Gases Problems 209 mol- The (a) Determine A so that f(v) = N(v)/N is a true probability density function. i.e.. fo f(v)do = 1. Sket 1. Sketch f(v) versus v. be of (b) Find v and Ums in terms of vo- the (c) Differentiate f(v) with respect to v and set the result equal to zero to find self- the most probable speed Um. mean (d) The standard deviation of the speeds from the mean is defined as con- where the bar denotes the mean value. Show that 51) 0 = [02 - (0)271/2 in general. What is o for this problem? 11-2 At standard temperature and pressure the mean speed of hydrogen molecules is 1.70 X 10 ms". What is the particle flux? 11-3 What is the number density of molecules at a temperature of 77 K in an ultra- high vacuum at 10- torr? 11-4 Compute the rms speed of helium atoms at 2 K; of nitrogen molecules at 27 C; and of mercury atoms at 100 C. 11-5 Compute the mean energy in electron volts and the rms speed in ms" of an elec- tron at 1000 K. At 10,000 K, what fraction of the speed of light is the rms speed? If- 11-6 At what values of the speed does the Maxwell speed distribution have half its maximum value? Give your answers as a constant time (kT/m)'/2. 11-7 (a) Show that the Maxwell speed distribution can be written as N(x) dx = - 4N = xe r dx, VT where x = v/Um. (b) Show that the number of molecules with speeds less than some specified speed vo is given by No-.x. = Nerf (x0) - 2xoe. nt ne where xo = vo/ Umm. Here erf (Xo) is the error function, defined by erf ( x ) = 2 e-" du . (c) Find the fractional number of molecules with speeds less than Um . 11-8 Compute Um, v, and Vims for an oxygen molecule at 300 K. What are the corre- sponding values at 1 11-9 (a) Show that the mean speed of a nitrogen molecule at 300 K is 476 ms ! he ratio of the probability of finding a nitrogen molecule whose (b) What is the rat speed is 476 ms ' at 300 K to the probability of finding a nitrogen molecule