Half-Life

Formula

Number remaining at time t:

N(t) = N₀ · e^(-λt) = N₀ · (1/2)^(t/τ_½)

where τ_½ = ln(2)/λ ≈ 0.693/λ.

Activity (decays/sec):

A(t) = λ · N(t) = A₀ · e^(-λt)

Mean lifetime (average for individual nucleus):

τ_mean = 1/λ = τ_½ / ln(2) ≈ 1.443 · τ_½

Famous half-lives

Isotope	Half-life	Decay mode	Use
Po-212	0.3 µs	α	—
F-18	110 min	β⁺ → O-18	PET imaging
Tc-99m	6 hr	γ → Tc-99	SPECT (most common medical)
I-131	8 days	β⁻ → Xe-131	Thyroid therapy/diagnostics
Co-60	5.27 yr	β⁻ → Ni-60	Cancer therapy, sterilization
Tritium (H-3)	12.3 yr	β⁻ → He-3	Glow-in-dark, fusion fuel
Sr-90	28.8 yr	β⁻ → Y-90	Reactor accident contaminant
Cs-137	30.2 yr	β⁻ + γ → Ba-137	Reactor accident contaminant
Ra-226	1600 yr	α → Rn-222	Used to define curie
C-14	5,730 yr	β⁻ → N-14	Carbon dating
Pu-239	24,100 yr	α → U-235	Weapons, reactor fuel
U-235	7 × 10⁸ yr	α → Th-231	Reactor fuel
U-238	4.5 × 10⁹ yr	α → Th-234	Most natural U; dating
Bi-209	2 × 10¹⁹ yr	α → Tl-205	"Stable" until 2003
Te-128	2.2 × 10²⁴ yr	2β⁻ → Xe-128	Longest known half-life

JavaScript — half-life calculations

// Number remaining after time
function fractionRemaining(t, halfLife) {
  return Math.pow(0.5, t / halfLife);
}

// Number after n half-lives
function afterNHalfLives(N_0, n) {
  return N_0 / Math.pow(2, n);
}

console.log(`After 5 half-lives: ${(fractionRemaining(5, 1) * 100).toFixed(2)}%`);  // 3.13%
console.log(`After 10 half-lives: ${(fractionRemaining(10, 1) * 100).toFixed(4)}%`); // 0.0977%

// Solve for age given fraction
function ageFromFraction(fraction, halfLife) {
  return -halfLife * Math.log(fraction) / Math.log(2);
}

console.log(`75% C-14 left: ${ageFromFraction(0.75, 5730).toFixed(0)} yr`); // ~2378
console.log(`50%: ${ageFromFraction(0.50, 5730).toFixed(0)} yr`);          // 5730
console.log(`10%: ${ageFromFraction(0.10, 5730).toFixed(0)} yr`);          // 19035

// Activity from mass
function activityFromMass(mass_g, atomicMass, halfLife_s) {
  const N = mass_g / atomicMass * 6.022e23;
  const lambda = Math.log(2) / halfLife_s;
  return N * lambda;  // Bq
}

// 1 µg of Tc-99m
const Tc_halflife = 6 * 3600;  // 6 hours in seconds
const Tc_activity = activityFromMass(1e-6, 99, Tc_halflife);
console.log(`1 µg Tc-99m: ${(Tc_activity / 1e6).toFixed(2)} MBq`);

// Decay chain (parent-daughter)
function parentDaughterRatio(t, lambda_p, lambda_d) {
  // Bateman equation for daughter from parent
  // N_d/N_p = lambda_p/(lambda_d - lambda_p) · (1 - exp(-(lambda_d - lambda_p)t))
  if (Math.abs(lambda_p - lambda_d) < 1e-15) return lambda_p * t;
  return lambda_p / (lambda_d - lambda_p) * (1 - Math.exp(-(lambda_d - lambda_p) * t));
}

// Multi-isotope mixture: total activity is sum
function mixedActivity(isotopes, t) {
  return isotopes.reduce((sum, iso) => {
    const remaining = iso.N_0 * Math.pow(0.5, t / iso.halfLife);
    return sum + remaining * Math.log(2) / iso.halfLife;
  }, 0);
}

// Carbon dating: charcoal sample, 25% C-14 remaining
const sampleAge = ageFromFraction(0.25, 5730);
console.log(`Sample age: ${sampleAge.toFixed(0)} years`);

Where half-life matters

• Radiometric dating. C-14 for organics < 50,000 yr; U-Pb for rocks billions of years.
• Medical isotopes. Match half-life to medical procedure timing.
• Nuclear waste. Spent fuel needs storage for many half-lives of dangerous isotopes.
• Drug pharmacokinetics. Biological half-life concept (different — drug clearance, not radioactive).
• Smoke detectors. Am-241 (433 yr) — long enough to last lifetime of detector.
• Industrial gauging. Cs-137 (30 yr) typical for thickness gauges.
• Particle physics. Particle lifetimes characterized by half-life or mean lifetime.

Common mistakes

• Linear vs exponential. After 2 half-lives, 25% remains, NOT 0%. Exponential decay never reaches zero.
• Confusing half-life with mean lifetime. Mean = 1.443 × half-life. Mean is the AVERAGE time per particle; half-life is when half of population gone.
• Believing temperature affects half-life. NO — nuclear, not chemical. Decay rate same at any T (with rare exceptions like Be-7 EC).
• Using "100 hour quarantine" for safety. 100 hours isn't safe relative to half-life. Need 5-10 half-lives for activity to drop to negligible.
• Treating "stable" as forever. Some "stable" isotopes have actually been observed to decay (Bi-209 has 2 × 10¹⁹ yr half-life). True stability is rare.
• Confusing physical and biological half-life. Physical half-life — radioactive decay. Biological — clearance from body. Effective half-life combines both — usually shorter than physical.