What is the mean lifetime of a radioactive nucleus and how does it differ from the half-life?

The mean (average) lifetime τ = 1/λ = t½/ln(2) ≈ 1.4427 × t½. It is the average time a nucleus survives before decaying, computed by integrating t·λe^(−λt) from 0 to ∞. After one mean lifetime τ, the fraction remaining is 1/e ≈ 36.79%, not 50% as after one half-life. Mean lifetime is commonly used in particle physics and in deriving the exponential decay law.

What are the half-lives of common isotopes used in medicine?

Technetium-99m: 6.0058 hours (SPECT imaging, most widely used). Fluorine-18: 109.8 minutes (PET scanning). Iodine-131: 8.02 days (thyroid ablation, cancer therapy). Iodine-123: 13.22 hours (thyroid imaging). Thallium-201: 73.01 hours (cardiac SPECT). Gallium-67: 3.26 days (infection/tumor imaging). Lutetium-177: 6.65 days (targeted radionuclide therapy).

What are the half-lives of isotopes used in geology and geochronology?

Uranium-238: 4.468 × 10⁹ years (U-Pb dating, oldest rocks). Uranium-235: 7.04 × 10⁸ years (U-Pb dating). Potassium-40: 1.248 × 10⁹ years (K-Ar dating). Rubidium-87: 4.923 × 10¹⁰ years (Rb-Sr dating). Carbon-14: 5,730 years (radiocarbon dating, organic material up to ~50,000 yr). Samarium-147: 1.07 × 10¹¹ years (Sm-Nd dating of ancient rocks).

What is the biological half-life and how does it differ from the physical half-life?

The physical half-life (t½phy) is the nuclear decay rate - fixed and unalterable. The biological half-life (t½bio) is the time for the body to eliminate half of a substance through metabolic processes - depends on physiology, not nuclear physics. The effective half-life combines both: 1/t½eff = 1/t½phy + 1/t½bio, or equivalently t½eff = (t½phy × t½bio) / (t½phy + t½bio). Effective half-life is always shorter than either individual half-life.

What is the shortest known half-life?

The shortest confirmed half-life is that of hydrogen-7 (⁷H), measured at approximately 23 × 10⁻²⁴ seconds (23 yoctoseconds) - it exists for only a few nuclear diameters' worth of time before decaying. Many other nuclides far from the stability line have half-lives in the picosecond to femtosecond range. In contrast, the longest known half-lives are for naturally occurring primordial nuclides like Te-128 (t½ ≈ 2.2 × 10²⁴ years).

How is the half-life measured experimentally?

For short half-lives (seconds to days): measure the activity A(t) = λN over time and fit an exponential decay curve; the time for activity to halve is t½. For long half-lives (millennia to billions of years): measure current activity A = λN, determine N (number of atoms via mass spectrometry), then λ = A/N and t½ = ln(2)/λ. Carbon-14's half-life was first measured precisely by Willard Libby in 1949 using this second method.

Half-Life Calculator

Q: How do you calculate the time to reach a given fraction remaining?

From N(t)/N₀ = e^(−λt), solving for t: t = −ln(N/N₀) / λ = −t½ × log₂(N/N₀). For example, to find when 10% remains: t = −t½ × log₂(0.1) = t½ × log₂(10) ≈ t½ × 3.322. So it takes 3.322 half-lives for 90% of the sample to decay.

Q: Why can't the half-life of a radioactive element be changed?

The half-life is determined by the nuclear force binding protons and neutrons inside the nucleus. Chemical state, temperature, pressure, and physical form have no significant effect on nuclear structure - the binding energy scale (~MeV) is a million times larger than chemical energy scales (~eV). Tiny effects exist for some electron-capture isotopes (where the decay involves orbital electrons), but these are negligible for practical purposes.

Convert between half-life, decay constant, and mean lifetime. Find time to reach any target fraction of a radioactive sample.

Half-Life (t½)

Decay Constant (λ)per second (s⁻¹)

Mean Lifetime (τ)

⏳ What is Half-Life?

The half-life (symbol t½) of a radioactive nuclide is the time required for exactly half of a given number of atoms to undergo spontaneous nuclear decay. It is the most widely used measure of radioactive decay rate and was first introduced by Ernest Rutherford in 1907. Half-life is a fixed, intrinsic property of each nuclide - it cannot be changed by temperature, pressure, chemical state, or any other external condition.

Half-lives span an enormous range in nature. Polonium-214 has a half-life of 164 microseconds - it vanishes almost instantaneously. Carbon-14 has a half-life of 5,730 years - useful for dating organic material up to ~50,000 years old. Uranium-238 has a half-life of 4.468 billion years - comparable to the age of the Solar System, which is why primordial uranium still exists on Earth. Tellurium-128 holds the record for the longest measured half-life: approximately 2.2 × 10²⁴ years.

Half-life is directly related to two other decay rate parameters. The decay constant λ (lambda) is the probability per unit time that any nucleus will decay: λ = ln(2)/t½ ≈ 0.6931/t½. The mean lifetime τ (tau) is the average time a nucleus survives before decaying: τ = 1/λ = t½/ln(2) ≈ 1.4427 × t½. All three are equivalent descriptions of the same physical process - this calculator converts freely among them.

Half-life has critical applications across science and medicine. In radiocarbon dating, archaeologists use C-14's t½ = 5,730 yr to calculate the age of organic samples. In nuclear medicine, physicians choose isotopes with half-lives matched to the procedure - short enough to minimise patient dose, long enough to complete imaging. In nuclear waste management, the half-lives of fission products determine how long waste must be safely stored. In geochronology, the half-lives of U-238, K-40, and Rb-87 are used to date rocks billions of years old.

📐 Formula

Half-Life from Decay Constant:

t½ = ln(2) / λ ≈ 0.6931 / λ

Decay Constant from Half-Life:

λ = ln(2) / t½ ≈ 0.6931 / t½

Mean Lifetime:

τ = 1 / λ = t½ / ln(2) ≈ 1.4427 × t½

t½ = half-life (any time unit)

λ = decay constant (same time unit as t½, but inverted: e.g. yr⁻¹ if t½ in yr)

τ = mean (average) lifetime of a nucleus

ln(2) = natural logarithm of 2 ≈ 0.69315

Time to reach fraction f remaining:

t = −t½ × log₂(f) = t½ × ln(1/f) / ln(2)

Example: 90% decayed (f = 0.10) → t = t½ × log₂(10) ≈ 3.322 × t½

Common Isotope Half-Life Reference Table

Isotope	Symbol	Half-Life	Use / Context
Technetium-99m	⁹⁹ᵐTc	6.01 hr	Medical imaging (SPECT)
Fluorine-18	¹⁸F	109.8 min	PET scanning
Iodine-131	¹³¹I	8.02 days	Thyroid therapy
Carbon-14	¹⁴C	5,730 yr	Radiocarbon dating
Tritium (H-3)	³H	12.32 yr	Nuclear weapons, fusion
Caesium-137	¹³⁷Cs	30.17 yr	Fission product, Chernobyl
Strontium-90	⁹⁰Sr	28.8 yr	Fission product, RTGs
Cobalt-60	⁶⁰Co	5.27 yr	Radiation therapy, sterilisation
Plutonium-239	²³⁹Pu	24,110 yr	Nuclear weapons, reactor fuel
Radium-226	²²⁶Ra	1,600 yr	Curie's discovery, historical
Potassium-40	⁴⁰K	1.248 Gyr	K-Ar geochronology
Uranium-235	²³⁵U	703.8 Myr	Reactor fuel, U-Pb dating
Uranium-238	²³⁸U	4.468 Gyr	U-Pb dating, primordial
Thorium-232	²³²Th	14.05 Gyr	Thorium fuel cycle

📖 How to Use This Calculator

Select the mode: Enter Half-Life, Enter Decay Constant (λ), or Enter Mean Lifetime (τ).

Type your known value and (for t½ and τ) select the time unit from the dropdown.

Click Calculate - the calculator returns t½, λ (in s⁻¹), τ, and the fraction remaining after 1, 5, and 10 half-lives.

Consult the isotope reference table above for common nuclides and their half-lives.

💡 Example Calculations

Example 1 - Carbon-14 (known half-life, find λ and τ)

C-14 has t½ = 5,730 years. Find λ and mean lifetime τ.

λ = 0.6931 / 5,730 yr = 1.2097 × 10⁻⁴ yr⁻¹ = 3.836 × 10⁻¹² s⁻¹

τ = 1.4427 × 5,730 = 8,267 years

λ = 3.836 × 10⁻¹² s⁻¹ | τ = 8,267 yr

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Example 2 - Technetium-99m (medical imaging isotope)

Tc-99m has t½ = 6.0058 hours. Find λ in s⁻¹ and mean lifetime.

t½ in seconds = 6.0058 × 3,600 = 21,621 s

λ = 0.6931 / 21,621 = 3.206 × 10⁻⁵ s⁻¹

τ = 1.4427 × 6.0058 hr = 8.663 hr

λ = 3.206 × 10⁻⁵ s⁻¹ | τ = 8.663 hr

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Example 3 - From decay constant to half-life (Cs-137)

Caesium-137 has λ = 7.30 × 10⁻¹⁰ s⁻¹. Find t½ and τ.

t½ = 0.6931 / (7.30 × 10⁻¹⁰) = 9.494 × 10⁸ s

Convert: 9.494 × 10⁸ s / 31,557,600 s/yr = 30.09 years

τ = 1/λ = 1.370 × 10⁹ s = 43.43 years

t½ ≈ 30.1 yr | τ ≈ 43.4 yr

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Frequently Asked Questions

What is the half-life of a radioactive element and what does it measure?+

The half-life (t½) is the time required for exactly half of the atoms in a radioactive sample to undergo decay. It is the most intuitive measure of a nuclide's decay rate. Short half-lives indicate highly radioactive, quickly decaying nuclides. Long half-lives indicate slowly decaying, persistently radioactive nuclides. Half-life is independent of sample size - 1 atom or 1 mole of C-14 both have t½ = 5,730 years.

How is half-life related to the decay constant λ?+

They are related by t½ = ln(2)/λ ≈ 0.6931/λ. The decay constant λ is the probability that any given nucleus will decay per unit time. A large λ means fast decay (short half-life); small λ means slow decay (long half-life). The relationship comes from solving N₀/2 = N₀e^(−λt½), giving λt½ = ln(2).

What is the mean lifetime and how does it differ from the half-life?+

The mean lifetime τ = 1/λ = t½/ln(2) ≈ 1.4427 × t½. It is the average survival time of a nucleus before decaying. After one mean lifetime τ, the fraction remaining is 1/e ≈ 36.79%, not 50% as after one half-life. Mean lifetime is commonly used in particle physics.

What are the half-lives of common medical radioisotopes?+

Technetium-99m: 6.01 hours (SPECT imaging). Fluorine-18: 109.8 minutes (PET). Iodine-131: 8.02 days (thyroid ablation). Iodine-123: 13.22 hours (thyroid imaging). Thallium-201: 73.01 hours (cardiac SPECT). Lutetium-177: 6.65 days (cancer therapy). Half-lives are chosen to match the duration of the clinical procedure.

How do you calculate the time to reach a given fraction remaining?+

From N(t)/N₀ = e^(−λt), solving for t: t = −ln(f)/λ = −t½ × log₂(f) where f is the target fraction. For 10% remaining (90% decayed): t = t½ × log₂(10) ≈ 3.322 × t½. For 1% remaining: t = t½ × log₂(100) ≈ 6.644 × t½.

What is the biological half-life and effective half-life?+

The physical half-life is the nuclear decay rate. The biological half-life is the time for the body to eliminate half of a substance metabolically. The effective half-life combines both: 1/t½eff = 1/t½phy + 1/t½bio, so t½eff = (t½phy × t½bio)/(t½phy + t½bio). Effective half-life is always shorter than either individual value.

Why can't the half-life of a radioactive element be changed?+

The half-life is determined by nuclear forces acting at the MeV energy scale - about a million times larger than chemical energies (eV). Temperature, pressure, and chemical bonding have no significant effect on nuclear structure. Tiny effects exist for some electron-capture isotopes, but these are negligible in practice.

How are half-lives used in geological dating?+

Geochronologists use long-lived parent isotopes and their daughter products: U-238 → Pb-206 (t½ = 4.468 Gyr), K-40 → Ar-40 (t½ = 1.248 Gyr), Rb-87 → Sr-87 (t½ = 49.23 Gyr). By measuring the ratio of parent to daughter atoms and applying the decay equation, they calculate the age of rock formations up to billions of years old.

🔗 Related Calculators

📌 Quick Tips

💡Half-life and decay constant are reciprocal measures: t½ = ln(2)/λ ≈ 0.6931/λ. If you know one you automatically know the other - this calculator does the conversion instantly.

💡The mean lifetime τ = 1/λ ≈ 1.4427 × t½. After one mean lifetime, 1/e ≈ 36.8% of the sample remains (not 50%). This is different from the half-life, where exactly 50% remains.

💡To find time for N% to decay: t = −t½ × log₂(1 − N/100) = t½ × ln(100/(100−N)) / ln(2). For 99% decay: t = t½ × ln(100) / ln(2) ≈ 6.644 × t½.

💡Common mistake: confusing half-life with the time for the activity to halve from its current level. They are the same thing - activity A(t) = A₀·e^(−λt) follows the same exponential as N(t).

💡Medical physics rule: for patient radiation safety after nuclear medicine procedures, wait 10 half-lives before considering the nuclide negligible (less than 0.1% remains).

Half-Life Calculator

⏳ What is Half-Life?

📐 Formula

Common Isotope Half-Life Reference Table

📖 How to Use This Calculator

💡 Example Calculations

Example 1 - Carbon-14 (known half-life, find λ and τ)

C-14 has t½ = 5,730 years. Find λ and mean lifetime τ.

Example 2 - Technetium-99m (medical imaging isotope)

Tc-99m has t½ = 6.0058 hours. Find λ in s⁻¹ and mean lifetime.

Example 3 - From decay constant to half-life (Cs-137)

Caesium-137 has λ = 7.30 × 10−10 s−1. Find t½ and τ.

Frequently Asked Questions

🔗 Related Calculators

📌 Quick Tips

Caesium-137 has λ = 7.30 × 10⁻¹⁰ s⁻¹. Find t½ and τ.