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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.
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A First Course in Differential Equations with Modeling Applications (MindTap Course List)
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