10
time unit
0.06931472
1/time unit
14.42695
time unit
2
25
%
75
%
10
time unit
0.06931472
1/time unit
14.42695
time unit
2
25
%
75
%
The Half-Life Calculator determines the half-life of a radioactive substance (or any quantity undergoing exponential decay) from two measurements: the initial and remaining amounts and the elapsed time. It also computes the decay constant, mean lifetime, and number of half-lives elapsed.
The half-life is one of the most fundamental properties of a radioactive nuclide. It is the time required for exactly half the initial number of atoms to decay. Unlike many physical properties, the half-life of a radioactive nucleus is unaffected by temperature, pressure, chemical state, or any external condition — it depends only on nuclear properties. This constancy makes it an ideal chronometer for dating geological and archaeological samples.
From the exponential decay equation N(t) = N0 * e^(-lambda*t), we can derive the half-life as t_1/2 = ln(2)/lambda = ln(2)*t/ln(N0/N(t)). The mean lifetime (also called the time constant tau) is related to the half-life by tau = t_1/2/ln(2) = 1.4427 * t_1/2. It represents the average time a nucleus exists before decaying.
This calculator is not limited to nuclear physics: the same mathematics applies to pharmacokinetics (drug half-lives in the body), chemical kinetics (first-order reaction half-lives), electronics (RC circuit discharge), population dynamics, and any process described by exponential decay.
In medicine, understanding drug half-lives is essential for dosing schedules. A drug with a short half-life must be taken frequently; one with a long half-life reaches steady state slowly. For most drugs, steady-state concentration is reached after 4-5 half-lives of regular dosing.
From N(t) = N0 * e^(-lambda*t): ln(N0/N(t)) = lambda*t. Therefore: lambda = ln(N0/N(t))/t. Half-life: t_1/2 = ln(2)/lambda = ln(2)*t/ln(N0/N(t)). Mean lifetime: tau = 1/lambda = t/ln(N0/N(t)). Number of half-lives: n = ln(N0/N(t))/ln(2).
If N(t) = N0/2, then n = 1 (exactly one half-life elapsed). If N(t) = N0/4, then n = 2. The ratio N0/N(t) must be greater than 1 (some material must have decayed). The half-life, decay constant, and mean lifetime are universal: t_1/2 = 0.6931/lambda = 0.6931*tau.
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If 1000 atoms decrease to 250 in 20 time units, exactly 2 half-lives have elapsed (1000 -> 500 -> 250). Half-life = 10 time units. Mean lifetime = 14.4 time units.
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A drug at 500 mg decreasing to 62.5 mg in 24 hours has undergone 3 half-lives (3 * 8 hours). After 5 half-lives (40 hours), only ~3% would remain.
The time for exactly half of a radioactive sample (or any quantity undergoing first-order exponential decay) to disappear. After n half-lives, the fraction remaining is (1/2)^n.
t_1/2 = ln(2)/lambda = 0.6931/lambda. A larger decay constant means faster decay and shorter half-life. The mean lifetime tau = 1/lambda = t_1/2/ln(2) = 1.4427 * t_1/2.
No. You need at least two measurements: the ratio N0/N(t) and the elapsed time t. The half-life is t_1/2 = t * ln(2) / ln(N0/N(t)).
Yes. Any first-order kinetic process has a half-life: drug clearance, chemical reactions (first-order), electrical circuit discharge (tau = RC), radioactive tracers in biology. The mathematics is identical.
The time to reduce to 10% is t_10% = ln(10)/lambda = t_1/2 * log10(2)^-1 = 3.322 * t_1/2. It is used in radiation protection (dose from contamination) and pharmaceutical half-life calculations.
For short half-lives: directly counting decay events (activity measurement) over time. For long half-lives (millions of years): counting activity and number of atoms (activity = lambda*N), then lambda = A/N gives t_1/2 = ln(2)/lambda.
t_eff = (t_physical * t_biological) / (t_physical + t_biological). The effective half-life combines radioactive decay and biological elimination. It determines the actual radiation dose to patients.
For K-Ar dating: t = (1/lambda) * ln(1 + Ar/K * lambda/(lambda_total)), adapted for branching decay. For Rb-Sr: t = (1/lambda) * ln(1 + (Sr87-Sr87_initial)/Rb87). These are applications of the basic N(t)/N0 = e^(-lambda*t) equation.
An empirical relationship between the alpha decay half-life and the decay energy Q: log(t_1/2) = A/sqrt(Q) + B, where A and B are constants. Higher Q values give shorter half-lives — more energetic alpha particles tunnel through the Coulomb barrier more easily.
When nuclide A decays to B which decays to C..., the amounts of each nuclide follow Bateman's equations: N_B(t) = (lambda_A * N_A0 / (lambda_B - lambda_A)) * (e^(-lambda_A*t) - e^(-lambda_B*t)). This governs nuclear fuel chemistry and decay chain analysis.
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