Problem 74 - Master Resource Book in JEE Main Physics

Question

Plutonium decays with half-life of 24000 yr. If plutonium is stored for \(7200 \mathrm{yr}\), the fraction of it that remains is (a) \(1 / 8\) (b) \(1 / 3\) (c) \(1 / 4\) (d) \(1 / 2\)

Step-by-Step Solution

Answer

None of the given options exactly match; approx 0.813 of plutonium remains.

Step 1: Understand Half-Life Formula

The half-life formula describes the time it takes for half of a sample to decay. If plutonium has a half-life of 24000 years, this means every 24000 years, half of the plutonium decays.

Step 2: Use the Exponential Decay Formula

The exponential decay formula is given by \( N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \), where \( N(t) \) is the remaining quantity, \( N_0 \) is the initial quantity, \( t \) is time elapsed, and \( T_{1/2} \) is the half-life.

Step 3: Substitute Known Values

Substitute the known values into the formula: \( N_0 = 1 \), \( T_{1/2} = 24000 \) years, and \( t = 7200 \) years. This gives us: \( N(7200) = 1 \times \left( \frac{1}{2} \right)^{\frac{7200}{24000}} \).

Step 4: Simplify the Fraction

The fraction \( \frac{7200}{24000} \) simplifies to \( \frac{3}{10} \). So, the equation becomes \( N(7200) = 1 \times \left( \frac{1}{2} \right)^{0.3} \).

Step 5: Evaluate the Power

Calculate \( \left( \frac{1}{2} \right)^{0.3} \). Using a calculator, this evaluates approximately to \( 0.813 \).

Step 6: Conclude the Fraction Remaining

The remaining fraction of plutonium is approximately \( 0.813 \), which indicates more than half is remaining. This is not among the options provided, but closer evaluation shows it might have rounded the wrong way or require more precision calculation.

Key Concepts

Understanding Half-Life

The concept of half-life helps us understand how radioactive substances decay over time. Half-life is the period required for half of the radioactive atoms in a sample to decay. This consistent measure makes it easier to predict the behavior of these substances over extended periods. For example, if plutonium has a half-life of 24,000 years, it means that in 24,000 years, only half of the original plutonium will remain. This process continues in successive intervals, always reducing the remaining quantity by half. Understanding half-life is crucial when dealing with any radioactive material as it applies universally among different elements.
• Key takeaway: Half-life describes the consistent reduction by half for any given radioactive material over a set time.

Exponential Decay Explained

Exponential decay is a mathematical concept that describes the reduction of a quantity at a consistent rate over time. In the context of radioactive decay, it describes how substances diminish as time passes. This is captured by the formula:

• \( N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \)

Here, \( N(t) \) is the remaining quantity after time \( t \), \( N_0 \) is the initial amount, and \( T_{1/2} \) is the half-life. This formula demonstrates that for every half-life period that passes, the amount of substance remaining is halved. The process continues sequentially, giving us an exponentially decaying graph. Exponential decay is a key concept not only in radioactive decay but also in other fields such as finance and biology, where similar decay principles apply.
• Tip: Remember, each cycle maintains the same fraction of reduction, making tracking over multiple periods consistent and predictable.

Exploring Plutonium Decay

Plutonium, a radioactive element often discussed due to its use in nuclear power and weaponry, decays over time through radioactive decay mechanisms. With a half-life of 24,000 years, plutonium doesn't decay quickly, making it a concern in long-term storage and environmental impact. Despite its slow decay, over millennia, even a small initial volume of plutonium can significantly reduce. This slow rate of change underscores the importance of understanding how radioactive elements behave over long durations and planning for their safe storage accordingly. Plutonium decay is a perfect example of the real-world application of half-life and exponential decay concepts.
• Fun fact: Due to its long half-life, any significant environmental cleanup or containment strategy must consider the extended duration over which plutonium remains active.

Steps for Remaining Fraction Calculation

Calculating the remaining fraction of a radioactive substance such as plutonium involves a few straightforward steps:

• Step 1: Identify your known values: the initial quantity \( N_0 \), the time \( t \) the substance has been decaying, and the half-life \( T_{1/2} \).
• Step 2: Use the exponential decay formula: \( N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \).
• Step 3: Plug in your values. For instance, if \( N_0 = 1 \) and \( t = 7200 \) years, for plutonium with a half-life of 24,000 years, you have \( N(7200) = 1 \times \left( \frac{1}{2} \right)^{\frac{7200}{24000}} \).
• Step 4: Simplify the fraction \( \frac{7200}{24000} \), which becomes \( 0.3 \).
• Step 5: Calculate \( \left( \frac{1}{2} \right)^{0.3} \) to find how much remains after 7200 years, approximately 0.813.

This example shows how a small step-by-step process lets you determine the remaining fraction of a radioactive substance after a certain period.