Free tool
For any first-order process, such as radioactive decay or a first-order reaction, find the amount remaining, the time that has passed or the half-life. It also gives the rate constant k and how many half-lives have gone by.
Built for General Chemistry IIGrams, mg, molarity, atoms or activity - any unit, as long as N and N0 match.
Use the same unit for the half-life and the time. k comes out per that unit.
Enter the known values.
Half-life shows up twice in Gen Chem II, in kinetics and again in nuclear chemistry. Encodr keeps both on a spaced-repetition schedule so the formulas are there on exam day.
Get started freeN = N0 (1/2)^(t / t1/2), which is the same as N = N0 e^(-kt) with
k = ln 2 / t1/2 = 0.693 / t1/2. Solving for time gives t = (t1/2 / ln 2) ln(N0 / N).
A first-order half-life does not depend on how much you start with. Every half-life cuts what is left in half, whether that is 100 g or 1 mg. Radioactive decay is always first order, and so are many reactions. Zero- and second-order reactions have half-lives that change as the concentration falls, so this tool does not apply to them.
Worked example: 100 g of carbon-14 (t1/2 = 5,730 years) after 11,460 years. That is 11,460 / 5,730 = 2 half-lives, so N = 100 x (1/2)^2 = 25 g, and k = 0.693 / 5,730 = 1.21 x 10^-4 per year. A reaction example: k = 2.50 x 10^-3 s^-1 gives t1/2 = 0.693 / (2.50 x 10^-3) = 277 s. The full walkthrough is in half-life and first-order kinetics, and the whole course is in the free General Chemistry II deck.
The first-order integrated rate law, why its half-life is constant, k = 0.693 / t1/2, and worked examples from reaction kinetics and radioactive decay.
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