hmwk4wAnswers - Chem2070 Fall 2011 Homework #4: Due...

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Unformatted text preview: Chem2070 Fall 2011 Homework #4: Due September 23 1. Suppose an atom in an excited state can return to the ground state in two steps. If first falls to an intermediate state emitting radiation of wavelength λ 1 , and then to the ground state, emitting radiation of wavelength λ 2 . The same atom can also return to the ground state in one step, emitting one photon of wavelength λ . How are λ,λ 1 , and λ 2 related? How are their frequencies related (in both cases, write an algebraic expression, not a word explanation). The energy emitted must be the same, whether or not it appears in one, two or N steps. So we can write E = E 1 + E 2 for the onestep process, or the two step process. Since we know E = hν = hc λ , we also know that (after eliminating all constants, meaning h and/or c ) ν = ν 1 + ν 2 and 1 λ = 1 λ 1 + 1 λ 2 2. In the Bohr model, the energy of an electron orbital is 1 hc E n ( Z ) = 1 . 097 × 10 7 m Z 2 n 2 = 1 91 . 16 nm Z 2 n 2 for nuclear charge Z , and the electron orbits the nucleus at a distance r n = n 2 Z a where a = 0 . 053 nm. It’s true that early versions didn’t have the factor 1 hc ; I apologize for the confu- sion....most physical chemists, and I count myself as one, think of all units, including inverse meters, inverse seconds and meters, as energy units of one kind or another since they only require products of h and c to get back to joules. My hope was you would solve without worrying about the units but that was an error....I’ve asked the TA’s to be gentle, though I don’t know if we can repay you for your time. (a) Without any detailed calculations, indicate which of the following energy transi- tions in He + requires absorption of the most energetic photon. i. from n i = 1 to n f = 2 ii. from n i = 2 to n f = 4 iii. from n i = 3 to n f = 9 iv. from n i = 10 to n f = 1 If the energies go as 1 n 2 then the largest energies are associated with transitions to/from the lowest n states; so any transition where either n i or n f = 1 will occur at larger energies than all the others. It requires energy to be absorbed to increase n , i.e where n f > n i ; energy is released in emission, where n i > n f . So the most energetic photon is absorbed if we are undergoing transition (i.) 1 (b) The absorption spectrum shown below is observed from a one-electron (hydrogen- like) atom in the gas phase. All lines are shown corresponding to transitions tolike) atom in the gas phase....
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This document was uploaded on 10/26/2011 for the course CHEM 207 at Cornell University (Engineering School).

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hmwk4wAnswers - Chem2070 Fall 2011 Homework #4: Due...

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