The rate of decay of a radioactive isotope obeys dN/dt = —λN, where N is the number of radioactive nuclei present at time t, and the value of the decay constant A is different for different nuclei. By analogy to the equations of first- order kinetics, we have N = N0e—λt and λt1/2 = 0.693, where N0 is the initial number of radioactive nuclei and t1/2 is the half-life. The activity A of a radioactive sample is defined as the number of disintegrations per second: A ≡ −dN/dt = λN. (a) Show that A = A0e—λt. (b) A 1.00 g sample of 226Ra emits 3.7 X 1010 alpha particles per second. Find λ and t1/2. Find A after 999 years. (c) The only radioactive naturally occurring isotope of K is 40K, with a half-life of 1.28 × 109 yr and a natural isotopic abundance of 0.00117 percent. Find the activity of a sample of 10.0 g of KCl and find its activity after 2.00 × 108 yr.
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