Chapter 36, Problem 68P

(III) A certain atom emits light of frequency f₀ when at rest. A monatomic gas composed of these atoms is at temperature T. Some of the gas atoms move toward and others away from an observer due to their random thermal motion. Using the rms speed of thermal motion, show that the fractional difference between the Doppler-shifted frequencies for atoms moving directly toward the observer and directly away from the observer is $\Delta f/f_0\approx 2\sqrt{3kT/m{c}^2}$; assume $m{c}^2\gg 3kT$. Evaluate $\Delta f/f_0$ for a gas of hydrogen atoms at 550 K. [This “Doppler-broadening” effect is commonly used to measure gas temperature, such as in astronomy.]