Waves and Radioactivity: Synthesis and Exam Skills

You have now met every part of Topic P4 — the two wave types, the wave equation, reflection and refraction, the electromagnetic spectrum with its uses and dangers, atomic structure and isotopes, radioactive decay and nuclear equations, and half-life with the uses of radiation. This final lesson pulls those threads together. The skill that earns the most marks in an exam is not knowing each fact in isolation but choosing the right equation, spotting the hidden steps, and stringing several ideas into one answer. This lesson, part of Topic P4 (Waves and radioactivity) of OCR Gateway Combined Science A, gathers the key equations and facts, works through problems that combine ideas, and warns of the mistakes that lose marks most often.

By the end of this lesson you should be able to select and apply the P4 equations confidently, recall the key facts about the EM spectrum and the three radiations, balance a nuclear equation as part of a longer problem, avoid the most common exam errors, and tackle multi-step questions from start to finish.

This synthesis lesson exercises all three objectives together: AO1 recall of the EM spectrum and the three radiations, AO2 application in multi-step calculations across $v = f\lambda$ , $T = \dfrac{1}{f}$ , half-life and nuclear equations, and AO3 analysis when you interpret decay curves and evaluate extended-answer scenarios.

Topic P4 is really two halves joined under one heading. The first half is about waves in general — what they are, how fast they go, how they behave at boundaries, and the electromagnetic family in particular. The second half is about the nucleus — what atoms are made of, why some are unstable, how they decay, and how we use and control the radiation they give out. The two halves meet in the electromagnetic spectrum, because gamma rays are both the highest-energy part of the spectrum and one of the three types of nuclear radiation. Keeping this map of the topic in your head helps you see where a question is pointing and which tools it is asking you to use.

The P4 Equations and Constants at a Glance

Two relationships and two rules do almost all the work in P4. Knowing what each means, and when to reach for it, is half the battle.

Relationship	What it finds	Watch out for
$v = f\lambda$	Wave speed from frequency and wavelength	Convert frequency to hertz ( $\text{kHz} \times 1000$ , $\text{MHz} \times 10^6$ )
$T = \dfrac{1}{f}$	Period from frequency (and back)	$f$ in $\text{Hz}$ , $T$ in $\text{s}$ ; check $f \times T = 1$
Half-life halving	Amount remaining after $n$ half-lives	Fraction remaining $= \left(\tfrac{1}{2}\right)^{n}$ ; find $n$ = time ÷ half-life
Balancing equations	The unknown nuclide or particle	Top numbers balance and bottom numbers balance

Rearranged forms worth knowing:

$v = f\lambda \qquad f = \dfrac{v}{\lambda} \qquad \lambda = \dfrac{v}{f} \qquad T = \dfrac{1}{f}$

One constant to remember: the speed of all EM waves in a vacuum is $3 \times 10^8 \text{ m/s}$ (and the speed of sound in air is about $340 \text{ m/s}$ ).

Exam Tip: Before you calculate, write down what you know with units, then pick the relationship that contains those quantities. For a half-life question, first find the number of half-lives (time ÷ half-life); for a nuclear equation, immediately check both the top and bottom numbers balance.

The Electromagnetic Spectrum at a Glance

It is worth fixing the order, properties and key uses/dangers in one place.

Order (low to high frequency): radio, microwave, infrared, visible, ultraviolet, X-ray, gamma.
Shared by all: transverse; transfer energy; travel at $3 \times 10^8 \text{ m/s}$ in a vacuum; can cross a vacuum.
Trend: from radio to gamma, frequency and energy increase, wavelength decreases.
Ionising (and most dangerous): ultraviolet, X-rays, gamma — they damage cells, mutate DNA and can cause cancer.

graph LR
  R[Radio] --> M[Microwave]
  M --> I[Infrared]
  I --> V[Visible]
  V --> U[Ultraviolet]
  U --> X[X-ray]
  X --> G[Gamma]

Reading left to right above, frequency and energy increase and wavelength decreases, with the ionising waves (ultraviolet, X-ray, gamma) at the right-hand end.

The Three Radiations at a Glance

The other half of the topic is nuclear. Keep this comparison sharp, because it drives the uses, the hazards and the equations.

Radiation	What it is	Charge	Mass	Stopped/reduced by	Ionising	Nuclear-equation change
Alpha ( $\alpha$ )	Helium nucleus	$+2$	$4$	Paper/skin	Strong	$A-4$ , $Z-2$
Beta ( $\beta$ )	Fast electron	$-1$	≈ 0	Few mm aluminium	Moderate	$A$ same, $Z+1$
Gamma ( $\gamma$ )	EM wave	$0$	$0$	Reduced by thick lead	Weak	No change

Two links tie this table to the rest of the topic. First, penetration and ionising power are inversely related — alpha ionises most but penetrates least. Second, the irradiation/contamination reversal: outside the body gamma is most hazardous (it penetrates in), but inside the body alpha is most hazardous (it is most ionising and cannot escape).

Notice how the same handful of facts keeps doing work in different questions. The charge and mass of each radiation decide how you write it in a nuclear equation. Its penetrating power decides what stops it and whether it is dangerous from outside or inside the body. Its ionising power decides how much damage it does to living cells and how strongly it registers in a detector. And the half-life of the isotope decides how long a source stays active, which in turn decides which isotope is chosen for a job. If you understand these connections — rather than memorising each fact separately — a single line in the table above can answer several different exam questions.

A Worked Multi-Step Problem: Waves

Exam questions often chain several ideas together. Here is one that uses the period and the wave equation in one go.

Worked Example 1 — A buoy on the sea

A buoy floats on the sea. The waves passing it have a wavelength of $6 \text{ m}$ , and the buoy rises and falls, completing one full bob every $3 \text{ s}$ .

(a) Calculate the frequency of the water waves. (b) Calculate the speed of the water waves.

Part (a) — frequency from the period:

Step 1 — the time for one full bob is the period: $T = 3 \text{ s}$ .

Step 2 — use $f = \dfrac{1}{T}$ : $f = \dfrac{1}{3} = 0.33 \text{ Hz}$ (to 2 significant figures).

Answer (a): the frequency is $0.33 \text{ Hz}$ .

Part (b) — speed from the wave equation:

Step 1 — use $v = f\lambda$ with $f = \dfrac{1}{3} \text{ Hz}$ and $\lambda = 6 \text{ m}$ .

Step 2 — substitute and calculate: $v = \dfrac{1}{3} \times 6 = 2 \text{ m/s}$ .

Answer (b): the water waves travel at $2 \text{ m/s}$ .

Notice that part (b) used the frequency from part (a). Carry the unrounded value ( $\tfrac{1}{3}$ ) forward rather than the rounded $0.33$ , and round only at the end, to avoid rounding errors building up.

Exam Tip: In a multi-part question, an answer from one part often feeds into the next (here, the frequency from (a) is used in (b)). Carry the unrounded value forward where you can, and only round at the very end.

A Worked Multi-Step Problem: Radioactivity

The nuclear half of the topic also chains ideas — here, a nuclear equation and a half-life calculation together.

Worked Example 2 — Decay and half-life

Bismuth-212, $^{212}_{\ 83}\text{Bi}$ , decays by alpha emission.

(a) Write the balanced nuclear equation and name the daughter's atomic number. (Thallium has atomic number 81.) (b) A sample of a different isotope has a count-rate of $960 \text{ Bq}$ and a half-life of $20$ minutes. Find the count-rate after $1$ hour.

Part (a) — the nuclear equation:

Step 1 — alpha decay: $A \to A - 4 = 212 - 4 = 208$ ; $Z \to Z - 2 = 83 - 2 = 81$ .

Step 2 — atomic number 81 is thallium (Tl), so the daughter is $^{208}_{\ 81}\text{Tl}$ :

$^{212}_{\ 83}\text{Bi} \rightarrow\ ^{208}_{\ 81}\text{Tl} + {}^{4}_{2}\text{He}$

Step 3 — check: mass numbers $212 = 208 + 4$ ✓; atomic numbers $83 = 81 + 2$ ✓.

Answer (a): $^{212}_{\ 83}\text{Bi} \rightarrow\ ^{208}_{\ 81}\text{Tl} + {}^{4}_{2}\text{He}$ .

Part (b) — the count-rate:

Step 1 — number of half-lives in $1$ hour: $60 \div 20 = 3$ half-lives.

Waves and Radioactivity: Synthesis and Exam Skills

Waves and Radioactivity: Synthesis and Exam Skills

The P4 Equations and Constants at a Glance

The Electromagnetic Spectrum at a Glance

The Three Radiations at a Glance

A Worked Multi-Step Problem: Waves

Worked Example 1 — A buoy on the sea

A Worked Multi-Step Problem: Radioactivity

Worked Example 2 — Decay and half-life

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