Radioactive Half-Life Calculator

N = N₀(½)^(t/t₁∕₂)

Calculate remaining radioactive atoms after time t. N = N₀ (½)^(t/t½) — exponential decay.

Calculate

Initial quantity N₀ (atoms (or Bq))

Time elapsed t (years)

Half-life t₁/₂ ((same as t))

Enter values above to calculate

How it calculates

N = N₀ × (½)^(t / t½)

After one half-life: N = N₀/2. After two: N₀/4, etc.

Use consistent time units for t and t½

Formula

N = N₀(½)^(t/t₁∕₂)

Variable Table

Symbol	Quantity	SI Unit
N	Remaining quantity	same as N₀
N₀	Initial quantity	atoms or Bq
t	Elapsed time	any unit
t₁/₂	Half-life	same time unit
λ	Decay constant = ln2/t½	s⁻¹

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How to Use This Calculator

This radioactive half-life calculator is built for quick physics checks and worked-problem review. Enter values in the units shown beside each input, then compare the result with the formula and variable table before using it in a longer solution. The calculator does the arithmetic, but the physics still depends on choosing a model that matches the situation.

Start by identifying the system, the known quantities, and the quantity you want to find. If a value is given in a non-SI unit, convert it before substitution. A correct numerical answer with mixed units can still be physically wrong, especially when squared units, inverse seconds, charges, temperatures, or distances are involved.

Assumptions and Limits

The formula N = N₀(½)^(t/t₁∕₂) is a model, not a universal description of every possible case. It assumes the quantities in the variable table are the relevant quantities for the problem and that hidden effects are either negligible or already included in the inputs. If friction, drag, relativistic speeds, changing fields, non-constant temperature, or geometry-specific effects matter, check whether a more complete model is needed.

Use the result as a magnitude and units check. Ask whether the answer has the right sign, whether it grows or shrinks when an input changes, and whether the limiting cases make sense. Setting an input to zero, doubling a quantity, or using a very large value is often enough to catch a formula choice or unit mistake before it reaches a final answer.

Worked Example

Carbon-14: t½ = 5,730 years. 1,000 atoms, t = 11,460 years. Find N.

Step 1: t / t½ = 11460/5730 = 2.0 half-lives

Step 2: N = 1000 × (½)² = 1000/4

Answer: N = 250 atoms

Common Mistakes

Use same time units for t and t½
This is a probability — actual atom count fluctuates
Activity A = λN decreases at the same rate as N

Particle PhysicsHalf-Life Deep DiveBeta Decay

Frequently Asked Questions

What is a half-life?

The time for exactly half of a radioactive sample to decay. After n half-lives, N = N₀/2ⁿ.

Can you use this for carbon dating?

Yes — carbon-14 (t½ = 5,730 years) is used to date organic materials up to ~50,000 years old.