Question

Crystalline silicon has a cubic structure. The unit cell edge length is \(543 \mathrm{pm}\). The density of the solid is 2.33 \(\mathrm{g} / \mathrm{cm}^{3} .\) Calculate the number of \(\mathrm{Si}\) atoms in one unit cell.

Short answer

There are 8 Si atoms in one unit cell of crystalline silicon.

Step-by-step solution

Understand the Unit Cell Structure

Crystalline silicon has a face-centered cubic (FCC) structure, also known as cubic close-packed (CCP). In this structure, there are 4 atoms per unit cell.

Convert Edge Length to Centimeters

The edge length of the unit cell is given in picometers (pm). To use the density in g/cm³, first convert the edge length from pm to cm: 543 pm = 543 x 10^{-12} m = 543 x 10^{-10} cm (since 1 m = 10^2 cm) 543 pm = 5.43 x 10^{-8} cm.

Calculate the Volume of the Unit Cell

The volume of a cube (unit cell) is calculated as side length cubed. \( V = (5.43 \times 10^{-8})^3 \text{ cm}^3 = 1.60 \times 10^{-22} \text{ cm}^3 \).

Use Density to Find Mass of Unit Cell

Use the density formula, density = mass/volume, to calculate the mass of the unit cell: \( 2.33 = \frac{m}{1.60 \times 10^{-22}} \)Solving for mass (m), \( m = 2.33 \times 1.60 \times 10^{-22} \text{ g} = 3.73 \times 10^{-22} \text{ g} \).

Convert Mass to Number of Atoms Using Molar Mass

Find the number of atoms using the molar mass of silicon, which is 28.085 u, or 28.085 g/mol. Using Avogadro's number (\(6.022 \times 10^{23} \text{ atoms/mol}\)): Mass of one atom of Si = \( \frac{28.085}{6.022 \times 10^{23}} \text{ g/atom} = 4.66 \times 10^{-23} \text{ g/atom} \).Number of atoms in unit cell = \( \frac{3.73 \times 10^{-22}}{4.66 \times 10^{-23}} \approx 8 \text{ atoms} \).

Confirm Consistency with FCC Structure

The calculated number of 8 atoms per unit cell needs to be consistent with the expected structure type. For crystalline silicon, taking into account the tetrahedral positioning of atoms in silicon's diamond cubic lattice (a derivative of FCC), this consistency shows the calculated number aligns with 8 effective Si atom positions per unit cell.

Key concepts

Face-Centered Cubic Structure

The face-centered cubic (FCC) structure is a type of crystalline arrangement where each unit cell is constructed with atoms located at each of the cube's corners, and one atom placed on each face of the cube. This special layout makes FCC quite dense and an efficient packing method.

In simpler terms, you can imagine this as having an atom right in the center of each face of the cube.

• Each corner atom is shared among eight different unit cells, contributing to only 1/8th of an atom per corner for any single unit cell.
• Each face-centered atom is shared between two unit cells, contributing 1/2 of an atom per face for one unit cell.

Adding up all these fractional contributions, in an FCC structure, there are effectively 4 whole atoms per unit cell. This is a very efficient way to pack atoms, which is why many metals and elements adopt this structure to fill space optimally and create strong bonds.

Density Calculation

Density is a measure of mass per unit volume. For a crystalline solid like silicon, you need to find out how much mass is packed into a certain volume. You can use the formula:

\[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]To calculate the density of a silicon unit cell, first, it's crucial to have the volume, which is done by cubing the edge length (making sure it's in compatible units like cm). Once you have the volume, you can apply the density formula to find the mass. For silicon:

• Convert the edge length from picometers (pm) to centimeters (cm) to match the density's units.
• Calculate the volume using the edge length to the power of three.
• Determine the mass using the density value and the calculated volume.

Thus, by using the density and the volume of the unit cell, you can find the mass that tells you how compact the material is.

Molar Mass of Silicon

The molar mass of a substance is the weight of a single mole of its entities, such as atoms or molecules. For silicon, this value is 28.085 grams per mole. Molar mass connects the macroscopic mass to the microscopic number of atoms for a given element.

In problems of crystal density, knowing the molar mass is crucial. It lets you convert between the mass of individual atoms and the total mass of a unit cell. Here's how you can use it:

• Determine the mass of a single silicon atom by dividing the molar mass by Avogadro's number (since a mole contains Avogadro's number of atoms).
• This calculation gives the mass in grams for one atom of silicon.

With this atom mass, you can then figure out how many atoms make up the unit cell by dividing the unit cell's mass by the mass of one atom. This relationship bridges the gap between atomic-scale and macroscopic-scale in chemistry.

Avogadro's Number

Avogadro's number, which is \[6.022 \times 10^{23}\] per mole, is a fundamental constant that tells you how many units (atoms, molecules, etc.) are in one mole of a substance. This number allows chemists to translate between the atomic and macroscopic worlds by providing a common reference point.Here's why Avogadro's number is important:

• It provides a way to convert mass into the number of atoms when you know the molar mass. This is especially useful in density and material property calculations.
• It gives a sense of scale. With such a large number, we can understand how many extremely small atoms together account for measurable amounts of material.
• In our problem, we've used it to find out how many silicon atoms comprise the mass of the unit cell. This number helps in finding how densely packed silicon atoms are in its crystalline formation.

Without Avogadro's number, relating everyday quantities of material to their elemental components would be nearly impossible.