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The half-life of a radioactive substance is the time it takes for half of a given quantity to decay. It tells us how stable or unstable a nuclide is, and is calculated using the formula:\[T_{1/2} = \frac{\ln(2)}{\lambda}\]where \(T_{1/2}\) is the half-life and \(\lambda\) is the decay constant. To perform a half-life calculation, you first need the decay constant. Once \(\lambda\) is determined, plug it into the formula above. In our example, the decay constant was found to be approximately \(1.28 \times 10^{-11} \text{ s}^{-1}\). Therefore, the half-life was calculated as approximately \(5.42 \times 10^{10} \text{ s}\). This formula works because the decay process follows a first-order kinetics, meaning the rate of decay is directly proportional to the number of nuclei present. Understanding how to calculate half-life is crucial for predicting how long a radioactive substance will remain active.

The decay constant, denoted as \(\lambda\), is a fundamental parameter in the study of radioactive decay. It signifies the probability per unit time that a nucleus will decay. Hence, it is essentially a measure of how quickly a substance undergoes radioactive decay. The decay constant has the units \(\text{s}^{-1}\), which indicates decay events are measured per second. To find this constant, we use the following relation:\[\lambda = \frac{-\Delta N / \Delta t}{N}\]In this equation:

• \(\Delta N / \Delta t\) is the decay rate, given in Becquerels (Bq), representing the number of decays per second.
• \(N\) is the number of undecayed nuclei.

In the example problem, with a decay rate of \(6.24 \times 10^{11}\) Bq and \(N\) being about \(4.87 \times 10^{22}\), we calculate the decay constant as \(1.28 \times 10^{-11} \text{s}^{-1}\). This step is essential for half-life calculation.

Avogadro's number is a key concept when calculating the number of particles in a given amount of substance. It states that one mole of any substance contains exactly \(6.022 \times 10^{23}\) entities, be it atoms, molecules, or nuclei. Utilizing Avogadro's number is fundamental in converting between the amount of substance (in moles) and the number of actual entities (in our case, radioactive nuclei). In the exercise, we began by calculating the number of moles from the given mass using the molar mass. Then, we multiplied the number of moles by Avogadro's number to find the total number of nuclei.For example:

• Given mass: 8.50 g.
• Molar mass: 105 g/mol.
• Moles of substance = \(\frac{8.50}{105} = 0.08095\) mol.
• Total nuclei = \(0.08095 \times 6.022 \times 10^{23} = 4.87 \times 10^{22}\).

This calculation is essential not only for finding \(N\) in half-life equations but also for understanding the scale of atomic and molecular processes.

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It allows for the conversion between the mass of a sample and the number of moles, which is crucial in chemical calculations, including radioactive decay studies.In the context of our exercise, the nuclide in question has a mass number of 105. This indicates a molar mass of approximately 105 g/mol. The molar mass links directly to Avogadro's number when calculating how many particles (atoms, molecules, or nuclei) are in a sample of a material.Consider:

• Given a sample mass of 8.50 g.
• Molar mass ≈ 105 g/mol.
• The number of moles = \(\frac{8.50}{105} = 0.08095\) mol.

By understanding molar mass, you can determine how much of a substance you have in terms of a quantity that chemists use regularly (moles), facilitating further calculations of things like half-life and decay rates.