In This Question We Will Focus On The Gaussian Integrals And Some Important Applications Thereof Recall The Definition 1 (53.04 KiB) Viewed 34 times
In This Question We Will Focus On The Gaussian Integrals And Some Important Applications Thereof Recall The Definition 2 (43.05 KiB) Viewed 34 times
In this question we will focus on the Gaussian Integrals, and some important applications thereof. Recall the definition of the Gaussian integrals: 8 In = fx*e*r*dx for neven; = 1. = ( x*ear dx for nodd - 0 1 (a Using a suitable substitution show that I, = ( xerar dx= jx dama 2a (3) (3) (b) Show that the following recursion relation holds for the Gaussian integrals for both n even and n odd: au, ha Hint: for a well-behaved function you may interchange integration and differentiation (4) The Maxwell-Boltzmann speed distribution for a paticle in a monoatomic ideal gas is given by the following probability density function: 3 m m p(v) dv= 41 ev 47vexp dv 2xk T 2k T v av , where v is the speed of a particle, mis the mass of a gas particle, and T is the absolute temperature
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 paticle p=nw and the second moment of this quantity p2 = (nu)? in terms of m and T. (12) (e) Use one of the two results obtained in pat (d) to determine the average kinetic energy of a gas particle in terms of m and 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 particles. Comment on the physical significance of your result for the kinetic energy of N paticles. (6)
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