1. Consider the three component decay where isotope 1 decays to isotope 2 with decay constant...

  

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1. Consider the three component decay where isotope 1 decays to isotope 2 with decay constant X₁, isotope 2 decays to isotope 3 with decay constant A2, and isotope 3 is stable. (a) Suppose isotope 1 is produced at a constant rate Qo and there are none of isotopes 1, 2, 3 initially. Derive an expression for the population of each of the three isotopes as a function of time and plot the population for isotopes 1 and 2 with Qo 106 s-¹, A₁ = 0.4 s ¹, and X₂ 0.05 s ¹ for a time interval that clearly shows the behavior of the population. Use the backward substitution method with integrating factor. (3 points) = = (b) Suppose isotope 1 has been produced at a constant rate for a very long time and the production stops. Find (numerically) the time it takes for the activity of all three isotopes to be 1% of its level at the time production stopped. (2 points) 1. Consider the three component decay where isotope 1 decays to isotope 2 with decay constant X₁, isotope 2 decays to isotope 3 with decay constant A2, and isotope 3 is stable. (a) Suppose isotope 1 is produced at a constant rate Qo and there are none of isotopes 1, 2, 3 initially. Derive an expression for the population of each of the three isotopes as a function of time and plot the population for isotopes 1 and 2 with Qo 106 s-¹, A₁ = 0.4 s ¹, and X₂ 0.05 s ¹ for a time interval that clearly shows the behavior of the population. Use the backward substitution method with integrating factor. (3 points) = = (b) Suppose isotope 1 has been produced at a constant rate for a very long time and the production stops. Find (numerically) the time it takes for the activity of all three isotopes to be 1% of its level at the time production stopped. (2 points)