Consider an ideal gas enclosed in a spectral tube. When a high voltage is placed across...

Consider an ideal gas enclosed in a spectral tube. When a high voltage is placed across the tube, many atoms are excited, and all excited atoms emit electromagnetic radiation at characteristic frequencies. According to the Doppler effect, the frequencies observed in the laboratory depend on the velocity of the emitting atom. The nonrelativistic Doppler shift of radiation emitted in the x direction is . The resulting wavelengths observed in the spectroscope are spread to higher and lower values because of the (respectively) lower and higher frequencies, corresponding to negative and positive values of v_x. We say that the spectral line has been Doppler broadened. This is what allows us to see the lines easily in the spectroscope, because the Heisenberg uncertainty principle does not cause significant line broadening in atomic transitions. (a) What is the mean frequency of the radiation observed in the spectroscope? (b) To get an idea of how much the spectral line is broadened at particular temperatures, derive an expression for the standard deviation of frequencies, defined to be

Standard deviation

Your result should be a function of f ₀, T, and constants. (c) Use your results from (b) to estimate the fractional line width, defined by the ratio of the standard deviation to f₀, for hydrogen (H₂) gas at T = 293 K. Repeat for a gas of atomic hydrogen at the surface of a star, with T = 5500 K.