vacuum. The rate of effusion is directly related to the molecular flux and average molecular speed, which in turn influences how pressure and temperature evolve when gases move between connected compartments.

Thermal Conduction and Equilibrium

Thermal conduction refers to the transfer of heat energy between regions of different temperatures within a material or between two contacting systems. In systems with a heat-conducting partition, this process ensures that both sides remain in thermal equilibrium even as other physical parameters like pressure evolve over time.

Adiabatic Processes

Adiabatic processes are those in which no heat is exchanged with the surroundings. When a partition does not conduct heat, the gas in each compartment undergoes adiabatic changes, leading to different thermodynamic behaviors compared to isothermal conditions. This affects the relationship between pressure, temperature, and volume during processes like gas effusion or expansion.

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