. co A radioactive sample intended for irradiation of a...

$.$ co A radioactive sample intended for irradiation of a hospital patient is prepared at a nearby laboratory. The sample has a halflife of $83.61$ h. What should its initial activity be if its activity is to be $7.4 \times 10^{8} \mathrm{~Bq}$ when it is used to irradiate the patient $24 \mathrm{~h}$ later?

Key Concepts

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This process follows a statistical, exponential decay law where the rate at which nuclei decay is proportional to the number of undecayed nuclei present. This concept is fundamental in understanding how the activity of a radioactive sample decreases over time.

Half-life

The half-life of a radioactive substance is the time required for half of the atoms in a sample to decay. It is a characteristic property of each radioactive isotope and is used to predict the decay behavior over time. Knowledge of the half-life allows one to calculate the remaining activity of a sample after any given period.

Exponential Decay Law

The exponential decay law quantitatively describes the decrease in the number of radioactive atoms over time. It is expressed mathematically by the formula A(t) = A?e^(??t), where A(t) is the activity at time t, A? is the initial activity, ? is the decay constant, and t is the elapsed time. This law is essential for calculating initial or remaining activity based on the time elapsed and the half-life.

Decay Constant

The decay constant is a parameter that quantifies the probability per unit time that a given nucleus will decay. It is directly related to the half-life by the relation ? = ln(2)/T?/?. Understanding the decay constant is crucial for applying the exponential decay formula to determine how much of the radioactive substance remains after a specific period.

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