Radioactive Half-Life Calculator
Calculate the remaining amount of a radioactive substance (and derived quantities like decay constant, mean lifetime, fraction decayed) after a given elapsed time, given the initial amount and the half-life.
Category: Physics
Radioactive Half-Life Calculator Inputs
Radioactive Half-Life Calculator Formula
Equation
N(t) = N_0 \, (\tfrac12)^t / T_1/2
Excel Formula
=N(t)=N_0({1}{2})^t/T_1/2}
Variables
- Initial amount N₀ — Starting quantity (atoms, Bq of activity, grams of substance — any consistent unit).
- Half-life T₁/₂ — Time for half the species to decay (3 days for many medical isotopes, 5730 years for C-14).
- Elapsed time t — Time that has elapsed since N₀ (same units as the half-life).
How the Radioactive Half-Life Calculator Works
Unstable nuclei decay with a probability per unit time that is independent of age and of any neighbors. This produces exponential decay, characterized by a "half-life" — the time for half the remaining atoms to disappear. Most radioactive dating methods rely on this rule.
The core relationship is N(t) = N_0 \, \left(\tfrac{1}{2}\right)^{t / T_{1/2}}. Typical inputs include Initial amount N₀, Half-life T₁/₂, Elapsed time t.
Enter your values in the radioactive half-life calculator above, review the step-by-step solution, and compare against the worked examples below so you can see how each input changes the result. This free online physics tool is built for homework, design checks, and professional verification.
Radioactive Half-Life Calculator Theory & Explanation
Exponential decay
If the per-capita decay rate is λ, then dN/dt = −λ N has solution:
N(t) = N_0 \, e^-\lambda t
Half-life form
Solving N(T₁/₂) = N₀/2 gives λ = ln 2 / T₁/₂. The decay can be written as:
N(t) = N_0 \, (\tfrac12)^t / T_1/2
Mean lifetime τ
The mean lifetime (1/λ) is longer than the half-life by a factor of 1/ln 2 ≈ 1.4427:
\tau = (1)/(\lambda) = \fracT_1/2\ln 2 ≈ 1.443 \, T_1/2
Activity
The activity A = λN is the rate of decay events. After one half-life it falls to A₀/2; after N half-lives to A₀/2^N.
Carbon-14 dating
C-14 is produced in the upper atmosphere and incorporated by living things. After death, no new C-14 replaces what decays. By measuring the C-14/C-12 ratio and comparing to a living reference, archaeologists estimate the time since death, calibrated against known-age samples.
Half-Life, Mean Lifetime and Decay Constant
Three quantities describe the same exponential process and are easy to confuse. The decay constant \lambda is the probability per unit time that any given nucleus decays. The mean lifetime \tau = 1/\lambda is the average survival time of a nucleus. The half-life t_1/2 = \ln 2 / \lambda ≈ 0.693\,\tau is the time for half a large population to decay.
The mean lifetime is always longer than the half-life — by a factor of about 1.44 — because the few long-lived nuclei in the tail of the distribution pull the average up. Published tables may quote either, so check which one you have before substituting: mistaking one for the other introduces a systematic 44% error in every subsequent calculation.
t_1/2 = (\ln 2)/(\lambda), \qquad \tau = (1)/(\lambda) = \fract_1/2\ln 2
Activity and the Becquerel
Activity is the number of decays per second, A = \lambda N, measured in becquerels (1 Bq = one decay per second). The older curie is 3.7 × 10¹⁰ Bq, the activity of one gram of radium-226. Because activity is proportional to the number of remaining nuclei, it decays with exactly the same half-life as the population itself.
This is why a short half-life means high activity for a small quantity: technetium-99m, the workhorse of medical imaging, has a 6-hour half-life, so a tiny administered mass gives a strong signal and then clears quickly. A long half-life means the opposite — uranium-238 at 4.5 billion years is only weakly radioactive per gram, which is why it can be handled with modest precautions while being genuinely long-lived waste.
A = \lambda N = (\ln 2)/(t_1/2) N
Reading the Numbers and Common Pitfalls
After n half-lives, the fraction remaining is (1/2)^n: 50% after one, 25% after two, 12.5% after three, and under 0.1% after ten. A practical rule of thumb is that ten half-lives reduce a sample to a thousandth of its original activity, which is the usual basis for storage and clearance decisions.
Three cautions. First, keep the time and half-life in the same units — mixing years with days is the most frequent error. Second, exponential decay is a statistical law about large populations; for a handful of atoms the actual behaviour is random and the formula gives only an expectation. Third, many real decays are chains rather than single steps, and a daughter nuclide with a longer half-life than its parent will accumulate rather than clear, so the simple two-parameter model does not describe the total activity of the sample.
N(t) = N_0 ((1)/(2))^t/t_1/2 = N_0 e^-\lambda t
Radioactive Half-Life Calculator Worked Examples
Worked Example
Inputs
- initialAmount: 100
- halfLife: 5730
- elapsedTime: 11460
Result: remaining: 25 decayConstant: 0.000121 fractionDecayed: 0.75
Explanation
Two C-14 half-lives (2 × 5730 = 11,460 years) have elapsed. The remaining fraction is (1/2)² = 0.25, so 25 units remain out of 100. The decay constant is ln 2 / 5730 ≈ 1.21×10⁻⁴ per year, and the mean lifetime is ≈ 8267 years.
Second Scenario
Inputs
- initialAmount: 126
- halfLife: 5730
- elapsedTime: 11460
Result: remaining: 25 decayConstant: 0.000121 fractionDecayed: 0.75
Explanation
This scenario uses different inputs (initialAmount = 126, halfLife = 5730, elapsedTime = 11460) to show how changing one variable affects the radioactive half-life result. Run the calculator above with these values to get the exact updated output with step-by-step work.
Common Radioactive Half-Life Calculator Use Cases
- Physics problem sets and labs
- Engineering design checks
- Unit and formula verification
- Mean lifetime
- Fraction decayed) after a given elapsed time
Radioactive Half-Life Calculator FAQs
Why use half-life rather than mean lifetime?
Half-life is convenient experimentally because half of a sample is a directly measurable quantity. Mean lifetime (1/λ) is also useful and is approximately 1.443 times the half-life. Both fully describe the decay.
Are very short half-lives meaningful?
Yes, but harder to measure. Some isotopes (e.g. ^8 Be → 2 α) have half-lives of 10⁻¹⁶ s. Such "resonances" are observed in particle physics with very different experimental methods than apply to long-lived samples.
Are very long half-lives certain?
Beyond a certain limit we infer very long half-lives from non-detection. ^238 U has been observed to decay with T₁/₂ = 4.5 × 10⁹ years; some isotopes are predicted to be stable but unmeasurable in any practical sense.
Does decay ever stop?
No — exponential decay approaches but never reaches zero. After any finite time there is some remaining N. After ~10 half-lives, less than 0.1% remains, often considered "background".
Can anything speed up or slow down radioactive decay?
For practical purposes no. Temperature, pressure and chemical state leave the half-life essentially untouched, because decay is a nuclear process and chemistry only involves the electrons. There are tiny exceptions — electron-capture nuclides shift by a fraction of a percent when their chemical environment changes — but nothing that would let you neutralise radioactive waste by heating or reacting it.