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handout06 - 2.22 The Maxwell Speed Distribution We now know...

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Unformatted text preview: 2.22 The Maxwell Speed Distribution We now know the velocity distribution of the particles in each direction v x , v y , v z . QUESTION: What is the probability density of the overall particle speed f s ( v ) regardless of direction, where v = q v 2 x + v 2 y + v 2 z ? ANSWER: All particles at a specific speed v lie on a three dimensional sphere with radius v . The infinitesimal volume of a thin shell of thickness d v of this sphere is d S ( v ) = 4 v 2 d v . (2.195) Then f s ( v ) d v = d S ( v ) f ( v ) (2.196) = 4 v 2 d v f ( v ) (2.197) f s ( v ) = 4 v 2 m 2 k B T 3/2 e- mv 2 2 k B T . (2.198) 289 200 400 600 800 1000 1200 500 1000 1500 2000 1000 2000 3000 4000 5000 6000 7000 f s (v) speed v (m/s) Particle Speed Probability Density Function at T=300K speed v (km/h) 12 a.m.u. 40 a.m.u. v ave = 2619 km/h v ave = 1435 km/h The Average Speed The average particle speed is v ave = Z vf s ( v ) d v (2.199) = 4 m 2 k B T 3/2 Z v 3 e- m v 2 2 k B T d v (2.200) = 4 m 2 k B T 3/2 1 2 2 k B T m 2 (2.201) = 2 2/3 2/3 m 2 4/3 k 4/3 B T 4/3 m 4/3 2 k B T ! 3/2 (2.202) 290 = 2 k 1/3 B T 1/3 1/3 m 1/3 ! 3/2 (2.203) = 8 k B T m 1/2 , (2.204) where we have used the following identity: Z e- ax 2 x 2 n + 1 d x = n ! 2 a n + 1 (2.205) for n = 1 : Z e- ax 2 x 3 d x = 1! 2 a 2 = 1 2 1 a 2 . (2.206) 291 2.23 Some Interesting Applications AIM: Show some cool applications that make use of the phase diagram of materials. The applications discussed here are: Thermosiphons Heat Pipes fi Vacuum Distillation fl Freeze-Drying Thermosiphon BACKGROUND: We can transfer heat from one place to another using a fluid circulating in a loop. IDEA: We can move fluid around a loop without using a pump. Heat Source (e.g. Solar Collector) Heat Sink (e.g. Boiler) Cold Fluid Falls Hot Fluid Rises Up hot fluid is less dense than cold fluid it rises (Convection). Disadvantage: heat source must be placed lower than heat sink. 292 Heat Pipe HOT END COLD END Wick evaporation zone condensation zone return in fluid form by wicking We place a working fluid (usually at low pressure) in a tube....
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handout06 - 2.22 The Maxwell Speed Distribution We now know...

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