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Chapter 21: Problem 30

Consider the decay series \(\mathrm{A} \longrightarrow \mathrm{B} \longrightarrow \mathrm{C} \longrightarrow \mathrm{D}\) where \(A, B,\) and \(C\) are radioactive isotopes with halflives of \(4.50 \mathrm{~s}, 15.0\) days, and \(1.00 \mathrm{~s},\) respectively, and \(\mathrm{D}\) is nonradioactive. Starting with 1.00 mole of A, and none of \(\mathrm{B}, \mathrm{C},\) or \(\mathrm{D},\) calculate the number of moles of \(\mathrm{A}, \mathrm{B}, \mathrm{C},\) and \(\mathrm{D}\) left after 30 days.

Short Answer

Expert verified
The number of moles of A, B, C, and D left after 30 days can be calculated using the radioactive decay law and the given half-lives. The number of moles will depend on the rate constants of the isotopes and the elapsed time. These can be calculated by applying the radioactive decay law to each isotope considering the rate of formation and decay at each step. The final results would then be evaluated using these developed expressions.

Step by step solution

01

Calculation of Rate Constants

The first step is to calculate the rate constant for each isotope using the relation between half-life (T) and rate constant (k), given by \(k = 0.693/T\). For isotope A with a half-life of 4.5 s, the rate constant \(k_A\) is \(0.693 /4.5s\). Similarly, we can calculate rate constants for isotopes B and C, denoted as \(k_B\) and \(k_C\) respectively.
02

Calculation of Remaining Moles of A

Following the radioactive decay law, the number of moles of A remaining after time t is given by \(N_A = N_{A0} e^{-k_At}\). Given that the initial number of moles \(N_{A0}\) is 1 and the time t is 30 days (converted to seconds), we can calculate the remaining moles of A.
03

Calculation of Moles of B

As A decays, it produces B. But B itself decays to C. So, the number of moles of B is given by the difference in the rates of formation (from A) and decay (to C). We calculate this using \(N_B = (N_{A0} k_A / (k_B - k_A)) * (e^{-k_A t} - e^{-k_B t})\). Here, N_{A0} k_A represents the rate of formation from A, and \(N_{A0} k_B / (k_B - k_A)\) is a factor that accounts for the difference in decay rates of A and B.
04

Calculation of Moles of C and D

Following similar reasoning, the number of moles of C and D after time t is calculated by considering the difference in the rates of formation and decay. For C this becomes, \(N_C = (N_{A0} k_A k_B/(k_C(k_B - k_A)(k_C - k_B)) * (e^{-k_A t} - e^{-k_B t} - (k_B - k_A)e^{-k_C t})\). For D, which does not decay, the moles are calculated as \(N_D = N_{A0} - N_A - N_B - N_C\).
05

Evaluation of Results

After calculating the number of moles of A, B, C and D after 30 days using the developed expressions. We can then evaluate these values to get the final results.

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