(a) Show that the root mean square velocty vrms is

Because <v2) = (vx2> + <vy2> + <vx2) and (vx2) = <vy2> = <vx2>, it follows that

<vx2>1/2 = (τ/M)1/2 = vrms/31/2.

The results cart also be obtained directly from the expression in chapter for the average kinetic energy of an ideal gas.

(b) Show that the most probable value of the speed *v*mp is

*vmp =*(2τ/M)1/2.

By most probable value of the speed we mean the maximum of the Maxwell distribution as a function of v. Notice that v∞p < vrms.

(c) Show that the mean sped c is

The mean speed may also be written as <|*v*|>. The ratio

*v*rms/c = 1.086.

(d) Show that cz, the mean of the absolute value of the z component of the velocity of an atom, is

Cz ≡ <|vz|> = 1/2 c = (2τ/πM)1/2.