Activities and Half-Lives

1. Activities and Half-Lives AP Physics B Montwood High School R. Casao

2. Suppose you need to dispose of radioactive waste that contains a certain number of radioactive nuclei. • If no more are produced, the number decreases in a simple manner as the nuclei decay. • The decrease is a statistical process; there is no way to predict when any individual nucleus will decay. • The rate of nuclear decay varies over a wide range for different nuclides.

3. Let N(t) be the number of radioactive nuclei in a sample at time t, and let dN(t) be the negative change in that number during a short time interval dt. • The number of decays during the time interval dt is –dN(t). • The rate of change of N(t) is • The decay rate or activity of a radioactive nuclide is given by

4. The larger the number of nuclei in the sample, the more nuclei decay during any given time interval. • Activity (A) is directly proportional to N(t); activity is equal to a constant  multiplied by N(t): • The constant  is called the decay constant and it has different values for different nuclides. • A large value of  corresponds to rapid decay. • A small value of  corresponds to slow decay.

5. Solving for  shows us that  is the ratio of the number of decays per time to the number of remaining radioactive nuclei: •  can then be interpretedas the probability per time that any individual nucleus will decay. • At t = 0 s, the number of radioactive nuclei is No; N(0) = No.

6. Over time, the number of radioactive nuclei decreases to 0. • Derivation: • Collect like terms: • This is a differential equation.

7. Integrate from N to No and from t to to: • Take the exponential of both sides:

8. The number of nuclei remaining after time t: • Half-life T1/2 is the time required for the number of radioactive nuclei to decrease to one-half the original number No. • Then half of the remaining radioactive nuclei decay during a second interval of T1/2. • The numbers remaining after successive half-lives are

11. To get the relationship between half-life T1/2 and the decay constant , set:

12. Tosolve for T1/2, take the natural log of both sides: • The mean lifetime Tmean, generally called the lifetime, of a nucleus or unstable particle is proportional to the half-life:

13. The half-life is: • independent of the physical state (solid, liquid, gas), temperature, pressure, the chemical compound in which the nucleus finds itself, and essentially any other outside influence. • It is independent of the chemistry of the atomic surface, and independent of the ordinary physical factors of the outside world. • The only thing which can alter the half-life is direct nuclear interaction with a particle from outside, e.g., a high energy collision in an accelerator.

14. Because the activity at any time equals N(t), then N(t) = Noe-t tells us that the activity also depends on time as e-t.

15. Activity • Equation: , where A(t) = activity at time t,  = decay constant, and Ao = activity at t = 0 s • Equation: A = N • A common unit of activity is the curie (Ci), which is defined as 3.7 x 1010 decays/second. • Approximately equal to the activity of one gram of radium. • SI unit of activity is the becquerel (Bq); 1 Bq is equal to 1 decay/second.

16. Relationship between curies and becquerels: 1 Ci = 3.7 x 1010 Bq = 3.7 x 1010 decay/s • Example: Activity of Co-57 The radioactive isotope Co-57 decays by electron capture with a half-life of 272 days. • Determine the decay constant and the lifetime. • If you have a radiation source containing Co-57 with an activity of 2 Ci, how many radioactive nuclei does it contain? • What will be the activity of your source after one year?

17. a. • b.

18. c.

19. Radioactive Dating • One application of radioactivity is the dating on archeological and geological specimens by measuring the concentration of radioactive isotopes. • Carbon dating: the unstable C-14 isotope, produced during nuclear reactions in the atmosphere that result from cosmic-ray bombardment, give a small proportion of C-14 in the CO2 in the atmosphere. • Plants that obtain their carbon from this source contain the same proportion of C-14 as the atmosphere.

20. Radioactive Dating • When a plant dies, it stops taking in carbon and its C-14 undergoes - decay to N-14 with a half-life of 5730 years. • By measuring the proportion of C-14 in the remains, you can determine how long ago the organism died. • Similar radioactive techniques are used with other isotopes for dating geological specimens. • Some rocks contain the unstable K-40 isotope, a beta emitter that decays to the stable Ar-40 nuclide with a half-life of 2.4 x 108 years. • The age of the rock can be determined by comparing the concentrations of K-40 and Ar-40.

21. Radioactive Dating • Example: before 1900 the activity per mass of atmospheric carbon due to the presence of C-14 averaged about 0.255 Bq per gram of carbon. • a. What number of carbon atoms were C-14? • b. In analyzing an archeological specimen containing 500 mg of carbon, you observe 174 decays in one hour. What is the age of the specimen, assuming that its activity per mass of carbon when it died was that average value of the air?

22. a.

23. b.

24. Biological Effects of Radiation • As alpha particles, beta particles, neutrons, and EM radiation such as gamma rays and x-rays, pass through matter, they lose energy, break molecular bonds, and create ions (which is why they are called ionizing radiation). • Excessive exposure to radiation, including sunlight, x-rays, and all the nuclear radiations can destroy tissues. • Mild cases result in a burn, like a sunburn. • Greater exposures can cause severe illness or death by a variety of mechanisms, including massive destruction of tissue cells, alterations of genetic material, and destruction of the components in bone marrow that produce red blood cells.

25. Calculating Radiation Doses • Radiation dosimetry is the quantitative description of the effect of radiation on living tissue. • Absorbed dose (AD) of radiation is defined as the energy delivered to the tissue per unit mass. • SI unit of absorbed dose, the J/kg, is called the Gray (Gy); 1 Gray = 1 J/kg. • The unit in more common use is the rad (radiation absorbed dose) , defined as 0.01 J/kg; 1 rad = 0.01 J/kg = 0.01 Gy.

26. Calculating Radiation Doses • Absorbed dose by itself is not an adequate measure of biological effect because equal energies of different kinds of radiation cause different extents of biological effect. • The variation in biological effect is described by a numerical factor called the relative biological effect (RBE), also called the quality factor (QF), of each specific radiation. • The values for RBE depend somewhat on the kind of tissue in which the radiation is absorbed and on the energy of the radiation.

27. Calculating Radiation Doses • X-rays with 200 keV of energy are defined to have an RBE of 1. • The biological effect is described by the product of the absorbed dose and the RBE of the radiation, this is called the biological equivalent dose (or the equivalent dose, ED). • SI unit of equivalent dose is the Sievart (Sv). • 1 Sv = 100 rem. • RBE units: Sv/Gy or rem/rad • 1 rad = 1 rem (Röngen equivalent for man) = 0.01 J/kg.

28. RBE for Several Types of Radiation

29. Equations and Example • Equations: • Example: During a diagnostic x-ray examination a 1.2 kg portion of a broken leg receives an equivalent dose of 0.4 mSv. • a. What is the equivalent dose in mrem? • b. What is the absorbed dose in J/kg? • c. If the x-ray energy is 50 keV, how many x-ray photons are absorbed?

30. a. • b. • c.

31. Radiation Hazards • An ordinary chest x-ray delivers about 0.2 to 0.4 mSv to about 5 kg of tissue. • Radiation exposure from cosmic rays and natural radioactivity in soils, etc, is about 1 mSv (0.001 J/kg) per year at sea level and twice that at an elevation of 5000 ft. • A whole-body dose of up to about 0.2 Sv (0.2 J/kg) causes no immediate detectable effect. • A short-term whole-body dose of 5 Sv (5 J/kg) or more usually causes death within a few days or weeks. • A localized dose of 100 Sv (100 J/kg) causes complete destruction of the exposed tissues.

32. Radiation Hazards • Long term exposure to radiation can cause various cancers and genetic defects. • U.S. government regulations are based on maximum yearly exposure, from all except natural resources, of 2 to 5 mSv. • Workers with occupational exposure to radiation are permitted 50 mSv per year. • Radiation levels from nuclear power plants is not negligible, but the health hazards from coal smoke are serious and the natural radioactivity in the smoke from a coal-fired power plant is believed to be 100 times as great as that from a properly operating nuclear power plant.

33. Beneficial Uses of Radiation • Radiation is widely used in medicine for intentional selective destruction of tissue such as tumors. • Medicinal isotopes used have shorter half-lives and greater activity. • Isotopes can be chosen that emit the type and energy of radiation desired. • Radioactive isotopes are used as tracers and to visualize organs and arteries in the body.