The result of the previous problem holds for other classes o
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The result of the previous problem holds for other classes of particles, for instance, molecules of an ideal gas, provid
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The result of the previous problem holds for other classes of particles, for instance, molecules of an ideal gas, provided the energies of the particles are mainly the kinetic energies of their translational motions. In that result, we can let N and M grow very large in such a way that the largest gap between adjacent values of j approaches zero in length. In the limit, the kinetic energy of each atom is a function of its mass m and speed v: D 1 2 mv2. Consider for the moment only the part of the kinetic energy of the particle due to its velocity ui in the x direction. The number of atoms for which the x-component of velocity is u will be given by a density function n.u/ satisfying, by the result of the previous exercise, n.u/ D A e?Bmu2=2: (a) Show that p.u/ D n.u/ N is a normally distributed probability density function. What are the values of the mean and variance of u? (See Definition 7 in Section 7.8 and the following discussion.) Express the value of A in terms of B, m, and N. (b) Find the expectation of u2 for the random variable u, and hence the expected value of the part of the kinetic energy of a random particle in our system due to its motion in the x direction. What is the expected value of the total kinetic energy of a random particle in the system, and of all the particles? (c) Use the formula E D 3 2 NkT (from Exercise 25 of Section 12.6 or the discussion preceding Example 4 in Section 12.8), expressing the energy of an ideal gas at absolute temperature T and consisting of N molecules, to find the value of B. (Here k is the Boltzmann constant.) Hence, show that the probability density function for the number of particles having velocity v in an ideal gas is p.v/ D m 2 kT 3 2 e? m.jvj 2/ 2kT : This is known as the Maxwell-Boltzmann distribution.
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