Half-Life in Physics: Formula and Calculation Steps

This calculator finds the half-life of a radioactive sample from its starting amount, remaining amount, and elapsed time. Enter the two quantities in the same unit and the elapsed time; it returns the half-life in that time unit.

Formula

For exponential radioactive decay, the half-life is calculated from the initial and remaining amounts:

T_1/2 = t × ln(2) / ln(N₀ / N)

• T_1/2 is the half-life, in the same time unit as t.
• t is the time elapsed.
• N₀ is the amount at the start of the interval.
• N is the amount remaining after time t.
• ln means natural logarithm.

Amounts may represent the number of radioactive nuclei, mass of a particular isotope, or measured activity, provided the starting and remaining values refer to the same isotope and are measured consistently. The equivalent decay equation is N = N₀(1/2)^t/T_1/2.

Worked Example

Use the calculator defaults: N₀ = 100, N = 25, and t = 20 seconds. The remaining fraction is 25 / 100 = 0.25, or 25%.

1. Calculate the amount ratio: N₀ / N = 100 / 25 = 4.
2. Take its natural logarithm: ln(4) ≈ 1.386294.
3. Use ln(2) ≈ 0.693147 in the formula: T_1/2 = 20 × 0.693147 / 1.386294.
4. Calculate and round to two decimal places: T_1/2 = 10.00 seconds.

The sample has reached one-quarter of its initial amount, which corresponds to two half-lives. Twenty seconds divided by two gives the same result: 10.00 seconds.

What You’ll Need

• The starting amount and the amount remaining after a measured interval. They must use the same unit and refer to the same sample or isotope.
• The elapsed time between those measurements.
• A calculator with a natural logarithm function, or the calculator above. For a direct count, use the counts as amounts; for activity measurements, use consistent activity units.

Step-by-Step

1. Choose the quantity you will compare. Decide whether you are tracking nuclei, mass of the radioactive isotope, or activity. Do not compare unlike quantities, such as a starting mass with a later detector count.
2. Record the starting amount. Write this as N₀. It is the amount at the beginning of the interval you are analysing, not necessarily the sample’s original amount before any earlier decay.
3. Measure the remaining amount and elapsed time. Record N after time t has passed. Use the same time unit throughout the calculation. The calculator’s time input is in seconds, but the formula works with seconds, minutes, hours, days, or years if t is expressed in that unit; the resulting half-life has that same unit.
4. Check that the values make physical sense. For a decaying sample, N₀ must be positive and N must be greater than zero and less than N₀. If N equals N₀, no decrease is recorded, so this pair of measurements cannot determine a finite half-life. If N is zero, the logarithm is undefined; use a more sensitive measurement or a detection-limit method instead.
5. Enter the values and calculate. Put N₀ in Starting amount, N in Amount remaining, and t in Elapsed time. The first result is the estimated half-life. The second result is the fraction of the starting amount that remains.
6. Interpret the answer in its stated unit. A half-life of 10 seconds means that, under the same decay conditions, the amount is expected to halve every 10 seconds. It does not mean every individual nucleus survives for exactly 10 seconds; decay is random at the level of individual nuclei.
7. Check the result using the half-life equation. Substitute the calculated half-life into N = N₀(1/2)^t/T_1/2. The predicted remaining amount should be close to your measured N, allowing for rounding and measurement uncertainty.

Common Mistakes

• Reversing the amounts: the ratio in the formula is N₀/N. For decay, it should be greater than 1. Reversing it makes the logarithm negative and produces an unphysical negative half-life.
• Mixing units: use the same unit for both amounts and a single unit for elapsed time. If measurements span 2 hours and 30 minutes, convert to one unit before calculating. The output time unit follows the unit entered for t.
• Using percentage points as amounts incorrectly: if 80% remains, enter 80 and 100, or 0.8 and 1. Both give the same ratio. Do not enter 80 as the remaining amount and 1 as the starting amount.
• Rounding too early: retain full precision in intermediate logarithms and round only the final half-life. The displayed answer is an estimate, not a guarantee of measurement precision.
• Assuming every observed decrease is radioactive decay: sample loss, detector changes, background signal, or contamination can affect measurements. Correct or account for these effects where relevant.

Tips

• Use measurements taken over a time interval long enough for the decrease to be distinguishable from measurement noise, but not so long that the remaining amount falls below the instrument’s detection limit.
• For activity measurements, subtract background consistently if needed. Activity is proportional to the number of undecayed nuclei for a given isotope, so it can be used in the same ratio formula.
• If you have several measurements, plot ln(N) against time. Exponential decay gives an approximately straight line with slope −λ; then T_1/2 = ln(2)/λ. A fit to multiple data points can be more reliable than using only two.
• Keep units attached when reporting the result, and include how the amounts were measured. “12 days” is clearer than simply “12.”

Troubleshooting

The result is negative: check that the starting amount is larger than the remaining amount and that you entered them in the correct fields. If the sample increased, it may have been replenished, measured inconsistently, or affected by background signal rather than following simple decay.

The calculator rejects a value or gives no useful answer: the starting and remaining amounts must be positive, and elapsed time must be positive. A zero remaining amount cannot be used in a logarithm. If it is below detection, report a bound based on the detection limit rather than treating it as an exact zero.

The calculated half-life seems implausible: verify the elapsed-time unit, decimal placement, and amount ratio. Confirm that both measurements concern the same isotope and that the interval is measured from the correct starting point. Repeat measurements or use a multi-point decay fit if the data are noisy.

Different pairs give different answers: small differences can result from random decay and measurement uncertainty. Large or systematic differences may indicate a changing background, sample loss, mixed isotopes, or non-exponential behaviour. Check the measurement method before treating one result as definitive.

Frequently Asked Questions

What does half-life mean in physics?
It is the time required for the number of undecayed nuclei in a large sample, or a proportional measure such as its activity, to fall to half its value through radioactive decay.

Can I calculate half-life from a percentage remaining?
Yes. Use the starting amount as 100 and the remaining percentage as N. For example, 25% remaining means two half-lives have elapsed, so the half-life is elapsed time divided by two.

Does the sample’s mass have to be measured in grams?
No. Any consistent positive amount works, including grams, kilograms, or a nucleus count. The ratio N₀/N is dimensionless, so the amount unit cancels.

What if I know the decay constant instead?
Use T_1/2 = ln(2)/λ. The decay constant λ has units of inverse time, so the half-life is returned in the corresponding time unit.

Why is the half-life not an exact prediction for a small sample?
Individual decay events are random. Half-life describes the statistical behaviour of a large population; small samples can show noticeable variation around the expected amount.

Can I use this formula for any kind of decay?
Use it when the amount follows a single exponential decay process over the measured interval. Mixtures of isotopes, changing sources, or other non-exponential processes may require a different model.

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in N0 | Starting amount | - | 100 | 0
in N | Amount remaining | - | 25 | 0
in t | Elapsed time | s | 20 | 0
out half_life | Half-life | s | t * ln(2) / ln(N0 / N) | 2
out fraction | Fraction remaining | - | N / N0 | 3