250,000,000,000.0002
0.25
0.75
2.196284e-8
s^-1
250,000,000,000.0002
0.25
0.75
2.196284e-8
s^-1
The Radioactive Decay Calculator computes the number of radioactive nuclei remaining after a given time and the fraction that has decayed, using the fundamental law of radioactive decay. Radioactive decay is a spontaneous process in which unstable atomic nuclei transform into more stable configurations by emitting particles or radiation.
The law of radioactive decay states N(t) = N0 * e^(-lambda*t), where N0 is the initial number of nuclei, lambda = ln(2)/t_1/2 is the decay constant (in units of inverse time), and t_1/2 is the half-life. The half-life is the time required for exactly half of any given amount of the radioactive isotope to decay. After one half-life, N = N0/2; after two half-lives, N = N0/4; after ten half-lives, N ≈ N0/1024 (less than 0.1%).
Radioactive decay is a statistical process: each nucleus has a fixed probability per unit time of decaying (the decay constant lambda), independent of age or environment. This leads directly to the exponential decay law. The activity (decays per second) is A = lambda * N, also decreasing exponentially with the same half-life.
Half-lives span an enormous range: polonium-214 has t_1/2 = 164 microseconds; carbon-14 has t_1/2 = 5730 years (used in radiocarbon dating); uranium-238 has t_1/2 = 4.47 billion years (comparable to Earth's age). This vast range reflects the different nuclear forces governing alpha, beta, and gamma decay processes.
Applications include radiocarbon dating (using 14C, t_1/2 = 5730 yr), nuclear medicine (technetium-99m, t_1/2 = 6 hours, used in diagnostic imaging), nuclear power (fission product management), radiation safety (shielding and dose calculations), and geological/cosmological age dating (uranium-lead, potassium-argon systems).
Decay constant: lambda = ln(2)/t_1/2. Remaining nuclei: N(t) = N0 * exp(-lambda*t) = N0 * exp(-ln(2)*t/t_1/2) = N0 * (1/2)^(t/t_1/2). Fraction remaining: f = N(t)/N0. Fraction decayed: 1 - f. Number of half-lives elapsed: t/t_1/2.
After 1 half-life: 50% remains. After 3.32 half-lives: ~10% remains. After 6.64 half-lives: ~1% remains. After 10 half-lives: ~0.1% remains. For practical purposes, a radioactive source is considered essentially decayed after 7-10 half-lives. Activity A = lambda*N decays at the same rate as N.
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After one C-14 half-life (5730 years = 1.807e11 s), exactly 50% remains. If an artifact shows 25% of original C-14, it is 2 half-lives = 11,460 years old.
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After 24 hours (4 half-lives of 6 h each), only 6.25% of Tc-99m remains. This rapid decay is desirable in nuclear medicine — the patient's radiation exposure drops quickly after imaging.
A spontaneous process in which unstable nuclei emit particles (alpha, beta) or energy (gamma rays) to reach a more stable configuration. The decay rate is proportional to the number of undecayed nuclei: dN/dt = -lambda*N.
The time required for half of any given amount of radioactive material to decay. After n half-lives, the fraction remaining is (1/2)^n. Half-lives range from 10^-22 s to 10^19 years depending on the nuclide.
lambda = ln(2)/t_1/2. It represents the probability per unit time that a given nucleus will decay. The mean lifetime of a nucleus (average time before decay) is tau = 1/lambda = t_1/2/ln(2) = 1.443 * t_1/2.
Carbon-14 (t_1/2 = 5730 yr) is continuously produced in the upper atmosphere and incorporated into all living organisms. After death, C-14 decays without replenishment. Measuring the remaining fraction gives the age of organic material up to about 50,000 years.
Alpha decay (emission of He-4 nucleus), beta-minus decay (neutron to proton + electron + antineutrino), beta-plus decay (positron emission), electron capture, and gamma decay (emission of high-energy photon from excited nucleus).
In a decay chain, secular equilibrium is reached when a long-lived parent produces a short-lived daughter at the same rate as the daughter decays. At equilibrium, the activities of parent and daughter are equal: lambda_parent * N_parent = lambda_daughter * N_daughter.
Because each nucleus has a fixed probability per unit time of decaying, independent of its age (memoryless process). This leads to dN/dt = -lambda*N, which has the exponential solution N(t) = N0*e^(-lambda*t).
The Bateman equations describe the populations of all nuclides in a decay chain over time. They are coupled differential equations accounting for production (from parent decay) and loss (own decay) of each nuclide in the chain.
Diagnostic nuclear medicine uses gamma-emitting isotopes (Tc-99m, I-131) imaged by gamma cameras. Therapy uses beta or alpha emitters to irradiate tumors (I-131 for thyroid cancer, Lu-177 and Ra-223 for bone metastases).
Activity per unit mass of a radioactive sample. SA = lambda*NA/M (Bq/g), where NA is Avogadro's number and M is molar mass. Short half-life nuclides have very high specific activity (Tc-99m: ~2*10^17 Bq/g); long-lived nuclides have low specific activity (U-238: ~12,400 Bq/g).
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