This tool applies the ideal decay model N = N0 × 2^(−t/T). You give the starting amount N0, the half-life T and the elapsed time t, and it returns the number of half-lives, the amount left, the percentage remaining, a table of successive halving and a decay curve.
An optional background count shows what a detector would read and how subtracting the background gives back the source-only value.
How do I use it?
- Enter the initial amount N0 in any units, as a number greater than zero.
- Enter the half-life T greater than zero, and the elapsed time t in the same time unit as T. Time can be zero.
- Enter a background count if you want one, in the same units as N. Leave it empty otherwise.
- Press Calculate. Reset restores the starting values.
- If an input is not valid, the tool names the problem so you can correct it.
Example walk-through
The tool opens with N0 = 80, T = 10 and t = 30. The number of half-lives is 30 / 10 = 3, so N = 80 × 2^(−3) = 10 units, which is 12.5% of the original. The table lists 0, 1, 2 and 3 half-lives with amounts 80, 40, 20 and 10 at times 0, 10, 20 and 30.
Change t to 25. Now t / T = 2.5 and N = 80 × 2^(−2.5), about 14.1 units, or about 17.7%. The value sits between 20 (two half-lives) and 10 (three), as it should.
Now add a background of 5 with t back at 30. A detector would read about 10 + 5 = 15.
Subtracting the background of 5 returns 10, the source-only value. The curve in the result marks the current time with a dot, so you can see where you are on the halving curve.
How do I read the result?
- Number of half-lives: t divided by T, which can be a fraction.
- Amount remaining: N in the ideal model, with its percentage of N0.
- Background line: appears only if you entered a background, and shows the measured reading and the corrected value.
- Table and curve: each equal step of T halves the amount, which is why the curve drops steeply at first and flattens.
What are the limits?
This is an ideal model. Real decay is random, so real counts scatter around the smooth curve, more so when the sample is small.
It is not a radiation exposure or health-risk calculator, and it says nothing about safe levels. Units for N0 and the background must match, and T and t must use the same unit of time.
Which lessons explain the output?
Start with applying successive halving to a stated interval, then subtracting background from supplied counts and determining half-life from a decay graph. For why counts scatter, read interpreting random count-rate data and why one atom has no predictable decay time. Keep model calculation separate from personal risk advice in mind.
Mixed questions are in the half-life and background practice set. Other aids are on the tools page.
When the halving method is clear but real graph questions still stall, our team can work through them with you. See online one-to-one Physics tuition for how that works.