Define decay constant or distintegration constant of a radioactive el

1.	Define decay constant or distintegration constant of a radioactive element. If λ is the decay constant of a radioactive element, show that about 37% of the original nuclei remains undecayed after a time interval of λ-1.
Answer» The decay constant or disintegration constant of a radioactive element is defined as the ratio of the disintegration rate at an instant to the number of undecayed nuclei of the element present at that instant. Let N₀ be the number of nuclei of a radioactive element present at time t = 0 and N, the number of undecayed nuclei at time t. From the radioactive law, N = N₀e^-λt where λ is the decay constant. At t = λ^-1, the fraction of undecayed nuclei is \(\cfrac{N}{N_0}\) = e^{-λ × λ-1} = e^-1= \(\cfrac1e\) Since e ≅ 2.718, \(\cfrac{N}{N_0}\) = \(\cfrac1{2.718}\) = 0.3679 Therefore, about 36.79% ≈ 37% of the original nuclei remains undecayed after a time λ^-1Since λ is the probability that a nucleus of the element will decay in one second, λ^-1 gives the mean-life or the mean life time τ of the radioactive element measured in second; τ = λ^-1.

Define decay constant or distintegration constant of a radioactive element. If λ is the decay constant of a radioactive element, show that about 37% of the original nuclei remains undecayed after a time interval of λ-1.

Answer»

The decay constant or disintegration constant of a radioactive element is defined as the ratio of the disintegration rate at an instant to the number of undecayed nuclei of the element present at that instant.

Let N₀ be the number of nuclei of a radioactive element present at time t = 0 and N, the number of undecayed nuclei at time t. From the radioactive law,

N = N₀e^-λt

where λ is the decay constant. At t = λ^-1, the fraction of undecayed nuclei is

\(\cfrac{N}{N_0}\) = e^{-λ × λ-1} = e^-1= \(\cfrac1e\)

Since e ≅ 2.718,

\(\cfrac{N}{N_0}\) = \(\cfrac1{2.718}\) = 0.3679

Therefore, about 36.79% ≈ 37% of the original nuclei remains undecayed after a time λ^-1Since λ is the probability that a nucleus of the element will decay in one second, λ^-1 gives the mean-life or the mean life time τ of the radioactive element measured in second; τ = λ^-1.

Define decay constant or distintegration constant of a radioactive element. If λ is the decay constant of a radioactive element, show that about 37% of the original nuclei remains undecayed after a time interval of λ-1.

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