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APPM 2360 Radioactive Decay Project

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Questions:

1. Solve (2) analytically for NA(t) given the initial condition NA(0) = A0. (Your answer should be expressed symbolically in terms of NA(t), kA, A0, and t.)

2. (a) Use the result of question 1 to solve for kA. (This is just a rearrangement of terms. Your answer should still be expressed symbolically.)
(b) Now use the result of 2(a) and the fact that NA(tA,1/2) = 1
2A0 to solve for kA. (Your answer should still be expressed symbolically.)
(c) What are the units of the proportionality constant kA (explain)? Use the result of 2(b) and the half life tA,1/2 = (5 ln 2) seconds to find a value for kA.

3. (a) Suppose you have a sample of element A that initially has A0 = 15, 000 atoms. Use Euler’s method to solve (2) for the step sizes h1 = 1, h2 = 0.1, and h3 = 0.01. Plot all numerical solutions and the exact solution on a single plot for an appropriate range of t. (Be sure to include a legend.)
(b) The absolute error of a numerical approximation can be defined by
Abs. Error = |Exact Solution − Approximate Solution|
On the same graph, plot the absolute errors of the numerical solutions for the range of t used in part (a). (Plotting with semilogy may be useful. Be sure to include a legend.)
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