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Chapter 38: Problem 22

Carbon-11-labeled acetate shows promise in PET scans for determining the extent of metastasized prostate cancer. (a) Given C-11's 20.4-min half-life, how long will it take an initial dose of \(2.0 \times 10^{9} \mathrm{Bq}\) to decay to \(7 \mathrm{kBq}\) (roughly the natural radioactivity of the human body)? (b) What nucleus remains after \(\mathbf{C}-11\) decavs by positron emission?

Short Answer

Expert verified
a) The initial dose will decay to 7 kBq after approximately 146 minutes. b) After Carbon-11 decays by positron emission, a nucleus of Boron-11 remains.

Step by step solution

01

Calculation of decay time

The formula to calculate the amount of a radioactive substance remaining after time \(t\) is \(N = N_0 e^{-\lambda t}\). The decay constant \(\lambda\) can be calculated from the half-life (\(T_{1/2}\)) using the formula \(\lambda = \frac{ln(2)}{T_{1/2}}\). In this case, \(T_{1/2} = 20.4 min\), \(N_0 = 2.0 \times 10^9 Bq\), and \(N = 7 kBq = 7 \times 10^3 Bq\). By substituting these values to the equation, we can solve for \(t\).
02

Solve for decay time

Substituting the given values into the formula, we have \(7 \times 10^3 Bq = 2.0 \times 10^9 Bq \times e^{-\frac{ln(2)}{20.4 min} \times t}\). Solving this equation for \(t\) gives \(t = \frac{20.4 min \times ln(2.0 \times 10^9 Bq/7 \times 10^3 Bq)}{ln(2)}\), which equals approximately 146 minutes.
03

Identify the decay product

Carbon-11 undergoes beta plus decay (or positron emission), where a proton in the nucleus is converted into a neutron, a positron and an electron neutrino. This reduces the atomic number by one but keeps the atomic mass number the same, creating a nucleus of Boron-11 (B-11).

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Most popular questions from this chapter

You measure the activity of a radioactive sample at \(2.4 \mathrm{MBq}\). Thirty minutes later, the activity level is 1.9 MBq. Find the material's half-life.

It's possible, though difficult, to realize alchemists' dreams of synthesizing gold. One reaction involves bombarding mercury198 with neutrons to produce, for each neutron captured, a gold-197 nucleus and another particle. Write the equation for this reaction.

The table below lists reported levels of iodine- 131 contamination in milk in three countries affected by the 1986 Chernobyl accident, along with each country's safety guideline. Given I-131's half-life of 8.04 days, how long did each country have to wait for I-131 levels to decline to a level deemed safe by its standards? $$\begin{array}{lcc} \hline & \multicolumn{2}{c} {\text { Level }(\mathrm{B} q / \mathrm{L})} \\ \hline \text { Country } & \text { Reported } & \text { Safety Guideline } \\ \hline \text { Poland } & 2000 & 1000 \\ \text { Austria } & 1500 & 370 \\ \text { Germany } & 1184 & 500 \\ \hline \end{array}$$

How do (a) the number of nucleons and (b) the nuclear charge compare in the two nuclei \(_{17}^{35} \mathrm{Cl}\) and \(_{19}^{35} \mathrm{K} ?\)

On an energy-release-per-unit-mass basis, by approximately what factor do nuclear reactions exceed chemical reactions?
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