(III) At t = 0, a pure sample of radioactive nuclei contains...

(III) At $t =$ 0, a pure sample of radioactive nuclei contains $N_0$ nuclei whose decay constant is $\lambda$. Determine a formula for the number of daughter nuclei, $N \mathrm{_D}$ as a function of time; assume the daughter is stable and that $N \mathrm{_D} =$ 0 at $t =$ 0.

Key Concepts

Differential Equations in Decay Processes

The rate of change of the number of parent nuclei can be described using differential equations, which link the decay process to the accumulated number of daughter nuclei. These equations provide a framework for deriving relationships such as the exponential decay law and the corresponding formation of daughter nuclei over time.

Accumulation of Daughter Nuclei

In a decay process where the daughter nucleus is stable, the accumulation of daughter nuclei can be determined by considering that every decay event of a parent nucleus creates one daughter. Thus, the number of daughter nuclei at time t is given by the total initial parent nuclei minus the number still remaining, often expressed as N_D(t) = N?(1 - e^(??t)).

Radioactive Decay

Radioactive decay is the process by which an unstable nucleus loses energy by emitting radiation. It is inherently probabilistic and is typically described by a statistical law in which the number of undecayed nuclei decreases exponentially over time.

Decay Constant (?)

The decay constant is a key parameter in radioactive decay, representing the probability per unit time that a given nucleus will decay. It quantifies the rate of decay and is directly used in the exponential decay formula to determine how quickly the sample's composition changes.

Exponential Decay Law

The exponential decay law mathematically describes the reduction of the parent nucleus population over time. It shows that the number of remaining parent nuclei decreases exponentially, following the relationship N(t) = N?e^(??t), where N? is the initial number of nuclei.

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