three nuclides X1, X2, X3 are a part of a radioactive decay chain such that X decays to X2 with a decay constant of lambda 1 and X2 decays to X3 with a decay constant of lambda 2. set up the differential equation
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Let \( N_1(t) \) be the number of nuclides of \( X_1 \) at time \( t \), \( N_2(t) \) be the number of nuclides of \( X_2 \) at time \( t \), and \( N_3(t) \) be the number of nuclides of \( X_3 \) at time \( t \). Show more…
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Radioactive Decay Series The following system of differential equations is encountered in the study of the decay of a special type of radioactive series of elements: $$\begin{aligned} &\frac{d x}{d t}=-\lambda_{1} x\\ &\frac{d y}{d t}=\lambda_{1} x-\lambda_{2} y. \end{aligned}$$ where $\lambda_{1}$ and $\lambda_{2}$ are constants. Discuss how to solve this system subject to $x(0)=x_{0}, y(0)=y_{0} .$ Carry out your ideas.
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In a simple case of chain radioactive decay, a parent radioactive species of nuclei, A, decays with a decay constant $\lambda_{1}$ into a daughter radioactive species of nuclei, B, which then decays with a decay constant $\lambda_{2}$ to a stable element C. a) Write the equations describing the number of nuclei in each of the three species as a function of time, and derive an expression for the number of daughter nuclei, $N_{2}$, as a function of time, and for the activity of the daughter nuclei, $A_{2},$ as a function of time. b) Discuss the results in the case when $\lambda_{2}>\lambda_{1}\left(\lambda_{2} \approx 10 \lambda_{1}\right)$ and when $\lambda_{2}>>\lambda_{1}\left(\lambda_{2} \approx 100 \lambda_{1}\right)$.
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