What is the radius (in cm) of a pure copper sphere that contains 1.14 * 10^24 copper atoms? [The volume of a sphere is (4/3)πr^3 and the density of copper is 8.96 g/cm^3.]

Verified step by step guidance

1. Calculate the molar mass of copper (Cu) using the periodic table, which is approximately 63.55 g/mol.
2. Determine the number of moles of copper atoms by dividing the given number of atoms (1.14 \times 10^{24}) by Avogadro's number (6.022 \times 10^{23} atoms/mol).
3. Calculate the mass of copper using the number of moles and the molar mass: \text{mass} = \text{moles} \times \text{molar mass}.
4. Use the density formula to find the volume of the copper sphere: \text{volume} = \frac{\text{mass}}{\text{density}}.
5. Solve for the radius of the sphere using the volume formula for a sphere: \text{volume} = \frac{4}{3}\pi r^3, and rearrange to find r.

Key Concepts

Avogadro's Number

Avogadro's number, approximately 6.022 x 10^23, is the number of atoms, ions, or molecules in one mole of a substance. It is essential for converting between the number of atoms and the amount of substance in moles, allowing us to relate the number of copper atoms in the sphere to its mass.

Density

Density is defined as mass per unit volume and is a critical property for understanding how much matter is contained in a given space. For copper, with a density of 8.96 g/cm^3, this concept allows us to calculate the mass of the copper sphere once we determine its volume, which is necessary for finding its radius.

Volume of a Sphere

The volume of a sphere is calculated using the formula V = (4/3)πr^3, where r is the radius. This formula is crucial for determining the size of the copper sphere based on its mass and density, as it connects the physical dimensions of the sphere to the amount of material it contains.