At room temperature sodium crystallizes in a BCC lattice with the cell edge (a) \(4.24\) A. Find the density of sodium. (Atomic wt. of \(\mathrm{Na}=23\) ) (a) \(2.01 \mathrm{~g} / \mathrm{cm}^{3}\) (b) \(1.002 \mathrm{~g} / \mathrm{cm}^{3}\) (c) \(3.003 \mathrm{~g} / \mathrm{cm}^{3}\) (d) \(2.004 \mathrm{~g} / \mathrm{cm}^{3}\)

The BCC, or body-centered cubic lattice, is one of several ways that atoms can be arranged in solids. This type of crystal structure is defined by atoms at the corners of a cube and an additional atom at the very center of the cube. Imagine a dice with a dot on each corner and one right in the middle – that's what the BCC lattice looks like on a microscopic scale.

Each corner atom is shared by eight other adjacent cubes, so when we calculate the number of atoms in one unit cell, we must take into account only a fraction of each corner atom (1/8 to be exact), along with the full atom in the center. This means a BCC unit cell effectively contains 2 whole atoms. In the context of calculating density, knowing the number of atoms per unit cell allows for a precise calculation of mass within a given volume, which is crucial for determining the material's density.

Understanding atomic weight is essential when working with elements on the atomic scale. The atomic weight (often referred to as atomic mass) is a relative measure of how heavy atoms are compared to a carbon-12 atom. For sodium (Na), this number is 23, which means a sodium atom is 23 times heavier than 1/12th of a carbon-12 atom.

This concept is critical when we're trying to calculate the mass of atoms in a BCC lattice to find out the density of a material, as in our exercise. Knowing that the atomic weight of sodium is 23, we can convert the weight from atomic mass units to grams for a mole of atoms, a step that bridges the microscopic world of atoms with the macroscopic world we measure in grams and centimeters.

Avogadro's Number, denoted by \(6.022 \times 10^{23}\), is a fundamental constant that defines the number of particles, usually atoms or molecules, in one mole of a substance. It is this fixed quantity that allows chemists and physicists to count atoms by weighing, bridging the gap between atomic scales and human scales.

In our exercise, Avogadro's Number is utilized to translate the atomic weight of sodium into a tangible mass for two atoms. This conversion is crucial for calculating the density, as mass must correspond to the volume of the BCC lattice. It's a beautiful example of how Avogadro's number provides a link from the abstract concept of moles to the practical requirement of grams, enabling us to calculate the density of substances like sodium.