Red phosphorus is formed by heating white phosphorus. Calculate the temperature at which the two forms are at equilibrium, given $$ \text { white } \text { P: } \Delta H_{\mathrm{f}}^{\circ}=0.00 \mathrm{~kJ} / \mathrm{mol} ; S^{\circ}=41.09 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K} $$ red \(\mathrm{P}: \Delta H_{\mathrm{f}}^{\circ}=-17.6 \mathrm{~kJ} / \mathrm{mol} ; \mathrm{S}^{\circ}=22.80 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}\).

Gibbs free energy, symbolized by the letter G, is a thermodynamic quantity that represents the maximum amount of work that a system can perform at constant temperature and pressure. It's a crucial concept for understanding chemical equilibrium because it helps determine whether a reaction will proceed spontaneously. The change in Gibbs free energy, denoted by \( \Delta G \), for a process is given by the equation:\[ \Delta G = \Delta H - T\Delta S \]

Here, \( \Delta H \) is the change in enthalpy, \( T \) is the absolute temperature in Kelvin, and \( \Delta S \) is the change in entropy. When \( \Delta G \) is negative, the process occurs spontaneously; when it's positive, the process is non-spontaneous, and when it's zero, the system is at equilibrium, meaning there is no net change in the composition of the reaction mixture over time.

Enthalpy change, represented by \( \Delta H \) in chemical equations, is a measure of the heat absorbed or released during a reaction at constant pressure. It's an important concept in determining the energy changes within a chemical system. Endothermic reactions absorb heat (\(\Delta H > 0\)), making the surroundings cooler, while exothermic reactions release heat (\(\Delta H < 0\)), heating up the surroundings.

For the transformation of white phosphorus to red phosphorus, the enthalpy change helps us understand part of the energy dynamics during the equilibrium. Calculating the difference in enthalpy between the two forms of phosphorus is one step in determining the equilibrium temperature, as seen in the original exercise.

Entropy, symbolized as S, is a measure of the disorder or randomness in a system. The greater the disorder, the higher the entropy. Entropy change, indicated by \( \Delta S \), occurs when a system transitions from one state to another. An increase in entropy (\(\Delta S > 0\)) means that the disorder of the system is increasing, often associated with the dispersal of energy and matter.

Within the context of the exercise, we compare the entropy of white phosphorus and red phosphorus to help understand how the system's disorder changes during the equilibrium process. The equilibrium condition reflects a balance between the enthalpy and entropy changes at a certain temperature, contributing to the overall Gibbs free energy change of zero.

Equilibrium temperature is the temperature at which a system reaches a state of equilibrium, where there is no net change in the concentrations of reactants and products over time. This concept is closely tied to the concepts of Gibbs free energy, enthalpy, and entropy, as the equilibrium temperature is a condition where the total free energy change of a reaction mixture is zero.

In the step-by-step solution provided, the equilibrium temperature for the transition between white phosphorus and red phosphorus is calculated. By setting the Gibbs free energy change for the reaction to zero (which occurs at equilibrium) and solving for temperature, we determine that equilibrium will occur at approximately 962 K. This calculation is vital for predicting the conditions under which the two forms of phosphorus can coexist without further conversion to one another.