Problem 115 - A Complete Resource Book in Chemistry for JEE Main

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: Substitute into Half-Life Formula

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

Step 4: Calculate the Exponent

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

Step 5: Simplify the Expression

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

Step 6: 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.

Step 7: 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.