Exercise2. Why is it extremely difficult to separate the two isotopes of copper, 63Cu and 65Cu? Those two isotopes are chemical identical and can only be separated by using their relatively small difference in mass. Exercise 3. If a nuclear reaction adds an extra neutron to the nucleus of 57Fe (a stable isotope of iron), it produces 58Fe (another stable isotope of iron). Will this change in the nucleus affect the number and arrangement of the electrons in the atom that’s built around this nucleus? Why or why not? No: the number of electrons = number of protons (= atomic number) and this hasn’t changed. Because the number of electrons doesn’t change and the charge in the nucleus doesn’t change, the arrangement of electron orbitals (standing waves) also doesn’t change. Exercise 4: If a nuclear reaction adds an extra proton to the nucleus of 58Fe (a stable isotope of iron), it produces 59Co (a stable isotope of cobalt). Will this change in the nucleus affect the number and arrangement of the electrons in the atom that’s built around this nucleus? Why or why not? Yes: since the nuclear charge has increased, an additional electron is needed to form a neutral atom. Because of both the change in number of electrons and the stronger Coulomb attraction from the nucleus, the arrangement of electron orbitals will change. Exercise 7. When a large nucleus is split in half during an experiment at a nuclear physics lab, the result is usually two medium-size nuclei with too many neutrons to be stable. These nuclei eventually fall apart. Why don’t these smaller nuclei need as many neutrons as they received from the original nucleus? Because they have fewer protons (with their mutual Coulomb repulsions), less of the “diluting” effect of neutrons is needed. Exercise 19: The most troubling radioactive isotopes in nuclear waste are those with half-lives of between a few years and a few hundred thousand years. Explain. Isotopes with shorter half-lives decay quickly and can be waited out, while those with much longer half-lives are relatively less likely to decay during a human lifetime. Exercise 32. Control rods are usually installed on top of the reactor, where their weights tend to pull them into the core. Why is this arrangement much safer than putting the control rods at the bottom of the reactor? If the mechanism which is holding them fails, the control rods will descend into the fuel, making it go subcritical and turning the reactor off. However, if the control rods were on the bottom, they would fall out of the fuel, making it go supercrititical, leading to a meltdown or explosion. Problem 1. Gallium 67 (67Ga) is a radioactive isotope with a half-life of 3.26 days. It’s used in nuclear medicine to locate inflammations and tumors. Accumulations of 67Ga in a patient’s tissue can be detected by looking for the gamma rays it emits when it decays. A radiologist usually begins looking for the 67Ga about 48 hours after administering it to a patient. What fraction of the original 67Ga nuclei remain after 48 hours? 0.613 N/N0 = (0.5)^(t/T1/2). T1/2 = 3.26 days, t = 2 days, so N/N0 = (0.5) = 0.65 (= 65%) Problem 2. Two weeks after 67Ga was administered to the patient (Problem 1), what fraction of the 67Ga nuclei remain? 4.29 Now t = 14 days, so t/T1/2 = 4.29 and N/N0 = (0.5) = 0.051 (= 5.1 %) Exercise 20. An X-ray technician can adjust the energy of the X-ray photons produced by a machine by changing the voltage drop between the X-ray tube’s cathode and anode. Explain. The larger the voltage drop between cathode and anode, the more kinetic energy each electron has by the time it reaches the anode and the more energetic the X rays it can produce. [As shown by a figure in the April 29 lecture, not only does the maximum energy increase (i.e. minimum wavelength decrease), but the total energy at all wavelengths increases. This is because with more kinetic energy when it hits the anode, the electron has more ways to decelerate.] Exercise 21. Lead, with 82 electrons per atom, is an excellent absorber of X-rays. Why? Because it has so many electrons, the chances of one of them catching the x-ray photon is large. In addition, because of the large charge (82e) on the nucleus, many of these electrons are tightly bound, so they escape the atom with little kinetic energy, increasing the probability of photoelectron absorption. Exercise 24. Magnetic resonance imaging (MRI) differs from computed tomography imaging in that it involves no “ionizing radiation.” What electromagnetic radiation is used in MRI, and why aren’t the photons of this radiation able to remove electrons from atoms and convert those atoms into ions? MRI uses radiowaves whose photons have far too little energy (E = hf) to cause chemical damage to molecules. Exercise 25. Magnetic resonance imaging isn’t good at detecting bone. Why not? MRI detects hydrogen. Bone contains relatively little hydrogen compared to soft tissue. Exercise 26. No magnetic metals such as iron or steel are permitted near a magnetic resonance imaging machine. In part, this rule is a safety precaution since those magnetic metals would be attracted toward the machine. However, the magnetic fields from these magnetic metals would also spoil the imaging process. Why would having additional magnetic fields inside the imaging machine spoil its ability to locate specific protons inside a patient’s body? Magnetic metals would change the local magnetic fields inside a patient and impair the imaging machine's ability to precisely control those fields. The machine needs to be able to know exactly what the fields are at each point in the patient in order to know where the protons it is detecting are located. I heard a funny story (possibly not true or very exaggerated) from the first director of UK’s MRI center. When the magnet was turned on, a (soft ferromagnetic) wrench that was lying on the floor was attracted to the magnet and slammed into the magnet’s metal container, with a loud crash. A security guard outside the room heard the crash and ran in with his gun out. The magnet pulled the gun out of his hand and the gun also crashed into the container. Exercise 27. Why does the strength of the magnetic field used in MRI affect the frequency of the radio waves used to detect the protons? The energy splitting between nuclei with spin-down and spin-up is proportional to the magnetic field, so the photon frequency needed to cause a transition is also proportional to the magnetic field. (f = ΔE/h, which is proportional to B.) Exercise 28. The stronger the magnetic field used in MRI, the larger the fraction of protons that align their spins with the field. Why does this increased alignment make it easier for the MRI to study the protons? If the number of (low energy) spin-up and (high energy) spin-down nuclei are equal, the stimulated emission (from the spin-down would equal the absorption from the spin-up – i.e. there would be no net absorption. The net absorption comes from the excess of spin-up nuclei. Problem: • What is the minimum wavelength of x-rays emitted by electrons accelerated through a voltage of 25 kV? The kinetic energy of these electrons = 25 keV = (25000 eV) (1.6 x 10-19 J)/(eV) = 4.0 x 10-15 J; this will also be the 18 maximum x-ray energy emitted. The corresponding frequency fmax = E/h = 6.0 x 10 Hz. Since the frequency and wavelength are inversely proportional, this corresponds to a minimum wavelength: -11 λmin = c/fmax = 5.0 x 10 m = 0.05 nm.