Give any three of remaining quantity, starting quantity, time and half-life to find the fourth, or convert between half-life, mean lifetime and decay constant.
Fill in three of the four boxes and leave the one you want to find empty.
Give any one of the three to get the other two.
The half-life of a substance is the time it takes for half of it to decay. After one half-life, 50% is left. After two, 25% is left, after three 12.5%, and so on, halving each time. The idea is most often used for radioactive atoms, such as carbon-14, which has a half-life of about 5,730 years, but it fits any process of exponential decay, including some medicines leaving the body.
The quantity left after a time t is the starting quantity multiplied by one half raised to the power t divided by the half-life: N(t) = N0 × (1/2)^(t / t½). The calculator rearranges this to find whichever of the four values you leave out. To find the half-life from the other three, it uses t½ = t × ln 2 ÷ ln(N0 ÷ Nt). The remaining quantity must be smaller than the starting quantity and bigger than zero, because a decaying substance never grows and never quite reaches zero.
Two other numbers describe the same decay. The decay constant λ is the fraction of the substance that decays per unit of time, and equals ln 2 ÷ t½. The mean lifetime τ is the average time an atom survives, and equals 1 ÷ λ, which is t½ ÷ ln 2, about 1.44 times the half-life. The second calculator converts between all three, so you only need to know one.
Use t½ = t × ln 2 ÷ ln(N0 ÷ Nt), where N0 is the starting amount, Nt is the amount left, and t is the time that passed.
One eighth, or 12.5%, is left, because the amount is halved three times.
The half-life is the time for half of the atoms to decay. The mean lifetime is the average time an atom lasts, and it is about 1.44 times longer than the half-life.
It is the rate of decay, equal to ln 2 divided by the half-life. A larger decay constant means a shorter half-life.
Time and half-life must be in the same unit. The result for time or half-life comes out in that unit, so use years, days or seconds as you prefer.
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