Chapter 16, first assignment

Answers to selected problems for Chapter 16 are found here.

Exercise 2. Can it be truthfully said that, whenever a nucleus emits an alpha or beta particle, it necessarily becomes the nucleus of a different element?
Yes. Elements are defined by the number of protons in their nuclei (the atomic number). Alpha emission throws away two protons; beta emission changes a neutron into a proton (or a proton into a neutron for beta-plus or electron-capture). Since the number of protons in the nucleus changes, we now have a nucleus of a different element.
Exercise 6. The alpha particle has twice the electric charge of a beta particle, but it is deflected less than the beta particle in a magnetic field. Why is this so?
The alpha particle has a mass of 6.6 × 10-27 kg, while the electron has a mass of 9.1 × 10-31 kg (values obtained via Google). While there is twice the electromagnetic force acting on the alpha particle, it acts on something with more than 6,000 times the mass! Since acceleration equals force divided by mass, we expect that twice the force on 6,000 times the mass would produce one three-thousandth the acceleration.
Exercise 8. In what way is the emission of gamma radiation from a nucleus similar to the emission of light from an atom?
Light emission from atoms results from electrons falling into lower (more stable) energy states within the atom. Gamma emission from nuclei results from nucleons falling into lower (more stable) energy states within the nucleus.
Exercise 14. What evidence supports the contention that the strong nuclear force is stronger than the electrical force at short internuclear distances?
At intranuclear distances, the evidence is that the repulsion between protons does not cause the nucleus to fly apart. At very short internuclear distances (considerably less than the diameter of an atom!) the evidence is that protons can be made to fuse with nuclei in spite of the electrostatic repulsion between two positively-charge objects.
Exercise 17. webelements When 226Ra decays by emitting an alpha particle, what is the atomic number of the resulting nucleus? What is the resulting atomic mass?

click the WebElements link for a periodic table

Radium has an atomic number of 88; an alpha particle has an atomic number of 2 (and atomic mass of 4). 88 - 2 = 86; the radium has become radon. When 226Ra decays by alpha-emission, it loses four mass units to become 222Rn.
Exercise 18. webelements When 218Po emits a beta particle, it transforms into a new element. What are the atomic number and atomic mass of this new element? What are they if 218Po instead emits an alpha particle?

click the WebElements link for a periodic table

Beta-minus emission changes a neutron into a proton, so the atomic number rises by one. The mass doesn't change, so if 218Po undergoes beta-minus emission, it becomes 218At (astatine), with an atomic number of 85 and an atomic mass of 218.

If 218Po undergoes alpha-emission (loss of 2 protons and 2 neutrons), the atomic number drops by two (from 84 to 82) and the atomic mass by four (from 218 to 214), and it becomes a nucleus of 214Pb -- which is also radioactive, having too many neutrons for its protons.
Exercise 19. webelements State the number of neutrons and protons in each of the following nuclei: 2H, 12C, 56Fe, 197Au, 90Sr, and 238U.

click the WebElements link for a periodic table

  • 2H (atomic number 1) has one proton and one neutron
  • 12C (atomic number 6) has six protons and six neutrons
  • 56Fe (atomic number 26) has 26 protons and 30 neutrons
  • 197Au (atomic number 79) has 79 protons and 118 neutrons
  • 90Sr (atomic number 38) has 38 protons and 52 neutrons
  • 238U (atomic number 92) has 92 protons and 146 neutrons
Exercise 21. Elements with atomic numbers greater than that of uranium do not exist in any appreciable amounts in nature because they have very short half-lives. Yet there are several elements with atomic numbers smaller than that of uranium that have equally short half-lives and that do exist in appreciable amounts in nature. How can you account for this?
Light radioactive elements with short half-lives are decay products of long-lived radioactives like uranium and thorium. Since the most stable (hence long-lived) isotopes of uranium and thorium do not decay by beta-emission, no elements heavier than uranium are found in nature.
Exercise 24. When we speak of dangerous radiation exposure, are we customarily speaking of alpha radiation, beta radiation or gamma radiation? Discuss.
Alpha and beta radiation have little penetrating power, so dangerous radiation from external sources is going to be gamma rays. However, it is much more dangerous to ingest an alpha source than a beta- or gamma-source because the latter have more chance of penetrating your body without doing damage.
Exercise 25. People who work around radioactivity wear film badges to monitor the amount of radiation that reaches their bodies. These badges consist of small pieces of photographic film enclosed in a light-proof wrapper. What kind of radiation do these devices monitor?
Gamma rays, because beta particles will not penetrate the wrapper (nor will alpha particles).
Problem 2. If a sample of a radioactive isotope has a half-life of one year, how much of the original sample will be left at the end of the second year? At the end of the third year? At the end of the fourth year?

After each half-life, ½ of the amount present at the beginning of the half-life has decayed. So after the first year, ½ of the original amount is gone. After the second year, ½ of that is gone, or ¼ of the original amount. After the third year, ½ of that or 1/8 the original amount. After the fourth year, ½ of that or 1/16 the original amount.

If you're comfortable with exponential expressions, after n half-lives, 1/2n of the original amount remains.
Problem 4. A sample of a particular radioisotope is placed near a Geiger counter, which is observed to register 160 counts per minute. Eight hour later, the detector counts at a rate of 10 counts per minute. What is the half-life of the material?

The amount of radiation has dropped to 1/16 the original level. 16 = 24, so four half-lives have elapsed. That means that the half life is 8 hr ÷ 4 = 2 hr.

Of course, you've let things go too long! 10 cpm is so close to background that you can't accurately measure it.
Problem 5. The isotope cesium-137, which has a half-life of 30 years, is a product of nuclear power plants. How long will it take for this isotope to decay to about one-sixteenth its original amount?

Again, 16 = 24, so 15/16 of the original amount will be gone after four half-lives. 4 × 30 = 120 years.