Question

Radioactive plutonium- $239(t_{1/2}=2.44\times {10}^5\mathrm{yr})$ is used in nuclear reactors and atomic bombs. If there are $5.0\times {10}^2\mathrm{g}$ of the isotope in a small atomic bomb, how long will it take for the substance to decay to $1.0\times {10}^2\mathrm{g}$, too small an amount for an effective bomb?

Short answer

It will take approximately $5.87\times {10}^5$ years.

Step-by-step solution

Understand the Problem

We are given the initial mass of plutonium-239 as $5.0\times {10}^2\text{ g}$ and the final mass as $1.0\times {10}^2\text{ g}$. We need to find the time it takes for the mass to decay from the initial mass to the final mass. The half-life of plutonium-239, $t_{1/2}$, is given as $2.44\times {10}^5$ years.

Use the Radioactive Decay Formula

The formula for radioactive decay is given by $N(t)=N_0{\left(\frac{1}{2}\right)}^{t/t_{1/2}}$, where $N(t)$ is the remaining mass, $N_0$ is the initial mass, $t$ is the time, and $t_{1/2}$ is the half-life. We need to solve for $t$.

Set Up the Equation

Substituting the given values into the decay formula, we have $1.0\times {10}^2=5.0\times {10}^2{\left(\frac{1}{2}\right)}^{t/(2.44\times {10}^5)}$.

Solve for Time $t$

First, divide both sides by $5.0\times {10}^2$: $$\frac{1.0\times {10}^2}{5.0\times {10}^2}={\left(\frac{1}{2}\right)}^{t/(2.44\times {10}^5)}$$ which simplifies to $0.20={\left(\frac{1}{2}\right)}^{t/(2.44\times {10}^5)}$. Take the natural logarithm of both sides: $\ln (0.20)=\ln ((\frac{1}{2}{)}^{t/(2.44\times {10}^5)})$. Using log laws, this becomes $\ln (0.20)=\frac{t}{2.44\times {10}^5}\cdot \ln (0.5)$.

Calculate $t$

Rearrange the equation to solve for $t$: $$t=\frac{\ln (0.20)}{\ln (0.5)}\times 2.44\times {10}^5$$ Calculate this using a calculator: $$t\approx \frac{-1.609}{-0.693}\times 2.44\times {10}^5\approx 5.87\times {10}^5\text{ years}$$

Interpret the Result

The calculation shows that it will take approximately $5.87\times {10}^5$ years for the plutonium-239 to decay from $5.0\times {10}^2$ grams to $1.0\times {10}^2$ grams.

Key concepts

Half-life

Half-life is a fundamental concept in nuclear physics. It represents the time required for half of a given quantity of a radioactive isotope to decay. This process is exponential, meaning that the time taken for each halving of the substance remains constant, irrespective of how much of the substance is left.
Imagine you start with 100 grams of a radioactive isotope. After one half-life, you're left with 50 grams. By the end of the second half-life, you'll have 25 grams, and so on.
The decay follows a predictable pattern which is crucial in fields like archaeology, geology, and medicine, allowing professionals to estimate the age of artifacts, date rocks, and administer precise doses of radioactive drugs.
Understanding the half-life of an isotope enables scientists to predict how long a radioactive element will remain active or hazardous, aiding in the planning of containment and safety measures. In the case of plutonium-239, a half-life of 244,000 years means it remains dangerous for a long time.

Plutonium-239

Plutonium-239 is a man-made radioactive isotope, crucial in nuclear physics due to its key role in nuclear reactors and weapons. Discovered in the 1940s, this isotope has a massive destructive potential when used in atomic bombs, yet can provide immense energy in nuclear power plants.
This isotope has a half-life of about 244,000 years, which implies its longevity and the challenges in managing its radioactive waste.

• In reactors, plutonium-239 undergoes fission—the process of splitting atoms which releases large amounts of energy.
• In weapons, this rapid chain reaction leads to an explosive release of energy.

These properties make it both a powerful energy source and a significant environmental hazard, requiring careful handling. Its radioactive decay leads not only to energy release but also the production of further radioactive isotopes, complicating its disposal.

Nuclear Physics

Nuclear physics is the study of atomic nuclei and their interactions. This field explores the fundamental particles and forces that bind the nucleus, energy production through fission and fusion, and the transformation of elements.
In nuclear reactors, the fission of isotopes like plutonium-239 provides a powerful energy source. This involves the splitting of the nucleus into smaller parts, releasing energy and more neutrons that perpetuate the reaction.

• Researchers use nuclear physics to understand the behavior of nuclear materials under extreme conditions.
• It aids the development of safer nuclear energy solutions and better handling of radioactive materials.

Every advancement in nuclear physics translates into practical applications, from generating clean energy to developing medical imaging technologies. It is the backbone of numerous modern technologies and scientific explorations. Understanding the principles helps in the responsible and productive employment of nuclear technology.

Radioactive Isotopes

Radioactive isotopes, or radioisotopes, are atoms with unstable nuclei that release radiation as they break down to a stable form. These isotopes are prevalent both naturally and in human-made materials.
When an isotope decays, it emits particles and energy, often transforming into a different element or isotope. This decay process includes:

• Alpha decay, where an isotope emits an alpha particle.
• Beta decay, involving emission of electrons or positrons.
• Gamma decay, the release of gamma radiation.

Radioactive isotopes have various applications:

• In medicine for imaging and treatment, such as in PET scans and radiotherapy.
• In industry for quality control and environmental tracing.
• In scientific research as tracers in biochemical pathways.

Their radioactive nature requires strict safety protocols to protect against harmful radiation exposure, posing significant challenges in their storage and disposal. Understanding the behavior and decay rates of these isotopes is essential for leveraging their benefits while minimizing risks.