Gold (Au) forms a substitutional solid solution with silver (Ag). Compute the weight percent of gold that must be added to silver to yield an alloy that contains \(5.5 \times 10^{21}\) Au atoms per cubic centimeter. The densities of pure Au and \(\mathrm{Ag}\) are \(19.32\) and \(10.49 \mathrm{~g} / \mathrm{cm}^{3}\), respectively.

When two or more metallic elements combine to form an alloy, and the atoms of added element take the places of the host metal atoms within the crystal lattice, it results in a substitutional solid solution. This usually occurs when the radii of the atoms of both elements are similar and the chemical affinities are not too dissimilar. In the exercise, gold (Au) atoms are substituting some of the lattice points that silver (Ag) atoms would normally occupy, creating the alloy.

The amount of gold to be added to the silver to achieve the desired number of atoms per unit volume is critical for determining the final properties of the alloy. For a substitutional solid solution like the Au-Ag alloy, attributes such as color, tensile strength, and electrical conductivity can be modified by altering the composition. This is significant in materials engineering where specific material characteristics are desired.

Atomic weight denotes the mass of an atom, typically expressed in atomic mass units (amu), whereas density is a measure of mass per unit volume. There is a direct relationship between these two properties in the context of an element's bulk material. We can compute the number of atoms in a given volume if we know the atomic weight and the density of the element.

Using the formula \(\text{Atoms} = \frac{\text{Density}}{\text{Atomic Weight}} \times Avogadro's\; Number\), we can find out the number of atoms per cubic centimeter. Density provides the mass of the substance per unit volume, atomic weight converts this mass to moles, and Avogadro's number then translates moles into the number of atoms. By applying this relationship, we determine the composition of the alloy in the problem, ensuring that the weight percent of gold added to silver results in the desired number of gold atoms per cubic centimeter of the alloy.

Avogadro's number, denoted by \(N_A\), is a fundamental constant in chemistry and physics that represents the number of constituent particles (usually atoms or molecules) in one mole of a substance. Its value is approximately \(6.022 \times 10^{23}\) particles per mole.

In the context of the exercise, we use Avogadro's number to convert between the amount of substance in moles and the number of atoms or molecules. This is crucial as it allows chemists and material scientists to calculate the exact amounts of substances needed in a reaction or an alloy, based on atomic or molecular quantities. By utilizing Avogadro's number in the calculation, we are able to ascertain the weight percentage of gold needed when combined with silver to form the required solid solution alloy. This understanding is essential for students to grasp the significance of Avogadro's number in real-world applications, particularly in the preparation of specific alloys with precise atom counts.