Question

Two crystalline forms of white phosphorus are known. Both forms contain \(\mathrm{P}_{4}\) molecules, but the molecules are packed together in different ways. The \(\alpha\) form is always obtained when the liquid freezes. However, below \(-76.9^{\circ} \mathrm{C},\) the \(\alpha\) form spontaneously converts to the \(\beta\) form: $$\mathrm{P}_{4}(s, \alpha) \longrightarrow \mathrm{P}_{4}(s, \beta)$$ a. Predict the signs of \(\Delta H\) and \(\Delta S\) for this process. b. Predict which form of phosphorus has the more ordered crystalline structure (has the smaller positional probability).

Short answer

a. Since the α form spontaneously converts to the β form and releases energy, the process is exothermic, making ΔH negative. As spontaneous processes increase the total entropy, ΔS is positive. b. The α form has a more ordered crystalline structure as it has a smaller positional probability compared to the β form.

Step-by-step solution

Determine whether the process is exothermic or endothermic

The α form spontaneously converts to the β form. This suggests that the β form is more stable than the α form and the transformation process releases energy. Therefore, the process is exothermic, and we can expect the enthalpy change, ΔH, to be negative.

Determine whether the process increases or decreases entropy

Since the transformation process happens spontaneously, the entropy change, ΔS, should be positive. This is because spontaneous processes tend to increase the total entropy of the system and its surroundings. #b. Predicting the more ordered crystalline structure#

Analyze the change in positional probability

Because the β form is more stable and the system's entropy increases when it transforms from the α form to the β form, we can assume that the positional probability (how the molecules are randomly organized) is higher in the β form than in the α form.

Determine the more ordered crystalline structure

Since the positional probability of the β form is higher than the α form, the α form must have a more ordered crystalline structure. Therefore, the form of phosphorus with the smaller positional probability (more ordered) is the α form.

Key concepts

Enthalpy Change

Enthalpy change, represented as \(\Delta H\), is an important concept in understanding phase transitions. It reflects whether a process absorbs or releases energy. A negative \(\Delta H\) indicates that the process is exothermic, meaning energy is released, while a positive \(\Delta H\) indicates that the process is endothermic, meaning energy is absorbed. In the case of white phosphorus transitioning from the \(\alpha\) to the \(\beta\) form, the process is exothermic, meaning \(\Delta H\) is negative. This tells us that the transformation releases energy, suggesting that the \(\beta\) form is more stable than the \(\alpha\) form. Stability is often a result of a lower energy state, which is why energy is released during the transition to the \(\beta\) form.

Entropy Change

Entropy, symbolized as \(\Delta S\), measures the degree of disorder or randomness in a system. A positive \(\Delta S\) signifies an increase in disorder, while a negative \(\Delta S\) indicates a decrease.In phase transitions, like the conversion from \(\mathrm{P}_4(\alpha)\) to \(\mathrm{P}_4(\beta)\), a positive \(\Delta S\) is typically associated with spontaneous processes. In this case, the spontaneous transformation suggests that the entropy change is positive. This means that the system becomes more disordered as it shifts to the \(\beta\) form, indicating an increase in molecular randomness.

Crystalline Structure

Crystalline structures are the arrangements of molecules within a solid state, and they can vary in their level of order. Generally, a more ordered crystalline structure implies a specific, repeating pattern of organization.In the context of the exercise, the \(\alpha\) form of phosphorus has a more ordered crystalline structure compared to the \(\beta\) form. This is determined by analyzing the positional probability and entropy changes. A more ordered structure is usually an indication of lower energy and lower entropy, aligning with the smaller positional probability.

Positional Probability

Positional probability relates to the likelihood of molecules arranging in various configurations. A higher positional probability means more possible arrangements of the molecules, indicating a less ordered state.During the transition from \(\mathrm{P}_4(\alpha)\) to \(\mathrm{P}_4(\beta)\), the increase in entropy suggests a higher positional probability in the \(\beta\) form. This form is thus less ordered, as there are more ways to arrange the molecules, contributing to the overall increase in disorder. As a result, the \(\beta\) form, with its higher positional probability, is less ordered than the \(\alpha\) form, making \(\alpha\) phosphorus the more crystalline ordered form.