Many radioactive decays occur within a sequence of decays for example,
\(^{234}_{92}U\) \(\rightarrow\) \(^{230}_{88}Th\) \(\rightarrow\) \(^{226}_{84}Ra\).
The half-life for the \(^{234}_{92}U\) \(\rightarrow\) \(^{230}_{88}Th\) decay is
\(2.46 \times 10^{5}\) y, and the half-life for the \(^{230}_{88}Th\)
\(\rightarrow\) \(^{226}_{84}Ra\) decay is \(7.54 \times 10^{4}\) y. Let 1 refer to
\(^{234}_{92}U\), 2 to \(^{230}_{88}Th\), and 3 to \(^{226}_{84}Ra\); let \(\lambda\)1
be the decay constant for the \(^{234}_{92}U\) \(\rightarrow\) \(^{230}_{88}Th\)
decay and \(\lambda\)2 be the decay constant for the \(^{230}_{88}Th\)
\(\rightarrow\) \(^{226}_{84}Ra\) decay. The amount of \(^{230}_{88}Th\) present at
any time depends on the rate at which it is produced by the decay of
\(^{234}_{92}U\) and the rate by which it is depleted by its decay to
\(^{226}_{84}Ra\). Therefore, \(d$$N_{2}\)(\(t\))/\(d$$t\) = \(\lambda$$_1$$N$$_1\)(\(t\))
- \(\lambda$$_2$$N$$_2\)(\(t\)). If we start with a sample that contains \(N_{10}\)
nuclei of \(^{234}_{92}U\) and nothing else, then \(N{(t)}\) =
\(N_{01}\)e\(^{-\lambda_{1}t}\). Thus \(dN_{2}(t)/dt\) =
\(\lambda\)1\(N_{01}\)e\(^{-\lambda_{1}t}\)- \(\lambda2N_{2}(t)\). This differential
equation for \(N_{2}(t)\) can be solved as follows. Assume a trial solution of
the form \(N_{2}(t)\) = \(N_{10} [h_{1}e^{-\lambda_1t}\) + \(h_{2}e^{-\lambda_2t}\)]
, where \(h_{1}\) and \(h_{2}\) are constants. (a) Since \(N_{2}(0)\) = 0, what must
be the relationship between \(h_{1}\) and \(h_{2}\)? (b) Use the trial solution to
calculate \(dN_{2}(t)/dt\), and substitute that into the differential equation
for \(N_{2}(t)\). Collect the coefficients of \(e^{-\lambda_{1}t}\) and
\(e^{-\lambda_{2}t}\). Since the equation must hold at all t, each of these
coefficients must be zero. Use this requirement to solve for \(h_{1}\) and
thereby complete the determination of \(N_{2}(t)\). (c) At time \(t = 0\), you
have a pure sample containing 30.0 g of \(^{234}_{92}U\) and nothing else. What
mass of \(^{230}_{88}Th\) is present at time \(t = 2.46 \times 10^{5}\) y, the
half-life for the \(^{234}_{92}U\) decay?
