CAIE A-Level Physics 23.2 Radioactive Decay

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: 10]

(a)

A radiation detector is placed close to a radioactive source. The detector does not surround the source.
Radiation is emitted in all directions and, as a result, the activity of the source and the measured count rate are different.

Suggest two other reasons why the activity and the measured count rate may be different.
1.
2.

[ 2 ]

(b)

The variation with time t of the measured count rate in (a) is shown in Fig. 12.1.

Fig. 12.1

[ 5 ]

(i)

State the feature of Fig. 12.1 that indicates the random nature of radioactive decay.

[ 1 ]

(ii)

Use Fig. 12.1 to determine the half-life of the radioactive isotope in the source.
half-life = hours

[ 4 ]

(c)

The readings in (b) were obtained at room temperature.

A second sample of this isotope is heated to a temperature of $500^{\circ} \mathrm{C}$ .
The initial count rate at time t=0 is the same as that in (b).
The variation with time t of the measured count rate from the heated source is determined.

State, with a reason, the difference, if any, in
1. the half-life,
2. the measured count rate for any specific time.

[ 3 ]

[Maximum number: 8]

(a)

Define radioactive decay constant.

[ 2 ]

(b)

Show that the decay constant $\lambda$ is related to the half-life $t_{\frac{1}{2}}$ of a radioactive isotope by the expression

\lambda t_{\frac{1}{2}}=\ln 2

[ 2 ]

$A=A_{0} \mathrm{e}^{-\lambda t}$

after one half-life, $1 / 2 A_{0}=A_{0} \mathrm{e}^{-\lambda t 1 / 2}$

M1

$\frac{1}{2}=\exp \left(-\lambda t_{\frac{1}{2}}\right)$ and hence taking logs, $\ln 2=\lambda t_{\frac{1}{2}}$

A1

(c)

A small volume of solution containing the radioactive isotope sodium-24 $\left({ }_{11}^{24} \mathrm{Na}\right)$ has an initial activity of $3.8 \times 10^{4} \mathrm{~Bq}$ . Sodium-24, of half-life 15 hours, decays to form a stable daughter isotope.

All of the solution is poured into a container of water. After 36 hours, a sample of water of volume $5.0 \mathrm{~cm}^{3}$ , taken from the container, is found to have an activity of 1.2 Bq .

Assuming that the solution of the radioactive isotope is distributed uniformly throughout the container of water, calculate the volume of water in the container.
$\mathrm{cm}^{3}$

[ 4 ]

activity $\quad=3.8 \times 10^{4} \exp (-\ln 2 \times 36 / 15)$

C1

$=7200 \mathrm{~Bq}$

C1

or activity $=3.8 \times 10^{4} / 2^{2.4}$

(C1)

$=7200 \mathrm{~Bq}$

(C1)

volume $\quad=(7200 / 1.2) \times 5.0$

C1

$=3.0 \times 10^{4} \mathrm{~cm}^{3}$

A1

OR activity of $5.0 \mathrm{~cm}^{3}=1.2 \times 2^{2.4}$

(C1)

$=6.3336 \mathrm{~Bq}$

(C1)

volume $=\left(3.8 \times 10^{4} / 6.3336\right) \times 5.0$

(C1)

$=3.0 \times 10^{4} \mathrm{~cm}^{3}$

(A1)

All question bank results loaded

Covered syllabus points

23.2 Radioactive decay question 1

Question (a)

Question (b)

Question (i)

Question (ii)

Question (c)

23.2 Radioactive decay question 2

Question (a)

Question (b)

Question (c)