# The half life of a radioactive substance is 30 days. What is the time taken for 3/4th of the original mass to disintegrate?

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Radioactive disintegration follows first order reaction kinetics, such that, `t=1/lambdaln(a/(a-x))`

where the terms have their usual significance.

Half-life is the time required to disintegrate into half the original mass of a radioelement.

So, `30=1/lambdaln(a/(a-a/2))`

`rArr lambda=1/30ln(a/(a/2))`

`=1/30ln2`

`=ln2/30 day^(-1)`

Put this value of disintegration constant in the second condition:

`t=1/(ln2/30)ln(a/(a-3a/4))`

`=30/ln2* ln(a/(a/4))`

`=30/ln2*ln4`

`=30/ln2*2*ln2`

`=60 ` days

Therefore, it would require 60 days to disintegrate 3/4 th of the original mass of the radioelement.

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