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phys 369 09 hw 11 solution

# phys 369 09 hw 11 solution - Phys 369 Fall 2009 Homework#11...

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Phys. 369, Fall 2009 Homework #11 Solution 1. (22 points) a) The diameter of the molecules is 10 2 radius 3.76 10 m . d = × = × The collision cross section is 2 19 2 4.44 10 m . d σ π = = × The number density of the gas is 5 24 3 23 0.5 1.013 10 7.75 10 m . 1.381 10 (273 200) P n kT × × = = = × × × + The mean free path is 7 24 19 1 1 2.05 10 m. 2 2 7.75 10 4.44 10 n σ = = = × × × × × A b) The average speed of the molecules is 1/2 1/2 23 27 8 8 1.381 10 473 m 598 . 28 1.661 10 s kT v m π π × × × = = = × × × The collision rate is 9 1 7 598 2 2.92 10 s . 2.05 10 coll v nv σ Γ = = = = × × A c) The heat capacity per mole at constant volume is 5 , 2 2 v f c R R = = so 5 5 . 2 2 v A A c R k N N = = The thermal conductivity is 24 7 23 5 1.227 1.227 2 W 1.227 7.75 10 598 2.05 10 2.5 1.381 10 0.0403 m K v t A c k nv nv k N = = = × × × × × × × × = A A d) The viscosity is 27 5 19 5 28 1.661 10 598 0.347 0.347 2.17 10 Pa s. 4.44 10 32 2 mv mv π η σ σ × × × = = = = × × e) The distribution function ( , ) f x t for the displacement of the particle in the x -direction obeys the diffusion equation 2 2 self f f D t x = with the initial condition ( ,0) ( ). f x x δ = As we showed in HW 9, problem 1, the solution is 2 4 1/2 1 ( , ) 4 self x D t self f x t e D t π =

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