CAIE A-Level Physics A2 23.2 Radioactive Decay Questions

Practise analysing random decay, activity, decay constant, half-life and exponential equations or graphs to extract radioactive quantities.

Exam points

interpret count-rate fluctuations and explain radioactive decay as spontaneous and random
use A = ?N to calculate activity, decay constant or number of radioactive nuclei
define half-life and use ? = 0.693/t�
use and interpret the exponential decay relationship x = x?e??t for activity, count rate or undecayed nuclei
extract decay quantities from radioactive-decay graphs and data

Covered syllabus points

23.2.1—Fluctuations in count rate provide evidence for the random nature of 23.2.2—Radioactive decay is both spontaneous and random 23.2.3—Activity and decay constant, and recall and use A = λN 23.2.4—Half-life 23.2.5—Λ = 0.693 / t 1/2 23.2.6—The exponential nature of radioactive decay, and sketch and use the

[Maximum number: 4]

Carbon-15 $\left({ }_{6}^{15} \mathrm{C}\right)$ is an isotope of carbon that undergoes radioactive decay to nitrogen-15 $\left({ }_{7}^{15} \mathrm{~N}\right)$ , which is a stable isotope of nitrogen.

Radioactive decay is both a random and a spontaneous process.

(a)

State what is meant by:

[ 2 ]

(i)

random

[ 1 ]

(ii)

spontaneous.

[ 1 ]

(b)

A small sample of carbon-15 decays. The mass M of carbon-15 in the sample decreases with time t.

Fig. 10.1 shows the variation with t of the value of $\ln \left(M / 10^{-16} \mathrm{~g}\right)$ .

Fig. 10.1

[ 2 ]

(i)

State how Fig. 10.1 demonstrates that radioactive decay is random.

[ 1 ]

(ii)

Use your answer in (b)(iv) to calculate the half-life of carbon-15.
half-life = s

[ 1 ]

half-life $=0.693 / \lambda$ $\text { = } 0.693 \text { / } 0.28$ $=2.5 \mathrm{~s}$

[Maximum number: 9]

Carbon-11 is radioactive and decays by $\beta^{+}$ emission to form boron-11. Carbon-11 has a half-life of 20 minutes. Boron-11 is stable.

(a)

Define half-life.

[ 1 ]

(b)

A sample contains $N_{0}$ nuclei of carbon-11 and no other nuclei at time t=0.

On Fig. 9.1, sketch the variation with t of the number of nuclei of boron-11 in the sample.

Fig. 9.1

[ 3 ]

sketch: line with positive gradient starting at (0,0) and extending to $t=80 \mathrm{~min}$
exponential curve, extending from t=0 to $t=80 \mathrm{~min}$ , with gradient of steadily decreasing magnitude
line passing through $(0,0),\left(20,0.5 N_{0}\right)$ and $\left(40,0.75 N_{0}\right)$

(c)

Explain, with reference to the random nature of radioactive decay, why the activity of the carbon-11 sample in (b) decreases with time.

[ 2 ]

(d)

State, with reasons, whether a radiation detector placed near to the sample of carbon-11 indicates a measured count rate from the sample that is less than, the same as or greater than the activity of the sample.

[ 3 ]

[Maximum number: 5]

(a)

Define half-life of a radioactive isotope.

[ 1 ]

(b)

Radioactive isotope X decays to isotope Y .

A sample contains only nuclei of X at time t=0. Fig. 9.1 shows the variation with t of the numbers of nuclei of X and of Y as the sample decays.

Fig. 9.1

[ 4 ]

(i)

State the name of the quantity represented by the magnitude of the gradient of line X in Fig. 9.1.

[ 1 ]

(ii)

State three conclusions about X or Y that may be drawn from Fig. 9.1. The conclusions may be qualitative or quantitative. Use the space below for any working that you need.

1
2
3

[ 3 ]

Y is a stable isotope

- total number of nuclei is constant

- half-life (of X ) is 13.6 s

- decay constant (of X ) is $0.051 \mathrm{~s}^{-1}$

- amount (of X ) at t=0 is 0.066 mol

- activity (of X ) at t=0 is $2.0 \times 10^{21} \mathrm{~Bq}$

Any three points, 1 mark each

[Maximum number: 1]

A small sample of carbon-15 decays. The mass M of carbon-15 in the sample decreases with time t.

Fig. 10.1 shows the variation with t of the value of $\ln \left(M / 10^{-16} \mathrm{~g}\right)$ .

Fig. 10.1

State how Fig. 10.1 demonstrates that radioactive decay is random.

[Maximum number: 2]

Carbon-11 is radioactive and decays by $\beta^{+}$ emission to form boron-11. Carbon-11 has a half-life of 20 minutes. Boron-11 is stable.

Explain, with reference to the random nature of radioactive decay, why the activity of the carbon-11 sample in (b) decreases with time.

[Maximum number: 1]

Radioactive isotope X decays to isotope Y .

A sample contains only nuclei of X at time t=0. Fig. 9.1 shows the variation with t of the numbers of nuclei of X and of Y as the sample decays.

Fig. 9.1

State the name of the quantity represented by the magnitude of the gradient of line X in Fig. 9.1.

[Maximum number: 1]

Define half-life of a radioactive isotope.

[Maximum number: 2]

Polonium-211 $\left({ }_{84}^{211} \mathrm{Po}\right)$ decays by alpha emission to form a stable isotope of lead (Pb).

The variation with time t of the number of unstable nuclei N in a sample of polonium-211 is shown in Fig. 9.1.

Fig. 9.1

At time t=0, the sample contains only polonium-211.

Use Fig. 9.1 to determine the decay constant $\lambda$ of polonium-211. Give a unit with your answer.
$\lambda=\ldots \ldots \ldots \ldots \ldots . \ldots \ldots . . . . . . . . .$ . unit ..................

(half-life found as) 0.52 s or correctly read points substituted into
$N=N_{0} e^{-\lambda t}$ $\lambda=\frac{0.693}{t_{1 / 2}}$ $\lambda=\frac{0.693}{0.52}$ $\lambda=1.3 \mathbf{~ s}^{-1}$

[Maximum number: 3]

Iodine-131 $\left({ }_{53}^{131} \mathrm{I}\right)$ is a radioactive isotope with a decay constant of $9.9 \times 10^{-7} \mathrm{~s}^{-1}$ .

Regulations require that the activity of iodine-131 in 1.0 kg of water is to be less than 170 Bq .

Calculate the time, in days, for the activity of the contaminated water in (b) to be reduced to 170 Bq .

time =

days

$\begin{aligned} A=A_{0} \mathrm{e}^{-\lambda t} 170=560 \exp \left(-9.9 \times 10^{-7} \times t\right) \end{aligned}$ $t=1.2 \times 10^{6} \mathrm{~s}$
= 14 days

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Covered syllabus points

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