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Question

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.

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

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