Question

One of the hazards of nuclear explosion is the generation of \(\mathrm{Sr}^{90}\) and its subsequent incorporation in bones. This nuclide has a half life of \(28.1\) years. Suppose one microgram was absorbed by a new born child, how much \(\mathrm{Sr}^{90}\) will remain in his bones after 20 years? (a) \(61 \mu \mathrm{g}\) (b) \(61 \mathrm{~g}\) (c) \(0.61 \mu \mathrm{g}\) (d) none

Short answer

After 20 years, approximately 0.61 micrograms of \(\mathrm{Sr}^{90}\) will remain, corresponding to option (c).

Step-by-step solution

Understand the Half-Life Formula

The half-life of a substance is the time required for the quantity to reduce to half its initial amount. The formula to calculate the remaining quantity after a given period is: \[ N = N_0 \left( \frac{1}{2} \right)^\frac{t}{T_{1/2}} \]where:- \(N\) is the remaining quantity,- \(N_0\) is the initial quantity,- \(t\) is the time elapsed,- \(T_{1/2}\) is the half-life of the substance.

Identify the Known Values

For this problem, the initial amount \(N_0\) is 1 microgram, the time \(t\) is 20 years, and the half-life \(T_{1/2}\) is 28.1 years. We need to find the remaining amount \(N\) after 20 years.

Substitute into Half-Life Formula

Substitute the known values into the half-life formula:\[ N = 1 \text{ } \mu\text{g} \times \left( \frac{1}{2} \right)^\frac{20}{28.1} \]

Calculate the Exponent

Calculate the exponent fraction:\[ \frac{20}{28.1} \approx 0.711 \]

Simplify the Expression

Use the exponent calculated in the previous step to simplify the expression:\[ N = 1 \times \left( \frac{1}{2} \right)^{0.711} \approx 1 \times 0.615 \]

Calculate the Remaining Amount

Perform the calculation:\[ N \approx 0.615 \mu\text{g} \]Thus, approximately 0.61 micrograms of \(\mathrm{Sr}^{90}\) will remain.

Select the Correct Option

Review the options given in the problem. The calculated amount is approximately 0.61 micrograms, which matches option (c).

Key concepts

Radioactive Decay

Radioactive decay is a fundamental concept in nuclear chemistry, describing how unstable atomic nuclei lose energy. This process occurs naturally. Subatomic particles or electromagnetic waves are emitted. As a result, the nucleus becomes more stable. The emitted particles can be alpha particles, beta particles, or gamma rays.
Radioactive decay follows a random process. However, it happens at a predictable average rate over time. This predictability allows scientists to use it in various applications:

• Medical Treatments: In cancer therapies, radioactive decay can target and destroy cancer cells.
• Dating Artifacts: Carbon dating uses the decay rate of carbon-14 to estimate the age of archaeological specimens.
• Energy Generation: In nuclear reactors, the decay process releases energy that we use for electricity.

To understand decay at a deeper level, one should know that each radioactive substance has a characteristic half-life, which ensures decay processes are predictable over time.

Half-Life Calculation

The half-life of a radioactive substance is the time it takes for half of its atoms to decay. This calculation is crucial for determining the remaining quantity of a substance after a certain time. The formula to calculate the remaining amount is:\[N = N_0 \left( \frac{1}{2} \right)^\frac{t}{T_{1/2}}\]- Here, \(N\) represents the remaining quantity.
- \(N_0\) is the initial quantity.
- \(t\) is the time elapsed.
- \(T_{1/2}\) is the given half-life.
Knowing the half-life helps in estimating how long a substance will remain active or hazardous.

• If you have a radioactive isotope with a half-life of 10 years, after 10 years, only 50% of it will remain.
• The following 10 years will see that amount reduced by half again, leaving 25%.

This method of calculation applies universally. It’s applicable to any radioactive material, including industrial and medical isotopes.

Strontium-90

Strontium-90 (\(\mathrm{Sr}^{90}\)) is a notable radioactive isotope within nuclear chemistry. It's commonly associated with nuclear fission and fallout from nuclear explosions or accidents. Strontium-90 poses significant health risks due to its ability to mimic calcium, causing it to incorporate into bones. Once in the bone, it continuously emits radiation.

• Health Effects: The inclusion of \(\mathrm{Sr}^{90}\) in bones can lead to bone cancer or leukemia, particularly in young or developing children because of rapid bone growth.
• Environmental Presence: Found in soil and water, it can enter the food chain and affect various organisms, amplifying its risks.
• Detection and Management: Monitoring environmental levels and using protective measures are crucial. In medical emergencies involving exposure, treatments often focus on limiting further absorption and reducing existing levels in the body.

\(\mathrm{Sr}^{90}\)'s half-life, around 28.1 years, means it remains active for several generations. This longevity underscores the importance of effective management in environments at risk of contamination.