How to Figure Half-Life from Two Measurements

The half-life of a decaying compound is the length of time required for the quantity to decrease by 50%. Radioactive isotopes of elements often decay at constant, measurable rates, and half-life is a convenient way to describe the rate at which the substance breaks down. Something with a longer half-life decays more slowly, while something with a shorter half-life decays more rapidly.

If you know the levels at two different points in time, you can use these numbers to determine the equation of exponential decay and the half-life. For a substance that decays exponentially at a constant rate, the equation for the quantity at time T is

Q(T) = AB^T or equivalently
Q(T) = Ae^kT ,

where e is the natural log base and the relation between B and K is Ln(B) = k. Whether you use the first or second equation is a matter of personal preference. Below we show how to find the exponential decay equation and half-life with an example; you can also use the convenient half-life calculator on the left.

Example

Suppose you measure the quantity of a substance at two points in time. At time = 3 minutes, the quantity is 145 grams. Then at time = 46 minutes, the quantity is 107 grams. This gives us two pairs of (T,Q) coordinates: (3, 145) and (46, 107).

Set up the equation Q = AB^T and plug in both sets of coordinates so that you get two equations in A and B:

145 = AB³
107 = AB⁴⁶

You can solve this system for B by dividing the first equation by the second. This gives you

145/107 = (B³)/(B⁴⁶)
145/107 = B^-43
(145/107)^(-1/43) = B
0.9929573 = B

And to find A:

145 = A(0.9929573)³
145 = A(0.9790205)
148.1072 = A

So the equation of exponential decay is Q(T) = 148.1072(0.9929573)^T, or Q(T) = 148.1072e^{-0.007067556T}. To find the half-life, you set Q equal to one half of A and solve for T.

74.0536 = 148.1072(0.9929573)^T
74.0536/148.1072 = 0.9929573^T
0.5 = 0.9929573^T
Ln(0.5) = Ln(0.9929573^T)
Ln(0.5) = T(Ln(0.9929573))
Ln(0.5)/Ln(0.9929573) = T
98.07452 = T

So the half-life of the substance is 98.07452 minutes. This means that it takes 1 hour, 38 minutes, and 4 seconds for the quantity to halve.

General Formula

The general formula for the half-life H is

H = Ln(0.5)/Ln(B) = Ln(0.5)/k

You can also generalize this to compute the third-life, the time it takes for the substance to decrease to one third of its original level. The third-life J is given by

J = Ln(1/3)/Ln(B) = Ln(1/3)/k.

© Had2Know 2010