Density of NaCl Crystal

To determine the volume of the unit cell of a NaCl crystal, we first need to understand some key concepts. NaCl, or sodium chloride, is an ionic compound that forms a crystal lattice structure. In this structure, each sodium ion (Na+) is surrounded by six chloride ions (Cl-) and vice versa, creating a repeating pattern throughout the crystal.

Crystal Lattice Structure

The crystal lattice structure of NaCl is a face-centered cubic (FCC) arrangement. In an FCC structure, the lattice points are located at the corners and the centers of each face of the unit cell. This arrangement is highly symmetric and efficient in terms of packing.

Calculating Volume of Unit Cell

To calculate the volume of the unit cell, we need to determine the distance between adjacent ions in the crystal lattice. This distance is known as the lattice constant, denoted by "a." For NaCl, the lattice constant is approximately 0.564 nm.

Using this lattice constant, we can determine the volume of the unit cell by considering that each lattice point contributes a volume equal to the volume of a sphere with a radius of half the lattice constant. Therefore, the volume of the unit cell (V) can be calculated as:

V = a^3

V = (0.564 nm)^3

V = 0.179 nm^3

However, the density of NaCl is given in kilograms per cubic meter (kg/m^3), so we need to convert the volume to cubic meters. Since 1 nm = 1 x 10^-9 m, we can convert the volume as follows:

V = (0.179 nm^3) x (1 x 10^-9 m/nm)^3

V = 1.79 x 10^-29 m^3

Now, we can use the given density of the NaCl crystal to find the mass of the unit cell. The density is given as 2189 kg/m^3, so the mass (m) can be calculated as:

m = density x volume

m = (2189 kg/m^3) x (1.79 x 10^-29 m^3)

m = 3.92 x 10^-26 kg

Summary

- The volume of the unit cell in a NaCl crystal is determined by the lattice constant, which is approximately 0.564 nm.
- The volume of the unit cell can be calculated as V = a^3, where a is the lattice constant.
- Converting the volume from nm^3 to m^3, we find V = 1.79 x 10^-29 m^3.
- The mass of the unit cell can be found using the given density (2189 kg/m^3) and the calculated volume, resulting in m = 3.92 x 10^-26 kg.