Radioactive decay is a fundamental process in which an unstable atomic nucleus loses energy to become stable. This instability arises because the forces holding the nucleus together are not balanced, often due to an excess of protons, neutrons, or both. To reach a more stable state, the nucleus undergoes radioactive decay. These elements are called radioactive isotopes.
The program below calculates the amount of radioactive material left after a number of years depending on its half-life. The motivation of this program is to use half-life concept to show the visitors how NumPy and Matplotlib is used to callculate, plot, and tabulate the outputs of the programs.
For more depth on radioactive isotopes and half-life, check the information at the end of this page.
Exponential Decay and Half-life
Radioactive decay is a fundamental process in which an unstable atomic nucleus loses energy to become stable. This instability arises because the forces holding the nucleus together are not balanced, often due to an excess of protons, neutrons, or both. To reach a more stable state, the nucleus undergoes radioactive decay. These elements are called radioactive isotopes
The rate at which a radioactive isotope decays is measured by what is called Half-life.
Half-life (symbol t½) is the length of time required for a quantity of the radioactive isotope to reduce to half of its initial value. This term (half-life) is used to describe how quickly unstable atoms undergo radioactive decay. The term is also used more generally to characterize any type of exponential decay.
Examples of a few radioactive isotopes and their half-life are in the table below:
| Isotope | Used in | Half-life |
| Oxygen-26 | 4.2 seconds | |
| Magnesium-19 | 5 seconds | |
| Sodium-22 | 18.6 seconds | |
| technetium-99m | nuclear medicine | 6 hours |
| gallium-67 | nuclear medicine | 80 hours |
| Iodine-121 | 8.06 days | |
| Polonium-210 | 138.4 days | |
| cobalt-60 | manufacturing | 5.3 years |
| Cesium-137 | 30.17 years | |
| americium-241 | construction | 432 years |
| Radium-226 | 1,600 years | |
| carbon-14 | archeological dating | 5,715 years |
| Plutonium-239 | 24,100 years | |
| Thorium-232 | 14*10^9 years | |
| Uranium-238 | atomic power | 4.5 billion years |
Exponential Decay: Radioactive decay is described by an exponential function equation:
A(t) = Ao* e^(kt)
Where
- A(t) is the substance remaining at time ‘t’.
- Ao is the original amount at time zero.
- ‘k’ is a constant (negative in the case of exponential decay; and positive in the case of exponential growth.
Half-life formula:
To find the half-life of a function describing exponential decay, we substitute A(t) = 0.5*Ao and replace variable ‘t’ by half-life symbol ‘T’ in the exponential decay equation and solve the equation:
A(t) = Ao*e^kt (1)
At half-life, A(t) = (1/2)*Ao
0.5Ao = Ao*e^kT
0.5 = e^kT
ln(0.5) = kT (2)
k = ln(0.5)/T (3)
Substituting the constant ‘k’ in the exponential decay equation (1), we get the exponential decay equation specific to half-time:
A(t)= Ao * e **(ln0.5/T) *t
A(t) = Ao * (0.5)**(t/T) (4)
To derive the formula for half-life, we refer to equation (2)
ln(1/2) = kT
T = -ln(2)/k (5)
Half-life, T = -0.69314718056/k (6)
Since k is negative in the case of exponential decay, the value of half-life, T is positive.
Also note that the half-life depends only on the constant ‘k’ and not on the initial quantity of the material, Ao.
