Problem 56 - Chemistry The Central Science In Si Units

Question

A certain reaction has \(\Delta H^{\circ}=+20.0 \mathrm{~kJ}\) and \(\Delta S^{\circ}=\) \(+100.0 \mathrm{~J} / \mathrm{K} .\) (a) Does the reaction lead to an increase or decrease in the randomness or disorder of the system? (b) Does the reaction lead to an increase or decrease in the randomness or disorder of the surroundings? (c) Calculate \(\Delta G^{\circ}\) for the reaction at \(298 \mathrm{~K} .(\mathbf{d})\) Is the reaction spontaneous at \(298 \mathrm{~K}\) under standard conditions?

Step-by-Step Solution

Answer

(a) The system's disorder increases. (b) The surroundings' disorder decreases. (c) \( \Delta G^{\circ} = -9.8 \mathrm{~kJ} \). (d) The reaction is spontaneous.

Step 1: Understanding Changes in System Disorder

The change in entropy \( \Delta S^{\circ} \) of the system is given as \( +100.0 \mathrm{~J/K} \). A positive entropy change indicates that the disorder or randomness of the system increases during the reaction.

Step 2: Effect of Reaction on Surroundings Disorder

The reaction releases heat to the surroundings if \( \Delta H^{\circ} \) is negative. Since \( \Delta H^{\circ} = +20.0 \mathrm{~kJ} \), the reaction absorbs heat, decreasing the entropy (randomness) of the surroundings.

Step 3: Calculating Gibbs Free Energy (\( \Delta G^{\circ} \))

Gibbs free energy is calculated using the formula: \[\Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ}\]Substitute \( \Delta H^{\circ} = +20.0 \mathrm{~kJ} = +20000 \mathrm{~J} \), \( T = 298 \mathrm{~K} \), and \( \Delta S^{\circ} = +100.0 \mathrm{~J/K} \) into the equation: \[\Delta G^{\circ} = 20000 \mathrm{~J} - 298 \times 100.0 \mathrm{~J/K}\]\[\Delta G^{\circ} = 20000 \mathrm{~J} - 29800 \mathrm{~J} = -9800 \mathrm{~J}\]Therefore, \( \Delta G^{\circ} = -9.8 \mathrm{~kJ} \).

Step 4: Determining Reaction Spontaneity

A reaction is spontaneous if \( \Delta G^{\circ} \) is negative. Since \( \Delta G^{\circ} = -9.8 \mathrm{~kJ} \), the reaction is spontaneous at \( 298 \mathrm{~K} \) under standard conditions.

Key Concepts

Entropy

Entropy is a concept in thermodynamics that measures the amount of disorder or randomness in a system. When a reaction occurs, the change in entropy, denoted as \( \Delta S \), indicates whether the disorder of the system increases or decreases. In the given exercise, the system has a \( \Delta S^{\circ} \) of \(+100.0 \mathrm{~J/K}\). This positive value tells us that the disorder or randomness of the system increases during the reaction.

Understanding entropy helps predict the natural tendency of systems to evolve towards more disordered states.

• Positive \( \Delta S \): Increase in disorder.
• Negative \( \Delta S \): Decrease in disorder.

Thus, when entropy increases, it often means the system is moving towards a state with more possible configurations, all of which are equally probable.

Reaction Spontaneity

A reaction's spontaneity can be determined by calculating the Gibbs free energy change \( \Delta G \), which combines enthalpy, entropy, and temperature into a single value. The formula is \( \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} \).

For a reaction to be spontaneous, \( \Delta G \) must be negative at a given temperature. Spontaneity means the reaction will proceed without needing additional energy once started.

• Negative \( \Delta G \): Reaction is spontaneous.
• Positive \( \Delta G \): Reaction is non-spontaneous.

In the given problem, we calculated \( \Delta G^{\circ} = -9.8 \mathrm{~kJ} \) at \( 298 \mathrm{~K} \), indicating that the reaction is spontaneous under these conditions.

Enthalpy

Enthalpy, denoted \( \Delta H \), is a thermodynamic property that expresses the heat content of a system. It is directly related to heat transfer during chemical reactions. If a reaction absorbs heat, the enthalpy change \( \Delta H \) is positive, indicating an endothermic process. Conversely, a negative \( \Delta H \) signifies an exothermic reaction, where heat is released.

For the provided exercise, \( \Delta H^{\circ} = +20.0 \mathrm{~kJ} \), showing that the reaction is endothermic. This means that the reaction takes in energy from its surroundings, possibly causing a decrease in surrounding entropy due to the energy absorption.

• Positive \( \Delta H \): Reaction absorbs heat (endothermic).
• Negative \( \Delta H \): Reaction releases heat (exothermic).

Understanding enthalpy helps predict energy changes that occur during reactions and their effects on surroundings.