A sample of F-18 has an initial decay rate of 1.5⨉10⁵/s. How long will it take for the decay rate to fall to 2.5⨉10³/s? (F-18 has a half-life of 1.83 hours.)

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Identify the initial decay rate \( R_0 = 1.5 \times 10^5 \text{ s}^{-1} \) and the final decay rate \( R = 2.5 \times 10^3 \text{ s}^{-1} \).

Use the half-life formula for radioactive decay: \( R = R_0 \times \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \), where \( t_{1/2} = 1.83 \text{ hours} \).

Rearrange the formula to solve for time \( t \): \( t = t_{1/2} \times \frac{\log\left(\frac{R}{R_0}\right)}{\log\left(\frac{1}{2}\right)} \).

Substitute the known values into the equation: \( t = 1.83 \times \frac{\log\left(\frac{2.5 \times 10^3}{1.5 \times 10^5}\right)}{\log\left(\frac{1}{2}\right)} \).

Calculate the value of \( t \) to find the time it takes for the decay rate to fall to the desired level.

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Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose 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.

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Rate of Radioactive Decay

Half-Life

Half-life is the time required for half of the radioactive nuclei in a sample to decay. For F-18, the half-life is 1.83 hours, meaning that after this time, half of the original amount of F-18 will have decayed. This concept is essential for determining the time it takes for the decay rate to decrease from an initial value to a specified lower value.

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Zero-Order Half-life

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^(-λt), where N(t) is the quantity at time t, N0 is the initial quantity, λ 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 specific value.

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