Problem 99 - Chemistry The Central Science

Question

Tin exists in two allotropic forms: Gray tin has a diamond structure, and white tin has a close-packed structure. One of these allotropic forms is a semiconductor with a small band gap while the other is a metal. Which one is which? Which form would you expect to have the longer \(\mathrm{Sn}-\mathrm{Sn}\) bond distance?

Step-by-Step Solution

Answer

In conclusion, gray tin is the semiconductor with a small band gap due to its diamond covalent structure, while white tin is the metal because of its close-packed structure with strong metallic bonds. Additionally, gray tin is expected to have a longer Sn-Sn bond distance than white tin, as weaker covalent bonding results in a larger bond distance compared to the stronger metallic bonding found in white tin.

Step 1: Identifying the Semiconductor and Metal based on Structure

Gray tin has a diamond structure, which is a type of covalent semiconductor. In this structure, covalent bonds are formed between tin atoms, leading to a small band gap due to the presence of localized electron states. White tin has a close-packed structure, which is typical of metals. In this structure, the tin atoms are densely packed with strong metallic bonds that allow free movement of electrons. This leads to strong electrical conductivity, which is a characteristic of metals. Thus, gray tin is the semiconductor with a small band gap, and white tin is the metal.

Step 2: Comparing Sn-Sn Bond Distances in Gray and White Tin

Next, we can compare the Sn-Sn bond distances in the two allotropic forms to determine which one has the longer bond distance. In general, a longer bond distance indicates weaker bonding between the atoms. For gray tin with a diamond structure, the covalent bonds formed between the tin atoms lead to a more rigid and localized bonding arrangement. On the other hand, the close-packed structure of white tin leads to strong metallic bonds formed by delocalized electrons. Since covalent bonds are typically formed in semiconductors like gray tin, and metallic bonds are typically formed in metals like white tin, we can infer that the Sn-Sn bond distance is longer in gray tin due to the weaker covalent bonding compared to the stronger metallic bonding in white tin. In conclusion, gray tin is the semiconductor with a small band gap while white tin is the metal. Gray tin is expected to have a longer Sn-Sn bond distance than white tin due to the weaker covalent bonding.

Key Concepts

Semiconductor Band Gap

The semiconductor band gap is a crucial concept in understanding materials like gray tin. Semiconductors are distinguished from metals by their band gaps, which are energy differences between the valence band and the conduction band. This gap restricts the flow of electrons under normal conditions but allows them to flow when energy is applied. Gray tin has a small band gap because of its diamond structure, where electrons are tightly bound within covalent bonds. The localized nature of these bonds results in electron states being localized too, creating this gap. In contrast, materials with no band gap, like white tin, are metals and allow free-flowing of electrons, resulting in excellent electrical conductivity. When examining gray tin, remember the band gap isn't as wide as in insulators, allowing it to conduct electricity once a specific energy level is overcome. This property is what makes gray tin function as a semiconductor.

Covalent and Metallic Bonds

Understanding the types of bonds within elements such as tin can reveal much about their properties. Tin illustrates both covalent and metallic bonding in its two allotropic forms. **Covalent Bonds in Gray Tin**: Gray tin exhibits covalent bonding similar to that in silicon or diamond. Atoms share electrons equally, forming strong directional bonds. These bonds create a rigid structure that's harder to deform. The localized electron states from these bonds result in a restricted electron flow, contributing to the small band gap characteristic of semiconductors. **Metallic Bonds in White Tin**: On the other hand, white tin is characterized by metallic bonds. In this structure, electrons are not localized, rather they move freely across atoms, contributing to high electrical and thermal conductivity. The close-packed atomic arrangement allows for a dense structure where these free electrons can flow easily, a hallmark of metal properties. This results in white tin's metallic nature, without a band gap.

Tin Bond Distance

The bond distance between atoms can tell us a lot about the strength and type of bonding. This is particularly insightful when comparing the allotropic forms of tin, gray and white. **Bonding in Gray Tin**: In gray tin, the diamond structure leads to longer bonds. This happens because the covalent bonds, while strong in their rigidity, result in weaker overall attraction between neighboring atoms compared to metallic bonds. Hence, the Sn-Sn bond distance in gray tin is longer. **Bonding in White Tin**: Conversely, white tin exhibits shorter Sn-Sn bond distances due to its structure dominated by metallic bonds. Here, atoms are tightly packed together, allowing strong metallic bonding. This results in a compact structure and shorter bond distances. In essence, discerning the bond distances in different allotropes of tin helps us understand why gray tin has weaker covalent bonds leading to longer Sn-Sn distances, whereas, white tin's metallic bonds result in a denser and tighter atomic arrangement.