Problem 92 - Chemistry: An Atoms-Focused Approach

Question

There are two crystalline forms of manganese(II) sulfide \((\mathrm{MnS}):\) the \(\alpha\) form has a rock salt structure, whereas the \(\beta\) form has a sphalerite structure. a. Describe the differences between the two structures of \(\mathrm{MnS}\) b. The ionic radii of \(\mathrm{Mn}^{2+}\) and \(\mathrm{S}^{2-}\) are 67 and \(184 \mathrm{pm}\), respectively. Which type of hole in a ccp lattice of sulfide ions could theoretically accommodate a \(\mathrm{Mn}^{2+}\) \(\sin 3\)

Step-by-Step Solution

Answer

In summary, MnS can crystallize into two structures: rock salt and sphalerite. In the rock salt structure, the Mn2+ cations occupy octahedral holes, whereas in the sphalerite structure, they occupy tetrahedral holes. Given the radius ratio of Mn2+ and S2- ions (0.364), the Mn2+ cations could theoretically accommodate octahedral holes in a ccp lattice of sulfide ions. However, since the radius ratio is close to the lower limit of the octahedral hole range, the crystal structure might exhibit some distortion to accommodate the Mn2+ ions.

Step 1: a. Differences between rock salt and sphalerite structures

The chief differences between the rock salt and sphalerite structures are related to the arrangement of the cations in the crystal lattice. Both structures are derived from the face-centered cubic (fcc) close-packed arrangement of anions. In a rock salt structure, A and B type ions form an edge-shared cubic structure. The anions (in this case, \(\mathrm{S}^{2-}\) ions) are at the vertices of a cube and form an fcc lattice. The cations (in this case, \(\mathrm{Mn}^{2+}\) ions) occupy octahedral holes, meaning each cation is surrounded by six anions, and each anion is surrounded by six cations. The coordination number for both ions is 6. In the sphalerite structure, the cations occupy tetrahedral holes, meaning each cation is surrounded by four anions, and each anion is surrounded by four cations. The coordination number for both ions is 4. In this structure, two types of equally spaced and alternating tetrahedral holes are taken by the cations.

Step 2: b. Determining the suitable type of hole in a ccp lattice

In a close-packed (ccp) lattice, there are three types of holes between the anions: tetrahedral, octahedral, and cubic holes. These holes are essentially spaces between the anions in the lattice which can accommodate a cation. The size of a hole is approximately given by the radius ratio of the cation to the anion: \(\frac{r_\mathrm{cation}}{r_\mathrm{anion}}\). The choice of the type of hole depends on the size of the cations relative to the anions as follows: 1. For radius ratios between 0.155 and 0.225, the tetrahedral hole is favorable. 2. For radius ratios between 0.414 and 0.732, the octahedral hole is favorable. 3. For radius ratios between 0.732 and 1, the cubic hole is favorable. In our problem, we are given the ionic radii of \(\mathrm{Mn}^{2+}\) and \(\mathrm{S}^{2-}\) as 67 pm and 184 pm, respectively. We calculate the radius ratio: \(\frac{r_\mathrm{Mn^{2+}}}{r_\mathrm{S^{2-}}} = \frac{67\,\text{pm}}{184\,\text{pm}} = 0.364\) As the radius ratio falls between 0.414 and 0.732, the \(\mathrm{Mn}^{2+}\) ion can theoretically accommodate an octahedral hole in the ccp lattice of sulfide ions. However, since the given radius ratio is slightly below the range, close to the lower limit, it is possible that the actual MnS structure might exhibit some distortion to accommodate the Mn2+ ions.

Key Concepts

Rock Salt Structure

The rock salt structure is a fascinating arrangement commonly seen in certain ionic compounds like sodium chloride (table salt) and the alpha form of manganese(II) sulfide (\(\alpha\text{-MnS}\)). At the heart of this structure is a face-centered cubic (fcc) lattice, where the anions form a repeating pattern. In the case of \(\alpha\text{-MnS}\), the sulfide ions \(\text{S}^{2-}\) create this structure by positioning themselves at each vertex of the cube.

Within the rock salt structure, cations like \(\text{Mn}^{2+}\) occupy what are called octahedral holes. In simpler terms, each manganese ion is surrounded by six sulfide ions, perfect for creating a stable and balanced environment for the manganese cations. Similarly, each sulfide ion is surrounded by six manganese cations. Therefore, the coordination number, which is the number of neighboring ions surrounding a specific ion, is 6 for both \(\text{Mn}^{2+}\) and \(\text{S}^{2-}\).

Overall, the rock salt structure is a robust lattice arrangement that allows for secure bonding due to this high symmetry and coordination between ions, often resulting in high melting points and distinct physical properties. Understanding this structure is essential for anyone venturing into the field of solid-state chemistry.

Sphalerite Structure

The sphalerite structure is another intriguing crystalline form, seen in the beta form of manganese(II) sulfide (\(\beta\text{-MnS}\)). Unlike the rock salt structure, the sphalerite structure features a different coordination environment for the cations. This time, the cations, such as \(\text{Mn}^{2+}\), dwell in tetrahedral holes.

In this structure, each \(\text{Mn}^{2+}\) is surrounded by only four \(\text{S}^{2-}\) ions, rather than six. Consequently, the coordination number drops to 4. The sulfide ions also coordinate with four manganese ions, maintaining a consistent coordination for both ion types. It's vital to note that the arrangement of the ions forms a repeating pattern resembling a tetrahedron, which gives rise to unique characteristics.

The shift from octahedral coordination in rock salt to tetrahedral coordination in sphalerite comes with its own set of structural properties. The more spacious tetrahedral arrangement can sometimes accommodate slight distortions, which means the lattice can subtly adjust to fit the needs of the ions, especially when there's a critical difference in size between the cations and anions as seen in manganese sulfide.

Ionic Radii

Ionic radii are a crucial concept when analyzing crystal structures because they essentially determine how ions fit into a lattice. Ionic radius is the measure from the center of an ion's nucleus to the outer edge of its electron cloud when in a crystalline lattice.

In our example, the ionic radii for \(\text{Mn}^{2+}\) and \(\text{S}^{2-}\) are given as 67 pm and 184 pm, respectively. These measurements help us understand how these ions can fit into the lattice structures like those of rock salt or sphalerite. The ratio of these radii plays a significant role in predicting which type of sites or 'holes' can accommodate a given ion within a lattice. This is particularly critical when dealing with close-packed arrangements where the correct fit ensures stability in the crystal.

Understanding ionic radii aids in rationalizing why certain compounds prefer specific crystalline forms and how alterations in these radii, perhaps through substitutions or ionic distortions, might affect the stability and properties of the material.

Coordination Number

The concept of coordination number is a cornerstone in understanding crystal structures. It refers to the number of nearest neighbor anions that surround a cation (or vice-versa) in a crystal lattice. This number is pivotal as it influences the stability and geometry of the lattice arrangement.

For the rock salt structure, the coordination number is 6, meaning each \(\text{Mn}^{2+}\) ion is equidistant from six \(\text{S}^{2-}\) ions, resulting in a highly symmetrical and balanced environment. Conversely, in the sphalerite structure, the coordination number is reduced to 4, with each cation adjoined by four anions in a less dense arrangement.

The change in coordination number not only influences the packing efficiency of the lattice but also affects the physical properties of the material. A higher coordination number often correlates with greater density and hardness, whereas a lower coordination number can lead to a more flexible lattice structure capable of subtle distortions, making coordination number a critical consideration for material scientists.