What is the difference between the decay constant and the half-life?

Both are equivalent measures of how fast a radionuclide decays. The half-life t½ is the time for exactly half the atoms to decay - it is intuitive and easy to use for rough calculations. The decay constant λ is the instantaneous probability of decay per unit time and appears naturally in the exponential decay formula. For practical calculations, use whichever is given; this calculator accepts both.

What is secular equilibrium in radioactive decay chains?

Secular equilibrium occurs in a decay chain when the parent nuclide has a much longer half-life than its daughters. After sufficient time (about 7 daughter half-lives), the activity of each daughter equals the activity of the parent: A_parent = A_daughter = A_granddaughter. For example, U-238 (t½ = 4.47 Gyr) in equilibrium with its decay chain daughters. The simple N(t) = N₀e^(−λt) applies to each individual step.

What is the difference between radioactive decay and nuclear fission?

Radioactive decay is spontaneous - a single unstable nucleus transforms into a different nuclide, emitting radiation. It requires no trigger and follows the exponential decay law. Nuclear fission is induced - a heavy nucleus (U-235, Pu-239) absorbs a neutron and splits into two medium-mass fragments, releasing 2–3 neutrons and ~200 MeV of energy. Fission can sustain a chain reaction; spontaneous decay cannot. Both release nuclear energy, but via fundamentally different mechanisms.

Radioactive Decay Calculator

Find remaining nuclei, activity, and fraction decayed for any radioactive isotope using the exponential decay law.

Initial Quantity (N₀)atoms, mol, Bq, or any unit

Half-Life (t½)

Elapsed Time (t)

Initial Quantity (N₀)atoms, mol, Bq, or any unit

Decay Constant (λ)per second (s⁻¹)

Elapsed Time (t)seconds

☢️ What is Radioactive Decay?

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable configuration, releasing ionising radiation in the process. Unlike chemical reactions, radioactive decay cannot be slowed, accelerated, or reversed by temperature, pressure, or chemical state - it depends only on the inherent instability of the nucleus. The phenomenon was discovered by Henri Becquerel in 1896 and further characterised by Marie and Pierre Curie, whose work on polonium and radium laid the foundation for nuclear science.

The defining feature of radioactive decay is that every individual nucleus of a given radionuclide has an identical, constant probability of decaying per unit time. This makes the process random at the individual level but precisely predictable in aggregate. The result is the famous exponential decay law: N(t) = N₀ · e^−λt. Examples include Carbon-14 (t½ = 5,730 yr, used in radiocarbon dating), Iodine-131 (t½ = 8.02 days, used in thyroid cancer therapy), Uranium-238 (t½ = 4.47 × 10⁹ yr, a geological clock), and Technetium-99m (t½ = 6.01 hr, the most widely used medical imaging isotope).

Three main types of spontaneous decay are commonly encountered. Alpha decay ejects a helium-4 nucleus, reducing the parent's atomic number by 2 and mass number by 4 - typical of heavy actinides. Beta decay converts a neutron to a proton (or vice versa), changing the element but not the mass number - common across the periodic table for nuclides far from the valley of stability. Gamma decay releases a high-energy photon from an excited nuclear state with no change in nucleon count - often follows alpha or beta decay.

This calculator applies the exponential decay law in either direction: given N₀, t½ (or λ), and t, it finds N(t), the fraction remaining, and the activity. It is used in nuclear physics coursework, NEET/JEE modern physics problems, radiocarbon dating exercises, medical physics dose calculations, and nuclear engineering decay heat estimates.

📐 Formula

Exponential Decay Law:

N(t) = N₀ · e^−λt

N(t) = number of undecayed nuclei at time t

N₀ = initial number of nuclei (at t = 0)

λ = decay constant (s⁻¹, yr⁻¹, etc.) - probability of decay per unit time

t = elapsed time (same units as λ⁻¹)

e = Euler's number ≈ 2.71828

Half-Life ↔ Decay Constant:

t½ = ln(2) / λ ≈ 0.6931 / λ

λ = ln(2) / t½ ≈ 0.6931 / t½

Mean Lifetime:

τ = 1 / λ = t½ / ln(2) ≈ 1.4427 × t½

Activity:

A(t) = λ × N(t) [becquerels, Bq = decays/second]

Fraction Remaining:

N(t)/N₀ = e^−λt = (1/2)^t/t½

📖 How to Use This Calculator

Choose input mode: Use Half-Life (most common) or Use Decay Constant (λ).

Enter the initial quantity N₀ - number of atoms, moles, becquerels, or any consistent unit.

Enter the half-life and select its time unit (seconds through years), or enter λ in per-second.

Enter the elapsed time t and select its unit.

Click Calculate to see remaining quantity, fraction remaining, % decayed, half-lives elapsed, λ, and mean lifetime τ.

💡 Example Calculations

Example 1 - Carbon-14 Radiocarbon Dating

An organic sample contains 250,000 C-14 atoms. How many remain after 11,460 years (two half-lives)?

Given: N₀ = 250,000 | t½ = 5,730 yr | t = 11,460 yr

λ = 0.6931 / 5,730 = 1.209 × 10⁻⁴ yr⁻¹

N(t) = 250,000 × e^{−(1.209×10⁻⁴)(11,460)} = 250,000 × e^−1.386 = 250,000 × 0.25

Remaining: 62,500 atoms (25% - exactly two half-lives elapsed)

Try this example →

Example 2 - Iodine-131 Medical Therapy

A patient receives 370 MBq of I-131 (t½ = 8.02 days) for thyroid ablation. What activity remains after 24 days?

Given: N₀ = 370,000,000 Bq | t½ = 8.02 days | t = 24 days

λ = 0.6931 / 8.02 = 0.08642 day⁻¹

A(t) = 370,000,000 × e^{−0.08642×24} = 370,000,000 × e^−2.074 = 370,000,000 × 0.1257

Remaining activity: ~46.5 MBq (≈ 2.99 half-lives elapsed, 12.57% remains)

Try this example →

Example 3 - Uranium-238 Geological Decay

A rock sample contains 1,000,000 U-238 atoms (t½ = 4.468 × 10⁹ yr). How many remain after 1 billion years?

Given: N₀ = 1,000,000 | t½ = 4.468 × 10⁹ yr | t = 1 × 10⁹ yr

λ = 0.6931 / (4.468 × 10⁹) = 1.551 × 10⁻¹⁰ yr⁻¹

N(t) = 1,000,000 × e^−0.1551 = 1,000,000 × 0.8564

Remaining: 856,400 atoms (85.64% - only 0.224 half-lives have elapsed in 1 Gyr)

Try this example →

Example 4 - Technetium-99m Medical Imaging

A Tc-99m dose of 800 MBq is prepared. The patient scan occurs 3 hours later. What activity is available? (t½ = 6.0058 hr)

Given: N₀ = 800,000,000 Bq | t½ = 6.0058 hr | t = 3 hr

λ = 0.6931 / 6.0058 = 0.1154 hr⁻¹

A(t) = 800,000,000 × e^−0.1154×3 = 800,000,000 × 0.7071

Available activity: ~565.7 MBq (70.7% remains - exactly half a half-life elapsed)

Try this example →

Example 5 - Caesium-137 Post-Accident Remediation

After a nuclear incident, contamination of 10,000,000 Cs-137 atoms is measured (t½ = 30.17 yr). What remains after 90 years?

Given: N₀ = 10,000,000 | t½ = 30.17 yr | t = 90 yr

λ = 0.6931 / 30.17 = 0.02297 yr⁻¹

N(t) = 10,000,000 × e^{−0.02297×90} = 10,000,000 × e^−2.067 = 10,000,000 × 0.1266

Remaining: 1,266,000 atoms (12.66% - 2.98 half-lives elapsed; contamination persists for centuries)

Try this example →

Frequently Asked Questions

What is the radioactive decay law and what does each symbol mean?+

The radioactive decay law is N(t) = N₀ · e^(−λt). N(t) is the number of undecayed nuclei at time t. N₀ is the initial number of nuclei at t = 0. λ (lambda) is the decay constant in units of per time (s⁻¹, yr⁻¹, etc.) - it represents the probability per unit time that any given nucleus will decay. e is Euler's number (≈ 2.71828). The law states that the number of undecayed nuclei decreases exponentially with time.

How do you convert half-life to the decay constant λ?+

The relationship is λ = ln(2) / t½ ≈ 0.6931 / t½. For example, Carbon-14 has a half-life of 5,730 years, so λ = 0.6931 / 5,730 = 1.21 × 10⁻⁴ per year. Conversely, t½ = ln(2) / λ. Both quantities describe the same thing - the rate of radioactive decay - just expressed differently.

What is activity in radioactivity and what unit is it measured in?+

Activity is the rate of decay: A = dN/dt = λN. It measures how many nuclear disintegrations occur per second. The SI unit is the becquerel (Bq): 1 Bq = 1 decay per second. The older unit, still common in medical contexts, is the curie (Ci): 1 Ci = 3.7 × 10¹⁰ Bq. Activity decreases over time as A(t) = A₀ · e^(−λt).

How many half-lives does it take for a radioactive material to become safe?+

A common rule of thumb is 10 half-lives, after which only 1/2¹⁰ ≈ 0.1% of the original activity remains. Iodine-131 (t½ = 8.02 days) is mostly decayed after 80 days. Caesium-137 (t½ = 30.17 years) requires ~300 years. Plutonium-239 (t½ = 24,110 years) requires over 240,000 years.

What is carbon-14 dating and how does radioactive decay enable it?+

Radiocarbon dating uses the known half-life of C-14 (5,730 years). Living organisms maintain a constant C-14/C-12 ratio by exchanging carbon with the atmosphere. When an organism dies, the exchange stops and C-14 decays. By measuring remaining C-14 and applying N(t)/N₀ = e^(−λt), scientists calculate time of death - up to ~50,000 years with modern accelerator mass spectrometry.

What is the difference between alpha, beta, and gamma decay?+

Alpha decay: the nucleus emits a helium-4 nucleus (2p + 2n), reducing Z by 2 and A by 4. Stopped by paper or skin. Beta decay: a neutron converts to a proton (β⁻) emitting electron + antineutrino, or proton to neutron (β⁺) emitting positron. Gamma decay: the nucleus releases a high-energy photon to shed excitation energy, with no change in Z or A. All three follow exponential decay kinetics.

How do you calculate the number of atoms in a radioactive sample?+

N₀ = (mass in grams / molar mass in g/mol) × Avogadro's number (6.022 × 10²³). For example, 1 μg of U-235 (molar mass 235 g/mol): N₀ = (10⁻⁶ / 235) × 6.022 × 10²³ = 2.56 × 10¹⁵ atoms. Use this as N₀ in the decay equation.

What is secular equilibrium in a radioactive decay chain?+

Secular equilibrium occurs when the parent nuclide has a much longer half-life than its daughters. After about 7 daughter half-lives, the activity of each daughter equals the parent activity. For example, U-238 (t½ = 4.47 Gyr) eventually reaches equilibrium with all decay-chain daughters, including Ra-226 and Rn-222. Each step still follows its individual N₀e^(−λt) equation.

What is the mean lifetime of a radioactive nucleus?+

The mean lifetime τ = 1/λ = t½/ln(2) ≈ 1.4427 × t½. It is the average time a nucleus survives before decaying. Note that after one mean lifetime τ, only 1/e ≈ 36.8% of nuclei remain - not 50% as in a half-life. The mean lifetime is used in particle physics and when deriving the exponential decay law from Poisson statistics.

Why does radioactive decay follow exponential kinetics?+

Because each nucleus has a constant, memory-less probability p of decaying per unit time - the nucleus does not age. If N nuclei each have probability λdt of decaying in time dt, the expected number decaying is λN dt. This gives dN/dt = −λN, a first-order ODE whose solution is N = N₀e^(−λt). This is the same mathematics as compound interest in reverse, and as first-order chemical reactions.

🔗 Related Calculators

📌 Quick Tips

💡The decay constant λ and half-life t½ are related by λ = ln(2) / t½ ≈ 0.6931 / t½. If you know the half-life in days, divide 0.6931 by it to get λ in per-day.

💡After one half-life, 50% of the original sample remains. After two half-lives, 25%. After ten half-lives, less than 0.1% - this is the practical definition of 'gone' for most applications.

💡Activity A = λN. If you know the number of atoms N and the decay constant λ, multiplying them gives the decay rate in becquerels (decays per second). This is crucial for radiation safety calculations.

💡Carbon-14 (t½ = 5,730 years) is used in radiocarbon dating of organic material up to ~50,000 years old. Beyond that, too little C-14 remains to measure accurately.

💡The mean (average) lifetime τ = 1/λ = t½ / ln(2) ≈ 1.443 × t½. An individual nucleus has an equal chance of decaying in any interval τ - it has no memory of how long it has existed.