In an ultrahigh vacuum system, the pressure is measured to...

In an ultrahigh vacuum system, the pressure is measured to be $1.00 \times 10^{-10}$ torr (where 1 torr $=133 \mathrm{~Pa}$ ). Assume that the gas molecules have a molecular diameter of $3.00 \times 10^{-10} \mathrm{~m}$ and that the temperature is $300 \mathrm{~K}$. Find (a) the number of molecules in a volume of $1.00 \mathrm{~m}^{3},(\mathrm{~b})$ the mean free path of the molecules, and (c) the collision frequency, assuming an average speed of $500 \mathrm{~m} / \mathrm{s}$

Key Concepts

Ideal Gas Law

The ideal gas law relates the pressure, volume, temperature, and number of molecules in a gas, providing a way to calculate the number density of gas molecules. It is a fundamental tool in understanding the behavior of gases and is often used to link macroscopic quantities (like pressure and temperature) with microscopic properties (like the number of molecules).

Molecular Number Density

Molecular number density is a measure of the number of molecules present per unit volume in a gas. It is crucial for calculating several kinetic theory properties, including the mean free path and collision frequency, and is typically derived from the ideal gas law under the assumption of ideal gas behavior.

Collision Cross-Section

The collision cross-section represents the effective area that a molecule presents as a target for collisions with other molecules. It is generally calculated from the molecular diameter and is a key parameter in determining the probability of collisions between molecules in a gas.

Mean Free Path

The mean free path describes the average distance a gas molecule travels between successive collisions. It is derived from the number density and the collision cross-section. A longer mean free path indicates fewer collisions, which is an important concept in understanding transport phenomena in gases and the effects of pressure on molecular motion.

Collision Frequency

Collision frequency quantifies the number of collisions experienced by a molecule per unit time. This concept relies on the mean free path and the average molecular speed, and it is fundamental in the kinetic theory of gases when analyzing how energy and momentum are transferred during molecular collisions.

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