Half-Life Calculator

The half-life formula

Exponential decay based on half-life uses:

N(t) = N₀ × (1/2)^(t / t½)

Where:

• N₀ = initial amount
• N(t) = amount remaining after time t
• t = elapsed time
• t½ = half-life (same time unit as t)
• t / t½ = number of half-lives elapsed

The amount decayed is simply N₀ − N(t), and the percent remaining is N(t) / N₀ × 100.

Worked example

Start with 100 g of an isotope whose half-life is 10 years, and let 30 years pass.

• Half-lives elapsed = 30 ÷ 10 = 3
• Remaining = 100 × (1/2)³ = 100 × 0.125 = 12.5 g
• Decayed = 100 − 12.5 = 87.5 g
• Percent remaining = 12.5 ÷ 100 × 100 = 12.5%

Half-lives vs. fraction remaining

After 10 half-lives, about 0.0977% remains. Whether that is negligible depends on the substance, measurement and purpose; the model does not say that the quantity is literally gone.

Common pitfalls

• Mismatched units: half-life and elapsed time must share the same unit. Convert one value before calculating if one is in hours and the other is in days.
• Zero or negative values: the initial amount and half-life must be greater than zero. Elapsed time may be zero, but it cannot be negative.
• Confusing half-life with mean lifetime: mean lifetime τ = t½ ÷ ln(2) ≈ 1.4427 × t½, so it is not the same as half-life.
• Assuming linear decay: this equation models a constant fraction lost per interval, not a constant amount.
• Using the model outside its scope: mixtures, multi-compartment pharmacokinetics, changing environmental conditions and zero-order processes may need a different model.

Very small results are shown in scientific notation. With an extreme elapsed-time-to-half-life ratio, finite browser arithmetic can eventually represent the remaining amount as zero even though the mathematical exponential approaches zero without reaching it.