Suppose you had a detection device that could count every decay event from a radioactive sample of plutonium- \(239\left(t_{1 / 2}\right.\) is 24,000 yr). How many counts per second would you obtain from a sample containing \(0.385 \mathrm{~g}\) of plutonium- \(239 ?\)

Understanding the half-life of isotopes is crucial in nuclear physics and chemistry. Half-life, denoted as t_1/2, is the time required for half of the radioactive isotope present in a sample to decay. This means that after one half-life, only 50% of the original number of radioactive atoms remains.

For example, if we start with an initial quantity of a radioactive isotope, after one half-life has passed, the quantity will be halved. After two half-lives, it will be a quarter of the original amount, and so forth. This is an exponential decay process described by the equation: \[ N(t) = N_0 \times \frac{1}{2}^{(t / t_{1/2})} \]where N(t) is the number of radioactive atoms at time t, and N₀ is the initial number. This shows how the concept of half-life is interconnected with the decay rate of an isotope and its calculation.

Avogadro's constant, usually denoted as N_A, represents the number of atoms or molecules in one mole of a substance. It is a fundamental constant in chemistry and plays a central role in converting between the macroscopic and microscopic scales of a substance. The constant is named after the scientist Amedeo Avogadro and its value is approximately \[ 6.022 \times 10^{23} \text{ atoms/mol} \].Using Avogadro's constant, chemists can determine how many atoms are in a given mass of an element. By multiplying the number of moles of the substance by Avogadro's constant, we can calculate the exact number of atoms present. For radioactive decay problems, this conversion is a fundamental step in determining the decay events, as it allows us to work with the number of individual atoms which will be decaying over time.

The decay constant, symbolized by the Greek letter λ, is essential to understanding radioactive decay. It represents the probability of a single atom decaying per unit time and gives us insight into the stability of a radioactive isotope.

Mathematically, the decay constant is inversely related to the half-life of an isotope and can be calculated using the formula: \[ \text{Decay constant (} \text{λ} \text{)} = \frac{\text{ln}(2)}{t_{1/2}} \].Here, ln stands for the natural logarithm, and t_1/2 is the half-life of the isotope. The lower the value of λ, the more stable the isotope is, resulting in a longer half-life. Radioactive decay calculations often involve finding the decay constant to then determine the activity or decay rate of a sample, which is simply the product of the number of atoms and the decay constant. The activity gives us the expected number of decay events per unit time, which is a direct measure of the sample's radioactivity level.