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 highenergy photon.
To calculate radioactivity, it is necessary to know the time it takes for the nucleus to decay.
Things You'll Need
 Radioactive Decay Problem
 Calculator
 Introductory Physics Textbook

Find the expression for the halflife 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 halflife 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. Radium226 has a halflife 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 halflife 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 = lambdaN = 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.
Tips & Warnings
 Radioactive elements are also referred to as radioactive isotopes, radioisotopes, or radionuclides.
References
 Conceptual Physics; Paul Hewitt; 2006
 Essential University Physics; Richard Wolfson; 2007
 Physics for Scientists and Engineers; Douglas Giancoli; 2000
 Photo Credit radioactivity image by Albert Lozano from Fotolia.com