The half-life period of a radioactive sample | vedclass

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(D) Let $N_0$ be the initial number of radioactive nuclei.At time $t$,the fraction of nuclei that have disintegrated is $x$.Therefore,the fraction of

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$1 - \sqrt{1 - x}$

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(D) Let $N_0$ be the initial number of radioactive nuclei.At time $t$,the fraction of nuclei that have disintegrated is $x$.Therefore,the fraction of nuclei remaining is $N(t)/N_0 = 1 - x$.According to the law of radioactive decay,$N(t) = N_0(1/2)^{t/T}$.So,$1 - x = (1/2)^{t/T}$.Now,we need to find the fraction decayed in time $t/2$,which is $1 - N(t/2)/N_0$.$N(t/2)/N_0 = (1/2)^{(t/2)/T} = [(1/2)^{t/T}]^{1/2}$.Substituting $1 - x = (1/2)^{t/T}$,we get $N(t/2)/N_0 = \sqrt{1 - x}$.The fraction that decays in time $t/2$ is $1 - N(t/2)/N_0 = 1 - \sqrt{1 - x}$.

The half-life period of a radioactive sample is $T$. If a fraction $x$ disintegrates in time $t$,what fraction will decay in time $\frac{t}{2}$?

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