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What fraction of the original activity of a radioactive substance will remain at the end of half-life?

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Half-life T is the time during which the number of decaying species decreases by half
Decay Law
N=Nₒ•exp(-λ•t)
Decay constant
λ=ln2/T.
Half-life T is the time during which the number of decaying species decreases by half
If t=T, then
N= Nₒ•exp(-λ•T)= Nₒ•exp(-ln2•T /t)= Nₒ•exp(-ln2)=0.5•Nₒ.

Activity of a given amount of radioactive material is calculated as the decay constant  (related to the half-life T) multiplied by the number of radioactive nuclei
A=λ•N, Aₒ=λ•Nₒ.

A=Aₒ exp(-λ•T) =>
λ•N= λ•Nₒ • exp(-λ•T).
A=Aₒ/2.

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