How this calculator works
- Step 1Count half-livest ÷ THow many half-lives fit in the time.
- Step 2Halve that often(½)^(t ÷ T)The fraction still left.
- Step 3ScaleN₀ × fractionMultiply by the starting amount.
In exponential decay, a fixed share disappears in each equal stretch of time. The half-life is the time for half of what is there to go.
The calculator rearranges the same formula to find any one of the four quantities from the other three.
Worked example: cobalt-60 after 15 years
- Decay
N = N₀ × (½)^(t ÷ T)
- N₀Starting amount 100 g
- THalf-life 5.27 years
- tTime 15 years
- Half-lives15 ÷ 5.272.8463
- Fraction left(½) to the 2.846313.905237%
- Amount left100 g × 0.13905213.905237 g
OpenStax Chemistry 2e works this case in Example 21.5. It rounds λ to 0.132 per year first and prints 13.8%; without that rounding the answer is 13.905237%.
How much is left after each half-life
Read the result
Radon-222 (half-life 3.823 days) takes 11.113043 days to fall from 0.750 g to 0.100 g, matching the book's 11.1 days.
To reach 2% of a cobalt-60 source takes 29.743122 years. The decay constant and mean life beneath the answer describe the same decay rate.
Assumptions and limits
- Steady decayA constant half-life, as in radioactive decay.
- Matching unitsHalf-life and time in the same unit.
- Decay onlyThe amount left cannot exceed the start.
- DisplaySix decimals; percentages to four.
Related growth and decay tools
Decay and compound growth use the same exponential maths, with the sign reversed.
To solve for an exponent directly, use the log calculator. The exponent calculator raises a number to any power, including the fractional ones half-life problems need.
Common mistakes
Mistake: Mixing days and years.
Instead: Convert the half-life and the time to the same unit.
Mistake: Subtracting half each time from the start.
Instead: Halve what is left, not the original amount.
Mistake: Rounding λ early.
Instead: Keep full precision until the end, as the calculator does.
Checking the method
Every golden case is re-worked in 40-digit decimal arithmetic, and the OpenStax answers are reproduced.
An independent script recomputes each case with Python’s decimal module. It also checks the printed answers of OpenStax Example 21.5 and its radon exercise, and finds a half-life of 6 hours for 100 falling to 25 in 12 hours.
Read how we test our calculators, or browse more maths calculators.