Half-life (t½) is the time required for half of a substance to decay. It applies to radioactive decay, first-order reactions, and drug metabolism.
Amount Remaining:
N(t) = N₀ × (½)t/t₁/₂
Exponential Form:
N(t) = N₀ × e-kt
Half-Life from Rate Constant:
t½ = 0.693 / k
Units: g, mol, atoms, Bq, Ci, or any amount unit
Amount of substance remaining after time t
Units: Same as N(t)
Starting amount at t = 0
Units: s, min, h, days, years (time units)
Time for N to decrease to ½N₀
💡 Independent of starting amount for first-order decay
Units: 1/time (s⁻¹, min⁻¹, etc.)
Relationship: k = 0.693 / t½ = ln(2) / t½
Units: Same as t½
Time since decay started
| Time Elapsed | Fraction Remaining | Percent Remaining | Amount (if N₀=100) |
|---|---|---|---|
| 0 half-lives | 1/1 | 100% | 100 |
| 1 half-life | 1/2 | 50% | 50 |
| 2 half-lives | 1/4 | 25% | 25 |
| 3 half-lives | 1/8 | 12.5% | 12.5 |
| 4 half-lives | 1/16 | 6.25% | 6.25 |
| n half-lives | (1/2)n | 100/(2n)% | 100/(2n) |
25% = 1/4 = (1/2)² → 2 half-lives have passed
t = 2 × t½ = 2 × 5,730 = 11,460 years
N(t) = N₀ × (1/2)t/t₁/₂
0.25N₀ = N₀ × (1/2)t/5730
0.25 = (1/2)t/5730
(1/2)² = (1/2)t/5730
2 = t/5730
t = 11,460 years
Answer: The fossil is 11,460 years old
After 1 half-life: 50% remains. After 2 half-lives: 25% remains.
Never reaches zero! After n half-lives, (1/2)ⁿ remains. It approaches zero but never gets there.
It's (1/2)^(t/t₁/₂), NOT 2^(t/t₁/₂). Half, not double!
t and t½ MUST use the same units. If t½ is in years, t must be in years too.
0.693 is ln(2) rounded. For precise work, use ln(2) = 0.6931 or your calculator's ln function.
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The time required for half of a substance to decay. After one half-life, 50% remains; after two, 25% remains, and so on.
No! For first-order decay, half-life is constant regardless of amount. Whether you start with 1g or 1000g, half-life is the same.
t½ = 0.693/k or k = 0.693/t½. They're inversely related. Faster decay (larger k) means shorter half-life.
Theoretically never! It approaches zero exponentially but never reaches it. After 10 half-lives, less than 0.1% remains.
These formulas are for first-order only. Zero-order and second-order reactions have different half-life equations that DO depend on concentration.