(a) The daughter nucleus formed in radioactive decay is...

(a) The daughter nucleus formed in radioactive decay is often radioactive. Let $N_{10}$ represent the number of parent nuclei at time $t=0, N_{1}(t)$ the number of parent nuclei at time $t,$ and $\lambda_{1}$ the decay constant of the parent. Suppose the number of daughter nuclei at time $t=0$ is zero, let $N_{2}(t)$ be the number of daughter nuclei at time $t,$ and let $\lambda_{2}$ be the decay constant of the daughter. Show that $N_{2}(t)$ satisfies the differential equation $$\frac{d N_{2}}{d t}=\lambda_{1} N_{1}-\lambda_{2} N_{2}$$ (b) Verify by substitution that this differential equation has the solution $$N_{2}(t)=\frac{N_{10} \lambda_{1}}{\lambda_{1}-\lambda_{2}}\left(e^{-\lambda_{2} t}-e^{-\lambda_{1} t}\right)$$ This equation is the law of successive radioactive decays. (c) $^{218}$ Po decays into $^{214}Pb$ with a half-life of $3.10 \mathrm{min},$ and $^{214} \mathrm{Pb}$ decays into $^{214} \mathrm{Bi}$ with a half-life of 26.8 min. On the same axes, plot graphs of $N_{1}(t)$ for 218 Po and $N_{2}(t)$ for $^{\text {214 }}$ Pb. Let $N_{10}=1000$ nuclei, and choose values of $t$ from 0 to 36 min in 2 -min intervals. The curve for $^{214} \mathrm{Pb}$ at first rises to a maximum and then starts to decay. At what instant $t_{m}$ is the number of $^{214} \mathrm{Pb}$ nuclei a maximum? (d) By applying the condition for a maximum $\frac{d N_{2}}{d t}=0,$ derive a symbolic formula for $t_{m}$ in terms of $\lambda_{1}$ and $\lambda_{2} .$ Does the value obtained in (c) agree with this formula?

(a) The daughter nucleus formed in radioactive decay is often radioactive. Let $N_{10}$ represent the number of parent nuclei at time $t=0, N_{1}(t)$ the number of parent nuclei at time $t,$ and $\lambda_{1}$ the decay constant of the parent. Suppose the number of daughter nuclei at time $t=0$ is zero, let $N_{2}(t)$ be the number of daughter nuclei at time $t,$ and let $\lambda_{2}$ be the decay constant of the daughter. Show that $N_{2}(t)$ satisfies the differential equation $$\frac{d N_{2}}{d t}=\lambda_{1} N_{1}-\lambda_{2} N_{2}$$
(b) Verify by substitution that this differential equation has the solution $$N_{2}(t)=\frac{N_{10} \lambda_{1}}{\lambda_{1}-\lambda_{2}}\left(e^{-\lambda_{2} t}-e^{-\lambda_{1} t}\right)$$ This equation is the law of successive radioactive decays. (c) $^{218}$ Po decays into $^{214}Pb$ with a half-life of $3.10 \mathrm{min},$ and $^{214} \mathrm{Pb}$ decays into $^{214} \mathrm{Bi}$ with a half-life of 26.8 min. On the same axes, plot graphs of $N_{1}(t)$ for 218 Po and $N_{2}(t)$ for $^{\text {214 }}$ Pb. Let $N_{10}=1000$ nuclei, and choose values of $t$ from 0 to 36 min in 2 -min intervals. The curve for $^{214} \mathrm{Pb}$ at first rises to a maximum and then starts to decay. At what instant $t_{m}$ is the number of $^{214} \mathrm{Pb}$ nuclei a maximum? (d) By applying the condition for a maximum $\frac{d N_{2}}{d t}=0,$ derive a symbolic formula for $t_{m}$ in terms of $\lambda_{1}$ and $\lambda_{2} .$ Does the value obtained in (c) agree with this formula?

Key Concepts

Differential Equations in Radioactive Decay

This concept deals with setting up mathematical models to describe the time-dependent behavior of radioactive nuclei. The differential equations encapsulate production and depletion rates of a given nuclide, which in decay chains include terms for both the ingoing yield from parent decay and the outgoing loss due to the nuclide’s own decay.

Decay Constant and Half-life Relationship

The decay constant is a parameter that quantifies the probability per unit time that a nucleus will decay, and it is directly related to the half-life of the radioactive substance. Understanding this relationship is crucial since it allows the conversion between the exponential decay model and time scales relevant to practical measurements.

Successive Radioactive Decay

In many decay processes, a parent nucleus decays into a daughter nucleus which may itself be radioactive and undergo further decay. This chain of decays is modeled by coupled differential equations where the daughter population increases due to the decay of the parent and decreases due to its own decay, leading to a characteristic behavior over time.

Verification by Substitution in Differential Equations

This method involves checking whether a proposed solution satisfies the differential equation by directly substituting the solution back into the equation. It is a fundamental technique for validating analytical solutions in mathematical physics and applied mathematics.

Conditions for Maximum Daughter Activity

To determine the time at which the daughter nuclide reaches its maximum concentration, one uses the concept of extrema in calculus. By setting the derivative of the daughter population with respect to time to zero, a symbolic formula involving the decay constants is obtained, illustrating the balance between production and decay rates.

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