Problem 36 - Fundamentals of Physics

Question

Plutonium isotope \({ }^{239} \mathrm{Pu}\) decays by alpha decay with a halflife of \(24100 \mathrm{y}\). How many milligrams of helium are produced by an initially pure \(12.0 \mathrm{~g}\) sample of \({ }^{239} \mathrm{Pu}\) at the end of \(20000 \mathrm{y} ?\) (Consider only the helium produced directly by the plutonium and not by any by-products of the decay process.)

Step-by-Step Solution

Answer

Approximately 1.98 milligrams of helium are produced.

Step 1: Understanding Alpha Decay

Alpha decay occurs when a nucleus emits an alpha particle, which consists of 2 protons and 2 neutrons. For plutonium-239, this process reduces its atomic number by 2 and its mass number by 4, forming uranium-235, but producing helium as a by-product.

Step 2: Alpha Decay Equation

The alpha decay equation for plutonium-239 is: \(^{239}_{94}\text{Pu} \rightarrow \, ^{235}_{92}\text{U} \, + \, ^{4}_{2}\text{He}\). Here, each decay produces a helium nucleus, equivalent to an alpha particle.

Step 3: Calculating Number of Decays

Calculate the number of plutonium nuclei that decay in 20000 years. Use the formula for decay, \(N(t) = N_0 imes (1/2)^{t/T_{1/2}}\), where \(N_0 = 12.0\,g\), \(t = 20000\,y\), \(T_{1/2} = 24100\,y\). Find the remaining quantity of plutonium first.

Step 4: Finding Decayed Plutonium

Calculate the remaining plutonium: \(N(20000) = 12.0 g \times (1/2)^{20000/24100}\). The leftover plutonium is \(x\,\text{g}\). Find decayed plutonium: \(12.0 g - x\,\text{g}\).

Step 5: Moles of Decayed Plutonium

Convert the decayed plutonium mass to moles using the molar mass of plutonium-239 (239 g/mol): Moles = Decayed mass / 239 g/mol.

Step 6: Helium Production

Since each decay produces one helium atom, the number of moles of produced helium is equal to the number of moles of decayed plutonium. Calculate helium mass: Moles of helium \(\times 4.00\,g/mol =\) Helium in grams. Convert to milligrams (1 g = 1000 mg).

Step 7: Final Result

Determine the final mass of helium produced after rounding to appropriate significant figures based on initial data precision.

Key Concepts

Plutonium-239

Plutonium-239, a radioactive isotope of plutonium, is well-known for its role in nuclear physics and energy production. Its symbol is \({ }^{239} \ ext{Pu}\), where the "239" represents its atomic mass number, a sum of protons and neutrons in its nucleus. Plutonium-239 is particularly significant due to its ability to undergo alpha decay, a process where it emits an alpha particle. This emission reduces both its atomic and mass numbers, resulting in the formation of a different element, uranium-235. The alpha particles emitted can be classified as helium-4 nuclei. Plutonium-239 is crucial for its potential energy yield in nuclear reactors and weapons. However, its radioactive properties also demand careful handling and storage to avoid environmental contamination or health hazards.

Half-life Calculation

The concept of 'half-life' is essential when studying radioactive decay. It represents the time taken for half the quantity of a radioactive substance to decay. For Plutonium-239, the half-life is notably long, about 24,100 years. To calculate the remaining quantity of a radioactive material after a given time, we use:\[ N(t) = N_0 \times \left(\frac{1}{2}\right)^{t/T_{1/2}} \]Where:

• \(N_0\) is the initial quantity.
• \(t\) is the time elapsed.
• \(T_{1/2}\) is the half-life of the material.

In practical applications, such as the exercise provided, this formula assists in determining how much material remains or has decayed, enabling us to calculate the resulting by-products, such as helium in the case of alpha decay.

Nuclear Physics

Nuclear physics is the branch of science that deals with the constituents, structure, behavior, and interactions of atomic nuclei. In the context of Plutonium-239, nuclear physics principles help us understand alpha decay, one type of radioactive decay. During alpha decay, a nucleus emits an alpha particle, thereby transforming into a different element with a reduced atomic number and mass. Important aspects include:

• Nuclear Binding Energy: the energy that holds a nucleus together.
• Radioactive Decay: processes through which unstable nuclei release particles or energy.
• Transmutation: the conversion of one chemical element or isotope into another.

Understanding these concepts allows us to predict the behavior and changes in materials like plutonium during decay processes, and the consequent production of new elements or isotopes.

Helium Production

In the alpha decay of Plutonium-239, helium production is a direct result. Each decay event results in the creation of one helium nucleus, or alpha particle, composed of 2 protons and 2 neutrons. This makes the decay a valuable source of helium, especially in scientific and industrial applications. To find the amount of helium produced: 1. Calculate decayed plutonium after a specified period using the half-life calculation. 2. Convert the mass of decayed plutonium into moles (using the molecular weight of Plutonium-239). 3. Each mole of decayed plutonium corresponds to a mole of helium produced. 4. Calculate the mass of helium in grams by multiplying by helium's molar mass (4 g/mol), then convert to milligrams for the final result. This process not only reveals the amount of helium produced but also quantifies the extent of nuclear decay over specific time scales.