Atomic model evolution & radioactivity

What you will learn

trace how the atomic model changed through the work of Dalton, Thomson, Rutherford, Bohr, and quantum physicists,
describe the structure of an atom in terms of protons, neutrons, and electrons,
define isotope and use mass number / atomic number notation,
describe alpha, beta, and gamma radioactive decay and write decay equations,
use the half-life formula $N = N_0 \left(\dfrac{1}{2}\right)^{t/t_{1/2}}$ for simple calculations.

Worked example 0 Real-world example: smoke alarm in your ceiling

A household smoke alarm contains a tiny amount of americium-241, an alpha emitter with half-life about 432 years.

Alpha particles ionise air molecules in a small chamber, allowing a tiny current to flow.
Smoke particles disrupt the ionisation and reduce the current — the alarm triggers.
Because the half-life is long, the source’s activity barely changes over the alarm’s 10-year lifetime.
Alpha particles travel only a few cm in air and are stopped by the plastic case, so external exposure is negligible.

Key idea: the right isotope for a job matches the type of radiation, the activity, and the half-life to the application.

1. How the model of the atom changed

Year	Scientist	Model	Key evidence
1803	Dalton	solid indivisible sphere	constant proportions in compounds
1897	Thomson	”plum pudding” — electrons in a positive cloud	cathode-ray tube showed negatively charged particles
1911	Rutherford	tiny dense positive nucleus, electrons around it	gold-foil experiment: most alpha particles passed through, a few deflected
1913	Bohr	electrons in fixed circular orbits (shells)	hydrogen line spectrum
1920s+	Schrodinger / quantum	electrons in orbitals (probability clouds)	wave-particle duality, Heisenberg uncertainty

Four key atomic models in sequence. Each was the best fit to the evidence of its time, and each was later revised when new evidence appeared.

2. Structure of an atom

Particle	Charge	Relative mass	Location
Proton	$+1$	1	nucleus
Neutron	$0$	1	nucleus
Electron	$-1$	$\dfrac{1}{1836} \approx 0$	orbitals around nucleus

Notation: $^{A}_{Z}\text{X}$ , where $Z$ is the atomic number (protons) and $A$ is the mass number (protons + neutrons).

Number of neutrons $= A - Z$ .
Neutral atom: number of electrons $=$ number of protons.

3. Isotopes

Isotopes are atoms of the same element (same $Z$ ) with different numbers of neutrons (different $A$ ).

Carbon-12 ( $^{12}_{6}\text{C}$ ): 6 protons, 6 neutrons — stable.
Carbon-14 ( $^{14}_{6}\text{C}$ ): 6 protons, 8 neutrons — radioactive.

All isotopes of an element have the same chemistry (because chemistry depends on electrons) but different nuclear properties.

Worked example 1 Identifying particles

For $^{238}_{92}\text{U}$ , state the number of protons, neutrons, and electrons (neutral atom).

Protons: $Z = 92$ .
Neutrons: $A - Z = 238 - 92 = 146$ .
Electrons: 92 (neutral).

4. Radioactive decay

Unstable (radioactive) nuclei spontaneously emit radiation and become more stable. Three common decay types.

Types of radioactive decay

Alpha decay ($\alpha$)

A helium-4 nucleus is emitted; $Z$ decreases by 2, $A$ decreases by 4.

^{A}_{Z}\text{X} \to \,^{A-4}_{Z-2}\text{Y} + \,^{4}_{2}\alpha

Beta decay ($\beta^{-}$)

A neutron converts to a proton + electron; the electron is emitted. $Z$ increases by 1, $A$ is unchanged.

^{A}_{Z}\text{X} \to \,^{A}_{Z+1}\text{Y} + \,^{0}_{-1}\beta

Gamma emission ($\gamma$)

A high-energy photon is released, often after alpha or beta decay. $A$ and $Z$ are unchanged.

Penetration and shielding:

Radiation	Charge	Stopped by
Alpha	$+2$	a sheet of paper or a few cm of air
Beta	$-1$	a few mm of aluminium
Gamma	$0$	dense lead or thick concrete

Worked example 2 Writing a decay equation

Uranium-238 undergoes alpha decay. Write the nuclear equation and identify the daughter nucleus.

Alpha decay: $A$ drops by 4, $Z$ drops by 2.
$^{238}_{92}\text{U} \to \,^{234}_{90}\text{Th} + \,^{4}_{2}\alpha$ .
Daughter: thorium-234.

Worked example 3 Beta decay of carbon-14

Carbon-14 undergoes beta-minus decay. Write the equation.

$A$ unchanged, $Z$ increases by 1.
$^{14}_{6}\text{C} \to \,^{14}_{7}\text{N} + \,^{0}_{-1}\beta$ .
Daughter: nitrogen-14.

Key idea: check that mass numbers and atomic numbers balance on both sides.

5. Half-life

The half-life $t_{1/2}$ of a radioactive isotope is the time for half of the nuclei in a sample to decay. It is a constant for each isotope.

Remaining nuclei after time $t$

N = N_0 \left(\dfrac{1}{2}\right)^{t / t_{1/2}}

where $N_0$ is the initial number, $N$ is the number after time $t$ , and $t_{1/2}$ is the half-life.

Worked example 4 Counting half-lives

Iodine-131 has a half-life of 8 days. A hospital has 240 mg of I-131. How much remains after 24 days?

Number of half-lives: $\dfrac{24}{8} = 3$ .
Remaining: $240 \times \left(\dfrac{1}{2}\right)^3 = 240 \times \dfrac{1}{8} = 30$ mg.

Worked example 5 Carbon dating

A wooden artefact has $\dfrac{1}{4}$ of the carbon-14 of a living sample. Estimate its age. ( $t_{1/2}$ for C-14 $= 5730$ years.)

$\dfrac{1}{4} = \left(\dfrac{1}{2}\right)^2$ , so $t = 2 \times t_{1/2} = 2 \times 5730 = 11\,460$ years.
The artefact is about 11 460 years old.

Key idea: exponential decay means repeated halving. Each half-life cuts the amount to half of what was there.

Practice: Year 9

Fluency

Atoms and isotopes

List the atomic models in order and name the scientist most associated with each.
Describe Rutherford’s gold-foil experiment and the conclusion he drew.
For $^{35}_{17}\text{Cl}$ , state the number of protons, neutrons, and electrons.
Define isotope. Give one example.
Why do isotopes of the same element have the same chemistry?
Write the symbol for: (a) 6 protons, 7 neutrons; (b) 20 protons, 22 neutrons.

Fluency

Decay equations

Write the alpha decay equation for radium-226 ( $^{226}_{88}\text{Ra}$ ).
Write the beta-minus decay equation for strontium-90 ( $^{90}_{38}\text{Sr}$ ).
Americium-241 decays by alpha emission. Write the equation and name the daughter.
State which radiation (alpha, beta, gamma) is (a) stopped by paper, (b) an electron, (c) a photon.
A nucleus loses 2 alpha and 1 beta particle. How do $A$ and $Z$ change overall?

Reasoning

Half-life

A sample starts with 1600 atoms. After 3 half-lives, how many remain?
Technetium-99m has half-life 6 hours. Starting with 80 mg, how much remains after 24 hours?
A sample has dropped to $\dfrac{1}{16}$ of its original activity. How many half-lives have passed?
Cobalt-60 has half-life 5.27 years. Starting with 100 g, how much remains after 10.54 years?
A patient is injected with 40 MBq of an isotope with half-life 2 hours. What is the activity after 6 hours?

Problem solving

Applications and reasoning

Why is a long half-life (thousands of years) suitable for radioactive dating but unsuitable for medical imaging?
An archaeologist finds that a bone contains $\dfrac{1}{8}$ of the C-14 in a living bone. Estimate the age. ( $t_{1/2} = 5730$ y.)
Explain why gamma radiation is used in cancer radiotherapy but alpha sources are not used externally.
Suggest a reason Bohr’s model, though successful for hydrogen, failed to predict the spectra of larger atoms.

Challenge

Reasoning

Harder reasoning

Rutherford’s team fired alpha particles at a thin gold foil. Most passed straight through, but a tiny fraction bounced back. Using this evidence, argue why the “plum pudding” model had to be replaced.
A decay chain: $^{238}_{92}\text{U}$ eventually becomes $^{206}_{82}\text{Pb}$ via multiple alpha and beta decays. If the total change in $A$ is $-32$ and in $Z$ is $-10$ , how many alpha and how many beta-minus decays are involved? Show your working.
A patient receives a technetium-99m scan with activity 800 MBq at injection. If the effective half-life in the body is 4 hours, what activity remains after 16 hours? Comment on why Tc-99m is chosen for imaging.
The number $N$ of undecayed nuclei follows $N = N_0 (1/2)^{t/t_{1/2}}$ . Rearrange to express $t$ in terms of $N$ , $N_0$ , and $t_{1/2}$ , and use this to estimate the age of a rock in which $\dfrac{N}{N_0} = 0.3$ for an isotope with half-life $1.3 \times 10^9$ years.

Answers

Answer key

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Year 9 answers

Fluency

Atoms and isotopes

Dalton (solid sphere) -> Thomson (plum pudding) -> Rutherford (nuclear) -> Bohr (shells) -> Schrodinger/quantum (orbitals).
Alpha particles were fired at a thin gold foil. Most passed straight through, a few were deflected, and a very small fraction bounced back. Conclusion: the atom is mostly empty space with a tiny, dense, positively charged nucleus at its centre.
Protons 17, neutrons $35 - 17 = 18$ , electrons 17.
Isotopes are atoms of the same element (same number of protons) with different numbers of neutrons. Example: carbon-12 and carbon-14.
Chemistry depends on electrons (and their arrangement). Isotopes have the same electron configuration because they have the same number of protons and electrons.
(a) $^{13}_{6}\text{C}$ (carbon-13), (b) $^{42}_{20}\text{Ca}$ (calcium-42).

Fluency

Decay equations

$^{226}_{88}\text{Ra} \to \,^{222}_{86}\text{Rn} + \,^{4}_{2}\alpha$ .
$^{90}_{38}\text{Sr} \to \,^{90}_{39}\text{Y} + \,^{0}_{-1}\beta$ .
$^{241}_{95}\text{Am} \to \,^{237}_{93}\text{Np} + \,^{4}_{2}\alpha$ . Daughter: neptunium-237.
(a) alpha, (b) beta-minus (an electron emitted from the nucleus), (c) gamma (high-energy photon).
Two alpha: $A$ drops by 8, $Z$ drops by 4. One beta: $Z$ rises by 1, $A$ unchanged. Overall: $\Delta A = -8$ , $\Delta Z = -3$ .

Reasoning

Half-life

$1600 \times (1/2)^3 = 1600 / 8 = 200$ atoms.
Half-lives in 24 h: $24/6 = 4$ . Remaining: $80 \times (1/2)^4 = 80/16 = 5$ mg.
$(1/2)^n = 1/16 = (1/2)^4$ , so 4 half-lives.
$10.54 / 5.27 = 2$ half-lives. Remaining: $100 \times (1/2)^2 = 25$ g.
$6/2 = 3$ half-lives. Remaining activity: $40 \times (1/2)^3 = 5$ MBq.

Problem solving

Applications and reasoning

Long half-lives work for dating because activity changes measurably over thousands/millions of years. For medical imaging the source should decay quickly after the scan so the patient is not exposed to radiation for long — a short half-life (hours) is better.
$1/8 = (1/2)^3$ , so 3 half-lives. Age $= 3 \times 5730 = 17\,190$ years.
Gamma rays penetrate tissue to reach tumours and can be aimed precisely. Alpha particles have very short range in tissue, so external alpha cannot reach tumours; their high ionising power also makes them dangerous inside the body but ineffective from outside.
Bohr’s model assumed fixed circular orbits and a single electron. In multi-electron atoms, electron-electron repulsion and subshell structure (s, p, d) affect spectra in ways Bohr could not predict. The quantum model with orbitals handles these correctly.

Reasoning

Challenge

If atoms were uniform positive “puddings”, alpha particles (positive, fast) should have deflected only slightly. The back-scattering of a few shows there is a concentrated positive mass inside — the nucleus. Most passing straight through shows the atom is mostly empty space. This contradicts the plum pudding and supports Rutherford’s nuclear model.
Each alpha: $\Delta A = -4$ , $\Delta Z = -2$ . Each beta: $\Delta A = 0$ , $\Delta Z = +1$ . Let $a$ alphas and $b$ betas. Then $4a = 32 \Rightarrow a = 8$ , and $-2a + b = -10 \Rightarrow -16 + b = -10 \Rightarrow b = 6$ . So 8 alpha and 6 beta-minus decays.
$16/4 = 4$ half-lives. Remaining: $800 \times (1/2)^4 = 50$ MBq. Tc-99m has a short half-life (about 6 hours physically, shorter effectively), giving enough time for imaging but minimal radiation dose afterwards; it also emits gamma rays detectable outside the body.
$N/N_0 = (1/2)^{t/t_{1/2}} \Rightarrow t = t_{1/2} \cdot \dfrac{\log(N/N_0)}{\log(1/2)}$ . For $N/N_0 = 0.3$ and $t_{1/2} = 1.3 \times 10^9$ y: $t = 1.3 \times 10^9 \times \dfrac{\log 0.3}{\log 0.5} = 1.3 \times 10^9 \times 1.737 \approx 2.26 \times 10^9$ years.

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