# A radioactive substance decays for an interval of time equal to its mean life. Find the fraction of the amount of the substance which is left undecayed after this time interval.

#### Let decay constant of Radio active Substance $$= \lambda$$$$t_{mem} = t_{avg} = \dfrac{1}{\lambda}$$

Let No is initial Amount $$80$$

Amount remains After time t

$$N_t = N_0 e^{-\lambda t}$$

$$\dfrac{N_t}{N_0} = e^{-\lambda t}$$ ___(I)

$$t = t_{avg} = \dfrac{1}{\lambda}$$

from equation (I)

$$\dfrac{N_t}{N_0} = e^{-\lambda \times \frac{1}{\lambda}} = e^{-1}$$

$$\dfrac{N_t}{N_0} = \dfrac{1}{e} = 0.367$$

$$\therefore $$ fraction of Amount left

$$\dfrac{N_t}{N_0} = 0.367$$