Question

Sulfur undergoes a phase transition between 80 and \(100^{\circ} \mathrm{C}$$$\mathrm{S}_{8}(\text { rhombic }) \rightarrow \mathrm{S}_{8}(\text { monoclinic })$$ $$\Delta_{\mathrm{r}} H^{\circ}=3.213 \mathrm{kJ} / \mathrm{mol}-\mathrm{rxn} \quad \Delta_{\mathrm{r}} S^{\circ}=8.7 \mathrm{J} / \mathrm{K} \cdot \mathrm{mol}-\mathrm{rxn}$$ (a) Estimate \)\Delta_{r} G^{\circ}\( for the transition at \)80.0^{\circ} \mathrm{C}\( and \)110.0^{\circ} \mathrm{C} .\( What do these results tell you about the stability of the two forms of sulfur at each of these temperatures? (b) Calculate the temperature at which \)\Delta_{\mathrm{r}} G^{\circ}=0$ What is the significance of this temperature?

Short answer

At 80°C, the rhombic form is more stable; at 110°C, the monoclinic form is more stable. The transition temperature is 96.16°C, where both forms are equally stable.

Step-by-step solution

Define the Gibbs Free Energy Equation

The change in Gibbs Free Energy, \( \Delta_{r} G^{\circ}\), for a reaction is related to the enthalpy and entropy changes by the equation: \[\Delta_{r} G^{\circ} = \Delta_{r} H^{\circ} - T \Delta_{r} S^{\circ}\]where \( T \) is the temperature in Kelvin.

Calculate \( \Delta_{r} G^{\circ} \) at 80°C

Convert 80°C to Kelvin: \[ T = 80 + 273.15 = 353.15 \text{ K} \]Insert the values into the Gibbs equation:\[\Delta_{r} G^{\circ} = 3.213 \text{ kJ/mol} - (353.15 \text{ K} \times 0.0087 \text{ kJ/K} \cdot \text{mol}) \]\[= 3.213 \text{ kJ/mol} - 3.073 \text{ kJ/mol} = 0.140 \text{ kJ/mol}\]So, \( \Delta_{r} G^{\circ} = 0.140 \text{ kJ/mol} \) at 80°C.

Calculate \( \Delta_{r} G^{\circ} \) at 110°C

Convert 110°C to Kelvin: \[ T = 110 + 273.15 = 383.15 \text{ K} \]Insert the values into the Gibbs equation:\[\Delta_{r} G^{\circ} = 3.213 \text{ kJ/mol} - (383.15 \text{ K} \times 0.0087 \text{ kJ/K} \cdot \text{mol}) \]\[= 3.213 \text{ kJ/mol} - 3.333 \text{ kJ/mol} = -0.120 \text{ kJ/mol}\]So, \( \Delta_{r} G^{\circ} = -0.120 \text{ kJ/mol} \) at 110°C.

Analyze Stability at Given Temperatures

At 80°C, \( \Delta_{r} G^{\circ} \) is positive, indicating that the rhombic form is more stable. At 110°C, \( \Delta_{r} G^{\circ} \) is negative, indicating that the monoclinic form is more stable.

Calculate Temperature at Which \( \Delta_{r} G^{\circ} = 0 \)

Set \( \Delta_{r} G^{\circ} = 0 \) and solve for T:\[0 = \Delta_{r} H^{\circ} - T \Delta_{r} S^{\circ}\]\[T = \frac{\Delta_{r} H^{\circ}}{\Delta_{r} S^{\circ}} = \frac{3.213 \text{ kJ/mol}}{0.0087 \text{ kJ/K} \cdot \text{mol}} = 369.31 \text{ K}\]Convert to Celsius:\[ T = 369.31 - 273.15 = 96.16^{\circ} \text{C} \]

Interpret the Temperature \( \Delta_{r} G^{\circ} = 0 \)

The temperature where \( \Delta_{r} G^{\circ} = 0 \) (96.16°C) is the equilibrium or transition temperature where both phases are equally stable.

Key concepts

Phase Transition

A phase transition occurs when a substance changes from one state of matter to another, such as solid to liquid or solid to another solid form.
In the context of sulfur, this transition is between two distinct crystalline forms: rhombic and monoclinic sulfur.
These are different allotropes, or structural forms, of the same element.During this transition, sulfur shifts its internal structure, affecting its stability and thermodynamic properties.
This can be observed in changes in Gibbs Free Energy ( \( \Delta_{r} G^{\circ} \)), where stability at different temperatures can identify which form prevails.
As demonstrated in the original exercise, rhombic sulfur is more stable at lower temperatures (80°C), while monoclinic sulfur becomes more stable at higher temperatures (110°C). The temperature at which both forms are equally stable indicates the transition point. This is where the enthalpy and entropy changes balance out, leading to a \( \Delta_{r} G^{\circ} \) of zero.
Beyond this point, the sulfur transitions into the more stable phase for those conditions.

Enthalpy Change

Enthalpy change ( \( \Delta_{r} H^{\circ} \)) refers to the heat absorbed or released during a chemical process at constant pressure.
It measures the total heat content of a system and plays a central role in phase transitions.In the phase transition of sulfur, the enthalpy change value given is 3.213 kJ/mol. This positive value indicates that energy is absorbed during the transition from the rhombic to the monoclinic phase.
In other words, the transition is endothermic, requiring heat to proceed. Understanding enthalpy change helps us comprehend the energy dynamics involved in phase transitions and predict material behavior under different conditions.
Alongside entropy, it determines the Gibbs Free Energy and thus influences the stability and feasibility of the phase transition.

Entropy Change

Entropy change ( \( \Delta_{r} S^{\circ} \)) is a measure of the disorder or randomness within a system.
For the sulfur transition, this change is given as 8.7 J/K·mol, suggesting an increase in entropy.An increase in entropy during the transition indicates that the monoclinic form of sulfur is more disordered compared to the rhombic form.
This additional disorder can be attributed to changes in arrangement at the atomic level as the structure becomes more complex or less symmetrical. Entropy change is crucial in determining how spontaneous a reaction can be, especially in conjunction with enthalpy change.
When both are considered in the Gibbs Free Energy equation, they provide a complete picture of the material's behavior during phase transitions. Spontaneity favors processes that increase the total entropy of the universe, making the sulfur phase transition more favorable as temperature increases.
This reflects why monoclinic sulfur is more stable at higher temperatures.

Sulfur Allotropes

Sulfur appears in several different structural forms, known as allotropes.
The two allotropes mentioned in the context of this exercise are rhombic sulfur and monoclinic sulfur. Rhombic sulfur is the most stable form at room temperature and is composed of eight sulfur atoms linked in a puckered ring structure.
As temperature increases, rhombic sulfur transitions to monoclinic sulfur, another crystalline structure with slightly different bonding and spatial arrangement. Allotropes exhibit varied properties, such as melting points, densities, and interatomic distances.
These differences arise from how atoms bond and orient in space, leading to different crystalline forms or amorphous structures. Understanding sulfur allotropes is essential for not only studying phase transitions but also for applications where sulfur's reactivity and stability are pivotal.
This knowledge is vital for industries ranging from pharmaceuticals to materials science, where sulfur's unique properties are harnessed.