Half-Life
What Half-Life Means
We can consider that an object is such that its “amount” decreases by half after every fixed interval of
$
time. For a quantity 𝑥, if its half life is 𝑡!.# , it will reduce to % after 𝑡!.# units of time. It further reduces to
$
&
after another 𝑡!.# units of time, and so on.
Such things actually exist in real life, for example, radioactive isotopes. Although their radioactive decay
is unpredictable and random, the pattern can be well approximated to the characteristics mentioned
above. A sample of radioactive isotope containing 𝑥 amount (either initial mass or initial number) of
parent nuclei, decays to half that amount after a certain time, which depends on the specific isotope.
For such things, we can say that the rate of decay is directly proportional to the variable amount.
𝑑𝑥
∝𝑥
𝑑𝑡
We turn it into an equation by introducing a constant.
𝑑𝑥
= 𝜆𝑥
𝑑𝑡
Here, we get a separable first order ordinary differential equation. We can solve it to get the general
solution.
𝑑𝑥
= 𝜆𝑥
𝑑𝑡
1
' 𝑑𝑥 = ' 𝜆 𝑑𝑡
𝑥
ln 𝑥 = 𝜆𝑡 + 𝑐
𝑥 = 𝑒 '()*
𝑥 = 𝑒 '( × 𝑒 *
As 𝑒 * is a constant, we can write it in a simplified manner 𝐴. So we get,
𝑥 = 𝐴𝑒 '(
Here, we got the general solution of our differential equation. For the sake of simplicity, we will write is
as a function of 𝑡,
𝑥(𝑡) = 𝐴𝑒 '(
We know that initially, 𝑥 had the maximum value which we can denote as 𝑥! . So, the initial condition is
such that 𝑥(0) = 𝑥! . If we 𝑡 = 0 in our general solution, we get,
𝑥(0) = 𝐴𝑒 '×!
𝑥! = 𝐴 × 1
𝑥! = 𝐴
So, the value of the constant 𝐴 is equal to the initial amount. We can now rewrite our equation, which is
now the particular solution of the differential equation for the given conditions.
𝑥(𝑡) = 𝑥! 𝑒 '(
This equation gives us the model of how the amount 𝑥 changes as time 𝑡 progresses. 𝜆 is known as the
decay constant, and is characteristic to the specific object which is undergoing such decay. However, we
must be aware that the object is “decaying”, which denotes that the object’s amount is reducing. This is
only possible if 𝜆 is negative. So, we get our equation,
𝑥(𝑡) = 𝑥! 𝑒 ,'(
For an object undergoing such decay, we can find out the half-life of the object by studying about 𝜆 a
little more. We can consider that the object reduces to half its initial amount after 𝑡!.# units of time. In
$
other words, (𝑡!.# ) = % . We can use the following conditions in our equation as such.
𝑥=
𝑥
× 𝑒 ,'(!.#
2
𝑒 ,'(!.# = 2
−𝜆𝑡!.# = ln 2
𝑡!.# = −
ln 2
𝜆
Here, we can clearly see how the decay constant is related to the half-life of an object. As the decay
constant is different for different objects and is a characteristic property of the objects, every object
undergoing such decay have distinct half-lives.