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

(c) 0.61 μg.

Step 1: Understanding Half-Life

The half-life of a radioactive substance is the time required for half of it to decay. Given the half-life of \(\mathrm{Sr}^{90}\) is 28.1 years, it means that every 28.1 years, the quantity of \(\mathrm{Sr}^{90}\) will be reduced to half.

Step 2: Applying the Half-Life Formula

To find out how much \(\mathrm{Sr}^{90}\) remains after 20 years, we use the formula for decay: \[N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}\]where \(N(t)\) is the quantity remaining after time \(t\), \(N_0\) is the initial quantity, and \(T_{1/2}\) is the half-life.

Step 3: Substituting Values into the Formula

Here, \(N_0 = 1 \mu \mathrm{g}\), \(t = 20\) years, and \(T_{1/2} = 28.1\) years. Substitute these values into the formula:\[N(t) = 1 \times \left(\frac{1}{2}\right)^{\frac{20}{28.1}}\]

Step 4: Calculating the Exponent

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

Step 5: Calculating the Remaining Quantity

Compute \(\left(\frac{1}{2}\right)^{0.711}\):\[\left(\frac{1}{2}\right)^{0.711} \approx 0.615\]So, the remaining amount of \(\mathrm{Sr}^{90}\) is:\[N(t) = 1 \mu \mathrm{g} \times 0.615 \approx 0.615 \mu \mathrm{g}\]

Step 6: Conclusion

The remaining \(\mathrm{Sr}^{90}\) after 20 years is approximately \(0.61 \mu \mathrm{g}\). This matches the option (c) \(0.61 \mu \mathrm{g}\).

Key Concepts

Understanding Half-Life

Half-life is a term commonly used in nuclear physics to describe the time it takes for half of a radioactive substance to decay. It's a critical concept that helps scientists predict how long it will take for a given sample of radioactive material to decrease by half its mass. This decay process is exponential, meaning the substance doesn't lose the same amount of material over time; instead, it reduces proportionally.

For example, if a material has a half-life of 28.1 years, as with Strontium-90 (\(^{90}\mathrm{Sr}\)), it means that after 28.1 years, half of the original radioactive atoms will have decayed into a more stable form. Understanding half-life is essential because it allows us to calculate how much of a radioactive substance remains over different time intervals. This is exactly what we see in our exercise with \(^{90}\mathrm{Sr}\) being absorbed by a child's bones.

To calculate the amount remaining after any time period, we use the formula:\[N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}\]where \(N(t)\) is the remaining quantity, \(N_0\) is the initial quantity, and \(T_{1/2}\) is the half-life of the substance.

Basics of Nuclear Chemistry

Nuclear chemistry plays a pivotal role in understanding atomic reactions and the stability of elements, especially those that are radioactive. It is a branch of chemistry that deals with the changes that occur in atomic nuclei.

When a nucleus of an atom undergoes radioactive decay, it transforms into a different element or a different isotope of the same element. This process releases particles and energy. One of the most critical concepts in nuclear chemistry is recognizing how these transformations affect matter and the environment.

Key concepts in nuclear chemistry include:

• **Radioactive decay:** The spontaneous transformation of an unstable atomic nucleus into a lighter nucleus, releasing radiation in the form of particles or electromagnetic waves.
• **Types of radiation:** Alpha, beta, and gamma radiation are common, with each having different properties and penetration abilities.
• **Applications:** From medical therapies and imaging techniques to power generation and archaeological dating.

Understanding these principles is crucial for safely handling radioactive materials and evaluating their role in both beneficial technologies and potential hazards, such as in the case of \(^{90}\mathrm{Sr}\) contamination.

About Strontium-90

Strontium-90 (\(^{90}\mathrm{Sr}\)) is a radioactive isotope of strontium. It is noteworthy for several reasons and poses particular interest and concern, especially in the realm of nuclear safety and health.

First, \(^{90}\mathrm{Sr}\) has a relatively long half-life of 28.1 years, meaning it remains in the environment for extended periods. It is a byproduct of nuclear fission reactions and is notably produced in large amounts during nuclear explosions and reactor accidents.

Some important features of Strontium-90 include:

• **Chemical behavior:** Mimics calcium, meaning that it can easily replace calcium in bones and teeth once absorbed by an organism. This makes it particularly dangerous because it can lead to bone cancer and blood disorders.
• **Radiation type:** It emits beta particles during its decay process. These particles can cause damage to cells and tissues, leading to radiation sickness or cancer.
• **Environmental and health impact:** Once released into the atmosphere, it can disperse over wide areas, falling back onto the soil and crops, thus entering the food chain.

These attributes make understanding and controlling the spread of \(^{90}\mathrm{Sr}\) essential for public health and safety.