Q: During World War I, radium-226 was used in luminous paint. If it takes $$1.2 \times 10^3$ days for the paint to degrade to 2.4$$\%$$ of its original activity, what is the decay constant?

$$1.51 \times 10$$^{-3}$$\ \text{days}^{-1}$$

Question 1

Which of the following best describes the order of radioactive decay reactions?

Accepted Answer

Radioactive decay reactions are always first order.

Question 2

Given the integrated rate law for radioactive decay: $$\ln\left( \frac{N_t}{N_0$} \right) = -kt$$, what does $$k$$ represent in this equation?

Accepted Answer

$$k is $$the decay constant, which determines the rate of decay.

Question 3

A sample of a radioactive isotope decays from $$9.3 	imes 10^7$ disintegrations per second to 2.7$$ 	imes $$10^7$$ disintegrations per second. If the decay constant is $$0.084\ 	ext{min}^{-1}$$, how much time has elapsed?

Accepted Answer

$$2.54\ 	ext{min}$$

Question 4

For the radioactive decay of lead-202, the decay constant is $$1.32 	imes 10$$^{-2}$$\ 	ext{hr}^{-1}. $$How long will it take for the amount to decrease to $$52\% of $$its initial value?

Accepted Answer

$$1.87\ 	ext{hr}$$

Question 5

During World War I, radium-226 was used in luminous paint. If it takes $$1.2 	imes 10^3$ days for the paint to degrade to 2.4$$\%$$ of its original activity, what is the decay constant?

Accepted Answer

$$1.51 	imes 10$$^{-3}$$\ 	ext{days}^{-1}$$

Question 6

If the decay constant for polonium-209 is $$8.00 	imes 10$$^{-3}$$\ 	ext{yr}^{-1}$$, what fraction of the original sample remains after $$1.0 	imes 10^2$ years?

Accepted Answer

$$0.449$$

Question 7

Which of the following equations correctly relates the half-life $$t_{1/2} of a $$first-order radioactive decay to its decay constant $$k$$?

Accepted Answer

$$t_{1/2} = \frac{0.693}{k}$$

Question 8

A radioactive sample has a half-life of $$5$$ hours. What fraction of the original sample remains after $$15$$ hours?

Accepted Answer

$$0.125$$

Question 9

If a radioactive isotope decays to $$25\% of $$its original amount in $$t$$ hours, what is the relationship between $$t$$ and the half-life $$t_{1/2}$$?

Accepted Answer

$$t = 2 t_{1/2}$$

Question 10

Which of the following statements about the decay constant $$k is $$correct?

Accepted Answer

A larger $$k$$ means the substance decays more quickly.

Question 11

A scientist finds that a radioactive sample has decayed to $$10\% of $$its original amount after $$100$$ years. What is the decay constant $$k$$? (Use $$\ln(0.10) = -2.302)$$

Accepted Answer

$$0.0230\ 	ext{yr}^{-1}$$

Question 12

Which of the following best explains why the activity of a radioactive sample decreases over time?

Accepted Answer

The number of radioactive nuclei decreases as they decay.

Question 13

A radioactive isotope has a decay constant of $$0.005\ 	ext{yr}^{-1}. $$How many years will it take for $$75\% of $$the sample to decay?

Accepted Answer

$$277.2\ 	ext{years}$$

Question 14

Which of the following is a real-world application of radioactive decay?

Accepted Answer

Radiocarbon dating of archaeological artifacts.

Question 15

If $$N_0 is $$the initial amount of a radioactive substance and $$N_t is $$the amount remaining after time $$t$$, which equation correctly expresses the relationship for first-order decay?

Accepted Answer

$$N_t = N_0$ e$$^{-kt}$$

Radioactive Decay and Integrated Rate Laws

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