The half life of a radioactive substance is T . The time taken, for...

It is given that $t\dfrac{1}{2} = T$.

After disintegration of ${\left(\dfrac{7}{8}\right)}^{th}$part, ${\left(\dfrac{1}{8}\right)}^{th}$ part is only left.

Using the decay formula,

$N = {N}_{0}\left(\dfrac{1}{2}\right)\dfrac{t}{{t}_{\dfrac{1}{2}}}$

$⇒ \dfrac{{N}_{0}}{8} = {N}_{0}\left(\dfrac{1}{2}\right)\dfrac{t}{T} ⇒ - \ln 8 = - \dfrac{t}{T}\ln 2 ⇒ 3\ln 2 = \dfrac{t}{T}\ln 2 ⇒ t = 3T$