Radioactivity: Mean-Life significance and derivation

Radioactivity is a phenomenon through which an amount of energy is produced from the spontaneous reactions of less stable nuclei. Mean-life is a very important and useful feature of any radioactive decay. Average or mean life of an unstable isotope is determined by the ratio of the total lifetime of all individual parental atoms to the total number of atoms in the sample, or the average time that any unstable radioactive isotope exhibits before it decays completely. It is a very critical quantity that is measured for a small number of atoms. Through this article, you will understand the mean-life of the radioactive substance.

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Definitions & understanding

Radioactivity is a phenomenon in which disintegration happens through the emission of radiations from the nucleus of the unstable radioactive isotope.

In radioactivity processes, the mean life is an average life span of all the nuclei of the specific non-stable radioactive isotope. Additionally, time travel is the sum of lifetimes of all non-stable nuclei in a radioactive isotope sample, divided by the total number of non-stable nuclei present in it.

The duration of the half-life of any unstable nuclei is always lesser than the mean life, which is 1.443 times longer. For instance, Lead (209) decays to Bismuth (209) have a mean life of 4.69 hours and a half-life of 3.25 hours.

In other words, the average or mean life of an unstable isotope is determined by the ratio of the total life of all individual parental atoms to the total number of atoms in the sample, or the average time that any unstable radioactive exhibit before it decays completely. It is a very critical quantity that is measured for a small number of atoms.

Mathematical expression

In mathematical terms, it can be represented as;

Mean life (τ) = sum of the life of all-atom / total no of atoms present

The formula for mean life

The average life of any radioactive isotope has equaled the half-life of the substance divided by the natural log 2 which is exactly 0.693, and it’s equal to the number of τ which is represented in the exponential term e−t/τ in the decaying. It is called the time constant.

Let ‘n’ be an active nucleus, the mean life is

τ = τ1 + τ3 + … + τ2 / n, in which τ1, τ2, ……. τn shows the observed lifetime of any nuclei and n is a very large number. It is calculated as a weighted average;  

τ = τ1N1 + τ3N2 + … + τ2Nn / N1 + … + Nn 

in this, N1 nuclei live for time τ1,

N2 nuclei live for time τ2……. and so on.

This is related to the γ.

Derivation

Half-life: In the process of radioactivity, the duration of time required by atomic nuclei of any radioactive isotope to decay by one-half amount from its initial stage is called the half-life.

The equation below is showing the relationship between decay constant (λ) and half-life (t),

λ = ln (2) t1/2 ≈ 0.693 t 1/2

Furthermore, for understanding this relationship,

Let t= t1/2 and put this in the equation

N = N0e−λt,

And we will get,

N = N0e−λt= N0e−0.693 = 0.500 N0

Mean life: It is the sum of the life span of a specific number of nuclei divided by the number of nuclei. Let dN be the number of nuclei disintegrate in the time interval dt.

τ =∫ t dN / ∫dN

by integrating both sides,

τ = 1/λ

For example, the mean life of a 14C nucleus with T1/2 = 5730  is 8267 years.

Significance

Mean-life plays important role in the area of radioactive isotope; these are mentioned below;

1. It helps in identifying the exact dates of artifacts (in calculating the age of any substance).
2. It helps in the calculation of time for storing any radioactive waste until they become safe.
3. It also supports the doctors in analysing and use of safe radioactive tracers.
4. It is a very useful method to assess the rate of the decay process.

However, it is different from the half-life of any radioactive nucleus.

Conclusion

Hence, radioactivity processes are very sudden, and energy liberated processes in nature. Mean-life is a very essential and useful feature of any radioactive decay. Average or mean life of an unstable isotope is determined by the ratio of the total life of all individual parental atoms to the total number of atoms in the sample or the average time that any unstable radioactive exhibit before it decays completely. It is a very critical quantity that is measured for a small number of atoms. Through this article, you will understand the mean life of radioactive substances.