How to Calculate Radioactivity

••• radioactivity image by Albert Lozano from

In certain materials, the nucleus of an atom is unstable and will emit particles spontaneously without any external stimulus. This process is called radioactivity or radioactive decay.

Elements with atomic number 83 have more than 82 protons, and so are radioactive. Isotopes, which are elements where the nuclei have different numbers of neutrons, may also be unstable. The nuclei of unstable elements emit alpha, beta, or gamma particles. An alpha particle is a helium nucleus, and a beta particle is an electron or a positron, which has the same mass as an electron but has a positive charge. A gamma particle is a high-energy photon.

To calculate radioactivity, it is necessary to know the time it takes for the nucleus to decay.

    Find the expression for the half-life t(half) of a radioactive sample. It is the time it takes for half of the amount of the nuclei in a sample to decay. The half-life is related to the decay constant lambda, which has a value dependent on the sample's material. The formula is t(half) = ln 2 / lambda = 0.693 / lambda.

    Study the equation for the total decay rate or activity of a radioactive sample. It is R = dN / dt = lambda N = N(0) e(-lambda*t). N is the number of nuclei, and N(0) is the original or initial amount of the sample before the decay at time t = 0. The unit of measurement for the activity is Bq or becquerel, which is one decay per second. Another unit is the curie, which is equal to 3.7 x 10 exp (10) Bq.

    Practice calculating the radioactive decay. Radium-226 has a half-life of 1,600 years. Calculate the activity of a one gram sample, where N = 2.66 x 10 exp(21). To do this, first find lambda. Simultaneously, convert the half-life from years to seconds. Then lambda = 0.693 / t(half) = 0.693 / (1600 * 3.156 x 10 exp(7) s/yr) = 1.37 x 10 exp(-11) / s. The rate of decay is therefore dN/dt = lambda*N = 1.37 x 10 exp(-11)/s * 2.66 x 10 exp(21) = 3.7 x 10 exp(10) decays / s = 3.7 x 10 exp(10) Bq. Note this is a curie. Note also that decay / s is written as 1 / s.

    Things You'll Need

    • Radioactive Decay Problem
    • Calculator
    • Introductory Physics Textbook


    • Radioactive elements are also referred to as radioactive isotopes, radioisotopes, or radionuclides.


  • Conceptual Physics; Paul Hewitt; 2006
  • Essential University Physics; Richard Wolfson; 2007
  • Physics for Scientists and Engineers; Douglas Giancoli; 2000

About the Author

Kim Lewis is a professional programmer and web developer. She has been a technical writer for more than 10 years and has written articles for businesses and the federal government. Lewis holds a Bachelor of Science, and occasionally teaches classes on how to program for the Internet.

Photo Credits

  • radioactivity image by Albert Lozano from