Problem 22 - General Chemistry: Principles and Modern Applications

Question

Explain why you would expect a reaction of the type \(\mathrm{AB}(\mathrm{g}) \longrightarrow \mathrm{A}(\mathrm{g})+\mathrm{B}(\mathrm{g})\) always to be spontaneous at high rather than at low temperatures.

Step-by-Step Solution

Answer

A reaction of the type AB(g) -> A(g) + B(g) is expected to be more spontaneous at higher temperatures because the positive entropy change (due to increase in number of gas molecules) leads to a greater decrement in Gibbs Free Energy at higher temperatures, making the reaction more spontaneous as per the definition of spontaneity based on Gibbs Free Energy change.

Step 1: Recognize the Reaction

This is a chemical reaction which describes the process of an AB molecule breaking down into individual A and B atoms. The process increases the number of gas molecules and hence, increases the entropy, making \( \Delta S \) positive.

Step 2: Identify the Factors Involved in Spontaneity of Reaction

The spontaneity of a reaction is determined by the sign of the Gibbs Free Energy change, \( \Delta G \). According to the Gibbs Free Energy equation, \( \Delta G = \Delta H - T\Delta S \), where 'Delta H' is the enthalpy change, 'Delta S' is the entropy change, and 'T' is the temperature in Kelvin.

Step 3: Analyzing the Impact of Temperature

From the Gibbs Free Energy equation, it can be noticed that for a given positive \( \Delta S \), as temperature 'T' increases, the value of \( T\Delta S \) also increases. Since\( \Delta G = \Delta H - T\Delta S \), an increase in \( T\Delta S \) means a decrease in \( \Delta G \). As a result, the reaction becomes more spontaneous with higher temperatures.

Key Concepts

Entropy

In simple terms, entropy is a measure of disorder or randomness in a system. When looking at the given reaction, \( \mathrm{AB}(\mathrm{g}) \rightarrow \mathrm{A}(\mathrm{g})+\mathrm{B}(\mathrm{g}) \), we observe that a single molecule splits into two separate entities. This process naturally leads to more disordered states because now you have two gas molecules moving freely instead of one consolidated structure.
Consequently, the entropy change \( \Delta S \) is positive. Positive \( \Delta S \) indicates an increase in disorder. In this context, an increase in entropy is significant because it directly influences whether a reaction might proceed spontaneously according to the Gibbs Free Energy formula.
In general:

• An increase in entropy (positive \( \Delta S \)) often favors spontaneity.
• Entropy tends to increase in reactions where a substance decomposes or where matter spreads out.

Understanding entropy helps in predicting how likely a process is to occur naturally.

Spontaneity

When we talk about spontaneity in chemical reactions, we refer to the inclination of a process to occur without external influence once it has started. The Gibbs Free Energy equation \( \Delta G = \Delta H - T\Delta S \) is vital in determining spontaneity.
In this equation:

• \( \Delta G \) is the change in Gibbs Free Energy.
• \( \Delta H \) represents the change in enthalpy.
• \( T \) is the temperature in Kelvin, and \( \Delta S \) is the change in entropy.

A negative \( \Delta G \) indicates that a reaction is spontaneous under those conditions.
The provided exercise shows that for reactions with a positive \( \Delta S \), raising the temperature \( T \) will lower \( \Delta G \) since \( T\Delta S \) becomes larger. This is why such reactions become more spontaneous at higher temperatures.
Always remember:

• A negative \( \Delta G \) means the reaction can proceed on its own.
• High temperatures with a positive \( \Delta S \) favor spontaneity.

Understanding these aspects allows us to predict and control chemical reactions effectively.

Enthalpy

Enthalpy, symbolized by \( \Delta H \), is a measure of the total energy change during a chemical reaction. It reflects the heat absorbed or released under conditions of constant pressure. In the context of the reaction \( \mathrm{AB}(\mathrm{g}) \rightarrow \mathrm{A}(\mathrm{g})+\mathrm{B}(\mathrm{g}) \), determining whether \( \Delta H \) is positive or negative can provide insight into how energy influences spontaneity.
In general:

• \( \Delta H \) is negative in exothermic reactions where heat is released.
• \( \Delta H \) is positive in endothermic reactions where energy is absorbed.

However, even if the enthalpy change \( \Delta H \) were positive, the reaction could still proceed spontaneously if the entropy change \( \Delta S \) were sufficiently positive. Here, high temperatures help by making the \( T\Delta S \) term large enough to ensure that \( \Delta G \) is negative despite a positive \( \Delta H \).
Therefore, understanding enthalpy provides critical information about the energy changes involved and helps in conjunction with entropy to predict spontaneity.