At room temperature, sodium crystallizes in a body centred cubic lattice with \(a=4.24 \AA\). Calculate theoretical density of sodium (At. wt. of \(\mathrm{Na}=23\) ).

The body-centered cubic (BCC) lattice represents a common crystal structure found in many metals such as iron at certain temperatures. In this type of lattice, atoms are located at each corner of a cube and with one atom at the exact center of the cube. Imagining multiple BCC cells stacked together shows that the corner atoms are shared among eight unit cells, while the center atom is exclusive to its particular unit cell. This layout for the arrangement of atoms within the crystal affects many of the material's properties, including its density.

Because of sharing, when we calculate the total number of atoms within any single BCC unit cell, we must consider that each corner atom contributes only an eighth of an atom to the cell since the corners are shared by eight neighboring cells. Therefore, summing up the contribution from all eight corners and the singular center atom results in a total of two atoms per BCC unit cell.

The volume of a unit cell is the space it occupies, which is crucial when determining properties like density. For a cubic lattice, such as the BCC, calculating the volume is straightforward. Since a cube has all sides of equal length, known as the lattice parameter 'a,' the volume is simply the cube of this length, expressed as \( V = a^3 \).

In our case with a sodium (Na) unit cell with a given edge length of 4.24 angstroms, this calculation gives the volume of the space taken up by two sodium atoms in the BCC structure. Converting the volume to a more manageable unit like cubic centimeters can be necessary, especially when comparing materials or evaluating the practical application in manufacturing or engineering contexts.

Avogadro's number, denoted by \( N_A \), is a fundamental constant in chemistry representing the number of particles, usually atoms or molecules, in one mole of a substance. The value of Avogadro's number is approximately \( 6.022 \times 10^{23} \) entities per mole. This incredibly large number helps in converting measurements made on the atomic scale to macroscopic scales that are more meaningful in everyday contexts.

For instance, when calculating the mass of atoms in a unit cell, as seen in the exercise, Avogadro's number enables us to determine the mass of a single atom or a mole of atoms in grams, linking the atomic weight scale to the gram scale. This is essential when we are interested in the amount of substance in terms of mass, which is a common scenario in material science, chemistry, and physics.

Atomic weight, also known as relative atomic mass, is a dimensionless physical quantity expressed in atomic mass units (amu). It represents the average mass of atoms of an element, taking into account the different isotopes and their abundances. For sodium (Na) specifically, the atomic weight is approximately 23 amu. This means that an average sodium atom is 23 times heavier than 1/12 of the mass of a carbon-12 atom.

The atomic weight helps in understanding the mass of individual atoms when considering their contribution to the overall mass of a compound or element in bulk form. When calculating theoretical densities, the atomic weight allows us to deduce the mass of atoms in a unit cell from the macroscopic mass that can be measured in laboratories. Furthermore, by combining atomic weight with Avogadro's number, we can convert atomic mass to grams, which is a standard unit for measuring mass in the scientific and everyday world.