Open Question
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.]
1. Intro to General Chemistry
Density of Geometric Objects
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Iron crystallizes with a body-centered cubic unit cell. The radius of an iron atom is 126 pm. Calculate the density of solid crystalline iron in grams per cubic centimeter.
A
9.12 g/cm³
B
7.87 g/cm³
C
5.24 g/cm³
D
6.45 g/cm³
Verified step by step guidance1
Identify the type of unit cell: Iron crystallizes in a body-centered cubic (BCC) unit cell.
Determine the relationship between the atomic radius and the edge length of the BCC unit cell. For a BCC structure, the body diagonal is equal to 4 times the atomic radius. Use the formula: \( \text{Body diagonal} = \sqrt{3} \times a = 4 \times r \), where \( a \) is the edge length and \( r \) is the atomic radius.
Calculate the edge length \( a \) of the unit cell using the atomic radius \( r = 126 \text{ pm} \). Rearrange the formula to solve for \( a \): \( a = \frac{4 \times r}{\sqrt{3}} \).
Calculate the volume of the unit cell. The volume \( V \) of a cubic unit cell is given by \( V = a^3 \). Convert the edge length from picometers to centimeters before calculating the volume.
Determine the density of iron. The density \( \rho \) is given by \( \rho = \frac{\text{mass of atoms in unit cell}}{\text{volume of unit cell}} \). For BCC, there are 2 atoms per unit cell. Use the molar mass of iron and Avogadro's number to find the mass of these atoms, then divide by the volume to find the density.
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