# Average Life

For finding the average  life of any radioactive element it is important to know about its half life. In 1904 Rutherford introduced a constant known as the half life period of the radio elements for evaluating their radioactivity or for the comparison of the radioactivity with the activities of other radioactive elements.The halflife period of a radioactive element is defined as the time required by  a given amount of the element to decay to its one half of original value.

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Derivation of the half life  period:

$\frac{N_{t}}{N_{0}}=\frac{1}{2}$

$N_{t}=\frac{1}{2}N_{0}$

Subtracting the value of  Nt in equation $N_{t}=N_{0}e^{-\lambda t}$

$\frac{1}{2}N_{0}=N_{0}e^{-\lambda T_{\frac{1}{2}}}$

$\frac{1}{2}=e^{-\lambda T_{\frac{1}{2}}}$

$\lambda{T_{\frac{1}{2}}}=-ln \frac{1}{2}$

${T_{\frac{1}{2}}}=\frac{0.693}{\lambda }$

This is the formula for finding the half life

Halflife period is the measure of the radioactivity of the element since shorter the half period of an element, greater is the number of the disintegrating atoms and hence greater its radioactivity.The half life periods or the half lives of different radio elements vary widely, ranging from a fraction of a second to millions of years. Nucleates emitting high energy alpha rays have shorter half lives. This is known as Geiger Nuttal rule.

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## Relation between Average Life and Half Life

Sincetotal decay period of any element is infinity, it  is meaningless to use the term total decay period for radio elements.Thus the term averagelife is used which is determined by

Average Life(T)= $\frac{Sum of lives of the nuclides}{Total number of nuclides}$

Relation between average life and half life .Average life of an element is the inverse of its decay constant.

T="$\frac{1}{\lambda" }$

Substituting the value of lambda in the above equation

$T=\frac{T_{\frac{1}{2}}}{0.693}$

So                                    Average life = 1.44 x Half Life

## Numerical examples.

Question:One gram of a radioactive substance loses 1 centigram in 50 seconds ,what are its half life and average life

Answer:N= 1 gm      ,N=1-0.01=0.99 gm    t="50" secs

We know that     $\lambda =\frac{2.303}{t}log\frac{N_{0}}{N_{t}}$

= $2.026\times 10^{-4}s^{-1}$

Now    $T_{\frac{1}{2}}=\frac{.693}{\lambda }$

$T_{\frac{1}{2}}=3240$secs

So average life(T) =$\frac{1}{\lambda }$

=4936 secs