How to Calculate Half-Life: Radioactive Decay Formula

Half-life is the time it takes for half of a substance to decay or transform. It appears in nuclear physics, pharmacology, chemistry, and archaeology — wherever something decreases exponentially.

The Half-Life Formula

N(t) = N₀ × (½)^(t/t½)

Or equivalently:

N(t) = N₀ × e^(−λt)

Where:

• N(t) = remaining quantity at time t
• N₀ = initial quantity
• t½ = half-life period
• λ = decay constant = ln(2) ÷ t½ ≈ 0.693 ÷ t½
• e = Euler's number (2.718...)

Basic Half-Life Calculation

How much remains after n half-lives?

Remaining fraction = (½)^n = 1 ÷ 2^n

Example: 200 g of a substance with a 10-day half-life, after 30 days:

• Number of half-lives = 30 ÷ 10 = 3
• Remaining = 200 × (½)³ = 200 × 0.125 = 25 g

Finding Remaining Amount at Any Time

N(t) = N₀ × (½)^(t/t½)

Example: 500 mg substance, half-life = 8 hours. How much remains after 20 hours?

• N(20) = 500 × (½)^(20/8)
• N(20) = 500 × (0.5)^2.5
• N(20) = 500 × 0.1768 = 88.4 mg

Finding Elapsed Time from Remaining Amount

t = t½ × log(N(t)/N₀) ÷ log(½)

Or: t = t½ × ln(N₀/N(t)) ÷ ln(2)

Example: Start with 1,000 g, half-life = 5 years. When does 62.5 g remain?

• 62.5/1,000 = 0.0625 = (½)^n → n = 4 half-lives
• t = 4 × 5 = 20 years

The Decay Constant

λ = ln(2) ÷ t½ ≈ 0.693 ÷ t½

The decay constant λ is the probability per unit time that a nucleus will decay. It's used in the exponential decay formula:

N(t) = N₀ × e^(−λt)

Example: Half-life = 20 minutes:

• λ = 0.693 ÷ 20 = 0.03466 per minute
• After 60 minutes: N = N₀ × e^(−0.03466 × 60) = N₀ × e^(−2.079) = N₀ × 0.125

This confirms: 60 minutes = 3 half-lives → 12.5% remaining ✓

Radioactive Isotope Half-Lives

Carbon Dating: Practical Application

Carbon-14 has a half-life of 5,730 years and is found in all living organisms. When an organism dies, it stops absorbing new C-14, so the ratio of C-14 to C-12 decreases predictably.

Age = t½ ÷ ln(2) × ln(N₀/N)

Example: A sample has 25% of its original C-14 remaining:

• 25% = (½)^n → n = 2 half-lives
• Age = 2 × 5,730 = 11,460 years old

Carbon dating is reliable for samples up to ~50,000 years old (approximately 8–9 half-lives, after which so little C-14 remains that measurement becomes unreliable).

Half-Life in Pharmacology

Drug half-life determines dosing frequency. After 4–5 half-lives, approximately 94–97% of a drug has been eliminated:

*Aspirin's effects on platelets last much longer than its own half-life due to irreversible binding.

Use our exponent calculator to compute (½)^n for any number of half-lives quickly.