Half-Life Calculator
Calculate remaining quantity after radioactive decay using half-life formulas.
Starting quantity before decay.
Time it takes for half to remain.
Time passed since the start.
Solved value
12.5 amount units
N = N₀ × (1/2)^(t / h)
Initial amount
100
Remaining amount
12.5
Half-life
10 days
Elapsed time
30 days
Half-lives elapsed
3
Remaining percentage
12.5%
Decayed amount
87.5
87.5% decayed
Decay constant
0.069315
per days
Half-Life Formulas
Remaining amount
N = N₀ × (1/2)^(t / h)
Initial amount
N₀ = N ÷ (1/2)^(t / h)
Elapsed time
t = h × log(N / N₀) ÷ log(1/2)
Half-life
h = t × log(1/2) ÷ log(N / N₀)
Decay constant
λ = ln(2) ÷ h
Exponential decay form
N = N₀ × e^(-λt)
Variable Explanations
N
Remaining amount after decay.
N₀
Initial amount before decay starts.
t
Elapsed time.
h
Half-life.
λ
Decay constant.
e
Euler's number, used in exponential equations.
t / h
Number of half-lives elapsed.
N / N₀
Fraction of the original amount remaining.
What Half-Life Means
1 half-life
50% remains.
2 half-lives
25% remains.
3 half-lives
12.5% remains.
Exponential
Half-life is proportional decay, not linear subtraction.
Never exactly zero
The ideal model approaches zero but does not reach it.
Half-life speed
Shorter half-life means faster decay.
Worked Examples
Remaining amount after 1 half-life
Formula: N = N₀ × (1/2)^(t / h)
Substitution: 100 × (1/2)^(1)
Answer: 50
After one half-life, half of the amount remains.
Remaining amount after 3 half-lives
Formula: N = N₀ × (1/2)^3
Substitution: 100 × 0.125
Answer: 12.5
After three half-lives, 12.5% remains.
Initial amount from remaining amount
Formula: N₀ = N ÷ (1/2)^(t / h)
Substitution: 25 ÷ (1/2)^2
Answer: 100
Work backward from the remaining amount.
Elapsed time from initial and remaining amount
Formula: t = h × log(N / N₀) ÷ log(1/2)
Substitution: 10 × log(25 / 100) ÷ log(1/2)
Answer: 20 days
The amount changed from 100 to 25 after two half-lives.
Half-life from measured decay
Formula: h = t × log(1/2) ÷ log(N / N₀)
Substitution: 30 × log(1/2) ÷ log(12.5 / 100)
Answer: 10 days
A drop to 12.5% over 30 days means 3 half-lives passed.
Carbon-14 style example
Formula: N = N₀ × (1/2)^(t / h)
Substitution: 100 × (1/2)^(5730 / 5730)
Answer: 50
This is a generic educational decay example, not a dating analysis.
Medicine clearance estimate
Formula: Remaining % = 100 × (1/2)^(t / h)
Substitution: 100 × (1/2)^(24 / 8)
Answer: 12.5%
Educational estimate only. Actual biological clearance varies.
Very small remaining amount
Formula: N = 1000 × (1/2)^20
Substitution: 1000 × 0.0000009537
Answer: ≈ 9.537e-4
Scientific notation keeps very small values readable.
Exponential Decay and Remaining Amount
Decay happens by repeated proportional reduction. Every half-life halves what remains, not the original amount.
This is why the curve gets flatter over time. Remaining percentage can be calculated as 100 × (1/2)^(t / h).
The number of half-lives is elapsed time divided by half-life.
Common Half-Life Use Cases
Common Half-Life Mistakes
Understanding Your Results
Remaining amount
Estimated quantity left after decay.
Initial amount
Starting quantity before decay.
Elapsed time
Time required to decay from initial to remaining amount.
Half-life
Time needed for the quantity to halve.
Remaining percentage
Fraction of starting amount still present.
Number of half-lives
Elapsed time divided by half-life.
Frequently Asked Questions
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