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

Understanding half-life is essential when studying radioactive decay. The half-life of a radioactive isotope, such as Tl-201, is the time it takes for half of the sample to disintegrate or decay. This is a constant value for each isotope. In our problem, Tl-201 has a half-life of 3.042 days, a period after which only half of the original number of atoms remain undecayed.

During the next 3.042 days, half of the remaining Tl-201 atoms will decay, and so on. This process continues, each interval resulting in half of the preceding amount decaying, creating a pattern where the quantity of the substance diminishes exponentially over time.

The decay rate reflects the number of disintegrations occurring in a sample of the radioactive material per unit time. Initially, for Tl-201, the decay rate is given as 5.88 x 10^4 disintegrations per second. As time progresses, the decay rate diminishes. The calculation requires understanding that the decay rate is directly proportional to the number of undecayed atoms.

Therefore, as the decay continues, fewer atoms are available to disintegrate, causing the decay rate to lower. It’s not a linear decrease but rather an exponential one, reflecting the nature of radioactive decay processes. To find out when the decay rate drops to a certain level, such as 287 disintegrations per second in our case, we calculate the time that corresponds to the number of half-lives transpired.

Radioactive decay follows an exponential decay model, where the remaining quantity of a radioactive isotope decreases to half its initial amount after each half-life. Mathematically, this process is described by the formula \( R = R_0 \cdot \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \), where \( R \) is the final decay rate, \( R_0 \) is the initial decay rate, \( t \) is the time elapsed, and \( t_{1/2} \) is the half-life of the substance.

Because the decay rate reduction occurs exponentially, the plot on a graph shows a steep decline that gradually levels off over time. This principle is fundamental in nuclear chemistry and has applications in fields ranging from medicine to archaeology.

The use of logarithms is a powerful tool in chemistry, particularly when dealing with exponential relationships like radioactive decay. To find out how many half-lives have passed in a decay process, we can rewrite the exponential decay formula to include a logarithm. The new formula would be \( n = \frac{\log(R/R_0)}{\log(1/2)} \), where \( n \) is the number of half-lives.

Logarithms help us work backwards from the observed decay rate to determine the time that has passed since the initial measurement. In real-world applications, this might involve calculating the age of archaeological finds (carbon dating) or determining the dosage timing for radiopharmaceuticals. Logarithm applications simplify what would otherwise be challenging calculations.