LECTURE 24 HALF-LIFE, RADIOACTIVE DATING, AND BINDING ENERGY Instructor: Kazumi Tolich
Lecture 24 2 Reading chapter 32.3 to 32.4 Half-life Radioactive dating Binding energy
Nuclear decay functions 3 If we started out with N " nuclei with a decay constant of λ, the number of nuclei remaining at some later time is N = N " e &'( The activity as a function of time is R = N t = λn = R " e &'( The half-life is defined as the time it takes for N and R to decrease by half. T -. = ln 2 λ = 0.693 λ T -. 2T -.
Quiz: 1 4 You have 400 g of a radioactive sample with a half-life of 20 years. How much in grams is left after 50 years?
Quiz: 24-1 answer 5 You have 400 g of a radioactive sample with a half-life of 20 years. How much is left after 50 years? 71 g T -. = <=. ' = ".>?@ ' λ = ".>?@ = ".>?@ A B C." DEFGH = 0.0347 y&- N = N " e &'( = 400 g e & "."@KL DMB N" D = 71 g
Quiz: 2 6 A radioactive isotope has a half-life of 10 s. You are observing a sample of this isotope. After approximately one minute of observation, there is only one nucleus of this isotope left in your sample. How many atoms of this isotope will be left in your sample 15 s later? A. Definitely 1. B. Approximately 0.35. C. Definitely 0. D. Possibly 1 but probably 0. E. Possibly 0 but probably 1.
Quiz: 24-2 answer 7 A radioactive isotope has a half-life of 10 s. You are observing a sample of this isotope. After approximately one minute of observation, there is only one nucleus of this isotope left in your sample. How many atoms of this isotope will be left in your sample 15 s later? Possibly 1 but probably 0. As with other quantum processes, the time any particular nucleus will decay cannot be predicted. T -. = <=. = ".>?@ ' ' λ = ".>?@ = ".>?@ = 0.0693 s&- A B C -" H N = N " e &'( = 1 e & ".">?@ HMB -N H = 0.35 But, nuclei are quantized, so you can either have 1 nucleus or 0 nuclei.
Quiz: 3 8 You have a radioactive sample A with a half-life of 100 years, and another sample B with a half-life of 1000 years. Both samples have the same number of radioactive nuclei. Which sample has the higher activity? A. sample A B. sample B C. both the same D. impossible to tell
Quiz: 24-3 answer 9 Sample A If a sample has a shorter half-life, this means that it decays more quickly (larger decay constant λ): T -. = <=. ' The activity is higher for the sample with a higher decay constant λ: R = Q = λn (
Example 1 10 A sample of a radioactive isotope is found to have an activity of 115.0 Bq immediately after it is pulled from the reactor that formed the isotope. Its activity 2 h 15 min later is measured to be 85.2 Bq. a) Calculate the decay constant and the halflife of the sample. b) How many radioactive nuclei were there in the sample initially?
Radioactive dating 11 If you know how much of a radioactive material has decayed, you can read the elapsed time from the decay curve. The half-lives of various nuclei can vary widely. T -. 2T -.
Carbon dating 12 -K > C β -decays with a half-life of 5730 years. -K Carbon dating uses > C produced in the upper atmosphere during nuclear reactions caused by cosmic rays. -K > C reacts just like -. C -K -. > chemically, and the ratio of > C to > C in a living organism is the same as the equilibrium ratio in the atmosphere. -K -. After an organism dies, it no longer absorbs carbon, so the ratio of > C to > C -K continually decreases since > C decays. The decay rate per amount of carbon in a sample therefore yields the age of the previously living organism.
Quiz: 4 13 Suppose you have a sample of old dead tree that is believed to be 100 thousand years old. Can you use carbon-14 to accurately date this sample? A. Yes. B. No.
Quiz: 24-4 answer 14 No The range of dates that can be determined depends on the half-life of the material. If the time is much shorter than the half-life, not enough of the material will have decayed. If the time is much longer than about ten half-lives, too much of the material will have decayed, and it will no longer be detectable.
Refining radioactive dating 15 There are also many nonradioactive methods of dating materials, such as tree ring analysis. The radioactive and nonradioactive methods may be used to check each other and refine the procedure, increasing accuracy of dating. -K The > C method gives lifetimes that are too short after about 10,000 years, probably due to fluctuations in the -K -. to C ratio. C > >
Age of Earth 16 The age of Earth is measured to be around 4.54 billion years using radioactivity of rocks and meteorites. The type of the rocks used is known to reject lead during its formation. The method uses two series of α and β decays:.@u?. U (T -. = 4.47 billion years ) to U. Pb = 704 million years ) to Pb.@N?. U (T -..">.."L U.. The age of Earth is determined from the Pb to U ratios.
Quiz: 5 17 If a stable nucleus breaks into its constituent nucleons, was energy added to the nucleus, or was energy released by the nucleus? A. Energy was added to the nucleus. B. Energy was released by the nucleus.
Quiz: 24-5 answer 18 Energy was added to the nucleus. The mass of the stable nucleus is less than the sum of the masses of the nucleons. This reduction in mass, m, multiplied by c., is called the binding energy. E = m c. If a nucleus could break apart spontaneously by releasing energy, it would not be stable.
Binding energy per nucleon 19 The flat curve around the nuclei with 50 A 75 indicates that these nuclei are the most stable.
Quiz: 6 20 One way to think about the low binding energy per nucleon for low A numbers is A. that there are few protons, hence there is less electrostatic repulsion. B. that there are few electrons to provide the electrostatic attraction. C. at low A, the number of nearest neighbors per nucleon is small compared to higher A numbers. D. at low A, the nucleus is less spherical, hence there is less surface tension. E. None of the above statements is correct.
Quiz: 24-6 answer 21 at low A, the number of nearest neighbors per nucleon is small compared to higher A numbers. For elements with low A, adding more nucleons mean there are more particles for strong force to act on, making it harder to break apart. The shallow slope for large A indicates that nucleons interact only with their neighboring nucleons because the range of strong nuclear force is short.