Solving Radioactive Decay Problems

Let me guess, you have just opened up your modern physics assignments and you are staring at some problems that talk about things like 'radioactivity', 'half life' and 'decay constant' and you have no clue about solving them. You need to go through the theory of radioactive decay. To start with, the basic formulas are listed, that will help you solve problems.

First discovered by Henry Becquerel in 1896, radioactivity provided the first clues of what lies inside the microcosm of the atomic nucleus. Radioactive decay is the emission of ionizing radiation by certain unstable elements, leading to their transformation into different elements. There are three types, which include beta, gamma, and alpha decay.

From a given set of atoms in a radioactive substance, it is impossible to predict which ones will undergo decay at any moment, as it is an entirely random process. So, the problem of decay lends itself, only to a statistical solution. The standard equation of radioactivity, from which all related formulas are derived, is based on the following fact.

Any radioactive atom may decay at any point of time and the probability that a fraction of atoms decay, is directly proportional to the total number of atoms. The equation is as follows:

- dN/ dt = λ N

Here 'dN' is the fraction of elements decaying in a fraction of time 'dt', N is the total number of radioactive atoms, and 'λ' is the decay constant, which is unique for every radioactive element. Solving this equation, one can derive various useful formulas, listed below.

Solution of Radioactive Decay Equation N = N0 e-λt

(here N0 is the amount of radioactive atoms at t = 0)

Half Life of a Radioactive Element t1/2 = ln 2 / λ = 0.693 / λ

(where t1/2 denotes half life of the radioactive element)

Mean Lifetime of Every Radioactive Atom τ = 1 / λ

Activity of a Radioactive Sample A = λ N

These are the prime formulas that you need to know.

To solve any problem in physics, one need to understand what exactly is expected to be found out in a problem and what is given. Here are some problems on half life and decay constants. For any data related to these problems, refer to the list of radioactive elements.

Problem 1: Carbon-14 is one of the isotopes of carbon, with a half life of 5,730 years. Find the decay constant (λ) for this element.
Solution 1: We know the half life of C-14, from which we are expected to compute its decay constant. The equation that connects these two quantities is:

t_1/2 = ln 2 / λ = 0.693 / λ 

Therefore, rearranging terms, we get:

λ = ln 2 / t_1/2 = 0.693 / t_1/2 = 0.693 / (5730 x 365 x 24 x 60 x 60) sec = 3.836 x 10^-12 per second. 

So the decay constant of Carbon-14 is 3.836 x 10^-12 per second.

Problem 2: Cobalt-60 has a half life of 5.27 years. Find the average lifetime of each cobalt-60 atom.
Solution 2: The half life of Co-60 is given to be 5.27 years. To find the average lifetime (τ) of one cobalt-60 atom, the following formula must be used. 

τ = 1 / λ

but we know that, λ = 0.693 / t_1/2

Therefore, the above equation becomes,

τ = t_1/2 / 0.693 = (Half life of C0-60) / 0.693 = (5.72 x 365 x 24 x 60 x 60 sec)/0.693 = 8.25 years 

Thus, the average lifetime of any cobalt-60 atom is 8.25 years.

These problems are quite simple and most are of the plug-in type. If you are still confused, go through the theory of radioactive decay again.