Question

Under what conditions will the reaction be spontaneous? $$ \text { (a) } \begin{aligned} \mathrm{Al}_{2} \mathrm{O}_{3}(s)+2 \mathrm{Fe}(s) \longrightarrow \mathrm{Fe}_{2} \mathrm{O}_{3}(s)+2 \mathrm{Al}(s) \\ \Delta S>0 \text { and } \Delta H>0 \end{aligned} $$ (b) \(\mathrm{CS}_{2}(g) \longrightarrow \mathrm{CS}_{2}(l)\) \(\Delta S<0\) and \(\Delta H<0\)

Short answer

Reaction (a) is spontaneous at high temperatures, while reaction (b) is spontaneous at low temperatures.

Step-by-step solution

- Review Gibbs Free Energy

The spontaneity of a reaction can be determined using Gibbs Free Energy, given by the equation \[ \Delta G = \Delta H - T\Delta S \.\] A reaction is spontaneous if \( \Delta G < 0 \).

- Analyze Reaction (a)

For the reaction \( \mathrm{Al}_{2} \mathrm{O}_{3}(s) + 2 \mathrm{Fe}(s) \longrightarrow\mathrm{Fe}_{2} \mathrm{O}_{3}(s) + 2 \mathrm{Al}(s) \), given \( \Delta S>0 \) and \( \Delta H>0 \), the reaction will be spontaneous at high temperatures where the positive entropy change \( \Delta S \) is able to overcome the positive enthalpy change \( \Delta H \), making \( \Delta G < 0 \). Specifically, the reaction will be spontaneous if \( T > \frac{\Delta H}{\Delta S} \).

- Analyze Reaction (b)

For the reaction \( \mathrm{CS}_{2}(g) \longrightarrow \mathrm{CS}_{2}(l) \), given \( \Delta S<0 \) and \( \Delta H<0 \), the reaction will be spontaneous at low temperatures. This is because the negative enthalpy change (exothermic reaction) will lead to \( \Delta G < 0 \) without the need for temperature to amplify the effect of entropy change, as it is also negative.

Key concepts

Spontaneity of Chemical Reactions

Understanding the spontaneity of chemical reactions is crucial because it tells us whether a reaction will occur without external intervention. A spontaneous reaction does not necessarily happen quickly; rather, it means that the reaction is thermodynamically favorable under a given set of conditions. To predict spontaneity, we often refer to Gibbs Free Energy (\textbf{G}), which combines the concepts of enthalpy (\textbf{H}), entropy (\textbf{S}), and temperature (\textbf{T}) into a single equation:
\[ \Delta G = \Delta H - T\Delta S \]
For a reaction to be spontaneous, the change in Gibbs Free Energy (\( \Delta G \)) must be negative.
The equation shows that both enthalpy and entropy changes impact spontaneity. If the reaction releases heat (\textbf{exothermic}, \( \Delta H<0 \)), or if there is an increase in disorder (\textbf{increased entropy}, \( \Delta S>0 \)), the chances of the reaction being spontaneous increase. Conversely, reactions absorbing heat (\textbf{endothermic}, \( \Delta H>0 \)) or resulting in decreased disorder (\textbf{decreased entropy}, \( \Delta S<0 \)) are less likely to be spontaneous unless compensated by other factors, such as temperature.

Entropy and Enthalpy

In chemical thermodynamics, entropy (\textbf{S}) is a measure of the disorder or randomness in a system. An increase in entropy (\textbf{\( \Delta S>0 \)}) generally favors the spontaneity of a reaction, as nature tends toward disorder. However, the total entropy change must be considered along with enthalpy changes.
Enthalpy (\textbf{H}) reflects the heat content of a system and is associated with bond energies within molecules. An exothermic reaction (\textbf{\( \Delta H<0 \)}) releases heat and increases the universe's total entropy, which is thermodynamically favorable. On the other hand, an endothermic reaction (\textbf{\( \Delta H>0 \)}) requires heat input, which can be unfavorable unless the entropy increase within the system outweighs the energy input.
The interplay between entropy and enthalpy is nuanced and temperature-dependent. For instance, a reaction with \( \Delta S>0 \) and \( \DeltaH>0 \) might only be spontaneous at high temperatures where the positive entropy change can compensate for the input of heat. Conversely, reactions with \( \Delta S<0 \) and \( \Delta H<0 \), like condensation, are typically spontaneous at low temperatures.

Chemical Thermodynamics

Chemical thermodynamics is the branch of chemistry that deals with the interrelation of heat and work with chemical reactions or physical transformations within a system. It provides a framework that relates the macroscopic properties of materials under various conditions to the spontaneity of reactions through concepts like Gibbs Free Energy, enthalpy, and entropy.
This field allows us to predict whether processes will occur and is fundamental in designing chemical reactions and processes, such as those in industrial synthesis and energy production. Understanding thermodynamics helps chemists and engineers to manipulate conditions such as temperature and pressure to make processes more efficient, cost-effective, and sustainable.
In a broader sense, chemical thermodynamics helps us grasp the flow of energy in our universe. It underscores key principles that govern the transformation of energy and materials, contributing to advancements in fields like materials science, biochemistry, and environmental science, where managing energy efficiently is critical.