Application of Differential Equations in Radioactivity

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Differential Equations and Their Applications

Differential equations are equations that involve derivatives of functions which are unknown as well as independent variables. These equations have a wide range of application in our day to day life since they have the capability of predicting happenings in the world around us. Their use is evident in a wide variety of fields ranging from chemistry, economics, engineering, biology, and physics. Among the areas these equations are applied is in predicting population growth, change in investment returns over a specified duration as well as describing proportional growth and decay.

Applications in Chemistry

In chemistry, differential equations are widely applied in determining the rate of decay of radioactive materials. How the use of these equations in radioactivity occur can be obtained from

Radioactive Decay and Half-life

Radioactive decay is a term used to refer to the nuclear conversion of atoms and is accompanied by the emission of either particles or gamma radiation. In the application presented, the differential equation is used to determine the half-life of a radioactive isotope. Half-life refers to the amount of time that is required to decay the mass of an isotope by half of its original mass. Time (t) and mass (N) are the variables used in the equation.

The Equation for Determining Half-life

The world has over 3,000 isotopes, and only 265 of these isotopes are stable. All the other isotopes are radioactive. Therefore, when we have a sample containing any of these radioactive isotopes, it is easy to determine the half-life of every single radioactive isotope using the equation below:

in this equation, No represents the original mass of the isotope, N is the new mass after decay, t represents time taken to half the mass of the isotope, λ is lambda while e is an exponential.

Derivation of the Differential Equation

The reasoning behind this differential equation is that the amount of isotope that will decay is dependent on the original amount of the isotope (N) as well as the time specified. The more the materials, the more the decay. Also, the longer the duration of decay, the more the amount decayed. By the use of the link provided above, the derivation of this equation used integral calculus as well as the natural logarithm. Therefore, it is easy to deduce that the results obtained from the Radioactive Decay Law are controlled by a decay constant. Since the decay constant differs from one radioactive material to another, the rate of decay also varies, and presentation of this information on a graph provides curves with the different slope. An illustration is provided in the graph below:

Graph showing the rate of decay for different isotopes

Rate of Decay and Decay Constants

By solving the differential equation in the radioactive decay of isotopes, it was possible to determine their rate of decay given when given a sample of known mass. From the results, it was easy to deduce that rate of decay differed from one element to another. The reason behind this was because the samples had different decay constants. The larger the decay constant, the faster the rate of decay while the smaller the decay constant, the slower the rate of decay.


Considering how useful radioactivity is in our day to day activities, application of this differential equation is very helpful since it becomes possible to determine the time taken by various isotopes to decay. By considering how radioactive materials are being widely used in multiple sectors worldwide, it is very appealing to me to understand how the process of decay happens.

September 25, 2023


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