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hw2_f10_sol - Astronomy C10 L&S C70U Fall 2010 Homework#2...

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Astronomy C10 / L&S C70U: Fall 2010 Homework #2 Solutions Maximum points possible = 50 1. Absorption of Photons by Atoms (11 points total). (a) (2 points.) Which transition has the same energy as #1 #2 ? Jumping from state #1 to #2 is a jump of 5 of our energy units. There are two other such jumps. One is from state #2 to #5, the other is from state #5 to #7. [Graders: Either one OK for full credit.] (b) (3 points.) What is the ratio of energies of the photons emitted by a pair of downward jumps? Jumping from state #6 #4 is a change of (13 - 9) = 4 units. Jumping from #4 #3 is a change of (9 - 7) = 2 units. The ratio of energies is thus 2:1. [Graders: 1/2 is also valid if the student flips the ratio.] (c) (3 points.) Can the photon emitted by 3 2 be absorbed by an atom with electron in state 4? The jump from #3 #2 emits a photon with (7 - 5) = 2 energy units. Looking at the energy levels surrounding state #4, we see that jumping up to state #5 would take (10 - 9) = 1 unit, and a jump to state #6 would require (13 - 9) = 4 units. Absorption can only occur when the incoming photon has exactly the same energy that one of the electron’s possible energy level jumps requires. No “par- tial” absorption is possible. The emitted photon with 2 energy units is thus the wrong size for the #4 #5 jump and is too weak to be considered for any of the greater leaps. This photon will NOT be absorbed. (d) (3 points.) If the jump from state #2 #3 requires a photon with wavelength 5000 ˚ A, what is the wavelength needed for the #2 #4 jump? The energy associated with the jump from #2 #3 is (7 - 5) = 2 units. The jump from #2 #4 is (9 - 5) = 4 units. Thus, the larger jump will require (4 / 2) = 2 times the energy. The energy of a photon is directly proportional to its frequency. ( E = hν, where h is “Planck’s constant” – Class Slide 25) However, the wavelength of a photon is inversely proportional to its frequency (recall wavelength times frequency = speed, so ν = c/λ ). This means that energy is likewise inversely proportional to wavelength: E = hc/λ . A photon with 2 times the energy of another thus has 1/2 the wavelength. In our case, the more energetic photon will then have a wavelength of 2500 ˚ A.
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