Problem 72 - Physics for Scientists and Engineers with Modern Physics

Question

Strontium-90 is produced as a nuclear fission product of uranium in both reactors and atomic bombs. Look at its location in the Periodic Table to see what other elements it might be similar to chemically, and tell why you think it might be dangerous to ingest. It has too many neutrons, and it decays with a half-life of about 29 yr. How long will we have to wait for the amount of \({ }_{38}^{90} \mathrm{Sr}\) on the Earth's surface to reach \(1 \%\) of its current level, assuming no new material is scattered about? Write down the decay reaction, including the daughter nucleus. The daughter is radioactive: write down its decay.

Step-by-Step Solution

Answer

It takes about 193 years for strontium-90 to decay to 1% of its original amount.

Step 1: Understanding Strontium-90 and Its Reactivity

Strontium-90 (symbol: \(^{90}_{38} \mathrm{Sr}\)) is located in Group 2 of the Periodic Table, which contains the alkaline earth metals, making it chemically similar to elements like calcium. Its similarity to calcium means it can accumulate in bones, making it particularly dangerous as it emits harmful radiation. Its radioactivity and long half-life of 29 years make it highly hazardous as a contaminant.

Step 2: Half-Life Decay Formula

To determine how long until \(^{90}_{38}\mathrm{Sr}\) decays to \(1\%\) of its initial quantity, we use the formula: \[ N(t) = N_0 \left( \frac{1}{2} \right)^{t/T} \]where \(N(t)\) is the remaining quantity, \(N_0\) is the initial quantity, \(t\) is the time elapsed, and \(T\) is the half-life.

Step 3: Solving for Time to Reach 1%

We set \(N(t) = 0.01N_0\) and \(T = 29\) years:\[ 0.01 = \left( \frac{1}{2} \right)^{\frac{t}{29}} \] Taking the natural logarithm on both sides:\[ \ln(0.01) = \frac{t}{29} \ln\left(\frac{1}{2}\right) \]Solving for \(t\):\[ t = 29 \cdot \frac{\ln(0.01)}{\ln(0.5)} \approx 192.7 \text{ years} \]

Step 4: Writing the Decay Reaction for Strontium-90

Strontium-90 decays via beta decay:\[ _{38}^{90}\mathrm{Sr} \rightarrow _{39}^{90}\mathrm{Y} + \beta^- \] This shows strontium-90 transforming into yttrium-90, a radioactive daughter nucleus.

Step 5: Decay of the Daughter Nuclide

Yttrium-90 (\(_{39}^{90}\mathrm{Y}\)) also undergoes beta decay:\[ _{39}^{90}\mathrm{Y} \rightarrow _{40}^{90}\mathrm{Zr} + \beta^- \] This decay transforms yttrium-90 into zirconium-90, which is stable.

Key Concepts

Strontium-90

Strontium-90 is a radioactive isotope of strontium, denoted by \(_{38}^{90}\text{Sr}\). Found in Group 2 of the Periodic Table, it shares chemical properties with other alkaline earth metals, like calcium.
This similarity is significant because it allows strontium-90 to replace calcium in biological processes. When ingested, it can accumulate in bones and teeth, posing serious health risks.
**Why is it hazardous?**

• Strontium-90 emits beta radiation, which can damage bone marrow and lead to diseases such as leukemia and bone cancer.
• Its long half-life of 29 years means it persists in the environment and biological systems for extended periods.

Understanding the chemical behavior of strontium-90 helps us grasp why nuclear contamination is a long-term environmental and health concern.

Half-life

The concept of half-life is crucial in understanding radioactive decay processes. It describes the time it takes for half of a radioactive substance to decay. For strontium-90, the half-life is approximately 29 years.
**What does this mean?**

• After 29 years, only half of the initial amount of strontium-90 will remain.
• This property allows us to predict how long it takes a given amount of strontium-90 to reduce to a certain fraction of its original mass, such as the calculation to reach 1% which is approximately 192.7 years.

When dealing with radioactive materials, being able to calculate decay periods helps in planning waste management and environmental protection. **Calculating with the half-life formula:**The formula \( N(t) = N_0 \left( \frac{1}{2} \right)^{t/T} \) outlines this decay process, where \(T\) is the half-life. Using calculus, we can determine time intervals for specific decays, as demonstrated in the case with strontium-90.

Beta decay

Beta decay is a type of radioactive decay in which a beta particle (an electron or a positron) is emitted from an atomic nucleus. This process transforms one element into another, a key mechanism seen in strontium-90 decay. **How does it work?**

• In the case of strontium-90, it undergoes beta decay by emitting an electron, transforming into yttrium-90.
• The reaction is: \(_{38}^{90}\text{Sr} \rightarrow _{39}^{90}\text{Y} + \beta^-\)
• Yttrium-90 further decays via beta decay into zirconium-90, which is stable.

Beta decay not only changes the identity of the nucleus but is also a source of ionizing radiation, which can lead to biological tissue damage if exposure occurs. Understanding beta decay is essential for managing the risks associated with radioactive materials, ensuring safety, and comprehending the transformation chain from unstable isotopes to stable ones.