Concept explainers
Potassium-40 Decay The chemical element potassium is a soft metal that can be found extensively throughout the Earth's crust and oceans. Although potassium occurs naturally in the form of three isotopes, only the isotope potassium-40 (K-40) is radioactive. This isotope is also unusual in that it decays by two different nuclear reactions. Over time, by emitting beta particles a great percentage of an initial amount K0 of K-40 decays into the stable isotope calcium-40 (Ca-40), whereas by electron capture a smaller percentage of K0 decays into the stable isotope argon-40 (Ar-40). Because the rates at which the amounts C(t) of Ca-40 and A(t) of Ar-40 increase are proportional to the amount K(t) of potassium present, and the rate at which K(t)decays is also proportional to K(t), we obtain the system of linear first-order equations
where λ1 and λ2 are positive constants of proportionality. By proceeding as in Problem 1 we can solve the foregoing mathematical model.
- (a) From the last equation in the given system of differential equations find K(t) if K(0) = K0. Then use K(t) to find C(t) and A(t) from the first and second equations. Assume that C(0) = 0 and A(0) = 0.
- (b) It is known that λ1 = 4.7526 × 10−10 and λ2 = 0.5874 × 10−10 Find the half-life of K-40.
- (c) Use C(t) and A(t) found in part (a) to determine the percentage of an initial amount K0 of K-40 that decays into Ca-40 and the percentage that decays into Ar-40 over a very long period of time.
Want to see the full answer?
Check out a sample textbook solutionChapter 3 Solutions
Student Solutions Manual For Zill's A First Course In Differential Equations With Modeling Applications, 11th
- Half-Life The half-life of plutonium-241 is approximately 13 years. How much of a sample weighing 4 g will remain after 100 years? How much time is necessary for a sample weighing 4 g to decay to 0.1 g?arrow_forwardRepeat Example 5 when microphone A receives the sound 4 seconds before microphone B.arrow_forwardIsauter ort 2000 Decomes then patwon 1. At a certain instant, 10 g of a radioactive substance are present. minutes, 2 g are present. Determine the decay coefficient and the half-life. After 30arrow_forward
- Atoms consist of a nucleus made up of protons and neutrons. Orbiting outside the nucleus are a "cloud" of electrons. The number of protons determines the element. A carbon atom contains six protons and (normally) six neutrons; however, certain isotopes of carbon can exist. An isotope occurs when an atom has a different number of neutrons. Isotopes of carbon include: Carbon-12, Carbon-13, and Carbon-14. 6 protons + 6 neutrons electron + proton neutron Carbon atom Figure 1: Depiction of a carbon atom Carbon is an element common in living (organic) matter with C-12 and C-14 occurring in similar abundances. These isotopes combine with Oxygen to form Carbon Dioxide which plants use in photosynthesis. When these plants die, the C-12 and C-14 decay at different rates. (Radioactive decay occurs when an atom decays into a different element. Carbon-14 decays to Nitrogen-14.) Radiocarbon dating is a common technique used to find the age of dead organic matter by measuring the difference between…arrow_forward://www.desmos.com/calculator ww.desmos.com/matrix 24 Formula Sheet here: FORMULA SHEET An oncologist is monitoring the cell growth in a developing malignent tumor of a patient. The initial count on a cell culture was 280 cells. After only 7 hours, a new culture was taken revealing a cell count of 385 cells. Assuming the cell growth is best modeled with an exponential function, estimate the number of cells in the count taken after a total of 12 hours. Number cells. (round to the nearest whole number)arrow_forwardAtoms consist of a nucleus made up of protons and neutrons. Orbiting outside the nucleus are a "cloud" of electrons. The number of protons determines the element. A carbon atom contains six protons and (normally) six neutrons; however, certain isotopes of carbon can exist. An isotope occurs when an atom has a different number of neutrons. Isotopes of carbon include: Carbon-12, Carbon-13, and Carbon-14. 6 protons + 6 neutrons electron + proton neutron Carbon atom Figure 1: Depiction of a carbon atom Carbon is an element common in living (organic) matter with C-12 and C-14 occurring in similar abundances. These isotopes combine with Oxygen to form Carbon Dioxide which plants use in photosynthesis. When these plants die, the C-12 and C-14 decay at different rates. (Radioactive decay occurs when an atom decays into a different element. Carbon-14 decays to Nitrogen-14.) Radiocarbon dating is a common technique used to find the age of dead organic matter by measuring the difference between…arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage