Co Radioactive element A A can decay to either element B B...

co Radioactive element $A A$ can decay to either element $B B$ or element $C C$. The decay depends on chance, but the ratio of the resulting number of $B B$ atoms to the resulting number of $C C$ atoms is always $2 / 1$. The decay has a half-life of $8.00$ days. We start with a sample of pure $A A$. How long must we wait until the number of $C C$ atoms is $1.50$ times the number of $A A$ atoms?

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Key Concepts

Radioactive Decay

This is the process by which unstable atomic nuclei lose energy by emitting radiation. It is characterized by an exponential decrease in the number of undecayed nuclei over time according to a specific decay constant. Understanding radioactive decay is essential for analyzing how quantities of a radioactive substance change with time.

Half-life

The half-life of a radioactive substance is the time required for half of the initial quantity of the substance to decay. It is a key parameter in exponential decay equations, relating directly to the decay constant. Knowing the half-life allows one to predict the remaining amount of a substance after a given period.

Exponential Decay Law

This law mathematically describes the decay of a substance as an exponential function of time. It is expressed in the form N(t) = N?e^(–?t), where N? is the initial quantity, ? is the decay constant, and t is time. The law is foundational in calculating the remaining number of undecayed nuclei at any time point.

Branching Ratios

Branching ratios represent the probabilities associated with a decaying nucleus following different possible decay paths. They describe how the total decay process is divided among various decay products, allowing one to calculate the relative amounts of each product formed from the original substance.

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