M In a certain particle accelerator, protons travel around a...

$ \mathrm{M}$ In a certain particle accelerator, protons travel around a circular path of diameter $23.0 \mathrm{~m}$ in an evacuated chamber, whose residual gas is at $295 \mathrm{~K}$ and $1.00 \times 10^{-6}$ torr pressure. (a) Calculate the number of gas molecules per cubic centimeter at this pressure. (b) What is the mean free path of the gas molecules if the molecular diameter is $2.00 \times 10^{-8} \mathrm{~cm}$ ?

Key Concepts

Ideal Gas Law

The ideal gas law relates pressure, volume, and temperature to the number of moles of a gas and is foundational in understanding gas behavior under various conditions. By using this law, one can estimate the number density—the number of particles per unit volume—which is crucial for calculating other properties in kinetic theory and for understanding how gases behave in environments like evacuated chambers.

Number Density

Number density refers to the number of gas molecules per unit volume. It is derived from the ideal gas law and is essential for connecting macroscopic properties (such as pressure and temperature) to microscopic behavior. This concept is used to determine how many particles are available for collisions in a given volume, which directly affects phenomena like diffusion and reaction rates.

Mean Free Path

The mean free path is the average distance a particle travels between collisions in a gas. It is a key concept in kinetic theory that provides insights into the microscopic dynamics of particles. The mean free path depends on both the number density of the particles and the effective collision cross section, influencing transport properties such as viscosity and thermal conductivity.

Collision Cross Section

The collision cross section represents the effective area that a molecule presents for collisions, treating molecules as if they were hard spheres. It is a geometric interpretation based on the size of the molecules and is critical for calculating the mean free path. A larger collision cross section leads to a shorter mean free path, as particles are more likely to interact with one another.

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