Radioactive Decay Law

It was in the year 1896 that A.H. Becquerel discovered the radioactivity accidentally. He was initially studying the fluorescence and phosphorescence of compounds irradiated with visible light, and during this period, he observed radioactivity. It is when the radioactive decay law came into existence. 

Radioactivity is a process by which the unstable atom’s nucleus loses energy by emitting radiation. First, some pieces of Uranium-Potassium-Sulphate were illuminated by A.H. Becquerel with visible light. Next, these pieces were wrapped in black paper and separated from the photographic plate by using a piece of silver. After several hours, it was found that there was blackening on the plate after developing the photograph. 

Further examination stated that the compound emitted something which penetrated silver and black paper and hit the plate. The same phenomenon was further referred to as radioactive Decay.

What are the different types of radioactive decay?

Before getting to know about the radioactive decay law, it is essential to know about three types of radioactive decay, and these are:

1. Alpha Decay: 

When an alpha particle emits its nucleus, the same process is called alpha decay. The numerical formula of the alpha decay is mentioned as:

E = (mi – mf – mp)c2

Here,

• mi denotes the initial mass of the nucleus
• mf  represents the mass of the nucleus after the emission of particles
• mp represents the mass of the particles that are emitted

1. Beta Decay:

A beta particle frequently is referred to as an electron, and sometimes it can also be a positron. This is because neutrons are shed by the nucleus in case the reaction involves positrons. 

1. Gamma Decay:

The nucleus has orbiting electrons that have some energy. So there is the emission of a photon when electrons jump from a high energy level to a lower energy level. A similar thing takes place in the nucleus; whenever it rearranges from a high energy level to a low energy level, a high-energy photon is emitted, which is referred to as gamma-ray.

Stating the Law of the Radioactive Decay:

When it comes to stating the law of radioactive decay, it basically specifies that when a radioactive material undergoes any of the three types of radioactive decay. The number of nuclei undergoing the decay, per unit time, will be proportional to the total number of nuclei in the provided sample material. 

Therefore, when talking about the state the law of radioactive decay in the form of numerical expression, it is,

Law of radioactivity decay = N(t) = N0e– λt

Derive the law of radioactivity law:

According to the radioactivity decay law, ‘the probability that a nucleus of an unstable atom, per unit time, is a constant, independent of time’.

Therefore, when it comes to deriving the law of radioactivity law, if N is equal to the total number of nuclei in the given sample material and ΔN is equal to the number of nuclei undergoing the decay per unit time, then.

ΔN / Δt N

Or, ΔN / Δt = λN ….. (1)

Here, λ is the symbol denoting the radioactive decay constant, also referred to as the disintegration constant. 

Now, the change in the nuclei of an unstable atom in the given sample can be represented as 

dN = – ΔN in time Δt

Hence, the rate of change of N (in the limit Δt 0) can be stated as,

dN / dt = – λN     [N is total number of nuclei]

or, dN / N = – λ dt 

Now, after integrating both sides of the equation mentioned above, the following mentioned equation can be obtained:

NN0∫ dN/N = λ tt0∫ dt ….. (2)

Or, ln N – ln N0 = – λ(t – t0) ….. (3)

Here, N0 is represented as the number of radioactive nuclei in the given sample at some arbitrary time t0, and N denotes the number of radioactive nuclei at any subsequent point of time t. 

Next, by setting t0 = 0 and rearranging the third equation, the following mentioned can be obtained,

ln (N/N0) = – λt

Or, N(t) = N0e– λt ….. (4)

Equation (4) is defined as the radioactive decay law.

Conclusion

The law of radioactivity decay is an important phenomenon exhibited by the nuclei of an atom because of the unstable nuclear energy. In simple words, the radioactive decay law specifies that radioactivity is a nuclear phenomenon when the nucleus of an unstable atom undergoes decay. There are three types of decay- alpha, beta, and gamma. To know more about the law of radioactivity decay, its derivation, types of decay, etc., it is recommended to go through the article mentioned above.