# Solving Radioactive Decay Problems

If you are looking for some tips on solving physics problems, related to radioactive decay, this article will be a useful reference. Besides providing an overview of basic theory, with formulas, some solved examples are also presented.

Omkar Phatak

*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.

Basic Formulas

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.

**- dN/ dt = λ N**

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.

Solved Problems

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.*

Therefore, rearranging terms, we get:

λ = ln 2 / t

So the decay constant of Carbon-14 is 3.836 x 10

**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 / λTherefore, 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.*

but we know that, λ = 0.693 / t

Therefore, the above equation becomes,

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.

**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**= (Half life of C0-60) / 0.693 = (5.72 x 365 x 24 x 60 x 60 sec)/0.693 = 8.25 years_{1/2}/ 0.693Thus, 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.