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The half-life of a radioactive substance is a key concept in understanding radioactive decay. It represents the time it takes for half of the initial amount of the substance to decay into another form. For example, if you start with 100 grams of a radioactive material with a half-life of 5 days, after 5 days, only 50 grams will remain. The remaining 50 grams will continue to decay, with half of it disappearing every subsequent half-life period.
To calculate how much of a substance remains after multiple half-life periods, use the formula:
\[ A = A_0 \times \left( \frac{1}{2} \right)^n \]
Where:

• \(A\) is the remaining amount of the substance.
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• \(A_0\) is the initial amount of the substance.
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• \(n\) is the number of half-lives that have passed.

Example: After 3 half-lives, the remaining substance is \(A = A_0 \times \left( \frac{1}{2} \right)^3\).

In the context of radioactive waste, reducing radioactivity makes disposal safer. If lab waste has a half-life of less than 65 days, it must be stored for 10 half-lives before it can be discarded as non-radioactive trash. This policy drastically lowers the radioactivity. After each half-life, the remaining radioactive material is halved.
For example, starting with an initial amount \(A_0\), after 1 half-life, only \(\frac{A_0}{2}\) remains. After 2 half-lives, it’s \(\frac{A_0}{4}\). After 10 half-lives, the waste is reduced to:
\[ \frac{A_0}{2^{10}} = \frac{A_0}{1024} \]
That’s less than 0.1% of the original radioactivity, meaning the material is significantly safer. This sharp decrease exemplifies how effectively time and proper storage reduce radioactive risks.

Radioactive decay follows an exponential pattern, meaning it decreases rapidly at first and then more slowly over time. Exponential decay can be described with the formula:
\[ A = A_0 \times e^{-\beta t} \]
Where:

• \(A\) is the amount remaining.
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• \(A_0\) is the initial amount.
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• \(e\) is the base of the natural logarithm (approximately 2.718).
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• \(\beta\) is the decay constant.
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• \(t\) is the time elapsed.

However, for half-life calculations, the simpler formula \(A = A_0 \times (\frac{1}{2})^n\) is usually used because it’s easier to apply. The principle behind both formulas is the same: the quantity decreases by a constant percentage over equal time periods. Over 10 half-lives, as calculated, the remaining amount is \(\frac{A_0}{1024}\), demonstrating how powerful exponential decay is at reducing radioactivity quickly at first and then more gradually.