Problem 57 - Chemistry and Chemical Reactivity

Question

Sulfur undergoes a phase transition between 80 and \(100^{\circ} \mathrm{C}.\) $$\begin{array}{c} \mathrm{S}_{8}(\text { rhombic }) \rightarrow \mathrm{S}_{8}(\text { monoclinic }) \\ \Delta_{\mathrm{r}} H^{\circ}=3.213 \mathrm{kJ} / \text { mol-rxn } \Delta_{\mathrm{r}} S^{\circ}=8.7 \mathrm{J} / \mathrm{K} \cdot \text { mol- rxn } \end{array}$$ (a) Estimate \(\Delta_{\mathrm{r}} G^{\circ}\) 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 \(\Delta_{r} G^{\circ}=0 .\) What is the significance of this temperature?

Step-by-Step Solution

Answer

Rhombic form is stable at 80°C; monoclinic at 110°C. Transition occurs at 96.16°C.

Step 1: Convert temperature to Kelvin

To use thermodynamic formulas, we need the temperature in Kelvin. Convert the temperatures:\[ T_1 = 80.0^{\circ}C + 273.15 = 353.15 \, K \]\[ T_2 = 110.0^{\circ}C + 273.15 = 383.15 \, K \]

Step 2: Estimate ΔrG° using the Gibbs free energy equation

The standard Gibbs free energy change is calculated using the formula \( \Delta_{r} G^{\circ} = \Delta_{r} H^{\circ} - T \Delta_{r} S^{\circ} \).For \( T_1 = 353.15 \, K \):\[ \Delta_{r} G^{\circ} = 3213 \, \text{kJ/mol} - 353.15 \, K \times 8.7 \, \text{J/(K mol)} \times \frac{1 \, \text{kJ}}{1000 \, \text{J}} = 3213 - 3.072 = 0.141 \, \text{kJ/mol} \]For \( T_2 = 383.15 \, K \):\[ \Delta_{r} G^{\circ} = 3213 \, \text{kJ/mol} - 383.15 \, K \times 8.7 \, \text{J/(K mol)} \times \frac{1 \, \text{kJ}}{1000 \, \text{J}} = 3213 - 3.333 = -0.120 \, \text{kJ/mol} \]

Step 3: Analyze stability based on ΔrG°

At \(80.0^{\circ}C\), \(\Delta_{r} G^{\circ} = 0.141 \, \text{kJ/mol} > 0\), indicating that the rhombic form is more stable.At \(110.0^{\circ}C\), \(\Delta_{r} G^{\circ} = -0.120 \, \text{kJ/mol} < 0\), indicating that the monoclinic form is more stable.

Step 4: Calculate the temperature where ΔrG° is zero

Set \( \Delta_{r} G^{\circ} = 0 \) and solve for temperature \( T \):\[ 0 = \Delta_{r} H^{\circ} - T \Delta_{r} S^{\circ} \]Rearranging gives:\[ T = \frac{\Delta_{r} H^{\circ}}{\Delta_{r} S^{\circ}} = \frac{3213 \, \text{kJ/mol}}{8.7 \, \text{J/(K mol)}} \times \frac{1000 \, \text{J}}{1 \, \text{kJ}} = \frac{3213000}{8.7} = 369.31 \, K \]Convert to Celsius:\[ T = 369.31 - 273.15 = 96.16^{\circ}C \]

Step 5: Interpret the significance of the transition temperature

At \(96.16^{\circ}C\), \( \Delta_{r} G^{\circ} = 0 \), indicating this is the equilibrium temperature at which both the rhombic and monoclinic forms of sulfur are equally stable. Above this temperature, the monoclinic form is more stable, while below it, the rhombic form is more stable.

Key Concepts

Phase Transition

A phase transition involves a substance changing from one state to another, such as solid to liquid, liquid to gas, or within solid phases. For sulfur, this transition is between two solid forms known as rhombic sulfur and monoclinic sulfur.
The transition depends on temperature changes, causing sulfur molecules to rearrange into a different structure.
This process is characterized by an associated change in enthalpy (\(\Delta_r H^{\circ}\)) and entropy (\(\Delta_r S^{\circ}\)).

• The change in enthalpy indicates the heat absorbed or released during the transition.
• The change in entropy reflects the disorder change as the molecular arrangement shifts.

Understanding phase transitions is crucial in chemistry because it affects the stability and properties of materials across different conditions. This is important for processes like crystallization, where correct phase control results in the desired material properties.

Stability of Sulfur

The stability of different forms of sulfur at various temperatures is dictated by the Gibbs free energy change (\(\Delta_r G^{\circ}\)).
This value helps determine which sulfur form is more stable under specific conditions.
At a temperature of 80°C, rhombic sulfur is more stable since \(\Delta_{r} G^{\circ}\) is positive, meaning the conversion to monoclinic sulfur is not favorable at lower temperatures.

• Rhombic sulfur has a more tightly packed structure, making it energetically favorable at lower temperatures.
• At higher temperatures, like 110°C, monoclinic sulfur becomes more stable due to the negative \(\Delta_{r} G^{\circ}\), indicating an energetically favorable transition.

The temperature where Gibbs free energy equals zero (\(\Delta_{r} G^{\circ} = 0\)) is the point where both forms have equal stability. At this equilibrium temperature, estimated at about 96.16°C, neither form is preferred over the other.

Thermodynamic Formulas

Thermodynamic formulas are essential for calculating changes in energy, enthalpy, and entropy during reactions or phase changes.
In the context of sulfur's phase transition, specific equations like \(\Delta_{r} G^{\circ} = \Delta_{r} H^{\circ} - T \Delta_{r} S^{\circ}\) are used to determine the change in Gibbs free energy.
This formula helps predict the direction and favorability of a phase transition.

• \(\Delta_{r} H^{\circ}\) represents the constant pressure enthalpy change during the reaction.
• \(\Delta_{r} S^{\circ}\) is the entropy change providing insight into disorder changes.
• The temperature \(T\) in Kelvin is crucial as it directly affects the balance between enthalpy and entropy contributions to free energy.

By setting \(\Delta_{r} G^{\circ} = 0\), we find the precise temperature where the two different sulfur phases reach equilibrium.
Understanding these formulas is key to solving complex chemical thermodynamics problems.