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Chapter 14: Problem 93

Radioactive plutonium- \(239\left(t_{\frac{1}{2}}=2.44 \times 10^{5} \mathrm{yr}\right)\) is used in nuclear reactors and atomic bombs. If there are \(5.0 \times 10^{2} \mathrm{~g}\) of the isotope in a small atomic bomb, how long will it take for the substance to decay to \(1.0 \times 10^{2} \mathrm{~g},\) too small an amount for an effective bomb? (Hint: Radioactive decays follow first-order kinetics.)

Short Answer

Expert verified
The time it will take for the substance to decay to \(1.0 \times 10^{2} \mathrm{~g}\) is found by evaluating the expression given in step 3. You can perform the calculations using a calculator, making sure to use natural logarithms (ln) rather than base-10 logarithms.

Step by step solution

01

Recognize the problem's variables

In this problem, the initial amount of the substance \(N_{0}\) is \(5.0 \times 10^{2} \mathrm{~g}\), the final amount of the substance \(N\) is \(1.0 \times 10^{2} \mathrm{~g}\), and the half-life of the substance \(t_{\frac{1}{2}}\) is \(2.44 \times 10^{5} \mathrm{~yr}\). Our task is to find the time it will take for the substance to decay, t.
02

Express the decay constant in terms of the half-life

The decay constant k can be expressed in terms of the half-life using the equation \(k = ln(2) / t_{\frac{1}{2}}\). Substituting the given half-life into this equation gives \(k = ln(2) / (2.44 \times 10^{5})\).
03

Use the first-order decay equation to find the time

We rearrange the formula \(N = N_{0}e^{-kt}\) to solve for t, yielding \(t = -ln(N / N_{0}) / k\). Substituting the values for N, \(N_{0}\) and k gives \(t = -ln((1.0 \times 10^{2}) / (5.0 \times 10^{2})) / (ln(2) / (2.44 \times 10^{5}))\).

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Most popular questions from this chapter

The reaction \(\mathrm{S}_{2} \mathrm{O}_{8}^{2-}+2 \mathrm{I}^{-} \longrightarrow 2 \mathrm{SO}_{4}^{2-}+\mathrm{I}_{2}\) proceeds slowly in aqueous solution, but it can be catalyzed by the \(\mathrm{Fe}^{3+}\) ion. Given that \(\mathrm{Fe}^{3+}\) can oxidize \(\mathrm{I}^{-}\) and \(\mathrm{Fe}^{2+}\) can reduce \(\mathrm{S}_{2} \mathrm{O}_{8}^{2-},\) write a plausible two-step mechanism for this reaction. Explain why the uncatalyzed reaction is slow.

The decomposition of dinitrogen pentoxide has been studied in carbon tetrachloride solvent \(\left(\mathrm{CCl}_{4}\right)\) at a certain temperature: \(2 \mathrm{~N}_{2} \mathrm{O}_{5} \longrightarrow 4 \mathrm{NO}_{2}+\mathrm{O}_{2}\) $$ \begin{array}{cc} {\left[\mathrm{N}_{2} \mathrm{O}_{5}\right](\mathrm{M})} & \text { Initial Rate }(\mathrm{M} / \mathrm{s}) \\ \hline 0.92 & 0.95 \times 10^{-5} \\ 1.23 & 1.20 \times 10^{-5} \\ 1.79 & 1.93 \times 10^{-5} \\ 2.00 & 2.10 \times 10^{-5} \\ 2.21 & 2.26 \times 10^{-5} \end{array} $$ Determine graphically the rate law for the reaction and calculate the rate constant.

Write an equation relating the concentration of a reactant \(\mathrm{A}\) at \(t=0\) to that at \(t=t\) for a first-order reaction. Define all the terms and give their units.

Use the Arrhenius equation to show why the rate constant of a reaction (a) decreases with increasing activation energy and (b) increases with increasing temperature.

What are the units for the rate constants of first-order and second-order reactions?
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