Argon has a cubic closest packed structure as a solid. Assuming that argon has a radius of \(190 . \mathrm{pm},\) calculate the density of solid argon.

One of the most crucial concepts in solid-state physics is the face-centered cubic (FCC) structure. This geometric arrangement is where atoms are positioned at each corner and the centers of all the cube faces of a unit cell, which is the smallest repeating unit in a crystal lattice. The significance of this structure lies in how densely the atoms are packed, leading to unique material properties.

In an FCC lattice, each corner atom is shared among eight neighboring unit cells and each face atom among two, translating to a total contribution of four atoms per unit cell. This is vital for understanding many physical properties of materials, including their density. For example, in the case of solid argon, the FCC structure provides a framework to determine how closely packed the argon atoms are within the crystal, serving as the foundation for density calculation.

Knowing the atomic radius is essential for identifying the dimensions of a crystal's unit cell and hence its density. The atomic radius can be thought of as the distance from the very core of an atom to the outer edge of its electron cloud, and is often derived from experimentally obtained values.

In the context of calculating the density of a solid, like argon, we apply the relationship between the atomic radius and the unit cell's edge length in a face-centered cubic lattice. Given the atomic radius, one can determine the edge length of the unit cell with the formula \(a = 2\sqrt{2}r\). After establishing the edge length, it's simply a matter of cubing this value to find the volume of the unit cell. This step is pivotal because it sets the stage for determining the density by relating the mass of the atoms to the volume they occupy.

Avogadro's number, \(N_A = 6.022 \times 10^{23} \, \mathrm{atoms/mol}\), is a fundamental constant in chemistry, representing the number of constituent particles, usually atoms or molecules, in one mole of a substance. It provides a crucial bridge between the macroscopic and atomic scales, allowing us to relate a material's molar mass to the mass of its individual atoms.

When we calculate the density of a substance like solid argon, we use Avogadro's number to find the mass of a single atom, employing the formula \(m_{atom} = \frac{molar\ mass}{N_A}\). In density calculations, knowledge of Avogadro's number enables us to convert from the scale of moles, which we use in laboratory measurements, to the scale of atoms, which we use in theoretical models for solids. This understanding is crucial when calculating the total mass of atoms in a unit cell and subsequently the density of the material.