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The half-life of a radioactive substance is the time it takes for half of the sample to decay or transform into another substance. In this exercise, we know the half-life is 3 hours. However, when working with calculations, it's often necessary to convert time units to make it compatible with other variables, like the rate constant.
A simple conversion from hours to seconds involves multiplying by 3600, since there are 3600 seconds in an hour. Thus, a 3-hour half-life becomes 3.00 \( \times \) 3600 seconds.
This conversion is crucial for accurately using the half-life in further calculations such as finding the rate constant.

The rate constant \( k \) is a crucial term in radioactive decay, indicating the probability per second of decay of a molecule. It's derived from the half-life and is defined using the formula: \( t = \frac{0.693}{k} \).
Solving for \( k \) provides: \( k = \frac{0.693}{t} \), which is used to express how quickly a radioactive material loses its radioactive property.
Once we have the half-life in seconds, like our 3-hour example, we substitute it into the formula to get \( k \approx 6.4 \times 10^{-5} \; \mathrm{s}^{-1} \). This constant is fundamental in predicting how the quantity of the radionuclide decreases over time.

The mole concept is an essential bridge in chemistry that relates the mass of a substance to the quantity of individual atoms or molecules. One mole is defined as \( 6.022 \times 10^{23} \) entities, which could be atoms, molecules, ions, etc., known as Avogadro's number.
In this exercise, the decay rate is calculated for 1 mole of the nuclide. By multiplying the rate constant by Avogadro's number, we determine the total decay rate for the entire mole of radioactive atoms.
This illustrates how macroscopic properties, like mole quantity, directly connect to microscopic behavior, such as the number of atoms decaying per second.

Truly fundamental to chemistry, Avogadro's number \( (6.022 \times 10^{23}) \) specifies how many atoms or molecules are in one mole of any substance. This number provides a vital scale for comparing material at macroscopic and molecular levels.
With Avogadro's number, we transformed 1 mole of a radioactive nuclide into \( 6.022 \times 10^{23} \) atoms. This conversion enables physicists and chemists to utilize countable units (atoms) in their calculations.
For the decay rate, this number allows the calculation to scale the effects of each individual atom's decay to the entire sample's decay rate, which was found to be about \( 3.85 \times 10^{19} \) decays per second for our nuclide.