(a) Show that the speed distribution function can be written...

(a) Show that the speed distribution function can be written $n(v)=$ $4 \pi^{-1 / 2}\left(v / v_{w}\right)^{2} v_{m}^{-1} e^{-\left(v / v_{-}\right)^{2}},$ where $v_{m}$ is the most probable speed. Consider 1 mole of $\mathrm{mol}$ ecules and approximate $d v$ by $\Delta v=0.01 v_{w}$. Find the number of molecules with speeds in $d v$ at $(b) v=0,(c) v=v_{m},(d) v=2 v_{m},$ and $(e) v=8 v_{m}$.

Key Concepts

Maxwell-Boltzmann Speed Distribution

This concept refers to the statistical distribution of speeds for particles in an ideal gas at a given temperature. It is derived from kinetic theory and provides the probability density that a molecule will have a certain speed. The expression involves factors accounting for the three-dimensionality of the gas (such as the v^2 factor) and an exponential decay term reflecting the Maxwellian energy distribution.

Most Probable Speed

The most probable speed, often denoted v?, is the speed at which the Maxwell-Boltzmann speed distribution function reaches its maximum value. It represents the speed most frequently observed among the molecules in the gas at a given temperature. This parameter helps to understand the typical kinetic behavior of the particles in a thermalized ensemble.

Probability Density and Small Interval Approximation

In the context of molecular speed distributions, the probability density function gives the likelihood that a molecule's speed falls within a specified range. By using a small speed interval (?v), one can approximate the number of molecules with speeds in that range by multiplying the probability density at a given speed by ?v and the total number of molecules. This approach is particularly useful when dealing with discrete approximations in continuous distributions, as seen in estimating the number of molecules traveling at specific speeds.

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