Question

A particular form of cinnabar (HgS) adopts the zinc blende structure. The length of the unit cell edge is $5.852\text{AA}$. (a) Calculate the density of $\mathrm{HgS}$ in this form. (b) The mineral tiemmanite ( $\mathrm{HgSe}$ ) also forms a solid phase with the zinc blende structure. The length of the unit cell edge in this mineral is $6.085\text{AA}$. 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

In summary, cinnabar (HgS) and tiemmanite (HgSe) both adopt the zinc blende structure, but with different unit cell lengths and densities. Cinnabar has a smaller unit cell length and hence higher density due to the smaller size of sulfur atoms compared to selenium atoms in tiemmanite. The larger unit cell length in tiemmanite is due to the larger size of selenium atoms compared to sulfur atoms.

Step-by-step solution

Calculate the density of cinnabar (HgS)

First, we need to determine the number of atoms per unit cell for HgS. Since it adopts the zinc blende structure, it has 4 molecules per unit cell (2 Hg atoms and 2 S atoms). Next, find the volume of the unit cell: Volume = (length of unit cell edge)^3 = $(5.852\times {10}^{-10}{)}^3$ m³ Now, find the total mass of HgS in one unit cell: Mass of 2 Hg atoms = 2 x (200.59 g/mol), Mass of 2 S atoms = 2 x (32.07 g/mol) Total mass of HgS in one unit cell = (2 x (200.59 + 32.07)) g/mol Since density = mass/volume, the density of cinnabar HgS can be calculated as follows: Density = (Total mass of HgS in one unit cell) / (Total volume of unit cell)

Calculate the density of tiemmanite (HgSe)

We will use a similar approach to calculate the density of tiemmanite (HgSe), which also adopts the zinc blende structure. Volume = (length of unit cell edge)^3 = $(6.085\times {10}^{-10}{)}^3$ m³ Mass of 2 Hg atoms = 2 x (200.59 g/mol), Mass of 2 Se atoms = 2 x (78.96 g/mol) Total mass of HgSe in one unit cell = (2 x (200.59 + 78.96)) g/mol Density = (Total mass of HgSe in one unit cell) / (Total volume of unit cell)

Compare the densities, account for larger unit cell length and differences in densities

Now that we have calculated the densities of both HgS and HgSe, we can compare them. If the density value of HgS turns out to be higher than HgSe, we can attribute it to the difference in the size of their constituent atoms, which determines the length of the unit cell edge. Sulfur has a smaller atomic size than Selenium due to the increased effective nuclear charge experienced by the outer electrons. This, in turn, should result in a shorter unit cell length for HgS and a higher density compared to HgSe. So we can sum up the differences in densities by the difference in unit cell lengths. Larger unit cell lengths in tiemmanite (HgSe) can be attributed to the larger atom size of Selenium compared to Sulfur in cinnabar (HgS).

Key concepts

Density Calculation

Density is a crucial factor that helps us understand how much mass is packed within a given volume. In the context of crystalline solids like cinnabar (HgS) and tiemmanite (HgSe), the density can be calculated when we know the mass and volume of a unit cell.
To find the density:

• Determine the number of molecules per unit cell. The zinc blende structure has 4 molecules per unit cell.
• Calculate the mass of these molecules. For example, HgS includes 2 Hg atoms and 2 S atoms, whose combined molar masses will give the total mass of the unit cell.
• Find the volume of the unit cell using the edge length, with the formula: $$\text{Volume}=(\text{edge length}{)}^3$$
• Apply the density formula: $$\text{Density}=\frac{\text{mass of unit cell}}{\text{volume of unit cell}}$$

By plugging in these values, we can compare the densities of HgS and HgSe, which hinge not only on mass differences but also on how compactly the atoms are arranged.

Unit Cell Volume

The unit cell volume is a vital part of understanding crystal structures as it tells us how much space is occupied by the repeating unit of a structure. For minerals with the zinc blende structure, the unit cell is a cube defined by its edge length.
To calculate this volume, you will:

• Use the edge length value specific to the mineral. For example, cinnabar (HgS) has an edge length of $5.852\text{ Å}$, while tiemmanite (HgSe) has $6.085\text{ Å}$.
• Convert this edge length from angstroms to meters by noting that $1\text{ Å}={10}^{-10}\text{ m}$.
• Compute the cube of the edge length to find the volume:$$\text{Volume}=(\text{edge length}{)}^3$$

By calculating the unit cell volume, we grasp how the physical dimensions impact various properties such as density and can see how variations in atomic sizes affect these volumes.

Atomic Size Comparison

When we compare atomic sizes, we look at how different elements, like sulfur (S) and selenium (Se), affect the structure of minerals such as HgS and HgSe. Their atomic sizes impact both the unit cell dimensions and the overall density of these crystals.
Key considerations include:

• Atomic size is influenced by effective nuclear charge, which is the net positive charge experienced by an electron in a multi-electron atom. Sulfur has a higher effective nuclear charge compared to selenium, making it smaller in size.
• This size difference results in a smaller unit cell for HgS compared to HgSe since atoms fit more tightly together.
• A more compact structure like HgS leads to a higher density because the mass is contained within a smaller volume.

Understanding these concepts helps explain why HgS is denser than HgSe, as the larger selenium atoms spread out more, enlarging the unit cell and reducing density.