Problem 65 - Chemistry and Chemical Reactivity

Question

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

Step-by-Step Solution

Answer

S₈ (rhombic) is more stable at 80°C; S₈ (monoclinic) is more stable at 110°C. Transition temperature is about 96.51°C.

Step 1: Convert temperature to Kelvin

First, convert the temperatures from Celsius to Kelvin. The formula to convert Celsius to Kelvin is: \[ T(K) = T(^{ullet}C) + 273.15 \] For 80.0°C, \[ T_1 = 80.0 + 273.15 = 353.15 \, K \] For 110.0°C, \[ T_2 = 110.0 + 273.15 = 383.15 \, K \]

Step 2: Calculate ΔG° for each temperature

The formula to calculate Gibbs free energy change is: \[ \Delta G^{\circ} = \Delta H^{\circ} - T \cdot \Delta S^{\circ} \](a) At 80.0°C (353.15 K):\[ \Delta G^{\circ}_{80} = 3213 \, \text{J/mol} - 353.15 \, \text{K} \cdot 8.7 \, \text{J/K} = 3213 - 3071.8055 = 141.1945 \, \text{J/mol} \](b) At 110.0°C (383.15 K):\[ \Delta G^{\circ}_{110} = 3213 \, \text{J/mol} - 383.15 \, \text{K} \cdot 8.7 \, \text{J/K} = 3213 - 3331.905 = -118.9055 \, \text{J/mol} \]

Step 3: Interpretation of ΔG° values

At 80.0°C, \( \Delta G^{\circ} = 141.1945 \, \text{J/mol} \) is positive, indicating that S₈ (rhombic) is more stable than S₈ (monoclinic) at this temperature. At 110.0°C, \( \Delta G^{\circ} = -118.9055 \, \text{J/mol} \) is negative, indicating that S₈ (monoclinic) is more stable at this temperature.

Step 4: Calculate equilibrium temperature where ΔG°=0

Set \( \Delta G^{\circ} = 0 \). Use the equation \[ 0 = \Delta H^{\circ} - T \cdot \Delta S^{\circ} \]Solve for \( T \):\[ T = \frac{\Delta H^{\circ}}{\Delta S^{\circ}} = \frac{3213 \, \text{J/mol}}{8.7 \, \text{J/K}} \approx 369.66 \, \text{K} \]Convert this temperature back to Celsius:\[ T(^{\bullet}C) = 369.66 - 273.15 \approx 96.51^{\circ}C \]

Step 5: Significance of equilibrium temperature

The temperature \( 96.51^{\circ}C \) is the transition point at which both phases of sulfur, rhombic and monoclinic, are in equilibrium, meaning neither phase is more stable than the other.

Key Concepts

Gibbs Free Energy

Gibbs free energy, denoted as \( \Delta G \), is a thermodynamic quantity that helps us understand the spontaneity of a process under constant temperature and pressure. When \( \Delta G \) is negative, the process is spontaneous, meaning it will occur without needing additional energy. Conversely, a positive \( \Delta G \) indicates that the process is non-spontaneous under the given conditions.
For the sulfur phase transition, we calculate \( \Delta G^{\circ} \) using the equation: \[ \Delta G^{\circ} = \Delta H^{\circ} - T \cdot \Delta S^{\circ} \]where \( \Delta H^{\circ} \) is the change in enthalpy and \( \Delta S^{\circ} \) is the change in entropy. In the case of sulfur transitioning from rhombic to monoclinic:

• At \( 80.0^{\circ} \mathrm{C} \), \( \Delta G^{\circ} \) is positive, suggesting that the rhombic form is more stable.
• At \( 110.0^{\circ} \mathrm{C} \), \( \Delta G^{\circ} \) becomes negative, demonstrating that the monoclinic form is more stable.

These results illustrate how Gibbs free energy changes can indicate phase stability.

Equilibrium Temperature

The equilibrium temperature is a critical point where the Gibbs free energy change \( \Delta G^{\circ} \) equals zero. At this temperature, the tendencies for forward and reverse reactions are equal, meaning neither phase has a stability advantage over the other.
In the given problem, the equilibrium temperature for the sulfur phase transition is calculated using the formula:\[ T = \frac{\Delta H^{\circ}}{\Delta S^{\circ}} \]Substituting our values gives us approximately \( 369.66 \text{ K} \), or \( 96.51^{\circ} \text{C} \). At this temperature, both the rhombic and monoclinic forms of sulfur are equally stable.
This concept of equilibrium temperature is helpful in defining the conditions under which a substance changes phase, helping us predict material behavior in response to temperature changes.

Enthalpy and Entropy

Enthalpy (\( \Delta H \)) and entropy (\( \Delta S \)) are two fundamental thermodynamic concepts that contribute to determining the Gibbs free energy.1. **Enthalpy (\( \Delta H \))**: This is the heat content of a system at constant pressure. It represents the energy absorbed or released during a phase transition. - In our sulfur example, \( \Delta H_{\text{rxn}}^{\circ} = 3213 \text{ J/mol} \) indicates the energy required for the phase change from the rhombic to monoclinic form.
2. **Entropy (\( \Delta S \))**: This is a measure of disorder or randomness in the system. - The given reaction entropy \( \Delta S_{\text{rxn}}^{\circ} = 8.7 \text{ J/K} \) suggests increased randomness from the rhombic to the monoclinic form.
Together, these quantities define how much energy is transformed into usable work or absorbed as heat, vital for assessing phase changes and equilibrium conditions.

Sulfur Polymorphs

Sulfur is unique because it can exist in different structural forms known as polymorphs. The most common polymorphs of sulfur are rhombic and monoclinic.- **Rhombic Sulfur**: It is the more stable form at lower temperatures. Its crystal lattice is characterized by a well-defined, orthogonal crystal structure which keeps the sulfur atoms tightly packed.
- **Monoclinic Sulfur**: This form becomes stable at higher temperatures. It has a less symmetrical crystal lattice, allowing more freedom of atomic movement and resulting in a higher entropy state.The phase transition from rhombic to monoclinic involves changes in both the enthalpy and entropy of the system. At specific temperatures, like the calculated equilibrium point at \( 96.51^{\circ} \text{C} \), these sulfur polymorphs coexist in balance.
Understanding these sulfur polymorphs provides insight into how temperature influences crystal structure stability and transitions in materials.