Problem 64 - Chemistry The Central Science In Si Units

Question

From the values given for \(\Delta H^{\circ}\) and \(\Delta S^{\circ},\) calculate \(\Delta G^{\circ}\) for each of the following reactions at \(298 \mathrm{~K}\). If the reaction is not spontaneous under standard conditions at \(298 \mathrm{~K}\), at what temperature (if any) would the reaction become spontaneous? $$ \begin{array}{l} \text { (a) } 2 \mathrm{PbS}(s)+3 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{PbO}(s)+2 \mathrm{SO}_{2}(g) \\ \Delta H^{\circ}=-844 \mathrm{~kJ} ; \Delta S^{\circ}=-165 \mathrm{~J} / \mathrm{K} \\ \text { (b) } 2 \mathrm{POCl}_{3}(g) \longrightarrow 2 \mathrm{PCl}_{3}(g)+\mathrm{O}_{2}(g) \\ \Delta H^{\circ}=572 \mathrm{~kJ} ; \Delta S^{\circ}=179 \mathrm{~J} / \mathrm{K} \end{array} $$

Step-by-Step Solution

Answer

(a) \( \Delta G^{\circ} = -794.83 \text{ kJ} \); spontaneous. (b) \( \Delta G^{\circ} = 518.658 \text{ kJ} \); non-spontaneous. (b) becomes spontaneous at \( T > 3195.53 \text{ K} \).

Step 1: Write the Gibbs Free Energy Equation

The change in Gibbs free energy for a reaction can be calculated using the equation: \( \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} \).

Step 2: Convert Units if Necessary

Make sure the units for \( \Delta S^{\circ} \) are consistent with \( \Delta H^{\circ} \). Since \( \Delta S^{\circ} \) is given in J/K, convert it to kJ/K by dividing by 1000. For example, for \( \Delta S^{\circ} = -165 \text{ J/K} \), it becomes \(-0.165 \text{ kJ/K} \).

Step 3: Calculate \( \Delta G^{\circ} \) for Reaction (a)

Given \( \Delta H^{\circ} = -844 \text{ kJ} \) and \( \Delta S^{\circ} = -0.165 \text{ kJ/K} \):\[ \Delta G^{\circ} = -844 \text{ kJ} - 298 \text{ K} \times -0.165 \text{ kJ/K} \]Calculate \( \Delta G^{\circ} \):\[ \Delta G^{\circ} = -844 \text{ kJ} + 49.17 \text{ kJ} = -794.83 \text{ kJ} \]

Step 4: Determine Spontaneity for Reaction (a)

Since \( \Delta G^{\circ} < 0 \) for reaction (a), it is spontaneous at 298 K.

Step 5: Calculate \( \Delta G^{\circ} \) for Reaction (b)

Given \( \Delta H^{\circ} = 572 \text{ kJ} \) and \( \Delta S^{\circ} = 0.179 \text{ kJ/K} \):\[ \Delta G^{\circ} = 572 \text{ kJ} - 298 \text{ K} \times 0.179 \text{ kJ/K} \]Calculate \( \Delta G^{\circ} \):\[ \Delta G^{\circ} = 572 \text{ kJ} - 53.342 \text{ kJ} = 518.658 \text{ kJ} \]

Step 6: Determine Spontaneity for Reaction (b)

Since \( \Delta G^{\circ} > 0 \) for reaction (b), it is not spontaneous at 298 K.

Step 7: Calculate Temperature for Reaction (b) to Become Spontaneous

To find the temperature at which reaction (b) becomes spontaneous: Set \( \Delta G^{\circ} = 0 \) and solve for \( T \).\[ 0 = \Delta H^{\circ} - T \Delta S^{\circ} \rightarrow T = \frac{\Delta H^{\circ}}{\Delta S^{\circ}} \]\[ T = \frac{572 \text{ kJ}}{0.179 \text{ kJ/K}} = 3195.53 \text{ K} \]Reaction (b) will become spontaneous at temperatures above 3195.53 K.

Key Concepts

Spontaneity of Reactions

In chemistry, the spontaneity of a reaction is determined by the sign of the change in Gibbs free energy (\( \Delta G \)).
An important rule is that if \( \Delta G < 0 \), the reaction is spontaneous, meaning it can occur without any external input of energy. Conversely, if \( \Delta G > 0 \), the reaction is non-spontaneous and would require energy to proceed.
This spontaneity can be influenced by various factors such as temperature and the specific conditions under which the reaction occurs. Understanding these factors is essential to predicting whether a chemical reaction will proceed on its own.

Thermodynamics

Thermodynamics is a branch of physics that studies energy transformations, especially concerning heat and work.
Within chemistry, thermodynamics provides a framework for understanding the energy changes that accompany chemical reactions. This includes concepts such as enthalpy (\( \Delta H \)), entropy (\( \Delta S \)), and Gibbs free energy (\( \Delta G \)).

• **Enthalpy (\( \Delta H \))**: This describes the total heat content of a system. It measures the change in heat during a reaction.
• **Entropy (\( \Delta S \))**: This expresses the degree of disorder or randomness within a system. A higher \( \Delta S \) signifies greater disorder.
• **Gibbs Free Energy (\( \Delta G \))**: This combines both enthalpy and entropy into one value that predicts spontaneity.

These concepts are closely intertwined and help predict the behavior and outcomes of chemical reactions under given conditions.

Chemical Reactions

Chemical reactions involve the transformation of substances through the breaking and forming of chemical bonds. During a reaction, reactants are converted into products.
The process encompasses different types of changes:

• **Physical Change:** This involves changes in state, such as from solid to liquid, without transforming the substances chemically.
• **Chemical Change:** This results in new substances being formed and is characterized by irreversible transformations.

Analyzing the energy changes, such as enthalpy and entropy shifts, is crucial for understanding reactions. This provides insights into the energy flows and the extent of change, ultimately predicting the spontaneity and feasibility of the reaction. These foundational concepts help categorize reactions and understand their mechanisms.