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Chapter 14: Problem 99

Cobalt-60 is used in radiation therapy to treat cancer. It has a first-order rate constant for radioactive decay of $k=1.31 \times 10^{-1} \mathrm{yr}^{-1}$. Another radioactive isotope, iron59, which is used as a tracer in the study of iron metabolism, has a rate constant of $k=1.55 \times 10^{-2}\( day \)^{-1}$. (a) What are the half-lives of these two isotopes? (b) Which one decays at a faster rate? (c) How much of a 1.00-mg sample of each isotope remains after three half-lives? How much of a \(1.00-\mathrm{mg}\) sample of each isotope remains after five days?

Short Answer

Expert verified
\(t_{1/2(Co-60)} \approx 5.27\,yr\), \(t_{1/2(Fe-59)} \approx 44.8\,days\); Iron-59 decays faster; After 3 half-lives: \(mass_{Co-60} \approx 0.125\,mg\), \(mass_{Fe-59} \approx 0.125\,mg\); After 5 days: \(mass_{Co-60(5 days)} \approx 0.959\,mg\), \(mass_{Fe-59(5 days)} \approx 0.597\,mg\).

Step by step solution

01

Calculate the half-life for Cobalt-60 and Iron-59

The half-life for a first-order reaction can be calculated using the formula: \[t_{1/2} = \frac{ln(2)}{k}\] For Cobalt-60, \(k = 1.31 \times 10^{-1}\,yr^{-1}\). Thus, \[t_{1/2(Co-60)} = \frac{ln(2)}{1.31 \times 10^{-1} yr^{-1}}\] For Iron-59, \(k = 1.55 \times 10^{-2}\,day^{-1}\). Thus, \[t_{1/2(Fe-59)} = \frac{ln(2)}{1.55 \times 10^{-2} day^{-1}}\]
02

Determine the isotope that decays faster

Compare the rate constants to determine which isotope has the faster decay rate. Cobalt-60: \(k_{Co-60} = 1.31 \times 10^{-1} yr^{-1}\) Iron-59: \(k_{Fe-59} = 1.55 \times 10^{-2} day^{-1}\)
03

Calculate the mass remaining after 3 half-lives

The remaining mass can be calculated using the half-life and the number of half-lives passed by using the formula: \[mass = initial \times \left(\frac{1}{2}\right)^n\] For both isotopes, we have an initial mass of \(1.00~mg\) and \(n = 3\). Cobalt-60: \[mass_{Co-60} = 1.00~mg \times \left(\frac{1}{2}\right)^3\] Iron-59: \[mass_{Fe-59} = 1.00~mg \times \left(\frac{1}{2}\right)^3\]
04

Calculate the mass remaining after 5 days

Use the decay rate and the time duration to calculate the mass remaining, using the formula: \[mass = initial \times e^{-kt}\] For 5 days, we need the decay constant to be in days: Cobalt-60: Convert the rate constant to days using the conversion factor \(1 yr = 365 days\): \[k_{Co-60(day)} = 1.31 \times 10^{-1} yr^{-1} \times \frac{1}{365}\] Calculate the mass remaining after 5 days: \[mass_{Co-60(5 days)} = 1.00~mg \times e^{-k_{Co-60(day)} \times 5}\] Iron-59: Calculate the mass remaining after 5 days: \[mass_{Fe-59(5 days)} = 1.00~mg \times e^{-1.55 \times 10^{-2} day^{-1} \times 5}\]

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Most popular questions from this chapter

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