Relationship Between Radioactive Decay and Half Life

There are certain naturally occurring isotopes that are unstable due to the imbalanced numbers of protons and neutrons they have in their nucleus of atoms. Therefore, in order to become stable, these isotopes undergo a spontaneous process called radioactive decay. The radioactive decay causes an isotope of a particular element to be converted into an isotope of a different element. However, the final product of radioactive decay is always stable than the initial isotope. The radioactive decay of a certain substance is measured by a special term known as the half life. The time taken by a substance to become half of its initial mass through radioactive decay is measured as the half life of that substance. This is the relationship between radioactive decay and half life.

Key Areas Covered

1. What is Radioactive Decay
      – Definition, Mechanisms, Examples
2. What is Half Life
      – Definition, Explanation with Examples
3. What is the Relationship Between Radioactive Decay and Half Life
      – Radioactive Decay and Half Life

Key Terms: Half Life, Isotopes, Neutrons, Protons, Radioactive Decay

Relationship Between Radioactive Decay and Half Life - Summary

What is Radioactive Decay

Radioactive decay is the process in which unstable isotopes undergo decay through emitting radiation. Unstable isotopes are atoms having unstable nuclei. An atom can become unstable due to several reasons such as the presence of a high number of protons in the nuclei or a high number of neutrons in the nuclei. These nuclei undergo radioactive decay in order to become stable.

If there are too many protons and too many neutrons, the atoms are heavy. These heavy atoms are unstable. Therefore, these atoms can undergo radioactive decay. Other atoms also can undergo radioactive decay according to their neutron: proton ratio. If this ratio is too high, it is neutron rich and is unstable. If the ratio is too low, then it is proton rich atom and is unstable. The radioactive decay of substances may occur in three major ways.

  • Alpha Emission/Decay
  • Beta Emission/Decay
  • Gamma Emission/Decay

Alpha Emission

An alpha particle is identical to a Helium atom. It is composed of 2 protons and 2 neutrons. Alpha particle bears a +2 electrical charge because there are no electrons to neutralize the positive charges of 2 protons. Alpha decay causes the isotopes to lose 2 protons and 2 neutrons. Hence, the atomic number of a radioactive isotope is decreased by 2 units and the atomic mass from 4 units. Heavy elements such as Uranium can undergo alpha emission.

Beta Emission

In the process of beta emission (β), a beta particle is emitted. According to the electrical charge of the beta particle, it can be either a positively charged beta particle or a negatively charged beta particle. If it is β emission, then the emitted particle is an electron. If it is β+ emission, then the particle is a positron.  A positron is a particle having the same properties as an electron except for its charge. The charge of the positron is positive whereas the charge of the electron is negative. In the beta emission, a neutron is converted into a proton and an electron (or a positron). Hence, the atomic mass would be not changed, but the atomic number is increased by one unit.

Gamma Emission

Gamma radiation is not particulate. Therefore, gamma emissions do not change either the atomic number or the atomic mass of an atom. Gamma radiation is composed of photons. These photons carry only energy. Therefore, gamma emission causes the isotopes to release their energy.

Relationship Between Radioactive Decay and Half Life - 1

Figure 1: Radioactive Decay of Uranium-235

Uranium-235 is a radioactive element that is found naturally. It can undergo all three types of radioactive decay at different conditions.

What is Half Life

The half life of a substance is the time taken by that substance in order to become half of its initial mass or concentration through radioactive decay. This term is given the symbol t1/2. The term half life is used because it is not possible to predict when an individual atom might decay. But, it is possible to measure the time taken to half the nuclei of a radioactive element.

The half life can be measure regarding either the number of nuclei or the concentration. Different isotopes have different half lives. Therefore, by measuring the half life, we can predict the presence or absence of a particular isotope. The half life is independent of the physical state of the substance, temperature, pressure or any other outside influence.

The half life of a substance can be determined using the following equation.

ln(Nt / No)   =    kt


Nt is the mass of the substance after t time

No is the initial mass of the substance

K is the decay constant

t is the time considered

Relationship Between Radioactive Decay and Half Life

Figure 02: A Curve of
Radioactive Decay

The above image shows a curve of radioactive decay for a substance. The time is measured in years. According to that graph, the time taken by the substance to become 50% from initial mass (100%) is one year. The 100% becomes 25% (one fourth of initial mass) after two years. Therefore, the half life of that substance is one year.

100%           →            50%             →            25%             →            12.5%            →   →  

             (1st half life)                 (2nd half life)            (3rd half life)

The above chart has summarized the details given from the graph.

Relationship Between Radioactive Decay and Half Life

There is a direct relationship between radioactive decay and half life of a radioactive substance. The rate of radioactive decay is measured in half life equivalents. From the above equation, we can derive another important equation for the calculation of the rate of radioactive decay.

ln(Nt / No)   =    kt

since the mass (or the number of nuclei) is half of its initial value after one half life,

Nt = No/2


ln({No/2}/ No)   =    kt1/2

ln({1/2}/ 1)   =    kt1/2

ln(2)   =    kt1/2


t1/2   =   ln2 / k

The value of ln2 is 0.693. Then,

t1/2   =   0.693 / k

Here, t1/2 is the half life of a substance and k is the radioactive decay constant. The above-derived expression tells that highly radioactive substances are spent quickly, and the weakly radioactive substances take a longer time to decay completely. Therefore, a long half life indicates fast radioactive decay while a short half life indicates a slow radioactive day. The half life of some substances cannot be determined since it may take millions of years to undergo radioactive decay.


Radioactive decay is the process where unstable isotopes undergo decay through emitting radiation. There is a direct relationship between the radioactive decay of a substance and half life since the rate of the radioactive decay is measured by the equivalents of half life.


1. “Half-Life of Radioactive Decay – Boundless Open Textbook.” Boundless. 26 May 2016. Web. Available here. 01 Aug. 2017. 
2.”The Process of Natural Radioactive Decay.” Dummies. N.p., n.d. Web. Available here. 01 Aug. 2017. 

Image Courtesy:

1. “Radioactive decay” By Kurt Rosenkrantz from PDF. (CC BY-SA 3.0) via Commons Wikimedia

About the Author: Madhusha

Madhusha is a BSc (Hons) graduate in the field of Biological Sciences and is currently pursuing for her Masters in Industrial and Environmental Chemistry. Her interest areas for writing and research include Biochemistry and Environmental Chemistry.

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