Half-Life Calculator
Enter initial amount and half-life to see quantity remaining at any time. Or input two measurement points to find the half-life. Shows decay table, decay constant λ, and time to any target percentage.
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How It Works
Enter initial amount A₀ and half-life t½. Enter the time t to evaluate. A(t) = A₀ × (0.5)^(t/t½). The result shows: remaining amount, percentage remaining, fraction remaining, and number of half-lives elapsed (t/t½). A decay table shows values at t½, 2×t½, 3×t½, ... 10×t½.
A(t) = A₀ × (½)^(t/t½) | λ = ln(2)/t½A₀=1000, t½=5yr, t=15yr → 3 half-lives → 125 remaining (12.5%)Switch to "Find Half-Life" sub-mode. Enter: measurement at time 1 (A₁, t₁) and measurement at time 2 (A₂, t₂). Formula: t½ = (t₂−t₁) × ln(2) / ln(A₁/A₂). Example: 800 units at t=0, 200 at t=100yr → t½ = 100×0.693/ln(4) = 50 years. Also shows decay constant λ.
t½ = (t₂−t₁) × ln(2) / ln(A₁/A₂)A₁=800 at t=0, A₂=200 at t=100yr → t½=50yr, λ=0.01386/yrSwitch to "Find Time" sub-mode. Enter: A₀, t½, and target A. Formula: t = t½ × log₂(A₀/A) = t½ × ln(A₀/A)/ln(2). Example: 1000 mg initial, t½=6hr, target=10mg. t = 6 × ln(100)/ln(2) = 6 × 4.605/0.693 = 39.9 hours.
t = t½ × ln(A₀/A) / ln(2) | = t½ × log₂(A₀/A)A₀=1000, t½=6hr, target=10mg → t=39.9hr (6.65 half-lives)λ = ln(2)/t½ ≈ 0.693/t½. The decay equation: dA/dt = −λA. Mean lifetime τ = 1/λ = t½/ln(2) ≈ 1.443×t½. The mean lifetime is the average time a nucleus/molecule survives before decay. It is longer than t½ because some particles last much longer. The calculator shows λ and τ alongside each result.
λ = ln(2)/t½ | τ = 1/λ = t½/ln(2) ≈ 1.443×t½C-14: t½=5730yr → λ=0.0001209/yr → τ=8267yrAfter 5 half-lives: (0.5)^5 = 3.125% remains — considered clinically negligible. A drug with t½=8 hours reaches <3% in 40 hours. Drugs with active metabolites may have different effective half-lives. Enter drug t½ in Remaining mode and read the time at which percentage drops below 3%.
After 5×t½: 3.1% remains | After 7×t½: 0.78%Aspirin t½=3.5hr: after 5×3.5=17.5hr → 3.1% remainsQuick Reference
Verify these in the calculator above.
Decay
After 1 half-life
50% remains
Decay
After 2 half-lives
25% remains
Drug rule
After 5 half-lives
3.1% remains
Decay
After 7 half-lives
0.78% remains
Carbon dating
C-14 t½=5730yr: after 11460yr
25% remains
Decay const
λ for t½=5730yr
0.0001209/yr
Find t½
A₁=800 at t=0, A₂=200 at t=100yr
t½ = 50yr
Mean life
Mean lifetime τ = ?
1.443 × t½
Tips & Shortcuts
The decay table shows values at every half-life interval — useful for quickly reading off remaining percentages without calculating each time.
The "Find Half-Life" sub-mode is essential for experimental data — two measurements at known times determine the half-life without needing to know the initial value.
For drug calculations: the 5 half-life rule (97% eliminated) is standard for washout periods. For more conservative estimates, use 7 half-lives (99.2% eliminated).
The decay constant λ and mean lifetime τ are useful for nuclear physics problems — some textbooks use these instead of t½. They are mathematically equivalent: λ=ln(2)/t½.
Carbon dating works back about 10 half-lives (57,300 years) with reasonable accuracy. Beyond this, the remaining C-14 is too small to measure reliably.
Common Mistakes
Using linear decay instead of exponential
Half-life decay is exponential, not linear. After 2 half-lives, 25% remains (not 0%). After 3 half-lives, 12.5% (not 50%). A quantity never reaches exactly zero in exponential decay — it approaches zero asymptotically.
Mixing time units for half-life and elapsed time
If t½ is in hours and you enter t in days, convert first. t=2 days = 48 hours. For t½=8 hours: n_halflives = 48/8 = 6. Remaining = (0.5)^6 = 1.5625%. Use the same unit throughout.
Applying the 5 half-life rule to active metabolites
Some drugs have active metabolites with longer half-lives. Fluoxetine (Prozac) has t½=1-4 days but its active metabolite norfluoxetine has t½=7-15 days — requiring 5-6 weeks for full washout. The calculator computes one substance; consider metabolites separately.
Confusing decay constant λ with rate r
In A=Pe^(−λt), λ is positive for decay. In A=Pe^(rt), r is negative for decay: r=−λ. The numerical value is the same — only the sign convention differs. Always check which convention your formula uses.
Forgetting that half-life is constant throughout decay
The half-life does not change as the quantity decreases. Whether you start with 1000 atoms or 10 atoms, the fraction decaying per unit time (λ) is the same. This is the defining property of exponential decay.
Frequently Asked Questions
Half-life (t½) is the time for a quantity to reduce to half its initial value. After 1 half-life: 50% remains. After 2: 25%. After 10: 1/1024 ≈ 0.1%. Applies to radioactive decay, drug elimination, bacterial die-off, and any exponential decay process.
A(t) = A₀ × (1/2)^(t/t½) = A₀ × e^(−λt) where λ = ln(2)/t½ ≈ 0.693/t½. λ is the decay constant. Example: C-14 has t½=5730 years, λ=0.693/5730=0.0001209 per year.
C-14 is absorbed by living organisms from atmospheric CO₂. After death, C-14 decays with t½=5730 years. Measuring the ratio of C-14 to C-12 in a sample gives the fraction remaining, then t = −ln(fraction)/λ. Accurate to about 50,000 years.
After 7 half-lives: 100%×(0.5)^7 = 0.78% remains. After 10: 0.098%. After 20: 0.000095%. For practical purposes, 7-10 half-lives is often considered "gone" (below 1%). Medical protocols use 5 half-lives for drug clearance (97% eliminated).
λ = ln(2)/t½ ≈ 0.693/t½. It is the fraction decaying per unit time. For C-14: λ=0.000121/year means 0.0121% decays per year. The decay equation is dA/dt = −λA, which gives A(t) = A₀e^(−λt).
If A₁ at t₁ and A₂ at t₂: t½ = (t₂−t₁)×ln(2)/ln(A₁/A₂). Example: sample had 800 units at t=0 and 200 at t=100 years. t½ = 100×ln(2)/ln(800/200) = 100×0.693/ln(4) = 100×0.693/1.386 = 50 years.
Drug half-life determines dosing frequency. For stable blood levels, dosing every t½ maintains concentration. After 5 half-lives, ~97% of a drug is eliminated — used to determine washout period. Example: fluoxetine t½=1-4 days (but active metabolite t½=7-15 days) → 5-6 week washout.
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