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/cm3. What would be the radius of a sphere of this material that has a critical mass?

Verified step by step guidance

1. First, convert the critical mass from kilograms to grams. Since 1 kg = 1000 g, multiply 60 kg by 1000 to get the mass in grams.
2. Next, use the formula for density, which is \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). Rearrange this formula to solve for volume: \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \). Substitute the mass in grams and the density in g/cm³ into this formula to find the volume in cm³.
3. Now, recall the formula for the volume of a sphere: \( V = \frac{4}{3} \pi r^3 \). Set this equal to the volume you calculated in the previous step.
4. Solve the equation \( \frac{4}{3} \pi r^3 = \text{Volume} \) for the radius \( r \). This involves isolating \( r^3 \) and then taking the cube root.
5. Finally, ensure that the radius is expressed in centimeters, as the density was given in g/cm³, and verify that all units are consistent throughout the calculation.

Key Concepts

Critical Mass

Critical mass is the minimum amount of fissionable material needed to maintain a nuclear chain reaction. It depends on factors like the material's properties and configuration. For neptunium-237, the critical mass is 60 kg, which is the threshold for initiating a self-sustaining reaction.

Density

Density is the mass per unit volume of a substance, expressed in g/cm³ for solids. Neptunium-237 has a density of 19.5 g/cm³, which helps determine the volume of the material needed to achieve the critical mass. This property is crucial for calculating the size of the sphere containing the critical mass.

Volume of a Sphere

The volume of a sphere is calculated using the formula V = (4/3)πr³, where r is the radius. To find the radius of a sphere with a given mass and density, you first calculate the volume using the mass and density, then solve for the radius. This concept is essential for determining the sphere's size containing the critical mass of neptunium-237.