Answer to Question #160166 in Atomic and Nuclear Physics for Terry

Question #160166
Two samples of the same radioactive nuclide are prepared. Sample G hadms twice the initial activity of sample H. How does the half-life of G compare with the half-life of H?
(A) it is two times larger than that of G
(B) it is thev same as that of G
(C) it is half of that of G
(D) it is larger than that of G 4 times.
1
Expert's answer
2021-02-17T11:10:06-0500

The relationship between the activity RR and the number of nuclei in the sample, NN, can be written as follows:


R=Nλ,R=N\lambda,

here, λ\lambda is decay constant.

Since, both samples have the same size (and, therefore, have the same initial number of nuclei), we can write:


RG=NoλG,R_G=N_o\lambda_G,RH=NoλH.R_H=N_o\lambda_H.

Dividing RGR_G by RHR_H, we get:


RGRH=λGλH,\dfrac{R_G}{R_H}=\dfrac{\lambda_G}{\lambda_H},2RHRH=λGλH,\dfrac{2R_H}{R_H}=\dfrac{\lambda_G}{\lambda_H},λG=2λH.\lambda_G=2\lambda_H.

The relationship between the half-life of a sample, T1/2T_{1/2}, and its decay constant λ\lambda can be written as follows:


T1/2=ln2λ.T_{1/2}=\dfrac{ln2}{\lambda}.

Then, we can write the half-life of each samples as follows:


T1/2,G=ln2λG,T_{1/2,G}=\dfrac{ln2}{\lambda_G},T1/2,H=ln2λH.T_{1/2,H}=\dfrac{ln2}{\lambda_H}.

Dividing T1/2,GT_{1/2,G} by T1/2,HT_{1/2,H} and substituting λG\lambda_G we get:


T1/2,GT1/2,H=λHλG=λH2λH=12,\dfrac{T_{1/2,G}}{T_{1/2,H}}=\dfrac{\lambda_H}{\lambda_G}=\dfrac{\lambda_H}{2\lambda_H}=\dfrac{1}{2},T1/2,H=2T1/2,G.T_{1/2,H}=2T_{1/2,G}.

As we can see from calculations, the half-life of sample H is two times larger than that of G. Therefore, the correct answer is (A).

Answer:

(A) it is two times larger than that of G.


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