A sealed (1)/(4) litre bottle filled with oxygen at a...

A sealed $\frac{1}{4}$ litre bottle filled with oxygen at a pressure of $10^{-4}$ atmospheres is left on the surface of the moon by an astronaut. At a time when the temperature of the bottle is 400 K the jar develops a leak at a thin part of a wall through a small hole of diameter 2 microns. How will the amount of gas in the bottle depend on time and about how long will it take for the gas to decrease to $\frac{1}{10}$ of its original amount? Show your work, estimate any constants needed besides Boltzmann's constant. You may assume that the temperature is maintained constant by the sunlight of the lunar day. $$ k=1.38 \times 10^{-16} \mathrm{erg} / \mathrm{K} . $$

A sealed $\frac{1}{4}$ litre bottle filled with oxygen at a pressure of $10^{-4}$ atmospheres is left on the surface of the moon by an astronaut. At a time when the temperature of the bottle is 400 K the jar develops a leak at a thin part of a wall through a small hole of diameter 2 microns. How will the amount of gas in the bottle depend on time and about how long will it take for the gas to decrease to $\frac{1}{10}$ of its original amount?

Show your work, estimate any constants needed besides Boltzmann's constant. You may assume that the temperature is maintained constant by the sunlight of the lunar day.
$$
k=1.38 \times 10^{-16} \mathrm{erg} / \mathrm{K} .
$$

Key Concepts

Effusion

Effusion is the process by which gas molecules escape from a container through a small hole into a region of lower pressure, typically a vacuum. The rate of effusion depends on the size of the hole, the temperature of the gas (which determines the molecular speeds), and the gas’s molar mass. In this context, effusion is used to model the leakage of oxygen from the jar on the moon.

Exponential Decay

In systems where the rate of change of a quantity is proportional to the amount present, the quantity decays exponentially over time. With a small, constant-efficiency leak, the number of gas molecules decreases proportionally to their current number, leading to an exponential decay behavior. This concept allows us to relate the time constant of decay to the properties of the leak and the initial conditions of the gas.

Gas Kinetic Theory

Gas kinetic theory provides a framework for understanding the behavior of gas molecules in terms of their motion and collisions. It relates macroscopic properties such as pressure and temperature to microscopic molecular dynamics. This theory facilitates the calculation of mean molecular speeds and fluxes that are crucial for determining the rate at which gas molecules effuse through a small opening.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the statistical spread of speeds among molecules in a gas at thermal equilibrium. It is used to derive average molecular speeds which, in turn, help determine the rate of effusion through the small hole. Understanding this distribution is key for estimating how quickly the gas will escape from the bottle.

Time Constant Estimation

The time constant in an exponential decay process characterizes how quickly a system loses a percentage of its original quantity. In this problem, estimating the time constant involves calculating the effusion rate using parameters such as the molecular speed, the area of the leak, and the initial pressure. This approach is central to determining how long it takes for the gas content to reduce to one-tenth of its initial value.

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