Sodium has a density of 0.971 g>cm3 and crystallizes with a body-centered cubic unit cell. What is the radius of a sodium atom, and what is the edge length of the cell in picometers?

Verified step by step guidance

1. Determine the number of atoms per unit cell in a body-centered cubic (bcc) structure. In a bcc lattice, there is one atom at each corner of the cube and one atom in the center. Each corner atom is shared among eight unit cells, so each contributes 1/8 of an atom to the cell, resulting in a total of 2 atoms per unit cell.
2. Calculate the volume of the unit cell using the density formula: Density = Mass/Volume. Here, the mass is the mass of two sodium atoms (since there are two atoms per unit cell), and the volume is the volume of the cubic unit cell.
3. Use the atomic mass of sodium and Avogadro's number to find the mass of one mole of sodium atoms, then calculate the mass of two sodium atoms.
4. Substitute the values into the density formula to solve for the volume of the unit cell. Then, calculate the edge length of the unit cell (a) since the volume of a cube is given by a^3.
5. Calculate the atomic radius of sodium using the relationship between the edge length of the unit cell and the atomic radius in a bcc lattice, where the diagonal across the body of the cube (which spans four radii) is equal to \(\sqrt{3} \times a\).

Key Concepts

Density and its Calculation

Density is defined as mass per unit volume, typically expressed in grams per cubic centimeter (g/cm³). To find the radius of an atom from density, one must relate the mass of the unit cell to its volume. The density formula can be rearranged to find the volume, which is essential for determining the dimensions of the unit cell.

Body-Centered Cubic (BCC) Structure

A body-centered cubic (BCC) unit cell is a type of crystal lattice structure where one atom is located at each corner of the cube and one atom is at the center. In a BCC structure, the relationship between the edge length (a) and the atomic radius (r) is given by the formula: a = 4r/√3. Understanding this relationship is crucial for calculating the radius of the sodium atom.

Unit Cell Volume and Atomic Radius

The volume of a unit cell can be calculated using the edge length (a) of the cube, where volume = a³. For a BCC unit cell, knowing the density allows us to find the mass of the unit cell, which can then be used to derive the edge length and subsequently the atomic radius. This process is fundamental in solid-state chemistry for determining atomic dimensions.