The pressure in interplanetary space is estimated to be of...

The pressure in interplanetary space is estimated to be of the order of $10^{-14}$ Pa. Calculate $(a)$ the average number of molecules per cubic centimeter, $(b)$ the collision frequency, and $(c)$ the mean free path in miles. Assume that only hydrogen atoms are present and that the temperature is 1000 K. Assume that $d=0.2 \mathrm{nm}$.

Key Concepts

Mean Free Path

The mean free path is the average distance a molecule travels between successive collisions. It is inversely related to the product of the number density and the collision cross-sectional area. In regions with very low density, like interplanetary space, the mean free path can be extremely large, illustrating the sparse nature of particle interactions in such environments. This measure is vital for understanding transport phenomena and diffusion in gaseous systems.

Ideal Gas Law

The ideal gas law (PV = NkT or its variants) is central to connecting macroscopic properties like pressure and temperature to microscopic properties such as the number density of gas molecules. In contexts like interplanetary space, where the gas is extremely rarefied, the ideal gas law provides a first approximation to determine the average number of molecules in a given volume by relating pressure, temperature, and particle density.

Kinetic Theory of Gases

Kinetic theory of gases explains the behavior of gas particles in terms of their motions and collisions. It allows us to derive quantities like the average speed of molecules, collision frequency, and mean free path based on parameters such as temperature, molecular mass, and effective molecular diameter. This theory forms the framework for understanding the impact of few, fast-moving particles in low-density environments like interplanetary space.

Number Density

Number density refers to the number of molecules per unit volume and is calculated using the ideal gas law by dividing the pressure by the product of the Boltzmann constant and temperature. This concept is crucial when transitioning from a macroscopic measure of pressure to a microscopic perspective that quantifies the concentration of molecules in a specified volume, such as converting to molecules per cubic centimeter for practical comparison.

Collision Frequency

Collision frequency quantifies the average number of collisions experienced by a molecule per unit time. It is computed using the number density, the collision cross-sectional area (which depends on the effective diameter of the molecules), and the average velocity of the molecules derived from thermal energy. This concept is fundamental in predicting how often molecules interact, which affects energy transfer and reaction rates.

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