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Date	May 2022	Marks available	2	Reference code	22M.2.HL.TZ1.9
Level	Higher level	Paper	Paper 2	Time zone	1
Command term	Show that	Question number	9	Adapted from	N/A

Question

Potassium-40 $({}_{19}^{40}K)$ decays by two processes.

The first process is that of beta-minus (β⁻) decay to form a calcium (Ca) nuclide.

Potassium-40 decays by a second process to argon-40. This decay accounts for 11 % of the total decay of the potassium-40.

Rocks can be dated by measuring the quantity of argon-40 gas trapped in them. One rock sample contains 340 µmol of potassium-40 and 12 µmol of argon-40.

Write down the equation for this decay.

[2]

Show that the initial quantity of potassium-40 in the rock sample was about 450 µmol.

[2]

b.i.

The half-life of potassium-40 is 1.3 × 10⁹ years. Estimate the age of the rock sample.

[3]

b.ii.

Outline how the decay constant of potassium-40 was determined in the laboratory for a pure sample of the nuclide.

[3]

Markscheme

${}_{20}^{40}{Ca}$ ✓

${}_{- 1}^{0}e + {\bar{ν}}_{e}$ OR ${}_{- 1}^{0}β + {\bar{ν}}_{e}$ ✓

Full equation ${}_{19}^{40}K \to {}_{20}^{40}{Ca} + {}_{- 1}^{0}e + {\bar{ν}}_{e}$

total K-40 decayed = $\frac{12 μmol}{0.11} = 109$ «μmol» ✓

so total K-40 originally was 109 + 340 = 449 «μmol»✓

b.i.

ALTERNATIVE 1

$λ = \frac{ln (2)}{t_{\frac{1}{2}}}$ used to give 𝜆 = 5.3 x 10^-10 per year ✓

$340 = (449) (e^{- 5.3 \times 10^{- 10} \times t})$

$\ln (\frac{340}{449}) = - 5.3 \times 10^{- 10} \times t$ ✓

t = 5.2 x 10⁸ «years» ✓

ALTERNATIVE 2

$p = \frac{340}{449} = 0.76$ «remaining» ✓

$n = \frac{\ln (p)}{0.693} = \frac{\ln (0.76)}{0.693} = 0.40$ ✓

t = 0.40 x 1.3 x 10⁹ = 5.2 x 10⁸ «years» ✓

ALTERNATIVE 3

$p = \frac{340}{449} = 0.76$ «remaining» ✓

$0.76 = {(\frac{1}{2})}^{\frac{t}{1.3 \times 10^{9}}}$ ✓

t = 0.40 x 1.3 x 10⁹= 5.2 x 10⁸ «years» ✓

Allow 5.3 x 10⁸ years for final answer.

Allow ECF for MP3 for an incorrect number of half-lives.

b.ii.

«use the mass of the sample to» determine number of potassium-40 atoms / nuclei in sample ✓

«use a counter to» determine (radio)activity / A of sample ✓

use A = λN «to determine the decay constant / λ» ✓

Examiners report

This question was very well done by candidates. The majority were able to identify the correct nuclide of Calcium and many correctly included an electron/beta particle and a properly written antineutrino.

This was a "show that" question that was generally well done by candidates.

b.i.

This was a more challenging question for candidates. Many were able to calculate the decay constant and recognized that the ratio of initial and final quantities of the potassium-40 was important. A very common error was mixing the two common half-life equations up and using the wrong values in the exponent (using half life instead of the decay constant, or using the decay constant instead of the half life). Examiners were generous with ECF for candidates who clearly showed an incorrect number of half-lives multiplied by the time for one half-life.

b.ii.

Describing methods of determining half-life continues to be a struggle for candidates with very few earning all three marks. Many candidates described a method more appropriate to measuring a short half- life, but even those descriptions fell far short of being acceptable.

Syllabus sections

Core » Topic 3: Thermal physics » 3.2 – Modelling a gas

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