Question

Plutonium decays with half life time 24000 yrs. if Plutonium is stored after 72000 yrs, the fraction of it that remain (A) \((1 / 2)\) (B) \((1 / 9)\) (C) \((1 / 12)\) (D) \((1 / 8)\)

Short answer

After 72,000 years, the remaining fraction of plutonium is \(\frac{1}{8}\) (Option D).

Step-by-step solution

Determine the number of half-life periods

Since 1 half-life of plutonium is 24,000 years, we will divide the total time passed, which is 72,000 years, by 24,000 years to find out the number of half-lives that have passed. Number of half-lives = 72000 yrs / 24000 yrs = 3

Apply the half-life formula

The half-life formula is given by: Remaining fraction = \((\frac{1}{2})^n\), where n is the number of half-lives passed.

Calculate the remaining fraction

Now, we can plug in the number of half-lives (n = 3) into the formula: Remaining fraction = \((\frac{1}{2})^3 = \frac{1}{8}\) So, the remaining fraction of plutonium after 72,000 years is \(\frac{1}{8}\), which corresponds to the provided option (D).

Key concepts

Half-life

Understanding the concept of half-life is crucial to grasping how radioactive materials decay over time. The half-life of a substance is the amount of time it takes for half of the radioactive atoms in a sample to decay. This means if you start with a certain amount of a radioactive material, after one half-life, only half of it will remain.
For example, in the context of the original exercise, plutonium has a half-life of 24,000 years. This indicates that every 24,000 years, the amount of plutonium halves. Thus, if you began with 100 grams, after one half-life (24,000 years), you would have 50 grams left. After another 24,000 years, leaving a total elapsed time of 48,000 years, you'd be left with only 25 grams. The process continues similarly, causing the nosedive of the material over several half-lives.
Half-life is a fixed characteristic unique to each radioactive element and does not change over time or with the environment. This consistent decay rate is key in fields such as radiometric dating and nuclear medicine.

Plutonium decay

Plutonium is a heavy and highly toxic radioactive metal. Its decay process is of particular interest in both nuclear power generation and nuclear weapons production. Understanding plutonium decay is important because it allows us to predict the longevity of radioactive waste and manage it accordingly.
In our exercise, the radioactive decay process for plutonium is measured over spans known as half-lives. Given a half-life of 24,000 years, we know plutonium decays fairly slowly compared to other radioactive elements, meaning it takes a long time before its radioactivity diminishes to safer levels.

• This slow decay affects how nuclear waste containing plutonium is stored, often needing secure facilities for thousands of years.
• Its applications are weighed against the environmental impact and the risk associated with handling and storing it.

Therefore, when calculating how much plutonium remains after multiple half-lives as in the problem provided, it essentially highlights the ongoing challenge of containing such long-lived nuclear waste safely.

Exponential decay

Exponential decay refers to a mathematical process where the quantity diminishes at a rate proportional to its current value. Radioactive decay, like the decay of plutonium, is an example of this.
The process can be represented using the exponential decay formula: \[ N(t) = N_0 e^{-\lambda t} \]where:

• \(N(t)\) is the remaining quantity after time \(t\)
• \(N_0\) is the initial amount
• \(\lambda\) is the decay constant
• \(t\) is the time elapsed

However, for simplicity, especially concerning half-life problems, the formula is often adapted to \[ (\frac{1}{2})^n \] where \(n\) represents the number of half-life periods that have passed.
Exponential decay is characterized by its rapid drop-off, leading to a small but persistent amount remaining over time. Real-world examples like the decay of plutonium highlight this principle in action, showcasing the persistent nature of some long-lived isotopes, despite the decreases they undergo.