A sample of T1-201 has an initial decay rate of \(5.88 \times 10^{4} / \mathrm{s}\). How long will it take for the decay rate to fall to \(287 / \mathrm{s} ?\) (T1-201 has a half-life of 3.042 days. \()\)

Half-life is a term that describes the time required for half of the radioactive atoms in a sample to decay. In other words, after one half-life, the activity, or decay rate, of the substance is reduced to half its initial value. This concept is crucial when dealing with radioactive materials, such as T1-201 in our exercise, because it helps us understand how quickly a substance loses its radioactivity.

The half-life of a substance is a fixed property; it remains constant regardless of the amount of the substance or its initial activity. For example, T1-201 has a half-life of 3.042 days. This means that after 3.042 days, any sample of T1-201 will have half the radioactive decay rate it originally had, no matter the size of the sample. By using half-life, scientists and engineers can make predictions about the behavior of radioactive substances over time, which is essential for various applications in medicine, archeology, and nuclear power.

The decay constant, represented by the Greek letter \(\lambda\), is a probability factor that measures the likelihood of decay for a radioactive atom per unit time. It is directly related to the half-life of the substance. The mathematical relationship is given by the formula \(T_{1/2} = \frac{\ln(2)}{\lambda}\), where \(\ln(2)\) represents the natural logarithm of 2. This formula shows that the decay constant can be derived if the half-life is known.

Calculating the Decay Constant

For T1-201 with a half-life of 3.042 days, the first step is to convert that time into seconds, because the decay rate in the exercise is given in per second. After converting the half-life into seconds, we apply the above formula to find the decay constant. This value is fundamental for further calculations and provides insight into how rapidly a radioactive isotope is decaying. The smaller the decay constant, the slower the decay process, and vice versa.

Exponential decay describes the process by which the quantity of a radioactive substance decreases over time at a rate proportional to its current value. This rate of decay can be expressed by the formula \(N(t) = N_0 e^{-\lambda t}\), where:\

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• \\(N_0\)\ is the initial decay rate,
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• \\(N(t)\)\ is the decay rate at time \(t\),
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• \\(e\)\ is the base of the natural logarithm, and
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• \\(\lambda\)\ is the decay constant.
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This model assumes that the decay process is continuous and does not change over time, which is a valid assumption for most radioactive substances. In practice, if we know the initial decay rate and the decay constant, we can predict the decay rate at any future time.

Application to Our Exercise

To find out how long it will take for T1-201's decay rate to fall to \(287 / \mathrm{s} \), we plug in the initial decay rate, the final decay rate, and the decay constant into the formula and solve for \(t\). The resulting value tells us the time in seconds that must pass for T1-201's decay activity to decrease to the desired level. The exponential nature of the decay means that the rate of decay drops off rapidly at first but slows down over time, reflecting a hallmark characteristic of radioactive decay processes.