Problem 28 - Fundamentals of Physics

Question

The plutonium isotope \({ }^{239} \mathrm{Pu}\) is produced as a by-product in nuclear reactors and hence is accumulating in our environment. It is radioactive, decaying with a half-life of \(2.41 \times 10^{4} \mathrm{y}\). (a) How many nuclei of Pu constitute a chemically lethal dose of \(2.00 \mathrm{mg} ?\) (b) What is the decay rate of this amount?

Step-by-Step Solution

Answer

(a) Approximately \(5.04 \times 10^{18}\) nuclei. (b) The decay rate is \(1.45 \times 10^{14}\) decays/year.

Step 1: Determine the Number of Moles

The first step is to find the number of moles in 2.00 mg of Pu-239. The molar mass of Pu-239 is approximately 239 g/mol.First, convert milligrams to grams:\[ 2.00 \text{ mg} = 0.002 \text{ g} \]Then, calculate the number of moles using the formula:\[ \text{moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} = \frac{0.002}{239} \approx 8.37 \times 10^{-6} \text{ mol} \]

Step 2: Calculate the Number of Nuclei

To find the number of nuclei, use Avogadro's number, which is approximately \(6.022 \times 10^{23}\) nuclei/mol. Multiply the number of moles by Avogadro's number:\[ \text{{Number of Nuclei}} = 8.37 \times 10^{-6} \text{ mol} \times 6.022 \times 10^{23} \frac{\text{nuclei}}{\text{mol}} \approx 5.04 \times 10^{18} \text{ nuclei} \]

Step 3: Find Decay Constant

The decay constant \(\lambda\) can be found using the half-life \(T_{1/2}\) given by the formula:\[ \lambda = \frac{\ln(2)}{T_{1/2}} = \frac{0.693}{2.41 \times 10^4 \text{ y}} \approx 2.88 \times 10^{-5} \text{ y}^{-1} \]

Step 4: Calculate the Decay Rate

The decay rate \(R\) is calculated by multiplying the decay constant \(\lambda\) with the number of nuclei \(N\):\[ R = \lambda N = 2.88 \times 10^{-5} \text{ y}^{-1} \times 5.04 \times 10^{18} \text{ nuclei} \approx 1.45 \times 10^{14} \text{ decays/year} \]

Key Concepts

half-life

The concept of half-life is crucial in understanding radioactive decay. Half-life is the time it takes for half of the radioactive nuclei in a sample to decay. This means if you start with a certain number of radioactive atoms, say 100, after one half-life, only 50 of those atoms would remain undecayed. The rest would have transformed into different elements or isotopes. Plutonium-239 (\(^{239}Pu\)), for example, has a half-life of approximately 24,100 years. This long half-life means that Pu-239 remains radioactive and hazardous for thousands of years. Understanding half-life helps scientists predict how long a radioactive substance will remain dangerous.

nuclear reactors

Nuclear reactors are facilities where controlled nuclear reactions are carried out to produce energy. In these reactors, nuclear fuel such as Uranium-235 or Plutonium-239 undergoes fission, a process in which the atomic nucleus splits into smaller parts, releasing a significant amount of energy.
The environment within a reactor is designed to maintain a chain reaction at a steady rate. This allows them to produce a continuous supply of energy, which is converted into electricity. However, a by-product of this process is the creation of radioactive waste materials like Pu-239, which are harmful if not properly managed.
Nuclear reactors thus represent a double-edged sword: they provide large amounts of energy with relatively low air pollution, but their waste poses challenges due to radioactivity and long half-lives.

Avogadro's number

Avogadro's number is a constant, approximately equal to \(6.022 \times 10^{23}\), that represents the number of atoms, molecules, or particles in one mole of a substance. This figure is pivotal in chemistry for converting between the macroscopic and microscopic scale.
When dealing with chemical amounts, Avogadro's number allows the calculation of the number of atoms or molecules from a given mass. For the Pu-239 isotope, knowing the mole's mass and applying Avogadro's number, we can calculate the total nuclei in a sample. In the provided exercise, by multiplying 8.37 \(\times 10^{-6}\) moles of Pu-239 by Avogadro's number, we determine there are approximately 5.04 \(\times 10^{18}\) nuclei present in a 2 mg sample.

decay constant

The decay constant, symbolized as \(\lambda\), is a key parameter in the study of radioactive decay. It provides a measure of the probability per unit time that a single radioactive nucleus will decay. \(\lambda\) is intimately linked to the half-life (\(T_{1/2}\)) of a substance through the equation: \(\lambda = \frac{\ln(2)}{T_{1/2}}\). For Pu-239, with a half-life of 24,100 years, the decay constant is about 2.88 \(\times 10^{-5}\) per year.
This small value reflects the slow decay rate of Pu-239, meaning it remains in the environment for extended periods. Understanding the decay constant is crucial since, along with the number of radioactive nuclei, it helps determine the sample's decay rate, measured in decays per second or year, crucial for assessing radioactivity levels and potential risks.