Sulfur has about 20 different allotropes. The most common are rhombic sulfur (the stable form at \(25^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) ) and monoclinic sulfur. They differ in their crystal structures. Given $$ \mathrm{S}(s, \text { monoclinic }): \Delta H_{\mathrm{f}}^{\circ}=0.30 \mathrm{~kJ} / \mathrm{mol}, S^{\circ}=0.0326 \mathrm{~kJ} / \mathrm{mol} \cdot \mathrm{K} $$ at what temperature are the two forms in equilibrium?

The Gibbs free energy equation is a powerful tool in predicting the spontaneity of a chemical process. It is given by the formula:
ewline
ewline ewline \[ \Delta G = \Delta H - T \Delta S \]
where \(\Delta G\) represents the change in Gibbs free energy, \(\Delta H\) is the enthalpy change, \(T\) is the absolute temperature in Kelvin, and \(\Delta S\) is the entropy change. If the Gibbs free energy change is negative, the reaction occurs spontaneously.
At equilibrium, there is no net change in the composition of the system, and the change in Gibbs free energy is zero. This allows us to determine the equilibrium temperature by setting \(\Delta G = 0\) and solving for \(T\), expressed as:
\[ T_{\text{eq}} = \frac{\Delta H}{\Delta S} \] Understanding how the Gibbs free energy equation relates enthalpy and entropy changes to temperature can help students predict the behavior of reactions under varying conditions.

Enthalpy change, \(\Delta H\), is a measure of heat change during a chemical reaction at constant pressure. It represents the energy absorbed or released, and when positive, the process is endothermic (absorbs heat); when negative, it is exothermic (releases heat).
Enthalpy is a state function, meaning its change depends only on the initial and final states of the system, not on the path taken. In the context of allotropic forms of sulfur, the enthalpy change is relevant when comparing different crystal structures, like rhombic and monoclinic sulfur.
The phase transition between these allotropic forms involves an enthalpy change, which can be calculated from their standard enthalpies of formation. This enthalpy change allows us to evaluate the heat exchange during the transition and is integral to finding the equilibrium temperature using the Gibbs free energy equation.

Entropy change, \(\Delta S\), represents the measure of disorder or randomness in a system. It is a fundamental concept in thermodynamics which quantifies the number of ways a system can be arranged.
During a phase transition, the entropy change captures the difference in orderliness between the initial and final states. Reactions favoring an increase in entropy (\(\Delta S > 0\)) tend to be spontaneous since nature inherently moves towards greater disorder.
In the case of sulfur allotropes, knowing the entropy of each form is crucial. Calculating the entropy change between rhombic and monoclinic sulfur provides insight into the system's tendency to shift between these forms as temperature varies.

Equilibrium temperature refers to the temperature at which a system reaches a state of dynamic balance. At this temperature, the forward and reverse processes occur at equal rates, resulting in no net change.
For allotropic forms of sulfur, the equilibrium temperature is the point where rhombic and monoclinic sulfur coexist without converting into each other. Students can determine this crucial temperature by utilizing the Gibbs free energy equation for phase transitions where \(\Delta G = 0\).
By rearranging the equation to solve for equilibrium temperature, we get \( T_{eq} = \frac{\Delta H}{\Delta S} \), which signifies the temperature at which the two forms are at equilibrium. Consequently, gaining a deeper understanding of the equilibrium temperature can elucidate the conditions under which different crystalline structures of an element are stable.