Radioactivity Calculator Half-Life & Decay
Calculate half-life, radioactive decay, activity, and nuclear equations with step-by-step solutions. Perfect for GCSE and A-Level Physics.
Remaining = Initial × (½)^(number of half-lives)
Solve for:
Quick examples:
Half-life decay
Amount remaining after time t
Half-life (n form)
Amount remaining after n half-lives
Activity
Activity = decay constant × number of nuclei
Decay constant
Decay constant from half-life
Count rate correction
True source count rate
Exponential decay
Activity decay over time (A-Level)
What is Radioactivity?
Radioactivity is the spontaneous emission of radiation from unstable atomic nuclei. These nuclei release energy to become more stable, a process called radioactive decay.
Key Concepts
- • Random process - cannot predict when individual nuclei decay
- • Spontaneous - happens without external cause
- • Measured in Becquerels - 1 Bq = 1 decay/second
- • Half-life is constant - unaffected by conditions
Types of Radiation
- • Alpha (α) - helium nuclei, stopped by paper
- • Beta (β) - electrons, stopped by aluminium
- • Gamma (γ) - EM waves, reduced by lead
- • Neutron - uncharged, stopped by concrete
Half-Life Explained
Half-life (t½) is the time for half of the radioactive nuclei to decay. It is constant for each isotope and cannot be changed.
N = N₀ × (½)n where n = t ÷ t½
1
half-life
50%
2
half-lives
25%
3
half-lives
12.5%
4
half-lives
6.25%
Types of Radiation
| Type | Symbol | Composition | Charge | Stopped by | Ionising |
|---|---|---|---|---|---|
| Alpha | α | 2p + 2n (He nucleus) | +2 | Paper, skin | Strong |
| Beta⁻ | β⁻ | Electron | -1 | Aluminium (few mm) | Medium |
| Beta⁺ | β⁺ | Positron | +1 | Annihilates with electron | Medium |
| Gamma | γ | EM radiation (photon) | 0 | Thick lead/concrete | Weak |
Activity and Decay Constant
A-LevelFor A-Level Physics, you need to understand the relationship between activity, decay constant, and half-life.
Activity Formula
A = λN
Activity = decay constant × nuclei
Decay Constant
λ = ln(2)/t½
λ ≈ 0.693 / t½
Key insight: The decay constant λ represents the probability of decay per unit time. A larger λ means faster decay (shorter half-life).
Background Radiation
Background radiation is always present and must be subtracted from measurements. Typical background is about 20-30 counts per minute.
Natural Sources
- • Cosmic rays - from space
- • Rocks (radon gas) - from ground
- • Food and drink - K-40 in bananas
- • Your own body - C-14, K-40
Man-made Sources
- • Medical - X-rays, CT scans
- • Nuclear testing - fallout
- • Nuclear power - small contribution
- • Building materials - concrete, bricks
Uses of Radioactivity
Medical
- • Tc-99m tracers (t½ = 6h)
- • Cancer treatment (Co-60)
- • Sterilisation of equipment
- • PET scans (F-18)
Dating
- • Carbon-14 (organic, <50,000y)
- • Uranium-Lead (rocks, billions y)
- • Potassium-Argon (volcanic)
- • Archaeological dating
Industry
- • Thickness gauges
- • Smoke detectors (Am-241)
- • Leak detection
- • Nuclear power (U-235)
Common Mistakes to Avoid
Forgetting to subtract background radiation from count rate.
Thinking a sample completely decays - it never does!
Confusing n (number of half-lives) with t (time).
Not converting time units (years/days/hours/seconds).
Mixing up mass number (A) and atomic number (Z).
Confusing activity (Bq) with count rate (counts/min).
Worked Examples
Example 1: Simple half-life
A sample has 800g of radioactive material. How much remains after 3 half-lives?
Example 2: Finding half-life
A sample decays from 1000 to 125 atoms in 30 minutes. What is the half-life?
t½ = 30 ÷ 3 = 10 minutes
Example 3: Decay constant
Cobalt-60 has a half-life of 5.27 years. Calculate its decay constant.
Or in seconds: 0.693 / (5.27 × 365.25 × 24 × 3600) = 4.17×10⁻⁹ s⁻¹
Example 4: Nuclear equation
Write the equation for U-238 undergoing alpha decay.
Mass: 238 = 234 + 4 ✓
Atomic: 92 = 90 + 2 ✓
Example 5: Carbon dating
A fossil has 25% of its original C-14. How old is it? (t½ = 5730 years)
Age = 2 × 5730 = 11,460 years
Frequently Asked Questions
What is half-life in physics?
The time for half of the radioactive nuclei to decay. It's constant for each isotope.
How do you calculate half-life?
Use t½ = t / log₂(N₀/N) or t½ = ln(2)/λ.
What is the decay constant?
The probability of decay per unit time (λ = 0.693/t½).
Why subtract background radiation?
It's always present and would give incorrect readings of the actual source.
What is activity measured in?
Becquerels (Bq) - one decay per second.
How long until a sample is "safe"?
Never completely, but after ~10 half-lives only 0.1% remains.
What is the half-life of C-14?
5,730 years - useful for dating up to ~50,000 years.
How do you balance nuclear equations?
Mass numbers (A) and atomic numbers (Z) must be equal on both sides.
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