2020 A/L Physics Past Paper · Question 43 (English Medium)

Half-life — number, age, ionisation

Consider the following statements regarding the half-life $(T_{1/2})$ of the radioactive atoms in a sample.

(A) $T_{1/2}$ changes with the number of radioactive atoms in the sample.

(B) $T_{1/2}$ changes with the date the sample was prepared.

Of the above statements,

(1) Only (A) is true.
(2) Only (B) is true.
(3) Only (C) is true.
(4) Only (A) and (B) are true.
(5) Only (B) and (C) are true.

Answer and explanation

Answer: (3) Only (C) is true.

What the question asks

We must find which of the three statements are true. The underlying question: is the half-life a property of the sample, or of the nucleus? Key concept: radioactive decay is a random nuclear process happening independently inside each nucleus.

Relevant theory

Since the radioactivity is related to the internal stability of nuclei, it is a phenomenon

which does not depend on external factors such as temperature and pressure.

The time taken for the initial number of nuclei in a sample of radioactive element, to

become half, is called the half life of that radioactive element.

The half-life does not

depend on the initial number of nuclei, and is a constant for a given kind of nuclei.

In this manner the

remaining number of nuclei is halved after each period of time T ½.

Reading the options

The options are combinations of the true statements: {A}, {B}, {C}, {A, B}, {B, C}. Let us resolve each statement separately.

Reasoning and calculation

$T_{1/2}$ changes with the number of radioactive atoms in the sample.

$T_{1/2}$ does not depend on the initial number of nuclei; it is a constant for the nuclide. Only the decay rate varies with $N$ — a large sample and a small sample both halve at the same time.

$T_{1/2}$ changes with the date the sample was prepared.

If the sample is older, fewer nuclei remain; but the remaining amount still halves every $T_{1/2}$ . The sample's age changes only $N$ — and, as in (A), $N$ does not change $T_{1/2}$ .

$T_{1/2}$ does not change even if the radioactive atoms are ionised.

Radioactivity is a process linked to the internal stability of the nucleus — it does not depend on external factors. Ionisation only removes electrons; the nucleus is unchanged, so $T_{1/2}$ is unchanged.

Why the false statements fail: both (A) and (B) tie the half-life to a property "of the sample" (how many nuclei, how long ago). But since each nucleus decays independently with the same probability, however many nuclei remain, it takes the same time for half of them to decay.

Why the other options fail

(A) is false.

(B) is false; (C) is left out.

Both (A) and (B) are false.

(B) is false.

Conclusion

The set of true statements is {C} only; the option that gives this is "only (C) is true" — the answer is (3). Since both (A) and (B) are false, options (1), (2), (4) and (5), which contain them, are ruled out.

The decay rate ( $\lambda N$ ) varies with the size of the sample; the half-life ( $T_{1/2} = \frac{\ln 2}{\lambda}$ ) depends only on the nucleus.

Exam tip

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