What is half-life?

Half-life (symbol T_1/2) is the time required for a quantity undergoing exponential decay to halve. It's a characteristic constant of every decay process: every radioisotope, every drug in the body, every first-order chemical reaction has its specific half-life.

Core formulas

1. Remaining quantity over time

N(t) = N₀ × (1/2)^t/T_½

or equivalently:

N(t) = N₀ × e^−λt

• N₀: initial quantity (at t=0)
• N(t): quantity at time t
• T_½: half-life
• λ: decay constant, λ = ln(2)/T_½ ≈ 0.693/T_½

2. Elapsed time

t = T_½ × log₂(N₀/N(t))

3. Mean lifetime

τ = 1/λ = T_½ / ln(2) ≈ 1.443 × T_½

τ is the mean time a particle lives before decaying; ~44.3% longer than T_½.

The 5-half-lives rule

In pharmacology, 5 half-lives is conventionally when a drug is considered "cleared" (97% gone).

Applications

1. Radioactive decay

Original application, studied by Henri Becquerel and Marie Curie. Each radioisotope has characteristic half-life:

2. Pharmacology — biological half-life

In the human body, drugs metabolize following first-order kinetics. Half-life determines:

• Dosing frequency
• Steady-state (4-5 half-lives to reach)
• Wash-out period before switching therapy (5-7 half-lives)

Common drug half-lives:

• Ibuprofen: 2-4 hours
• Acetaminophen: 2-3 hours
• Diazepam (Valium): 20-100 hours
• Fluoxetine (Prozac): 4-6 days
• Amiodarone: 25-110 days (among longest known)
• Caffeine: 3-5 hours

3. Carbon-14 dating

C-14 decays with T_½ = 5,730 years. Living organisms continuously absorb C-14 from atmosphere. At death, absorption ceases:

Age = 5,730 × log₂(N₀/N_current) years

Example: artifact with 25% original C-14 = 2 half-lives = 11,460 years old.

Discovered by Willard Libby (1949, Nobel Chemistry 1960).

4. First-order chemical reactions

For A → products with first-order kinetics:

[A](t) = [A]₀ × e^−kt

T_½ = ln(2)/k, independent of initial concentration.

5. Environmental persistence

• PLA bioplastic: 2-3 years industrial compost
• PET in ocean: 450+ years
• Atmospheric CO₂: hundreds of thousands of years

6. Biology — protein/cell half-lives

• Hemoglobin: 120 days (RBC turnover)
• Albumin: 20 days (plasma)
• IgG antibodies: 21 days
• IgM antibodies: 5 days
• mRNA (various): minutes-hours

7. Finance — depreciation

Half-life applied to percentage value loss. Car losing 20%/year has T_½ = ln(2)/ln(1/0.80) ≈ 3.1 years.

Examples

1. Carbon-14 archaeology

Artifact with 12.5% original C-14 = 1/8 = (1/2)³. So 3 half-lives = 3 × 5,730 = 17,190 years.

2. Iodine-131 therapy

100 mCi dose, 8-day half-life. After 32 days (4 half-lives): 100/16 = 6.25 mCi.

3. Drug half-life

80 mg ibuprofen, 2-hour half-life. After 8 hours (4 half-lives): 80/16 = 5 mg.

4. Generic decay

Substance goes from 400 g to 50 g in 30 hours. 50/400 = 1/8 = (1/2)³ → 3 half-lives. T_½ = 30/3 = 10 hours.

Common misconception

Intuition error: "after 1 half-life 50% remains, after 2 half-lives 0%". FALSE! After 2 half-lives 25% remains.

Decay is exponential: the halved quantity halves again, never exactly reaching zero (asymptotic).

Log-linear plot

Plotting log(N) vs t on log scale gives a straight line with slope −λ. Experimental method to measure half-lives: linear fit of ln(N) vs t.

Decay constant λ

λ is instantaneous decay probability per unit time. Atom with λ = 0.1/year has 10% probability of decaying in 1 year (memoryless).

Fundamental: λ × T_½ = ln(2) ≈ 0.693

Effective vs biological vs physical half-life

For radiopharmaceuticals, decay occurs for two reasons:

1. Physical decay (radioactive, λ_phys)
2. Biological elimination (metabolic, λ_bio)

Effective half-life: 1/T_eff = 1/T_phys + 1/T_bio

Iodine-131: T_phys = 8 days, T_bio in thyroid = 80 days → T_eff ≈ 7.3 days.

History

• 1896 — Henri Becquerel: discovers radioactivity in uranium
• 1898 — Marie and Pierre Curie: isolate polonium, radium; coin "radioactivity"
• 1903 — Rutherford and Soddy: formulate exponential decay law
• 1907 — Rutherford: introduces "half-life period"
• 1949 — Willard Libby: develops C-14 dating, Nobel 1960

Common calculation errors

• Confusing λ and T_½: λ = ln(2)/T, not 1/T
• Inconsistent time units: always convert
• Linear assumption: 2 half-lives is 25%, not 0%
• Forgetting metabolism: pharmacology needs effective half-life
• Non-exponential processes: only first-order kinetics follows this law

FAQ

Why exactly halving?

Mathematical consequence of exponential decay: after T_½ quantity is exp(−λ × T_½) × N₀ = N₀/2 when λ × T_½ = ln(2).

Half-life vs mean lifetime?

Half-life more intuitive for communication. Mean lifetime more natural mathematically (in e^−t/τ). τ = T_½ × 1.443.

Does it ever reach zero?

Mathematically no. Practically after 10-15 half-lives, indistinguishable from zero.

Does temperature affect half-life?

Radioactive: NO, nuclear property. Chemical/biological: yes (Arrhenius).

Why 5 half-lives = practical zero?

After 5 T_½, 3.125% remains. Negligible for pharmacology; 10-20 half-lives needed for long-term nuclear safety.

Updated July 2026