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Half-Life Calculator

Radioactive decay and half-life

Radioactive decay is a single-molecule random process that, taken over Avogadro-scale populations, becomes a clean exponential. The half-life t_half is the time for the population to drop by half, and it parameterizes the whole curve. The two equivalent forms of the decay equation are:

N = N0 × (1/2)^(t / t_half) and N = N0 × e^(-lambda × t)

with lambda = ln(2)/t_half. The second form drops out of integrating dN/dt = -lambda × N, the rate law for first-order decay. Both predict the same numbers; the first is faster when t is a clean multiple of t_half, and the second is the algebraic form you keep when t isn’t.

This calculator solves the equation for whichever of the four variables — N0, N, t_half, or t — you leave blank. It also reports lambda, the number of half-lives elapsed, and the fraction remaining. When solving for time, the logarithmic algebra (taking ln of both sides) is shown explicitly.

What the calculator does

  1. Fill in any three of N0, N, t_half, t. Leave the fourth blank.
  2. The calculator solves for the missing value, shows the steps including any logarithm step, and reports lambda and the number of half-lives.
  3. Same time unit on t_half and t. Doesn’t matter which — seconds, days, years — but it must match.

Worked examples

Amount remaining. 100 mg of I-131 (t_half = 8.02 d) decays for 24.06 d.

  • 24.06 / 8.02 = 3.00 half-lives
  • N = 100 × (1/2)^3 = 12.5 mg

Time to decay. 500 g of Co-60 (t_half = 5.27 y) → 62.5 g.

  • 62.5/500 = 0.125 = (1/2)^3, so 3 half-lives
  • t = 3 × 5.27 = 15.81 y

Finding the half-life. Activity drops from 800 Bq to 200 Bq in 10.0 hours.

  • 200/800 = 0.25 = (1/2)^2, so 2 half-lives elapsed
  • t_half = 10.0/2 = 5.0 hours

Non-integer half-lives. 50.0 g of P-32 (t_half = 14.3 d) decays for 20.0 d.

  • 20.0/14.3 = 1.399 half-lives
  • N = 50.0 × (1/2)^1.399 = 19.0 g

Decay constant. C-14 has t_half = 5,730 y.

  • lambda = ln(2)/5730 = 1.21 × 10^-4 / y

Notable half-lives

IsotopeHalf-LifeUse
C-145,730 yRadiocarbon dating
I-1318.02 dThyroid treatment
Co-605.27 yRadiation therapy
U-2384.47 × 10^9 yGeological dating
Tc-99m6.01 hMedical imaging
P-3214.3 dDNA labeling

Where the same math shows up

The same exponential governs first-order chemical reactions, drug elimination kinetics (the biological half-life is computed identically), capacitor discharge, and any process where the rate of change is proportional to the current amount. The “half-life” terminology is nuclear, but the formula is universal to first-order decay.

Frequently Asked Questions

What is a half-life?
A half-life is the time it takes for half of a radioactive sample to decay. After one half-life, 50% remains; after two, 25%; after three, 12.5%, and the pattern continues geometrically. Each radionuclide has its own characteristic half-life, ranging from microseconds for some heavy isotopes to billions of years for U-238.
What is the half-life formula?
Two equivalent forms: N = N0 × (1/2)^(t/t_half) using half-life directly, or N = N0 × e^(-lambda × t) using the decay constant. They describe the same exponential, just parameterized differently. The first form is intuitive when the elapsed time is a clean multiple of the half-life; the second is the form you get from integrating the rate equation dN/dt = -lambda × N.
How is the decay constant related to half-life?
lambda = ln(2)/t_half ≈ 0.693/t_half. It is the fractional decay rate — the probability per unit time that any given nucleus decays. A larger lambda means a shorter half-life and a more unstable isotope. Activity in becquerels equals lambda × N, so the same constant connects the population to the count rate you actually measure.
Can this calculator handle any time unit?
Yes, with one rule: the half-life and the elapsed time must be in the same unit. Seconds with seconds, years with years. The formula is dimensionless inside the exponent because it depends only on the ratio t/t_half, so any consistent unit works. The calculator does not convert across units, so pick one and stay there.
What is radioactive activity?
Activity is the rate of decay, in disintegrations per second (becquerel, Bq) or curies (1 Ci = 3.7 × 10^10 Bq). It equals lambda × N, the decay constant times the current population. Because N decays exponentially, activity decays with the same half-life — measure A at two times and you can extract t_half without ever weighing the sample.