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The decay constant, often denoted by \( \lambda \), is a crucial parameter in understanding radioactive decay. It represents the probability per unit time of a single nuclide decaying, and it is directly related to the stability of the radioactive substance.
For any radioactive element, the decay constant is linked to its half-life through the equation:

• \( \lambda = \frac{\ln(2)}{t_{1/2}} \)

where \( t_{1/2} \) is the half-life, which is the time it takes for half of the radioactive atoms to decay.
This constant helps us predict how quickly a radioactive substance will lose its activity. A larger decay constant indicates a faster rate of decay. In the context of the problem, the decay constant for \( {}^{24}Na \) can be determined using its 15-hour half-life, highlighting just how quickly this isotope undergoes radioactive decay.

Half-life is a fundamental concept in radioactivity and represents the amount of time required for half of a radioactive substance to decay. It is a measure of how quickly or slowly a radioactive isotope transforms into a stable element.

• For \( {}^{24}Na \), the half-life is given as 15 hours.
• This means every 15 hours, the amount of \( {}^{24}Na \) remaining is halved.

The half-life is crucial in calculations involving radioactive decay because it provides a time scale over which the decay process occurs. Knowing the half-life, one can determine how much of the original isotope remains active over time, allowing scientists and technicians to make precise calculations on a substance's behavior through time.

Activity concentration refers to the radioactivity of a sample per unit volume, measured in units like millicuries per milliliter (mCi/mL).
The initial activity concentration can be altered by processes such as dilution, important in understanding the net activity in a new solution.

• The problem starts with an initial activity concentration of 2.5 mCi/mL before dilution.
• After dilution, the activity concentration decreases and must be recalculated based on the final volume.

This concept is key in applications and situations where precise dosage or activity levels are essential, like in medical or laboratory settings. In our scenario, after diluting the stock solution, the activity concentration reduces, allowing us to calculate the adjusted activity of the sample after a specific decay time.