1. Carbon dating is a technique used to determine the ages of carbon-containing objects that obtain their carbon from contemporary plants. It’s based on a radioactive isotope of carbon, 14C (carbon–14)

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Cases – Chapter 15

1. Carbon dating is a technique used to determine the ages of carbon-containing objects that obtain their carbon from contemporary plants. It’s based on a radioactive isotope of carbon, 14C (carbon–14), that’s present in our atmosphere in small quantities.

a. The 14C in our atmosphere is produced indirectly by cosmic rays—high-energy particles that enter our atmosphere from elsewhere in the universe. When these cosmic rays collide with nuclei in the atmosphere, they often produce neutrons, and some of these neutrons interact with the nuclei of nitrogen atoms. Why is it easy for a neutron to approach a 14N nucleus in order to stick to it?

b. When it sticks to a 14N nucleus, a neutron causes that nucleus to rearrange and release a proton. The result is a 14C nucleus. Adding a neutron and subtracting a proton leaves the number of nucleons in the nucleus unchanged, but it converts one chemical element into another. Why is 14C chemically different and distinguishable from 14N?

*c. 14C is radioactive with a half-life of 5730 years. That means there’s a 50% chance of each 14C nucleus spontaneously falling apart during a period of 5730 years. After 28,650 years, what fraction of the original 14C nuclei in an object will remain?

d. From the 5730-year half-life of 14C, it’s possible to calculate the fraction of 14C nuclei that will survive 1 year without decaying: 99.9879%. The fraction of nuclei that survive 2 years without decaying is 99.9879% of 99.9879%, or 99.9758%. And so on. If you know how many 14C nuclei an object contained when it was new, how could you determine how old the object is?

e. Most carbon nuclei are not radioactive. The carbon in rocks and petroleum is about 99% 12C and about 1% 13C. But because 14C is introduced into the atmosphere at a steady rate and plants obtain their carbon from the atmosphere, 14C nuclei make up a small but important fraction of the carbon nuclei in a living plant. This fraction can be measured in a modern plant to determine its value in an ancient plant while that plant was alive. After the plant died, the fraction of carbon nuclei that were 14C decreased as the 14C nuclei decayed. How does measuring the fraction of 14C nuclei in a plant’s carbon allow scientists to determine how long ago it lived?

f. Scientists can also determine how long ago an animal lived if that animal ate plants to obtain carbon. But if the animal obtained a substantial fraction of its carbon from minerals, that dating process wouldn’t work. Why not?

2. Radon is a colorless, odorless gas that’s extremely radioactive. It has three naturally occurring isotopes—219Rn, 220Rn, and 222Rn—that are made when radioactive elements in the earth’s crust decay. While the nuclei of these isotopes each contain 86 protons, they contain different numbers of neutrons. Radon is thought to be the single most important cause of lung cancer in nonsmokers. It’s particularly abundant in indoor air above geological formations that include radon-producing minerals.

a. 219Rn, 220Rn, and 222Rn nuclei are found in the atoms of chemically inert gases. In fact, radon is one of the noble gas elements, which include helium, neon, argon, krypton, and xenon. It’s not surprising that the atoms of all three isotopes are gases. It would be much more surprising if one isotope were normally a solid, one a liquid, and the third a gas. Why?

*b. The first isotope, 219Rn, is produced when the radioactive element actinium (227Ac) decays. The half-life of 219Rn is extremely short, only 3.92 s. If there were 800 of these 219Rn nuclei in a box, how long would it be before only about 25 of them were left?

c. The second isotope of radon, 220Rn, is produced when the radioactive element thorium (232Th) decays. This isotope has a half-life of 54.5 s and decays into yet another radioactive nucleus. The ensuing series of radioactive decays usually produces a stable isotope of lead (208Pb). In the long series of unstable nuclei that appear briefly as thorium turns into lead, the only nucleus that forms a gaseous atom is 220Rn. From that information, why can you safely assume that very little 220Rn is found in building air, even if it’s built above rocks that are rich in thorium?

d. The third radon isotope, 222Rn, is produced when uranium (238U) decays. This isotope has a half-life of 3.8 days. Why is this isotope much more likely than the others to be found in building air, particularly if that building is built above uranium-containing rocks?

3. There are two types of smoke detectors in common use: photoelectric and ionization. A photoelectric smoke detector uses a beam of light to “see” the smoke particles. Unfortunately, it isn’t sensitive to the smallest smoke particles and requires considerable electric power to operate. An ionization smoke detector is more sensitive and consumes less power. It uses a radioactive material to ionize the air so that a small amount of electric current can pass through that air. When smoke is present, the ionized air molecules stick to the smoke particles, less current flows through the air, and the detector’s alarm goes off. The radioactive material in an ionization smoke detector is an isotope of americium, a radioactive element that’s not found in nature. The specific isotope of americium is 241Am. This isotope is made by adding neutrons to 238U in a nuclear reactor. In fact, 241Am is made in exactly the same way that 239Pu is made, except that 241Am has been exposed to a few more neutrons.

a. 241Am has a half-life of 458 years. What would happen if an ionization smoke detector used a radioactive isotope with a half-life of 458 days instead?

b. Many 241Am nuclei were made during the supernova explosion that produced the earth’s uranium. Why aren’t there any naturally occurring 241Am nuclei around today?

c. When uranium nuclei undergo spontaneous fission in the earth’s crust, their fragments are often radioactive nuclei. Why aren’t any of these fragments 241Am nuclei?

d. An 241Am nucleus decays into a 239Np nucleus, an isotope of neptunium with a half-life of 2 million years. Both americium and neptunium are radioactive solids. Why is it important that the radioactive isotope used in the smoke detector not decay into a radioactive gas such as radon?

4. Physicists use accelerators to study the smallest objects in the universe: subatomic particles. They need these accelerators because enormous energies are required to resolve extremely fine details. Since a particle’s wavelength decreases as its energy increases, higher energy particles are able to resolve tinier features.

a. In an electron accelerator, electrons pass through a long chain of resonant cavities, powered by strong microwave generators. The electric fields in these cavities push the electrons forward. Why must the electrons be bunched together in packets rather than sent into the cavities continuously?

b. A packet of electrons takes the same amount of time to pass through each cavity of the chain. The first few cavities of the chain are shorter than the rest. Why?

c. The remaining cavities of the chain are essentially equal in length. Why don’t the cavities continue to get longer?

d. Some accelerators also work with positrons. They inject packets of positrons into the chain of cavities so that each positron packet is exactly one cavity behind an electron packet. Explain why this arrangement leads to the acceleration of both the electrons and the positrons.

e. When the packets of electrons and positrons emerge from the chain of cavities, they pass through a magnetic field. The electrons turn one way while the positrons turn the other way. Explain why these two particles turn in different directions.

f. The electrons and positrons are steered with magnets until they collide with one another head on. When they hit, an electron and positron annihilate each other in a burst of energy that makes new particles from empty space. Why is it that the more energy the electron and positron have, the more mass the particles they create can have?

5. Police forensic laboratories often use the neutron activation technique to look for trace elements on objects left at a crime scene or belonging to a suspect. In this technique, the objects are made temporarily radioactive by exposing them to neutrons. Instruments then study the radioactive decays of the activated atoms to determine what atoms they were.

a. Neutron activation is done most easily at a nuclear reactor facility. Why can’t a lamp or an X-ray tube be used to add neutrons to an object’s nuclei?

b. A neutron can attach itself to almost any nucleus in an object, even if the neutron is only moving with thermal energy. Why doesn’t the nucleus push the neutron away before the neutron makes contact with it?

c. Once the neutron has attached itself to a nucleus, the nucleus may become radioactive. Detectors studying an object that has been exposed to neutrons detect electrons and gamma rays emerging from that object. Describe one likely mechanism by which neutron-activated nuclei may produce these electrons and gamma rays.

d. By measuring the energies of the gamma rays emitted by an object following neutron activation, the detectors can determine what elements are present in that object, even if those elements are there in extremely small quantities. For example, they can determine whether trace amounts of barium, residue of the primer in a bullet cartridge, are present on a suspect’s glove. Why are the gamma rays emitted by a neutron-activated barium nucleus different from those emitted by a neutron-activated iron nucleus?

e. While neutron activation studies can tell what elements are present in an object, they can’t tell what chemicals those elements are part of. Why can’t neutron activation tell whether the barium atoms it detects are present as barium metal or as a barium compound such as barium chloride?

f. How can neutron activation techniques be used to detect an art forgery made with pigments that were not known to the painting’s alleged artist?

g. An object is radioactive following neutron activation, but that radioactivity fades quickly. The object is normally safe in a few weeks or months; you just have to wait. Why won’t heating or chemically treating the object speed the decrease in radioactivity?

6. Magnetic resonance imaging (MRI) is an outgrowth of a technique used by scientists to identify the chemical and magnetic environments of nuclei—usually hydrogen nuclei, which are individual protons. The original technique is called nuclear magnetic resonance, or NMR. When nuclear magnetic resonance was used in imaging, the word “nuclear” was dropped from its title because that word made the imaging technique sound far more dangerous than it actually is.

a. When a sample of a chemical containing hydrogen atoms is placed in a magnetic field, it will absorb only specific electromagnetic waves. How are these electromagnetic waves affecting the protons in the sample when the sample is absorbing them?

b. If the magnetic field in which the sample resides is increased, the frequency of the electromagnetic waves must also be increased or the sample will stop absorbing them. Why?

c. The electrons in the sample itself affect the magnetic field that the protons experience. In and around each proton are a number of electrons, which can move somewhat in response to electric fields. When the sample is placed in a magnetic field, the field induces a response that is analogous to a current in the sample, even if the sample isn’t actually metallic. This current partially cancels the magnetic field that the proton experiences so that it absorbs electromagnetic waves at a lower frequency than it would absorb if it had no nearby electrons. Why does the “induced” current tend to cancel the magnetic field being applied to the sample rather than strengthen it?

d. The frequency of electromagnetic waves that a particular proton will absorb also depends on any nearby protons and whether they are spin-up or spin-down. How are these nearby protons able to change this frequency?

7. Nuclear power plants inevitably convert some of the uranium in their fuel rods into plutonium. One of the great concerns with the presence of nuclear reactors and power plants in so many countries is that those countries might try to extract plutonium from the used reactor fuel and use that plutonium in nuclear weapons.

a. Instead of worrying about plutonium, perhaps we should be worried about the uranium fuel rods themselves. Fortunately, the fuel rods themselves aren’t a danger. Why can’t a typical nuclear reactor’s uranium fuel be used to make nuclear weapons?

b. Why is it much easier to extract 239Pu from the used fuel rods than it is to extract 235U from those same fuel rods?

c. In most nuclear power reactors, the plutonium that’s produced isn’t pure 239Pu. About 25% of the plutonium is actually 240Pu, which is about 4 times as radioactive as 239Pu (240Pu has a half life of only 6,580 years, compared to 239Pu’s half life of 24,400 years). Some people argue that plutonium from nuclear power reactors is “safer” than weapons grade plutonium because the large fraction of 240Pu in the reactor grade plutonium makes assembling a supercritical mass of reactor grade plutonium very difficult. Why would it be hard to assemble a supercritical mass of reactor grade plutonium?

d. Reactor grade plutonium also contains a percentage or two of 238Pu. This highly radioactive isotope (with a half life of 88 years) contributes to the problem discussed in c and adds a new problem: heat. Even without a chain reaction, the core of a nuclear weapon built from reactor grade plutonium would need a cooling system like that in a nuclear reactor. This cooling system would have to run all the time while the weapon was “sitting on the shelf.” Why?

e. Why is reactor grade plutonium, which has these relatively large proportions of 238Pu and 240Pu, a more serious health risk to the people working with it than weapons grade plutonium, which is relatively pure 239Pu?

8. Scientists often use electromagnetic waves to study the structure of atoms and nuclei. But while there are many ways of producing electromagnetic waves at the long wavelength end of the spectrum, it’s relatively difficult to produce gamma rays. One technique that’s used to prepare gamma rays is Compton scattering.

a. When an X-ray photon collides with a stationary electron, energy and momentum are transferred between the two particles. Why is the energy carried by the photon after the collision always less than the energy it had before the collision?

b. Suppose an X-ray photon collides with a moving electron. What must happen to the electron in order for the photon to leave the collision with more energy than it had before the collision?

c. Suppose that an X-ray machine emits an X-ray photon directly behind an extremely high-energy electron that’s just emerged from a particle accelerator. The electron has far more energy than the X-ray photon, yet the X-ray photon is able to overtake the electron. Why can the X-ray photon always catch up to the electron?

d. As it overtakes the electron, the X-ray photon can undergo Compton scattering from the electron. The photon can emerge from this collision with considerably more energy than it had before. Instead of being an X-ray photon, it’s now a gamma ray photon. However, because momentum is also conserved, the electron must retain a substantial fraction of its original energy. In fact, the electron continues onward at almost the same speed it had before the collision. How can it give up a substantial fraction of its energy to the photon without slowing down dramatically?

9. When a positron enters matter, it quickly finds an electron and the two annihilate one another. In the brief moments before that annihilation, however, they often form a strange atom consisting only of an electron and a positron. This short-lived atom is called positronium.

a. A positronium atom resembles a hydrogen atom because both have one electron and one positively charged particle. In a hydrogen atom that positively charged particle is a proton, but in a positronium atom it’s a positron. While it’s sensible to think of the electron in a hydrogen atom as orbiting the massive proton, that’s not the case for positronium. In positronium, the electron and positron orbit one another as equals. Why doesn’t the positron remain essentially stationary while the electron orbits it?

b. The electron and positron in a positronium atom can only annihilate one another when they actually touch. If they orbited one another as particles, they would never touch. How does quantum physics make it possible for these two orbiting objects to touch one another?

c. When the electron and positron in a stationary positronium atom annihilate one another, they turn into two or more gamma ray photons and nothing else. Noting that a photon carries with it both energy and momentum, explain why a stationary positronium atom can’t turn into a single photon when its electron and positron annihilate.

d. When a stationary positronium atom turns into two gamma ray photons, the momentums of those two photons are exactly equal in magnitude but opposite in direction. Why must these two photons have equal but oppositely directed momentums?

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