Question

It is quite common for a solid to change from one structure to another at a temperature below its melting point. For example, sulfur undergoes a phase change from the rhombic crystal structure to the monoclinic crystal form at temperatures above \(95^{\circ} \mathrm{C}.\) a. Predict the signs of \(\Delta H\) and \(\Delta S\) for the process \(S_{\text {rhombic }}(s) \longrightarrow S_{\text {monoclinic }}(s).\) b. Which form of sulfur has the more ordered crystalline structure (has the smaller positional probability)?

Short answer

a. For the process \(S_{\text {rhombic }} (s) \longrightarrow S_{\text {monoclinic }} (s)\), \(\Delta H >0\) and \(\Delta S >0\). b. The rhombic form of sulfur has the more ordered crystalline structure (smaller positional probability).

Step-by-step solution

Predict the signs of \(\Delta H\) and \(\Delta S\) for the phase change

As the temperature increases, sulfur undergoes a phase change from rhombic crystal structure to monoclinic crystal structure. Generally, during a phase change, if the new phase requires more energy to be stable, then the heat of transformation is positive, i.e., \(\Delta H >0\). However, if the new phase requires less energy to be stable, then the heat of transformation is negative, i.e., \(\Delta H <0\). Now let's consider entropy. If the new phase is less ordered, the entropy increases, i.e., \(\Delta S >0\). If the new phase is more ordered, the entropy decreases, i.e., \(\Delta S <0\). In this case, the phase change occurs when the temperature increases. A higher temperature suggests that sulfur in the monoclinic crystal form can absorb more energy than the sulfur in the rhombic crystal form. So, during the phase change, the heat transferred to the system is more significant in the monoclinic form, which means that \(\Delta H >0\). Similarly, the higher temperature in the monoclinic form means that the crystal structure of sulfur in the monoclinic form is less ordered than in rhombic form, which implies that \(\Delta S >0\). Hence, for the process \(S_{\text {rhombic }} (s) \longrightarrow S_{\text {monoclinic }} (s)\), the sign of \(\Delta H\) is positive, and the sign of \(\Delta S\) is positive.

Determine the more ordered crystalline structure

We have already established that the monoclinic form has a less ordered crystalline structure than the rhombic form due to the positive \(\Delta S\). This means that the rhombic form has the more ordered crystalline structure and smaller positional probability. Answer: a. The signs of changes in enthalpy and entropy are positive: \(\Delta H >0\) and \(\Delta S >0\). b. The rhombic form of sulfur has a more ordered crystalline structure (smaller positional probability).

Key concepts

Solid-State Transformations

Solid-state transformations involve a change in the structure of a solid material without a change in its state, i.e., the material remains solid. These transformations can be induced by various factors such as changes in temperature, pressure, or the introduction of impurities. During such a transformation, a solid can rearrange its atoms or molecules to form a different crystal structure. This process typically involves a change in energy, displayed as a difference in enthalpy, and an increase or decrease in order, represented by entropy changes.

For example, sulfur experiences such a transformation when it changes from rhombic to monoclinic crystals. This process is fascinating because it happens while the substance is still in the solid-state, which allows us to observe changes in physical properties connected to the crystal structure, such as density and electrical conductivity, without the material transitioning to a liquid or gas.

Crystalline Structure

A crystalline structure is a highly ordered arrangement of atoms, ions, or molecules in a solid material. This structure extends in a repeating pattern, both in two and three-dimensional space. The regularities in crystal structures are often reflected in the external symmetry of the crystals themselves, with flat faces and sharp edges. However, there are many types of crystal lattices, each with its own specific geometrical arrangement and properties.

Different polymorphs of the same material, such as sulfur, have different crystalline structures. These structural differences directly influence the material's physical properties, like melting point, solubility, and hardness. Therefore, understanding an element's crystalline structure isn't just an abstract concept; it's crucial for predicting and making use of its physical behaviors.

Enthalpy (ΔH)

Enthalpy (ΔH) is a measure of the total heat content of a thermodynamic system. It reflects the energy required for a transformation at constant pressure and temperature. When analyzing phase changes, especially solid-state transformations like the one between different forms of sulfur, ΔH indicates whether the process is endothermic (absorbing heat, ΔH > 0) or exothermic (releasing heat, ΔH < 0).

In the case of the phase change from rhombic to monoclinic sulfur, the process is endothermic; sulfur absorbs heat to rearrange its structure, resulting in a positive value for ΔH. Monitoring changes in enthalpy allows scientists and engineers to predict how a substance will respond to different conditions and to design processes accordingly, ensuring safety and efficiency.

Entropy (ΔS)

Entropy (ΔS) is a fundamental concept in thermodynamics, representing the measure of disorder or randomness of a system. A key point to understand is that natural processes tend to move towards a state of higher entropy. In a solid-state transformation, the change in entropy indicates whether the atoms or molecules become more or less orderly. An increase in entropy (ΔS > 0) signifies a transition towards a more disordered state, while a decrease (ΔS < 0) indicates a move to a more ordered structure.

With the sulfur phase change from rhombic to monoclinic, the disorder increases as the temperature rises, implying a positive change in entropy. This concept teaches us that the inherent chaos within materials can actually be quantified and helps to grasp why certain physical changes happen under specific conditions.

Rhombic vs Monoclinic Sulfur

Rhombic and monoclinic sulfur are two allotropes of the element sulfur, meaning they are forms of the same element with different molecular structures and physical properties. Rhombic sulfur, also known as alpha-sulfur, is stable at room temperature and has an orthorhombic lattice, which is highly ordered. Monoclinic sulfur or beta-sulfur forms at higher temperatures and has monoclinic lattice crystals.

The key difference between the two relates to their crystalline structures and stability at various temperatures. The transition from rhombic to monoclinic sulfur is accompanied by an increase in both enthalpy and entropy, signifying an endothermic reaction and a rise in disorder. Between the two, rhombic sulfur is more ordered, as confirmed by its smaller positional probability and more structured lattice, which has a significant impact on the physical and chemical behavior of sulfur in practical applications.