Answer to Question #126049 in Atomic and Nuclear Physics for Muhammad qasim

Question #126049

 Two radioactive nuclides, S and M, both have the same activity X at time t = 0. S and M have a half-life of 26 years and 16 years respectively. Now both samples S and M are mixed. Find out the total activity of the mixture at t = 50 years?


1
Expert's answer
2020-07-13T11:42:59-0400

Let initial number of atom of both S and M is N0N_0 .

Since,

A=dNdt=λNA=\bigg|\frac{dN}{dt}\bigg|=\lambda N

Thus, initially activity is

X=λSN0+λMN0X=\lambda_SN_0+\lambda_MN_0

Also,

λ=ln2t1/2    λS=ln226λM=ln216\lambda=\frac{\ln2}{t_{1/2}}\\ \implies \lambda_S=\frac{\ln2}{26}\\ \lambda_M=\frac{\ln2}{16}

Hence,

N0=208X21ln2N_0=\dfrac{208X}{21\ln2}

Since,

N=N0(12)t/t12N=N_0\bigg(\frac{1}{2}\bigg)^{t/t_{\frac{1}{2}}}

Thus,

NS=N0(1/2)50/26=N0(1/2)25/13NM=N0(1/2)50/16=N0(1/2)25/8N_S=N_0(1/2)^{50/26}=N_0(1/2)^{25/13}\\N_M=N_0(1/2)^{50/16}=N_0(1/2)^{25/8}

Thus, activity at t=50 is

A=λSNS+λMNM    A=ln226N0(1/2)25/13+ln216N0(1/2)25/8    A=(821(1/2)25/13+1321(1/2)25/8)X0.17XA=\lambda_SN_S+\lambda _MN_M\\ \implies A=\frac{\ln2}{26}N_0(1/2)^{25/13}+\frac{\ln2}{16}N_0(1/2)^{25/8}\\ \implies A=\bigg(\frac{8}{21}(1/2)^{25/13}+\frac{13}{21}(1/2)^{25/8}\bigg)X\approx0.17X


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