Serway & Jewett — The Kinetic Theory of Gases: Problem 45.

Equations 20.43 and 20.44 show that \(v_{rms} > v_{avg}\) for a collection of gas particles, which turns out to be true whenever the particles have a distribution of speeds. Let us explore this inequality for a two-particle gas. Let the speed of one particle be \(v_1 = av_{avg}\) and the other particle have speed \(v_2 = (2 - a)v_{avg}\). (a) Show that the average of these two speeds is \(v_{avg}\). (b) Show that \(v_{rms}^2 = v_{avg}^2 (2 - 2a + a^2)\) (c) Argue that the equation in part (b) proves that, in general, \(v_{rms} > v_{avg}\). (d) Under what special condition will \(v_{rms} = v_{avg}\) for the two-particle gas?