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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