All answers have been checked against the answer key, and should be presumed to be correct. You should ask for help in the recitations if you are unable to obtain these results.
2. L = 1.1 x 1028 m
8. The mass uncertainty is 1.30 x 10-54 kg.
10. Use the radius as an estimate of the uncertainty in the position. Use Heisenberg's uncertainty principle to get the uncertainty in momentum, and use this as an estimate of the momentum to calculate the kinetic energy, KE = 21 MeV.
12. The uncertainty in position is 7.13 x 10-10 m.
24. (a)
1s2 2s2 2p6 3s2 3p6
3d7 4s2 (Note that 4s is filled before 3d.)
(b) 2s2 2p6 3s2
3p6 3d10 4s2 4p6
(c) 2s2 2p6 3s2
3p6 3d10 4s2 4p6 5s2
(Note that 5s is filled before 4d.)
28.
All but one charge +e of the nucleus is screened by the innermost
electrons, so the energy levels are like hydrogen:
E2 = -13.6 eV/22 = -3.4 eV.
30. Add the values of ml and ms for each state. Since all of the values are filled, the sums are zero, giving total angular momentum zero.
2. Bohr's theory had the electron in fixed orbits at some radius, traveling with a fixed velocity and momentum. This is incompatible with the uncertainty principle, which does not permit the position and velocity to be specified simultaneously.
3. The uncertainty in the future position depends on the uncertainty in the current position and in the current velocity, since x = x0 + vt. The uncertainty in the momentum cannot be reduced without making the position more uncertain. However, the uncertainty in the velocity can be reduced by making the object heavier, because Dv = Dp/m.
4. If the momentum is known to Dp, the position is unknown to Dx = h/(2pDp. For a baseball, uncertainty in momentum is relatively large on the scale set by Planck's constant, even if the velocity is measured accurately, because baseballs are heavy. This makes the uncertainty in their position unobservably small. For electrons, which are much lighter, the uncertainty in momentum is normally not so large compared to Planck's constant, so the uncertainty in position is big enough to be observable on an atomic scale.
5. No. To be balanced precisely on its point, the needle's position would have to be exactly over the point, since no variation is allowed at all without causing the needle to tip. It would also have to be perfectly stationary, since any movement to the side would cause it to tip. That is inconsistent with the uncertainty principle, since both the position and momentum cannot be exactly zero.
6. No, because some heat is transfered from the soup to the thermometer, cooling the soup slightly. The thermometer changes the temperature of the object being measured, unless they are at the same temperature to begin with.
7. It is possible to avoid the escape of the air. I cannot think of any reason why the uncertainty principle would say anything on that matter. However, you still cannot measure the pressure with perfect accuracy, because there will be small fluctuations in the position of the meter, and also its motion (momentum).
8. Yes. To measure an energy to perfect accuracy takes an infinitely long time, by the uncertainty principle. The electron will stay in its ground state forever, so you can measure it as long as you want. The excited states will not stay there forever, since eventually a photon will be spontaneously emitted so the atom can go back to its ground state. Therefore, the energy can be measured only to an accuracy (width) given by the uncertainty principle and the lifetime of the excited state.
9. The quantum-mechanical model predicts that the electron spends more time near the nucleus, since there is some probability distribution close to the nucleus. The Bohr model places the electron at a fixed distance from the nucleus.
10. The size of the outermost orbits is determined by the net charge the electrons there feel, which is the nuclear charge screened by all of the other electrons. That means that the size of all of the atoms will be comparable to the size of the hydrogen atom, since the potential felt by the outer electrons is comparable to that.
11. The energies of the quantum states are different, even though the quantum numbers are the same. That is because one electron only partly screens the other one from the nuclear charge in the helium atom. In the excited state, the screening is fairly complete, meaning that the excited states of helium and hydrogen should have comparable wavelengths, but the energy of the ground states will be significantly different due to the different effective charges seen by the electrons.
12. This question is beyond the scope of the course, I think, but here is the answer. The excited state producing the yellow line should have l = 1, which gives three possible values for the orbital angular momentum. In the rest frame of the excited electron, the other electrons are moving about with this angular momentum, forming a "current loop" which produces a magnetic field. This magnetic can either align with the spin of the excited electron or not, depending on the value of ml. This will create a splitting between the different possible magnetic quantum numbers of the excited states. This is an example of a "spin-orbit" interaction.
13 (a) is an allowed excited state. (b) is forbidden due to too many 2p electrons. (c) is an allowed excited state.
14.
The configuration of the 92 electrons in Uranium is
1s2 2s2 2p6 3s2 3p6
3d10 4s2 4p6 4d10 4f14
5s2 5p6 5d10 5f3 6s2
6p6 6d1 7s2.
The periodic table in the back of the book shows the outermost electrons, and
the rest just fill in the deeper shells.
15. (a) second column, (b) right column (noble gas), (c) next column in from the right (one missing electron from full p shell) 16. Neon is a noble gas with a filled spherically symmetric electron shell. This is a very stable configuration, so it takes a lot of energy to pull an electron out. Sodium has one loosely bound electron outside a closed shell, which is relatively easy to remove. Due to the screening of the nucleus by the other electrons, it feels a Coulomb potential only a factor of 3 larger than a 3s electron in hydrogen would.
17. Both are missing just a single electron from having a full shell, which makes them more likely to accept an electron from another atom.
18. Potassium and sodium both have one additional s electron outside a full shell, which is loosely held and easily shared to form chemical bonds.
19. The chemistry is dominated by their two loosely held s electrons, as for the transition metals.
The rest of the questions are beyond the scope of the course, since we did not discuss these topics, but the answers are included for completeness.
20. The Bohr theory is incomplete, and only takes into account the principle quantum numbers, not the angular momentum or spin.
21. The photon frequencies are proportional to (1/n1 - 1/n2) for the two levels. These values can be calculated for the various transitions, and the relative spaces between the spectral lines compared to the results to see which arrangement matches.
22. These electrons are more tightly bound because they are not screened as much from the nucleus, and are closer to it. This gives them bigger binding energies and bigger binding energy differences, which produces shorter wavelengths in transitions.
23. Spontaneous emission occurs when an excited atom returns to its ground state on its own, through random emission of a photon. Stimulated emission occurs when a photon interacts with an excited electron and induces a transition matching its frequency.
24. Laser light is monochromatic and coherent (entirely in phase). Normal light is a mixture of frequencies and phases.
25. Light from a 1000W street lamp spreads out in all directions, with the intensity dropping as the inverse square of the distance. Laser light is emitted in a single direction, and spreads very little over distance, so the power loss is much less.
| Physics 222 | Department of Physics | University of Tennessee |