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(C) The radioactive decay law is given by $N(t) = N_0 e^{-\lambda t}$,where $\lambda$ is the decay constant.Given the half-life is $T$,we have $N(T) =

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$1/\sqrt{2}$

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(C) The radioactive decay law is given by $N(t) = N_0 e^{-\lambda t}$,where $\lambda$ is the decay constant.Given the half-life is $T$,we have $N(T) = N_0/2 = N_0 e^{-\lambda T}$.This implies $e^{-\lambda T} = 1/2$,so $\lambda T = \ln 2$,or $\lambda = (\ln 2)/T$.We need to find the fraction remaining after time $t = T/2$,which is $N(T/2)/N_0 = e^{-\lambda (T/2)}$.Substituting $\lambda = (\ln 2)/T$ into the expression:$N(T/2)/N_0 = e^{-[(\ln 2)/T] \cdot (T/2)} = e^{-(\ln 2)/2} = e^{-\ln(2^{1/2})} = e^{\ln(2^{-1/2})} = 2^{-1/2} = 1/\sqrt{2}$.Therefore,the fraction of the substance remaining is $1/\sqrt{2}$.

The half-life of a radioactive sample is $T$. What fraction of the substance will remain after a time $T/2$?

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