Problem 62 - College Physics

Question

Radioactive fallout. One of the problems of in-air testing of nuclear weapons (or, even worse, the use of such weapons!) is the danger of radioactive fallout. One of the most problematic nuclides in such fallout is strontium-90 \(\left(^{90} \mathrm{Sr}\right),\) which breaks down by \(\beta^{-}\) decay with a half-life of 28 years. It is chemically similar to calcium and therefore can be incorporated into bones and teeth, where, due to its rather long half- life, it remains for years as an internal source of radiation. (a) What is the daughter nucleus of the \(^{90}\) Sr decay? (b) What percent of the original level of 90 \(\mathrm{Sr}\) is left after 56 years? (c) How long would you have to wait for the original level to be reduced to 6.25\(\%\) of its original value?

Step-by-Step Solution

Answer

(a) Yttrium-90; (b) 25%; (c) 112 years.

Step 1: Understand the Decay Process

Strontium-90 undergoes beta decay. During beta decay, a neutron in the nucleus is transformed into a proton, and a beta particle (electron) is emitted. This process leads to the formation of a new element with an atomic number increased by 1 but with the same mass number. In mathematical terms, if the parent nucleus is represented as \( ^{A}_{Z}X \), the daughter nucleus will be \( ^{A}_{Z+1}Y \). Hence, for \( ^{90}_{38}\text{Sr} \) undergoing beta decay, the daughter nucleus will be \( ^{90}_{39}\text{Y} \) (Yttrium-90).

Step 2: Apply Half-Life Concept for Percent Calculation

The half-life of Strontium-90 is 28 years. To find out what percent remains after 56 years:1. Determine the number of half-lives: \[ \frac{56}{28} = 2 \text{ half-lives} \]2. Calculate the remaining fraction after two half-lives: \[ \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]3. Convert the fraction to a percentage (multiply by 100): \[ \frac{1}{4} \times 100 = 25\% \]Thus, 25% of the original Strontium-90 remains after 56 years.

Step 3: Determine Time for Reduction to 6.25%

To find out how long it takes for Strontium-90 to decay to 6.25% of its original concentration:1. Recognize that 6.25% is equivalent to \( \frac{1}{16} \) of the original quantity.2. Calculate the number of half-lives to reach \( \frac{1}{16} \), which is \( \log_{2}(16) = 4 \) half-lives.3. Calculate the time for 4 half-lives, given a half-life of 28 years: \[ 4 \times 28 = 112 \text{ years} \]Therefore, it would take 112 years for the Strontium-90 level to be reduced to 6.25% of its original value.

Key Concepts

Strontium-90

Strontium-90 is a radioactive isotope produced from nuclear reactions, such as the detonation of nuclear weapons. It is particularly dangerous due to its chemical similarity to calcium, which allows it to easily enter biological systems. Once inside the body, it tends to accumulate in the bones and teeth, replacing calcium in these structures.

This accumulation poses a problem, as Strontium-90 will emit radiation internally for a long time, being a potential source of harm. Understanding its behavior helps us mitigate risks associated with nuclear fallout and its presence in the environment. Its decay and the resultant transformation to another element make it a subject of concern for scientists and health professionals alike.

Beta Decay

Beta decay is a radioactive decay process where a neutron in a nucleus transforms into a proton, releasing a beta particle, which is essentially an electron. This emission not only changes the neutron into a proton but also increases the atomic number by one, while the mass number stays the same.

For Strontium-90, the beta decay means turning into Yttrium-90, which is represented as follows:

• Parent nucleus: \( ^{90}_{38} \text{Sr} \)
• Daughter nucleus after beta decay: \( ^{90}_{39} \text{Y} \)

The beta decay process is crucial in the study of nuclear reactions and affects how we handle and store radioactive materials.

Half-Life

The half-life of a radioactive isotope indicates the time required for half of the given isotope to decay. Strontium-90 has a half-life of 28 years, meaning that every 28 years, half of a sample will have decayed into another element. This constant rate of decay makes it possible to predict the decrease in quantity over time.

Calculating changes in a substance's level due to half-life involves considering how many half-lives have passed. In two half-lives (56 years for Strontium-90), about 25% of the original is left. After four half-lives (112 years), only 6.25% is left. Understanding half-life is essential in managing and predicting the behavior over time of radioactive materials.

Radiation

Radiation refers to the energy emitted by radioactive substances as they decay. It comes in various forms, including alpha, beta, and gamma radiation. In the case of beta decay of Strontium-90, beta particles, or electrons, are emitted. These small, high-energy particles can penetrate the skin and may damage living tissues, leading to health risks.

Being internally exposed to beta radiation is particularly harmful since the emitted particles are very damaging at close range, especially if radioactive material has been absorbed by the bones or other critical organs over time. Understanding radiation types and their effects helps us to effectively protect ourselves from potential exposure in different environments.