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 \(\mathrm{S}_{\text { rhombic}}(s)\longrightarrow \mathrm{S}_{\text { monoclinic }}(s)\) b. Which form of sulfur has the more ordered crystalline structure (has the smaller positional probability)?

Short answer

a. For the phase change from rhombic to monoclinic crystal structures, ∆H is positive (endothermic process) and ∆S is positive (increased disorder). b. The rhombic crystal structure has a more ordered crystalline structure (smaller positional probability) than the monoclinic crystal structure.

Step-by-step solution

Understanding the relationship between enthalpy and entropy

Enthalpy (H) represents the heat content of a system, while entropy (S) represents the disorder or randomness of a system. The change in enthalpy (∆H) and change in entropy (∆S) depict the changes in heat content and disorder during a process or reaction. Positive ∆H indicates an endothermic process (absorption of heat), while negative ∆H indicates an exothermic process (release of heat). Positive ∆S indicates increased randomness or disorder, while negative ∆S indicates increased order.

Analyzing the phase change

During the phase transition, sulfur transforms from the rhombic crystal structure to the monoclinic crystal structure at temperatures above 95°C. As the temperature increases, this phase change occurs to minimize the overall energy of the system, implying that heat is absorbed in this process.

Predicting the signs of ∆H and ∆S

Since heat is absorbed during the phase change, ∆H will be positive, indicating an endothermic process. The crystal structure transformation suggests a change in the positional probability, or order, of the system. As the temperature increases, the increased kinetic energy of the atoms usually results in a more disordered system. Therefore, we can predict that ∆S is positive, indicating increased disorder. #b. Which form of sulfur has the more ordered crystalline structure (has the smaller positional probability)?#

Comparing the entropy values

A more ordered crystalline structure will have lower entropy or smaller positional probability. Since we predicted that ∆S is positive for the phase transition, this implies that the initial rhombic crystal structure has a lower entropy value compared to the final monoclinic crystal structure.

Identifying the more ordered crystalline structure of sulfur

Based on the provided information, the rhombic crystal structure has a lower entropy value or smaller positional probability, and thus, it possesses the more ordered crystalline structure.

Key concepts

Enthalpy

Enthalpy is a concept used in thermodynamics to express the total heat content of a system at constant pressure. When discussing enthalpy, it's crucial to understand how it affects phase transitions. In a phase transition, such as when sulfur changes structure below its melting point, the change in enthalpy (\(\Delta H\)) indicates whether the process is absorbing or releasing heat. - A positive \(\Delta H\) reflects an endothermic process. This means that the system takes in heat. In our example, sulfur absorbs heat while transitioning from the rhombic to the monoclinic structure.- Conversely, a negative \(\Delta H\) suggests an exothermic process, where the system releases heat.Enthalpy changes are particularly evident in phase changes because they involve breaking and forming of bonds at the atomic level, altering the heat content. Understanding this concept helps us predict how a material like sulfur will behave when it experiences different environmental temperatures.

Entropy

Entropy is a measure of disorder or randomness in a system. It gives insight into the degree of disorder gained or lost during a process, such as a phase transition. Entropy is denoted by the symbol \(S\), and changes in entropy are indicated by \(\Delta S\).- A positive \(\Delta S\) means the system becomes more disordered. This occurs when sulfur changes from a rhombic to a monoclinic structure, as the increased atomic motion at higher temperatures leads to greater disorder.- A negative \(\Delta S\) would indicate a process leading to increased order in the system.During phase transitions, entropy changes because the arrangement of atoms or molecules in the structure varies. The outcome of a phase transition, such as the one experienced by sulfur, can often be predicted by examining the changes in entropy, as more disorder often accompanies an increase in temperature.

Crystalline Structure

Crystalline structures define how atoms are arranged in a solid. The arrangement not only affects the physical appearance of the material but also dictates many of its physical properties. When substances undergo phase transitions, their crystalline structures change. - The rhombic crystal structure of sulfur is more ordered. Hence, it has a smaller positional probability and lower entropy. - The monoclinic crystal structure is less ordered compared to the rhombic structure. As sulfur changes from one form to another, the arrangement of sulfur atoms becomes slightly more disorderly. The difference in crystalline structure can be seen in the spatial arrangement and repetition of unit cells, which are the smallest units defining the structure's geometry. Detailed knowledge of crystalline structures aids in understanding how materials behave under various temperature and pressure conditions, and why some phase transitions require input or output of energy.