Probability density functions on ℝ

Introduction

A liter bottle filled with air (at room temperature and pressure) holds approximately 2.7 · 1022 molecules of N2, O2, CO2, … that fly about at high-speed colliding with one another and with the walls of the bottle. We cannot hope to keep track of every particle in such a large ensemble, but in the mid-19th century, Maxwell showed that we can deduce many statistical properties about such a system. For example, he found the probability density

for the speed V of a given gas molecule, see Fig. 12.1. (The parameter

depends on the mass m of the molecule, the absolute temperature T and Boltzmann's constant k = 1.3806 … · 10-23 joule/K.) We use the integral

to find the probability that a molecule will have a speed in the interval v1 < V < v2.

Example Use Maxwell's density to find the fraction of N2 molecules (with m = 4.65 · 10-26 kg) in a warm room (with T = 300 K) that have speeds less than v0 = 1000 km/hr = 278 m/sec.

Solution We use (1) with the parameter

and the Maclaurin series for the exponential to compute

Approximately one-sixth of the N2 molecules have speeds less than 1000 km/hr! ▪

We can use the density function (1) to compute the average (or expected) value of certain functions of the molecular speed V.

Example Use Maxwell's density to find the average speed and the average kinetic energy for an N2 molecule when T = 300 K.