Equation derivation for mean life of radioactive element
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Mean Life
Mean-Life
Mean Life is an important topic of radioactivity. Just as the half-life of a radioactive substance is important, formula to calculate mean life is also equally important. The IIT JEE often picks up questions on the calculation of average life of radioactive substances. It is important to have expertise in this area as well in order to remain competitive in the JEE.
Mean life of radioactive elements is expected to be somewhat longer than the half-life. If in a given radioactive element, half of its elements have decayed after one half life, some well-defined average life expectancy can be assumed which is the mean life of the atoms.
Formulas of Calculating Radioactivity Mean Life
The mean life of an element equals the half-life of the substance divided by the natural logarithm of 2 which is about 0.693. In fact, the mean life turns out to equal the number τ which appears in the exponential term e−t/τinvolved in the description of decay or growth. It is termed as the time constant.
Mean lifetime is a very significant quantity that can be measured directly for small number of atoms. If there are ‘n’ active nuclei, (atoms) (of the same type, of course), the mean life is
τ = τ
1 + τ3 + ... + τ2 / n
Where τ1, τ2,....... τn represent the observed lifetime of the individual nuclei and n is a very large number. It can also be calculated as a weighted average:
τ = τ1N1 + τ3N2 + ... + τ2Nn / N1 + ... + Nn
Where N1 nuclei live for time τ1,
N2 nuclei live for time τ2....... and so on
hope it will help you....
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