Question

In the fall of 2002, scientists at Los Alamos National Laboratory determined that the critical mass of neptunium-237 is about 60 kg. The critical mass of a fissionable material is the minimum amount that must be brought together to start a nuclear chain reaction. Neptunium-237 has a density of 19.5 g/cm\(^3\). What would be the radius of a sphere of this material that has a critical mass?

Short answer

The radius of the sphere is approximately 8.59 cm.

Step-by-step solution

Convert Mass to Grams

The critical mass of neptunium-237 is given as 60 kg. First, convert this mass into grams since the density is given in grams per cubic centimeter:\[ 60 \text{ kg} = 60,000 \text{ g} \]

Calculate Volume using Density Formula

Using the density formula, \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \), rearrange to solve for volume:\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} = \frac{60,000 \text{ g}}{19.5 \text{ g/cm}^3} \] Solve for volume:\[ \text{Volume} \approx 3076.92 \text{ cm}^3 \]

Use Volume of a Sphere Formula

The volume of a sphere is given by \( V = \frac{4}{3} \pi r^3 \). Rearrange to solve for the radius \( r \):\[ r^3 = \frac{3V}{4\pi} \] Substitute the calculated volume from Step 2 into the formula:\[ r^3 = \frac{3 \times 3076.92}{4 \pi} \]

Solve for the Radius

Now solve for \( r \) by taking the cube root:\[ r = \sqrt[3]{\frac{3 \times 3076.92}{4 \pi}} \] Calculating gives approximately \( r \approx 8.59 \text{ cm} \).

Key concepts

Nuclear Chain Reaction

A nuclear chain reaction occurs when a single nuclear reaction causes additional nuclear reactions, leading to a self-sustaining series of reactions. This process is fundamental in nuclear reactors and atomic bombs. The basic idea is that when a nucleus undergoes fission, it splits into smaller fragments, releasing energy and more neutrons.

These additional neutrons can then induce fission in nearby nuclei, propagating the reaction. For a chain reaction to be sustained, a critical mass of the fissionable material is needed. This is the minimum amount required so that the neutrons produced can effectively cause further fissions without escaping the system or being absorbed without causing fission.

Key aspects to consider in a nuclear chain reaction include:

• The type of fissionable material used (e.g., uranium-235, plutonium-239, neptunium-237).
• The presence of a moderator to slow down neutrons, enhancing their probability of causing additional fissions.
• The geometry and arrangement of the material to ensure sufficient neutron capture.

Density Formula

The density formula is a core concept in understanding how mass and volume relate to each other for a given material. Density (\( \rho \)) is defined as the mass (\( m \)) of an object divided by its volume (\( V \)). This relationship is expressed mathematically as:

\[ \rho = \frac{m}{V} \]
This formula helps determine how much space a material's mass occupies. It's especially useful when calculating the volume required to achieve a specific mass when the density is known.

In the context of the exercise, the density of neptunium-237 was crucial for determining the volume of the sphere at critical mass. By rearranging the formula, we find the volume (\( V \)) when the mass (\( m \)) and density (\( \rho \)) are given:

\[ V = \frac{m}{\rho} \]
This calculation provides the sphere's volume through which the critical mass of the material can be achieved.

Volume of a Sphere

The volume of a sphere is a fundamental concept in geometry, essential for understanding how much space a spherical object occupies. The formula for the volume (\( V \)) of a sphere with radius (\( r \)) is given by:

\[ V = \frac{4}{3} \pi r^3 \]
This equation calculates the three-dimensional space enclosed within the sphere. Understanding this concept allows us to determine the required radius when the volume is known, as in this exercise.

After calculating the volume using the density formula, we can rearrange the sphere volume formula to find the radius:

\[ r^3 = \frac{3V}{4\pi} \]
Finally, by taking the cube root of both sides, we arrive at the radius. This step is critical for determining the physical dimensions necessary to achieve a particular volume. The formula is applicable whenever one needs to find the sphere's radius using a given volume.

Fissionable Material

Fissionable materials are elements capable of sustaining a nuclear fission chain reaction. These materials contain heavy atomic nuclei that can split into smaller nuclei when struck by a neutron. This splitting releases a significant amount of energy, along with additional neutrons that can propagate a chain reaction.

Examples of fissionable materials include uranium-235, plutonium-239, and neptunium-237. In the exercise, neptunium-237's capability to reach a critical mass and maintain a nuclear chain reaction was analyzed. Key properties of fissionable materials include:

• Their ability to absorb neutrons and undergo fission.
• The critical mass, which is the smallest amount needed to maintain a chain reaction.
• The energy released during the fission process, contributing to nuclear power or weaponry.

Understanding these materials is crucial in both peaceful applications like nuclear power generation and more destructive uses such as atomic bombs. Their management involves not only physics but also significant considerations of safety and environmental impact.