69.314718
time
0.01
1/time
100
time
0.125
12.5
%
0.875
69.314718
time
0.01
1/time
100
time
0.125
12.5
%
0.875
The Half-Life Calculator determines the half-life of a radioactive isotope from its decay constant or from experimental measurements of remaining fraction over time. Half-life is the single most important characteristic of a radioactive substance, defining the timescale on which it decays and determining its applications in medicine, dating, power generation, and research.
The half-life (\(t_{1/2}\)) is defined as the time required for exactly half of the radioactive atoms in a sample to undergo decay. After one half-life, 50% remains; after two, 25%; after three, 12.5%; and so on in a geometric progression. This elegant mathematical pattern arises from the constant probability of decay per atom per unit time.
This calculator operates in two modes. In Mode 1, you provide the decay constant (\(\lambda\)) directly, and the calculator finds the half-life using \(t_{1/2} = \ln 2 / \lambda\). In Mode 2, you provide the fraction of material remaining after a known elapsed time, and the calculator works backward to determine the half-life—this is how half-lives are measured experimentally.
The calculator also computes the mean lifetime (\(\tau = 1/\lambda\)), which is the average time an atom survives before decaying. The mean lifetime is always longer than the half-life by a factor of \(1/\ln 2 \approx 1.443\). Additionally, you can specify a number of half-lives to see what fraction of the original sample remains.
Half-life values span an extraordinary range in nature. The shortest known half-life belongs to hydrogen-7 at about \(2.3 \times 10^{-23}\) seconds, while tellurium-128 has a half-life of approximately \(2.2 \times 10^{24}\) years—trillions of times the age of the universe. This 47-order-of-magnitude range reflects the diverse quantum mechanical processes governing nuclear decay.
Knowledge of half-lives is essential for radiation safety (determining safe handling times), nuclear medicine (choosing isotopes with appropriate decay rates for diagnosis or therapy), radiocarbon dating (calibrating the carbon-14 clock), nuclear waste management (planning storage timescales), and astrophysics (understanding element synthesis in stars and supernovae).
Mode 1: From Decay Constant
$$t_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693147}{\lambda}$$
Mode 2: From Remaining Fraction
Starting from \(N/N_0 = e^{-\lambda t}\), we solve for \(\lambda\):
$$\lambda = -\frac{\ln(f)}{t}$$
Then: \(t_{1/2} = \ln 2 / \lambda\)
Mean lifetime:
$$\tau = \frac{1}{\lambda} = \frac{t_{1/2}}{\ln 2}$$
Fraction remaining after n half-lives:
$$f(n) = \left(\frac{1}{2}\right)^n$$
The half-life tells you the timescale of the decay process. Short half-lives (seconds to days) indicate highly radioactive but quickly depleting sources—useful in medical imaging (Tc-99m: 6 hours). Long half-lives (thousands to billions of years) indicate low activity but persistent radiation—relevant for geological dating (U-238: 4.47 billion years) and long-term waste storage. The mean lifetime is always 44.3% longer than the half-life.
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A decay constant of 1.21×10⁻⁴ per year gives a half-life of ~5728 years—very close to carbon-14's half-life of 5730 years. After 3 half-lives, only 12.5% remains.
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If 25% of a sample remains after 100 time units, the half-life is 50 time units (since 25% = two half-lives). After 5 half-lives, only 3.125% would remain.
They are inversely proportional: t½ = ln(2)/λ ≈ 0.693/λ. The decay constant λ represents the probability of decay per unit time for a single atom, while the half-life is the macroscopic timescale for half the sample to decay. A larger λ means a shorter half-life and faster decay.
Mean lifetime (τ = 1/λ) is the average time an atom survives before decaying. It equals the half-life divided by ln(2), so τ ≈ 1.443 × t½. After one mean lifetime, about 36.8% of the original sample remains (compared to 50% after one half-life). Mean lifetime is widely used in particle physics.
Under normal conditions, nuclear half-lives are essentially immutable constants—they cannot be changed by temperature, pressure, chemical reactions, or electromagnetic fields. However, extreme conditions can have small effects: electron capture decay rates can be altered by ionizing the atom (changing electron density near the nucleus), and very strong gravitational fields cause time dilation effects.
For short half-lives (seconds to years), scientists measure the decay rate over time and fit an exponential curve to determine λ. For very long half-lives, they measure the activity of a known mass of material: since A = λN and N can be calculated from the mass, λ (and thus t½) can be determined from a single activity measurement.
The shortest measured half-life is approximately 2.3 × 10⁻²³ seconds (hydrogen-7). The longest directly measured half-life is about 2.2 × 10²⁴ years for tellurium-128. Some theoretically predicted decays (like proton decay) would have half-lives exceeding 10³⁴ years.
Half-lives depend on the quantum mechanical details of the nuclear decay process—the energy available for decay, the angular momentum change, the nuclear structure, and the barrier the emitted particle must tunnel through. Even small changes in these factors can produce enormous changes in half-life, which is why the range spans over 50 orders of magnitude.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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