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, understand that the critical mass is the mass required to sustain a nuclear chain reaction. For neptunium-237, this is given as 60 kg.
2. Convert the mass from kilograms to grams because the density is given in grams per cubic centimeter. Use the conversion: 1 kg = 1000 g. Therefore, 60 kg = 60000 g.
3. Use the formula for density, which is \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). Rearrange this formula to solve for volume: \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \).
4. Substitute the known values into the volume formula: \( \text{Volume} = \frac{60000 \text{ g}}{19.5 \text{ g/cm}^3} \). Calculate this to find the volume of the sphere in cubic centimeters.
5. Use the formula for the volume of a sphere, \( V = \frac{4}{3} \pi r^3 \), where \( V \) is the volume and \( r \) is the radius. Rearrange this formula to solve for the radius: \( r = \left( \frac{3V}{4\pi} \right)^{1/3} \). Substitute the volume you calculated in the previous step to find the radius.

Key Concepts

Critical Mass

Critical mass is the minimum amount of fissile material needed to maintain a nuclear chain reaction. It depends on factors like the material's properties, shape, and density. Understanding critical mass is crucial for nuclear physics, as it determines the conditions under which a nuclear reaction can be sustained.

Density

Density is a measure of mass per unit volume, expressed in units like g/cm³. It is essential for calculating the volume of a material when its mass is known. In this context, knowing the density of neptunium-237 allows us to determine the volume of a sphere with a critical mass, which is necessary for finding its radius.

Volume of a Sphere

The volume of a sphere is calculated using the formula V = (4/3)πr³, where r is the radius. This formula is vital for determining the size of a sphere when its volume is known. By calculating the volume from the mass and density, we can use this formula to find the radius of the sphere of neptunium-237 with a critical mass.