Question

From the information on the temperature stability of white and gray tin given in this chapter, which form would you expect to have the more ordered structure (have the smaller positional probability)?

Short answer

Based on the temperature stability information, we would expect gray tin to have the more ordered structure and smaller positional probability compared to white tin. This is because gray tin is stable at lower temperatures, where the motion of atoms is less, which corresponds to a more ordered structure and smaller positional probability.

Step-by-step solution

Review properties of white and gray tin

White tin (also called beta tin) is the stable form of tin at room temperature and above, which is metallic and has a tetragonal crystal structure. Gray tin (alpha tin) is stable at low temperatures and is a non-metallic, diamond-cubic crystal structure.

Consider temperature stability

As temperature increases, white tin remains stable, while gray tin converts into white tin. It suggests that white tin has a higher temperature stability.

Determine possible ordered structures

White tin has a higher temperature stability, which means it can maintain its structure even at higher temperatures. On the other hand, gray tin is stable at lower temperatures, where the motion of atoms is less. Since ordered structures usually correspond to less motion of atoms, it is more likely for gray tin to have a more ordered structure.

Compare positional probability

In an ordered structure, atoms have a smaller positional probability (i.e., they are confined to specific positions within the lattice). Given our rationale above that gray tin is likely to have a more ordered structure, we can conclude that gray tin would have a smaller positional probability compared to white tin. So, based on the temperature stability information, we would expect gray tin to have the more ordered structure and smaller positional probability.

Key concepts

White Tin

When we discuss white tin, we're referring to the form of tin that is stable at room temperature and above, known colloquially as beta tin. Its metallic characteristics and tetragonal crystal structure enable it to be used in a variety of applications, from coatings for food cans to solder components.

White tin's structure is less dense than gray tin, and the atoms are arranged in a manner that makes it resistant to changes, even as temperatures rise. As a result, white tin does not easily transform into other structures, which highlights its temperature stability. Understanding the behavior of white tin in various conditions is essential for materials science and industrial applications where temperature variations can be significant.

Gray Tin

Gray tin, known also as alpha tin, possesses a very different structure and properties compared to white tin. Stable at lower temperatures, it exhibits a diamond-cubic crystal arrangement similar to that found in diamond and silicon, leading to its non-metallic appearance.

Unique Transformation

Notably, gray tin can transform into white tin upon heating, this transition is referred to as the tin pest. It's essential for students to understand that gray tin's stability is less robust compared to white tin, partly due to its more ordered crystal structure which is easily disrupted by increased thermal energy.

Temperature Stability of Tin

The stability of tin's various allotropes under changing temperatures is a critical area of focus in material science.

Varying Stability

White tin proves robust as temperatures rise, resisting the transformation into other structures. Conversely, gray tin can only maintain its diamond-cubic structure at cooler temperatures. In fact, below 13.2°C (55.76°F), gray tin is the more stable allotrope.

However, once temperatures push past this threshold, a transformation into white tin occurs, rendering gray tin's low-temperature stability limited. This temperature-responsive behavior is key for those designing products with tin, as the wrong phase at a given temperature can lead to material failure.

Ordered Crystal Structures

Ordered crystal structures refer to the highly organized arrangement of atoms within a crystal lattice. Each atom has a well-defined position, contributing to the overall symmetry and properties of the material.

Significance of Order

An ordered structure means that there is a consistent pattern that repeats itself throughout the material. The degree of order can affect a range of material properties, from electrical conductivity to melting point. In the context of tin, gray tin has a more ordered structure, meaning its atoms exhibit less movement and are more fixed, especially evident at lower temperatures where thermal energy is minimized.