Problem 87 - Chemistry: An Atoms-Focused Approach

Question

Two allotropes ( \(A\) and \(B\) ) of sulfur interconvert at \(369 \mathrm{K}\) and 1 atm pressure:$$\mathrm{S}_{\mathrm{s}}(s, \mathrm{A}) \rightarrow \mathrm{S}_{\mathrm{g}}(s, \mathrm{B})$$.The enthalpy change in this transition is \(297 \mathrm{J} / \mathrm{mol}\). What is the entropy change?

Step-by-Step Solution

Answer

Answer: The entropy change for the transition between the two allotropes of sulfur is 8.05 x 10^-4 kJ/mol·K.

Step 1: Convert given enthalpy change to kJ/mol

To convert the given enthalpy change from J/mol to kJ/mol, divide it by 1000: $$\Delta H = \frac{297 \mathrm{J/mol}}{1000} = 0.297 \mathrm{kJ/mol}$$

Step 2: Use the equation for Gibbs free energy change to find the entropy change

Since we are at equilibrium, the Gibbs free energy change is zero, so we can write the equation as: $$0 = \Delta H - T \Delta S$$ Now plug in the known values for enthalpy change and temperature: $$0 = 0.297 \mathrm{kJ/mol} - 369 \mathrm{K} \Delta S$$

Step 3: Solve the equation for entropy change

To solve the equation for the entropy change, first divide both sides of the equation by \(-369 \mathrm{K}\): $$\Delta S = \frac{0.297 \mathrm{kJ/mol}}{369 \mathrm{K}}$$

Step 4: Calculate the entropy change

Finally, perform the calculation: $$\Delta S = \frac{0.297 \mathrm{kJ/mol}}{369 \mathrm{K}} = 8.05 \times 10^{-4} \mathrm{kJ/mol \cdot K}$$ Thus, the entropy change for this transition between the two allotropes of sulfur is \(8.05 \times 10^{-4} \mathrm{kJ/mol \cdot K}\).

Key Concepts

Gibbs Free Energy

Gibbs free energy is a concept in thermodynamics that helps us understand whether a chemical process or reaction will occur spontaneously. It is denoted by the letter \( G \) and is defined as \( G = H - TS \), where \( H \) is enthalpy, \( T \) is temperature, and \( S \) is entropy.
In simple terms, Gibbs free energy tells us about the balance between the system's heat content and the disorder it creates.

• If \( \Delta G \) is negative, the reaction proceeds spontaneously.
• If \( \Delta G \) is positive, the reaction is non-spontaneous.
• If \( \Delta G \) is zero, the system is at equilibrium.

In the exercise above, we found that we were at equilibrium because the Gibbs free energy change was zero. This situation occurs when the enthalpy change is perfectly balanced by the entropy change times the temperature, meaning no net change in energy is driving the reaction forward. Understanding how Gibbs free energy operates helps us predict and control chemical processes effectively.

Enthalpy Change

Enthalpy change, represented as \( \Delta H \), is the heat change at constant pressure during a chemical reaction. It's a measure of total energy within a system, accounting for both internal energy and the energy to "make space" for molecules against atmospheric pressure.
In our example, the enthalpy change was given as 297 J/mol, which we converted to 0.297 kJ/mol. This conversion makes it easier to calculate the entropy change later, as temperatures in thermodynamics are often given in large units that match better with kJ/mol.

• Enthalpy change can be positive, indicating an endothermic reaction where heat is absorbed.
• It can also be negative, representing an exothermic reaction where heat is released.

For the transition between the two allotropes of sulfur, a small positive enthalpy change suggests that the reaction slightly absorbs heat. This energy absorption is necessary for the rearrangement of sulfur’s molecular structure between its different forms.

Allotropes of Sulfur

Allotropes are different structural forms of the same element. Sulfur, being a highly versatile element, has several allotropes, each with a unique arrangement of its atoms.
The two allotropes mentioned in the exercise, \( A \) and \( B \), represent different arrangements or phases of solid sulfur. This interconversion of allotropes is a fascinating process due to the unique properties each one holds.

• One well-known allotrope of sulfur is rhombic sulfur, also known as \( \alpha \)-sulfur, which is stable at room temperature.
• Another form is monoclinic sulfur, or \( \beta \)-sulfur, with a higher stability temperature.

Understanding the nature of sulfur allotropes is crucial in industries like vulcanization, which uses sulfur to harden rubber.Recognizing these changes at a structural level deepens our understanding of why elements behave as they do under different environmental conditions.