The radioactive isotope 198 Au has a half-life of 64.8 h. A sample containing this isotope has a

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The radioactive isotope 198 Au has a half-life of 64.8 h. A...

The radioactive isotope 198 $\mathrm{Au}$ has a half-life of 64.8 $\mathrm{h.} \mathrm{A}$ sample containing this isotope has an initial activity $(t=0)$ of 40.0$\mu \mathrm{Ci}$ . Calculate the number of nuclei that decay in the time interval between $t_{1}=10.0 \mathrm{h}$ and $t_{2}=12.0 \mathrm{h} .$

Key Concepts

Integration of Decay Rate Over Time

To determine the total number of decays occurring between two time points, one integrates the decay rate (or activity) over that interval. This approach accounts for the continuously decreasing number of radioactive nuclei and is used to compute decays over a specified time period.

Activity

Activity is a measure of the number of decays per unit time in a radioactive substance, directly proportional to the number of undecayed nuclei and the decay constant. It connects the physical decay of nuclei to measurable quantities like decay rates in practical scenarios.

Exponential Decay Law

The exponential decay law describes how the number of radioactive nuclei decreases over time, following the formula N(t) = N0 • e^(–?t). This law underpins all calculations involving the decay process, including the time-dependent activity of a sample.

Decay Constant

The decay constant (?) represents the probability per unit time that a given nucleus will decay. It is mathematically related to the half-life by the formula ? = ln(2) / (half-life) and is essential in formulating the exponential decay law.

Half-life

The half-life is the period of time required for half of a sample of radioactive nuclei to decay. It is a fundamental property of a radioactive isotope and is used to determine how quickly or slowly the isotope undergoes decay.

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