A sample of Tl-201 has an initial decay rate of 5.88⨉10⁴/s. How long will it take for the decay rate to fall to 287/s? (Tl-201 has a half-life of 3.042 days.)

Verified step by step guidance

1. Identify the initial decay rate \( R_0 = 5.88 \times 10^4 \text{ s}^{-1} \) and the final decay rate \( R = 287 \text{ s}^{-1} \).
2. Use the decay formula \( R = R_0 \times \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \), where \( t_{1/2} \) is the half-life.
3. Substitute the known values into the decay formula: \( 287 = 5.88 \times 10^4 \times \left( \frac{1}{2} \right)^{\frac{t}{3.042}} \).
4. Solve for \( t \) by taking the natural logarithm of both sides to isolate \( t \).
5. Calculate \( t \) using the equation \( t = 3.042 \times \frac{\ln(287/5.88 \times 10^4)}{\ln(1/2)} \).

Key Concepts

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This decay occurs at a characteristic rate for each isotope, often described by its decay constant, which is related to the half-life. Understanding this concept is crucial for calculating how the decay rate changes over time.

Half-Life

The half-life of a radioactive isotope is the time required for half of the radioactive atoms in a sample to decay. For Tl-201, the half-life is 3.042 days, meaning that after this period, half of the original sample will remain. This concept is essential for determining the time it takes for the decay rate to decrease to a specific value.

Exponential Decay Formula

The exponential decay formula describes how the quantity of a radioactive substance decreases over time. It is expressed as N(t) = N0 * e^(-kt), where N(t) is the quantity at time t, N0 is the initial quantity, k is the decay constant, and e is the base of the natural logarithm. This formula is fundamental for calculating the time required for the decay rate to reach a certain level.