91Ó°ÊÓ

Exponential decay is a mathematical process describing how quantities decrease over time. In our context, it refers to the decrease in the amount of a radioactive substance. When a substance decays exponentially, the amount remaining at any time can be expressed using the formula:

• \( N(t) = N_0 e^{-kt} \)

Here:

• \( N(t) \) is the amount of substance left after time \( t \).
• \( N_0 \) is the initial quantity of the substance.
• \( e \) is the base of natural logarithms, approximately equal to 2.71828.
• \( k \) is the decay constant.

This formula reflects how the decay rate depends on the specific characteristics of the substance. By using this model, you can predict how much of the substance remains at various times, which is particularly useful in fields like physics and environmental science.

The half-life of a radioactive substance is the time it takes for half of the substance to decay. This is a critical concept for understanding how long a substance will remain active. When viewing exponential decay, the half-life \( T_{1/2} \) can be found using:

• \( T_{1/2} = \frac{\ln(2)}{k} \)

Where \( \ln(2) \) is the natural logarithm of 2, approximately 0.693147. The half-life tells us not only about the stability but also the longevity of a substance's activity. By calculating the half-life, we can determine when a substance will have diminished to a point where it is less of a concern or no longer effective. In our example, we determined the half-life to be approximately 24.07 days using the decay constant we calculated.

The decay constant \( k \) is a crucial factor in the exponential decay equation. It represents the probability of decay of a single atom per unit time. The decay constant directly impacts how fast or slow a radioactive substance will decay. To find \( k \), rearrange the exponential decay formula:

• \( 0.75 = e^{-10k} \)

This step requires isolating \( k \) by solving the equation using logarithms:

• \( k = -\frac{\ln(0.75)}{10} \)

In the provided example, the decay constant was found to be approximately 0.0287682. This tells us how quickly the substance decays under the conditions given, which is vital for assessing the potential risks or uses of the material. Understanding \( k \) allows for accurate predictions of how substances behave over time.