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625_Physics ProblemsTechnical Physics

625_Physics ProblemsTechnical Physics - Chapter 21 P21.64(a...

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Chapter 21 627 P21.64 (a) The effect of high angular speed is like the effect of a very high gravitational field on an atmosphere. The result is: The larger-mass molecules settle to the outside while the region at smaller r has a higher concentration of low-mass molecules. (b) Consider a single kind of molecules, all of mass m. To cause the centripetal acceleration of the molecules between r and r dr + , the pressure must increase outward according to F ma r r = . Thus, PA P dP A nmAdr r + = − a f b g e j ω 2 where n is the number of molecules per unit volume and A is the area of any cylindrical surface. This reduces to dP nm rdr = ω 2 . But also P nk T B = , so dP k Tdn B = . Therefore, the equation becomes dn n m k T rdr B = ω 2 giving dn n m k T rdr n n B r 0 2 0 z z = ω or ln n m k T r n n B r a f 0 2 2 0 2 = F H G I K J ω ln n n m k T r B 0 2 2 2 F H G I K J = ω and solving for n : n n e mr k T B = 0 2 2 2 ω . P21.65 First find v av 2 as v N v N dv v av 2 = z 1 2 0 . Let a m k T B = 2 . Then, v N a N v e a a a k T m av dv B av 2 = = = z 4 4 3 8
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