Question

A particular form of cinnabar (HgS) adopts the zinc blende structure. The length of the unit cell edge is 5.852 A. (a) Calculate the density of HgS in this form. (b) The mineral tiemannite (HgSe) also forms a solid phase with the zinc blende structure. The length of the unit cell edge in this mineral is 6.085 A. What accounts for the larger unit cell length in tiemmanite? (c) Which of the two substances has the higher density? How do you account for the difference in densities?

Short answer

The density of cinnabar (HgS) is 7.72 g/cm³. Tiemannite (HgSe) has a larger unit cell length due to the larger size of the selenium atom compared to sulfur in HgS. HgSe has a higher density of 8.23 g/cm³, which can be attributed to the higher atomic weight of selenium compared to sulfur, resulting in a greater mass for the unit cell.

Step-by-step solution

Calculate the molecular weight of HgS and HgSe

First, let's calculate the molecular weight of the substances HgS (cinnabar) and HgSe (tiemannite). This will be useful for calculating their densities. For HgS: Molecular weight (HgS) = Atomic weight of Hg + Atomic weight of S = 200.59 g/mol (Hg) + 32.07 g/mol (S) = 232.66 g/mol (HgS) For HgSe: Molecular weight (HgSe) = Atomic weight of Hg + Atomic weight of Se = 200.59 g/mol (Hg) + 78.96 g/mol (Se) = 279.55 g/mol (HgSe)

Calculate moles per unit cell

In the zinc blende structure, there are 4 formula units of the compound per unit cell. Thus, the moles per unit cell for both HgS and HgSe are: Moles per unit cell = 4 × (1 formula unit / Avogadro's number) = 4 × (1 / (6.022 x 10^23)) = 6.644 x 10^(-24) moles

Calculate the volume of the unit cell

Now, we will calculate the volume of the unit cell for both compounds, using the unit cell length provided in the exercise. For HgS: a (unit cell edge) = 5.852 Å Convert Å to cm: 1 Å = 10^(-8) cm 5.852 Å = 5.852 x 10^(-8) cm Volume (HgS) = (5.852 × 10^(-8) cm)^3 = 2.002 x 10^(-22) cm^3 For HgSe: a (unit cell edge) = 6.085 Å 6.085 Å = 6.085 x 10^(-8) cm Volume (HgSe) = (6.085 × 10^(-8) cm)^3 = 2.257 x 10^(-22) cm^3

Calculate the densities of HgS and HgSe

Now, we will calculate the densities of both compounds using the molecular weight, moles per unit cell, and volume of the unit cell. Density = (mass of unit cell) / (volume of unit cell) Mass of unit cell = moles per unit cell × molecular weight For HgS: mass (HgS) = (6.644 x 10^(-24) moles) × (232.66 g/mol) = 1.545 x 10^(-21) g Density (HgS) = (1.545 x 10^(-21) g) / (2.002 x 10^(-22) cm^3) = 7.72 g/cm^3 For HgSe: mass (HgSe) = (6.644 x 10^(-24) moles) × (279.55 g/mol) = 1.857 x 10^(-21) g Density (HgSe) = (1.857 x 10^(-21) g) / (2.257 x 10^(-22) cm^3) = 8.23 g/cm^3

Analyzing the results

a) The density of HgS is calculated to be 7.72 g/cm^3. b) Tiemannite has a larger unit cell length than cinnabar because the selenium atom in HgSe is larger than the sulfur atom in HgS. This causes an increase in the unit cell edge length to accommodate the larger atom in the zinc blende structure. c) The density of HgSe was calculated to be 8.23 g/cm^3, which is higher than the density of HgS. The difference in density can be accounted for by the fact that selenium has a higher atomic weight than sulfur (78.96 g/mol for Se versus 32.07 g/mol for S). As a result, the unit cell of HgSe has more mass, leading to a higher density.

Key concepts

Zinc Blende Structure

The zinc blende structure is a type of crystal lattice arrangement common in many semiconductors and minerals. It's named after the mineral sphalerite, which is a form of zinc sulfide (ZnS). In zinc blende, each atom is tetrahedrally coordinated, meaning each atom is surrounded by four other atoms in the shape of a tetrahedron.

This structure is characterized by its face-centered cubic (FCC) lattice in which there are typically two different types of atoms. Within the lattice:

• Each atom of one type is bonded to four atoms of the other type, forming a sturdy 3D network.
• It's known for having excellent optical and electronic properties, which make it important in the field of materials science.

The typical axes of this structure are around 5 Å (angstroms), but this can vary depending on the atoms involved, such as sulfur versus selenium.

Molecular Weight

Molecular weight, also known as molecular mass, is the sum of the atomic weights of all atoms in a given molecule. It is commonly measured in grams per mole (g/mol). For compounds like cinnabar (HgS) and tiemannite (HgSe), calculating molecular weight is a crucial step in understanding their properties.

Here's how we determine molecular weight:

• Add up the atomic weights of the constituent elements in the compound. For instance, Hg (mercury) weighs around 200.59 g/mol.
• For sulfur (S), the weight is approximately 32.07 g/mol, leading to the molecular weight of HgS being 232.66 g/mol.

Similarly, the presence of selenium (Se) in HgSe increases its molecular weight to about 279.55 g/mol because selenium itself has a weight of 78.96 g/mol. Understanding molecular weight is essential for calculating other properties, such as density.

Unit Cell

A unit cell is the smallest repeating pattern in a crystal lattice that shows the entire structure's symmetry and properties. It serves as the basic building block of the crystal. By repeating this unit cell in three dimensions, you can recreate the entire crystal structure. Understanding unit cells helps in discerning characteristics such as density and packing efficiency.

The unit cell's main features include:

• The edge lengths (distances between the corners of the cell) dictate the size and shape of the cell.
• Having precise angles between these edges ensures uniformity in repeating the pattern across the crystal.

In zinc blende structures, each unit cell contains four formula units of the compound, contributing to its overall properties when it expands into a larger crystal. For properties like density, the unit cell size is key because a larger edge length can result in a larger, but possibly less dense, structure depending on the atoms involved.

Atomic Weight

Atomic weight is the average mass of an atom of an element, typically expressed in atomic mass units (u or amu). This concept is foundational for calculating both molecular weight and, subsequently, the properties of compounds.

Key points about atomic weight include:

• It considers the isotopic composition of an element, which is why the atomic weight of mercury (Hg) is noted as approximately 200.59 u.
• Sulfur (S) has an atomic weight of about 32.07 u, while selenium (Se) has a higher atomic weight of 78.96 u.

These atomic weights directly affect the molecular weight calculation and influence properties like density. In compounds, heavier atoms like selenium lead to higher molecular weights relative to those with lighter atoms like sulfur. This also affects the physical properties, such as the density of the resulting materials.