In this question we will focus on the Gaussian Integrals, and some inportant applications thereof. Recall the definition
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In this question we will focus on the Gaussian Integrals, and some inportant applications thereof. Recall the definition of the Gaussian integrals 00 00 1, = ſx"e-dx for neven; = 1. = { x'eer dx for nodd. -30 0 00 1 (a) Using a suitable substitution show that I, = ( xe ex dx=; = 2a 0 (3) (b) Show that the following recursion relation holds for the Gaussian integrals for both n even and n odd: ol I-2 ca Hint: for a well-behaved function you may interchange integration and differentiation (4) The Maxwell-Boltzmann speed distribution for a paticle in a monoatonic ideal gas is given by the following probability density function Nico תר p(v)dy= m 47v ATV exp va dv 2xk,T 2k, T where v is the speed of a particle, mis the mass of a gas particle, and T is the absolute tenperature
(C) Find the most probable speed of a single gas particle in terms of mand T. (5) (d) Determine the magnitude of the average classical linear momentum of a single gas particle p=nw and the second moment of this quantity p? = (mv) * in terms of m and T (12) (e) Use one of the two results obtained in part (d) to determine the average kinetic energy of a gas particle in terms of mand T. Justify your choice of the result that you use! Then use the average kinetic energy of a gas particle to find the kinetic energy of a gas containing N paticles. Comment on the physical significance of your result for the kinetic energy of N particles. (6)