91Ó°ÊÓ

Density is a fundamental property that describes how much mass is contained in a given volume. It is a crucial concept in understanding how heavy or light a material is for its size. The formula for density is expressed as:

• \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \)

In our problem, the density of silver is given as \(10.5 \text{ g/cm}^3\). This tells us that for every cubic centimeter of silver, the mass is 10.5 grams.
To find the mass of the cube of silver, we multiply the density by the volume of the cube. Since each edge of the cube measures 1.000 cm, the volume of the cube is\(1.000 \text{ cm} imes 1.000 \text{ cm} imes 1.000 \text{ cm} = 1.000 \text{ cm}^3\). The resulting mass is therefore \(10.5 \text{ g/cm}^3 \times 1.000 \text{ cm}^3 = 10.5 \text{ g}\). This mass is then used to find out how many atoms are present in the cube.

Avogadro's Number is a key concept in chemistry, providing a bridge between the atomic scale and the macroscopic world. It is defined as the number of atoms or molecules in one mole of a substance, and it is approximately \(6.022 \times 10^{23} \text{ atoms/mol}\).
In the exercise, to determine how many atoms are in the silver cube, we first find how many moles of silver are present in the cube.
This is done by dividing the mass of the cube by the molar mass of silver. Given the mass of the cube is 10.5 g and the molar mass of silver is \(107.87 \text{ g/mol}\), we calculate the moles as:

• \( \frac{10.5 \text{ g}}{107.87 \text{ g/mol}} \approx 0.0974 \text{ mol} \)

Then using Avogadro's Number, the number of atoms is calculated by multiplying the number of moles by Avogadro's constant:

• \(0.0974 \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 5.86 \times 10^{22} \text{ atoms}\)

This technique is fundamental for converting between measurements on an atomic scale and measurements we can observe macroscopically.

Understanding the volume of a sphere is important when dealing with the space occupied by atoms, which are often modeled as spheres. The formula for the volume of a sphere is:

• \( V = \frac{4}{3}\pi r^3 \)

Where \(V\) is the volume and \(r\) is the radius of the sphere. In our exercise, each silver atom is considered a sphere. The calculation of the single atom's volume is done by dividing the total volume the atoms occupy by the number of atoms.
Since we know only 74% of the cubic space is filled by silver atoms, the volume occupied by atoms is \(0.74 \times 1.000 \text{ cm}^3 = 0.740 \text{ cm}^3\).
Each atom then fills \( \frac{0.740 \text{ cm}^3}{5.86 \times 10^{22} \text{ atoms}} \approx 1.26 \times 10^{-23} \text{ cm}^3\) of space. Finally, using the spherical volume formula, we rearrange to solve for the radius.

The molar mass of a substance is the mass of one mole of its entities (atoms, molecules, etc.). For silver, the molar mass is given as \(107.87 \text{ g/mol}\). This value is crucial when converting between mass and number of atoms using Avogadro's Number.
In this exercise, we are provided with the mass of the silver cube, and we need to find out how many atoms it contains. This is where the molar mass comes into play.
By dividing the given mass of the silver by its molar mass, we can determine the moles of silver present. Knowing the moles allows us to use Avogadro's Number to find the total number of atoms.
This gives us a coherent way to link the given macroscopic mass to a microscopic count of atoms, essential in solving questions involving atomic scale dimensions and quantities.