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Homework 5 Key - CHEMISTRY 201X Homework#5"Wavefunctions...

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CHEMISTRY 201X: Homework #5 Due : Monday, October 15, 2007 -1- “Wavefunctions and Electronic Transitions” (1) (8 pts) Using the Bohr model for the hydrogen atom, (a) (2 pts) Derive an expression for the speed v n of an electron in orbit n in terms of h , m , and e , where h is Planck’s constant, m and e are the mass and charge of the electron. 2 2 2 n n 2 2 n nh h nh 2 π e 1 mvr = ; r = n ; v = = 2 π 4 π me 2 π mr h n (b) (2 pts) Calculate the speed (in m/sec) of an electron in the ground state and the two lowest excited states of the H atom. One challenge is to obtain the factor 2 2 2 π e h in units m/sec. We can use the Rydberg to help us here: 2 4 -18 2 2 π me 2.18×10 J = h . Therefore, 1/2 2 4 -18 2 -18 6 2 -31 π e 1.09×10 J 2 π e 1.09×10 J = and = 2 = 2.19×10 m/sec h m h 9.11×10 kg n = 1: ( ) 6 6 1 1 v = 2.19×10 m/sec = 2.19×10 m/sec 1 n = 2: ( ) 6 6 2 1 v = 2.19×10 m/sec = 1.09×10 m/sec 2 n = 3: ( ) 6 5 3 1 v = 2.19×10 m/sec = 7.29×10 m/sec 3 (c) (2 pts) In a transition from the ground state of H that involves moving an electron through a distance that is approximately equal to the orbit diameter of the ground state, what characteristic time would be required for the transition?
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