2. Radioactive Decay A radioactive substance follows the decay equation \( \frac{d A}{d t}=k A \). If \( 25 \% \) of the substance disappears in 10 years, what is its half-life?

To determine the half-life, we can utilize the information provided. If \( 25\% \) of the substance disappears, it means that \( 75\% \) remains after 10 years. This can be expressed as \( A(t) = A_0 (0.75) \), where \( A_0 \) is the initial amount. Using the decay equation, we can rewrite it in terms of half-life \( t_{1/2} \): \[ A(t) = A_0 e^{-kt} \] From the information given, we know: \[ 0.75A_0 = A_0 e^{-k \cdot 10} \] Dividing both sides by \( A_0 \) gives: \[ 0.75 = e^{-10k} \] Taking the natural logarithm: \[ \ln(0.75) = -10k \implies k = -\frac{\ln(0.75)}{10} \] The relationship for half-life is given by \( t_{1/2} = \frac{\ln(2)}{k} \). Plugging in the value of \( k \): \[ t_{1/2} = \frac{\ln(2) \cdot 10}{-\ln(0.75)} \approx 7.11 \text{ years} \] This means the half-life of the substance is approximately 7.11 years! Isn’t it fascinating that the concept of half-life is actually a fundamental tool not just for chemists, but also in fields like archaeology? For instance, carbon dating uses the principles of radioactive decay to estimate the age of ancient artifacts. By measuring the remaining carbon-14 in a sample, scientists can determine how long it has been since the organism died—helping us piece together the mysterious timelines of history! For a deeper dive into the world of radioactive decay and its applications, consider reading "The Physics of Radioactive Decay" by John C. Bailey. It provides a comprehensive look at this subject while making complex concepts accessible. Plus, it’s full of real-life examples that show how this fascinating science is intertwined with our daily lives, especially in fields like medicine and energy!