All calculatorsRadioactive Decay Calculator
Predict the remaining amount, time, half-life, decay constant, or activity using the exponential decay law: N(t) = N₀·e−λt, with λ = ln 2 / t₁/₂. Student-friendly steps and common isotope quick picks included.
Background
Radioactive decay follows first-order kinetics. The decay constant λ (s⁻¹) and half-life t₁/₂ are related by λ = ln 2 / t₁/₂. Activity is A(t) = λN(t). As long as you use the same units for N and N₀ (g, mol, atoms, etc.), the ratio N(t)/N₀ is consistent.
How to use this calculator
- Use half-life or λ: Pick a parameterization—enter t₁/₂ or λ. They are related by λ = ln 2 / t₁/₂.
- Solving for N(t): Provide N₀, time, and t₁/₂ (or λ).
- Solving for t: Provide N₀, target N(t), and t₁/₂ (or λ).
- Solving for t₁/₂ or λ: Provide the other parameter (λ or t₁/₂).
- Activity: A(t) = λN(t). Enter N(t) directly, or let us compute N(t) from time.
Keep units consistent. If you enter grams and provide molar mass, we’ll show atoms/mol conversions in the steps.
Formulas & Equations Used
Core relations (first-order decay):
- Decay law: N(t) = N₀·e−λt
- Half-life relation: λ = ln 2 / t₁/₂
- Time from amounts: t = (1/λ)·ln(N₀/N(t))
- Activity: A(t) = λ·N(t) (1 Bq = 1 s⁻¹)
Example Problems & Step-by-Step Solutions
Example 1 (Remaining amount)
C-14, t₁/₂=5730 y. For N₀=1.00 (any unit), t=11,460 y (2 half-lives): N = N₀·(1/2)² = 0.25.
Example 2 (Time to target)
I-131, t₁/₂=8.02 d. N₀=100 mg. When N=12.5 mg, that’s 1/8 ⇒ 3 half-lives ⇒ t ≈ 24.1 d.
Example 3 (Activity)
If λ=0.693/t₁/₂ and N(t)=2.0×10¹² atoms, then A=λN(t) in s⁻¹. Convert to Bq if desired (1 Bq = 1 s⁻¹).
Frequently Asked Questions
Q: Do N and N₀ need special units?
No—any consistent unit works (g, mol, atoms). The ratio N/N₀ is unitless.
Q: How do I convert between grams and atoms?
Use molar mass: mol = g / (g·mol⁻¹); atoms = mol × NA. We show this if you provide molar mass.
Q: What’s the relation between t₁/₂ and λ?
λ = ln 2 / t₁/₂. At any time, N(t) = N₀·e−λt.
