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The half-life is a crucial concept in understanding radioactive decay. It represents the time it takes for half of the radioactive substance to decay and transform into another element or isotope.
For Polonium-210, the half-life is 138.4 years. This means that every 138.4 years, half of the Polonium-210 present will have decayed.
Using half-life, we can easily predict how much of a radioactive material remains after a certain period. For instance, if you start with 0.1 mg of Polonium-210, then after one half-life (138.4 years), only 0.05 mg will remain.
After two half-lives (276.8 years), only a quarter of the initial amount will remain, which is 0.025 mg.
This repetitive halving continues, showing an exponential decrease in the amount of the substance.

The exponential decay formula used for radioactive substances is a powerful mathematical tool. It helps us understand and predict how the amount of a substance decreases over time.
The formula is: \[A(t) = A_0 \times \bigg(\frac{1}{2}\bigg)^{\frac{t}{T_{1/2}}}\bigg\bigg)\text{where:}\biggr\]

• A(t) is the amount of substance left at time t.
• A_0 is the initial amount of substance.
• T_{1/2} is the half-life period.

This formula accounts for the continuous halving of radioactive material over each half-life.
In the given problem, the formula is: \[A(t)=0.1 \times \bigg(\frac{1}{2}\bigg)^{t/138.4}\bigg\bigg)\]The initial amount \(A_0\) is 0.1 mg and the half-life \(T_{1/2}\) is 138.4 years.By substituting different values of \(t\), we can see how much Polonium-210 remains:

• When \(t=138.4\) years, we have \[A(138.4) = 0.1 \times \bigg(\frac{1}{2}\bigg)^{1} = 0.05 \text{ mg}\]
• For \(t=276.8\) years, we get \[A(276.8) = 0.1 \times \bigg(\frac{1}{2}\bigg)^{2} = 0.025 \text{ mg}\]
• And at \(t=500\) years, it becomes \[A(500) ≈ 0.1 \times \bigg(\frac{1}{2}\bigg)^{3.613} ≈ 0.008 \text{ mg}\]

Interpreting the results from calculations using the exponential decay formula can give us clear insights into how radioactive materials behave over time.
For example, let's take the given calculations from the problem:

• After one half-life (138.4 years), the material decays to 0.05 mg from 0.1 mg. This confirms that exactly half of the Polonium-210 decays in this period.
• After two half-lives (276.8 years), the amount further reduces to 0.025 mg, showing the continued halving pattern as expected from the definition of half-life.
• Lastly, after a longer duration like 500 years, only about 0.008 mg remains. This significant reduction demonstrates the exponential nature of radioactive decay.

These interpretations are vital as they help in understanding real-world applications such as dating archaeological finds, medical treatments, and nuclear waste management.
By understanding such decay patterns, we can predict how a substance will diminish over time, which is essential for both scientific research and practical applications.