Solve half-life problems for radioactive decay and exponential processes. Find the remaining quantity after a given time, or determine the half-life from initial and final amounts.
Half-life is the time it takes for half of a quantity undergoing exponential decay to disappear. The concept comes from nuclear physics — half of a radioactive sample's atoms have decayed after one half-life, half of what's left has decayed after two, and so on — but the math applies to anything that decays exponentially: drug clearance from the body, pharmaceutical degradation, capacitor discharge, microbial population decline, radio-isotope dating, and even some economic and ecological processes.
After 1 half-life: 50% remains. After 2: 25%. After 3: 12.5%. After 10: 0.098%. After 20: 0.0001%. The decay is rapid at first because there's more material to decay, then slows because each remaining particle has the same decay probability per unit time. This produces the characteristic "exponential curve" that's hard to see by eye but obvious on a logarithmic plot, where it becomes a straight line.
This calculator handles three common half-life problems: given the initial amount, half-life, and elapsed time, find what remains; given the initial and final amounts plus elapsed time, find the half-life (used in dating); given initial amount, half-life, and final amount, find elapsed time. Use the same time units throughout — half-life and elapsed time must match (years/years, seconds/seconds, etc.).
**Scenario:** A wooden artifact has 22% of the C-14 found in a living tree. How old is it? **Calculation:** N/N₀ = 0.22. t = t₁/₂ × ln(N₀/N)/ln(2) = 5730 × ln(1/0.22)/ln(2) = 5730 × 1.514/0.693 = 12,520 years old. **Result:** Approximately 12,500 years old — Late Pleistocene era, possibly from a Clovis culture site. C-14 dating works reliably from a few hundred to about 50,000 years old; beyond that, too little C-14 remains to measure.
**Scenario:** Ibuprofen has a half-life of ~2 hours. You took 400 mg at 8 AM. How much remains at noon (4 hours later)? **Calculation:** t/t₁/₂ = 4/2 = 2 half-lives. N = 400 × (1/2)² = 400 × 0.25 = 100 mg remaining. **Result:** 100 mg of active ibuprofen at noon — 25% of the original dose. By 4 PM (3 half-lives, 8 hours): 50 mg. By 8 PM (5 half-lives, 12 hours): 12.5 mg, essentially cleared. Standard dosing interval (every 4-6 hr) brings concentration back up while previous dose is still active.
**Scenario:** A hospital receives 10 GBq (gigabecquerels) of Tc-99m for medical imaging at 8 AM. Patient scan at noon (4 hr later). How much activity remains? **Calculation:** Tc-99m half-life = 6.0 hours. t/t₁/₂ = 4/6 = 0.667. N = 10 × (1/2)^0.667 = 10 × 0.630 = 6.3 GBq. **Result:** 6.3 GBq remaining at scan time — about 63% of the morning shipment. This is why nuclear medicine departments time isotope orders carefully; the short half-life means waste if delivery is delayed. By 8 PM (12 hr, 2 half-lives): 2.5 GBq remaining. By next morning (24 hr, 4 half-lives): 0.625 GBq — essentially unusable for imaging.
**Use half-life calculations for:**
- **Radioactive decay problems**: nuclear chemistry, isotope dating (C-14, U-Pb, K-Ar), medical imaging dose planning, radiation safety after accidents. - **Pharmacokinetics**: drug dosing intervals, washout periods before swapping medications, predicting steady-state concentrations during chronic use. - **Geology and archaeology**: dating samples, sediment cores, tree rings, glaciers, lunar rocks. - **Environmental science**: pollutant breakdown rates (e.g., DDT, PCBs, atmospheric methane). - **Microbiology**: bactericidal kinetics, sterilization cycle design. - **Forensics**: estimating time since death from K consumption, pesticide degradation timelines. - **Engineering**: capacitor discharge (RC circuits), thermal cooldown of objects, isotope battery output decay.
**Rules of thumb worth memorizing:**
- After 1 half-life: 50% remains. - After 2: 25%. - After 3: 12.5%. - After 4: 6.25%. - After 5: 3.13% — usually considered the "practical clearance" point in pharmacology and many other contexts. - After 10: ~0.1% remains. - After 20: ~10⁻⁶ — effectively gone.
**The "5 half-lives" rule for drug clearance:** after 5 half-lives, less than 3% of a drug remains in your system. Most pharmacology references use this as the "effectively cleared" threshold. Drugs with long half-lives (fluoxetine: 4 days) take weeks to wash out; short-half-life drugs (caffeine: 5 hr) wash out in a single day.
**Equivalent decay constants:**
- λ = ln(2) / t₁/₂ ≈ 0.693 / t₁/₂ - Mean lifetime τ = 1/λ = t₁/₂ / ln(2) ≈ 1.443 × t₁/₂
**Carbon-14 dating limits:**
- Younger than ~300 years: too much modern carbon contamination. - Older than ~50,000 years: too little C-14 left to measure reliably. - Between: works well for organic materials (wood, bone, charcoal, shell). - For older samples, use longer-lived isotopes (U-Pb, K-Ar, Ar-Ar).
Calculate pH from H+ concentration and convert between pH, pOH, [H+], and [OH-].
Calculate molarity (moles per liter) and perform dilution calculations (C1V1 = C2V2).
Calculate molar mass from element atomic weights and their quantities in a compound.
Calculate dilution using C1V1 = C2V2 equation. Find any missing variable.
Calculate titration endpoint including molarity, volume, and equivalence point.
Calculate boiling point elevation using ΔTb = Kb x m x i.
Remaining
25.00
% Remaining
25.00%
Half-Lives
2.00
| Parameter | Value |
|---|---|
| Initial Amount | 100 |
| Remaining Amount | 25 |
| Percent Remaining | 25.0000% |
| Amount Decayed | 75 |
| Half-Life | 5,730 |
| Time Elapsed | 11,460 |
| Half-Lives Elapsed | 2.0000 |
| Decay Constant (λ) | 1.209681e-4 |
| Mean Lifetime (τ) | 8,266.6426 |
| Formula | N(t) = N₀ × (½)^(t/t₁/₂) |