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In radioactive decay, the decay constant \( \lambda \) is a crucial value that indicates how quickly an isotope decays. It gives us the probability per unit time that a single atom will decay. This constant is linked with the activity of a sample, which is the number of decay events per second. To find the decay constant, you can use the formula:\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]where \( t_{1/2} \) is the half-life of the substance. This formula arises because an isotope reduces to half its original amount in one half-life, corresponding to the natural logarithm of 2 (approximately 0.693). By calculating \( \lambda \), as we did for sulfur-35 with a half-life of 88 days, we convert to seconds and use \( \lambda = \frac{\ln(2)}{7603200} \approx 9.118 \times 10^{-8} \text{ s}^{-1} \). Understanding the decay constant is essential in predicting how much of the isotope will remain after a given time.

The half-life of a radioactive isotope is the time taken for half its atoms to decay. This is a specific characteristic of each isotope, providing insight into its stability and rate of decay over time. For example, the half-life of sulfur-35 is 88 days, meaning every 88 days, half of any sulfur-35 sample would have decayed into a different element. Half-life helps in calculating how long an isotope will remain active or hazardous, and it can be illustrated in the formula connecting it with the decay constant as \( \lambda = \frac{\ln(2)}{t_{1/2}} \). Recognizing the half-life of an element like sulfur-35 is crucial in real-world applications, such as dating archaeological finds or managing radioactive waste.

Sulfur-35, denoted as \(^{35}\text{S}\), is a radioactive isotope of sulfur. Unlike the more stable forms of sulfur, \(^{35}\text{S}\) undergoes radioactive decay, emitting radiation as it transforms into chlorine-35. Used in biological and medical research, \(^{35}\text{S}\) has practical applications in tracing processes like protein synthesis. The decay of \(^{35}\text{S}\) is measurable by its half-life—88 days—which makes it suitable for experiments needing precise timing. In the context of the given exercise, understanding \(^{35}\text{S}\)'s properties enables better calculations related to the sample's disintegration rate and mass. Knowing its decay behavior assists researchers in optimizing its use while ensuring safety protocols are followed.

The concept of moles and molar mass bridges the micro world of atoms with the macro world of observable matter. A mole represents \(6.022 \times 10^{23}\) atoms, known as Avogadro's number. Molar mass is the mass of one mole of an element or compound, expressed in grams per mole (g/mol). For sulfur-35, the molar mass is simply its atomic mass—35 g/mol. To find the mass of a radioactive material in a sample, one must first calculate the number of moles from the number of atoms, using the formula:\[ \text{moles} = \frac{N}{N_A} \]where \(N\) is the number of atoms, and \(N_A\) is Avogadro's number. In the exercise, converting \(^{35}\text{S}\) atoms to moles and then to mass allowed us to determine that the mass of sulfur-35 in the sodium thiosulfate sample was approximately \(4.90 \times 10^{-4} \text{ grams}\). This step underscores the importance of relating atomic-level measurements to tangible quantities in chemical calculations.