Radioactive half-life calculations become straightforward when you identify the starting amount, the remaining amount, the half-life, and the elapsed time. This guide provides the formulas, a reusable decay table, worked examples, unit checks, and a practical checklist for solving quantity, time, half-life, and activity problems.
Overview
A half-life is the time required for half of a radioactive sample, or half of its activity, to remain. The process is exponential rather than linear: the sample loses half of what is present during each half-life, not a fixed physical amount.
The main radioactive decay formula is:
N = N0(1/2)t/T
- N is the amount remaining after time t.
- N0 is the initial amount.
- t is the elapsed time.
- T is the half-life.
The amount may be measured in grams, atoms, moles, or another consistent quantity. If the question concerns radioactive activity, use the same equation with activity symbols:
A = A0(1/2)t/T
Here, activity is often expressed in becquerels (Bq), where one becquerel represents one decay per second. The same fractional decay applies to the activity of a sample when other conditions remain appropriate.
Remaining fraction after whole half-lives
| Half-lives elapsed | Fraction remaining | Percentage remaining | Fraction decayed |
|---|---|---|---|
| 0 | 1 | 100% | 0% |
| 1 | 1/2 | 50% | 50% |
| 2 | 1/4 | 25% | 75% |
| 3 | 1/8 | 12.5% | 87.5% |
| 4 | 1/16 | 6.25% | 93.75% |
| 5 | 1/32 | 3.125% | 96.875% |
| 6 | 1/64 | 1.5625% | 98.4375% |
For example, after three half-lives, the remaining amount is N0/8. If the original sample was 80 milligrams, 10 milligrams remain after three half-lives.
Checklist by scenario
To calculate the amount remaining
- Write down the initial amount, N0.
- Identify the elapsed time, t, and the half-life, T.
- Convert both time values to the same unit.
- Calculate the number of half-lives: t/T.
- Substitute into N = N0(1/2)t/T.
- Round only at the end, using the precision requested by the problem.
Example: A sample begins with 200 grams and has a half-life of 4 hours. How much remains after 12 hours? The number of half-lives is 12/4 = 3. Therefore, N = 200(1/2)3 = 25 grams.
To calculate elapsed time
When the initial and final amounts are known, rearrange the formula:
t = T log2(N0/N)
If your calculator does not have a base-2 logarithm, use:
log2(x) = ln(x)/ln(2)
Example: A sample decreases from 160 units to 20 units, and its half-life is 6 days. The ratio is 160/20 = 8, which equals 23. Three half-lives have passed, so the elapsed time is 3 × 6 = 18 days.
To calculate the initial amount
Rearrange the equation as:
N0 = N ÷ (1/2)t/T
Equivalently, multiply the final amount by 2t/T. If 15 grams remain after four half-lives, the initial amount was 15 × 24 = 240 grams.
To calculate the half-life
When the initial amount, final amount, and elapsed time are known, use:
T = t log(1/2) ÷ log(N/N0)
The logarithm base can be any consistent base, provided it is used in both places. For instance, if a sample falls from 100 units to 25 units in 10 hours, the sample has gone through two half-lives, so its half-life is 10/2 = 5 hours.
Using the decay constant
Some problems use the exponential form:
N = N0e-λt
The decay constant is related to half-life by:
λ = ln(2)/T
If T is measured in seconds, λ has units of inverse seconds. If the half-life is given in days, the decay constant is in inverse days unless you convert it first.
What to double-check
- Time units: A half-life of 3 hours cannot be divided directly into 2 days. Convert 2 days to 48 hours first.
- What the quantity represents: Mass, number of atoms, moles, and activity can use the same fractional formula, but they are not interchangeable units.
- Initial versus final values: Label the starting value as N0 and the later value as N before entering numbers.
- Decay versus amount decayed: The formula gives the amount remaining. The amount that has decayed is N0 − N.
- Fractional half-lives: The exponent t/T does not need to be a whole number. A time equal to half of one half-life leaves (1/2)1/2, or about 70.7%, remaining.
- Significant figures: Match the final answer to the least precise relevant input rather than reporting unnecessary digits.
For help with powers of ten, rounding, and scientific notation, see the scientific notation and significant figures reference. When different quantities use different units, the science unit converter guide can help organize conversions before you calculate.
Common mistakes
Treating decay as linear: Subtracting half the original amount during every interval is incorrect. Each half-life removes half of the amount currently present.
Using the wrong exponent: The exponent is elapsed time divided by half-life, t/T, not the reverse.
Confusing remaining percentage with percentage decayed: After two half-lives, 25% remains but 75% has decayed.
Forgetting calculator mode: Logarithm formulas require careful parentheses. Enter ratios such as N0/N as a complete expression before applying the logarithm.
Rounding too early: Keep extra digits during intermediate steps, then round the final result. This matters especially for fractional half-lives.
Assuming a half-life changes during an ordinary classroom calculation: These exercises generally assume a constant half-life and ideal exponential decay. Real measurements can include detection limits, background radiation, sampling uncertainty, or mixtures of materials with different half-lives.
When to revisit
Revisit this checklist whenever a problem changes the measured quantity, time unit, or requested output. Before a laboratory exercise, confirm whether the task asks for remaining amount, activity, elapsed time, or amount decayed. Before an exam or homework submission, redo the unit check and compare the result with the decay table for a quick order-of-magnitude test.
For repeated calculations, keep a small worksheet with these fields: initial amount, final amount, half-life, elapsed time, unit conversion, formula used, and rounding rule. Update the worksheet if a course changes its notation conventions or if a reference table supplies a revised half-life for the material being studied. The underlying method remains the same: make units consistent, identify the number of half-lives, apply exponential decay, and check whether the result is physically sensible.
