Question

The entropy change for a certain nonspontaneous reaction at \(50^{\circ} \mathrm{C}\) is \(104 \mathrm{~J} / \mathrm{K}\) (a) Is the reaction endothermic or exothermic? (b) What is the minimum value of \(\Delta H\) (in \(\mathrm{kJ}\) ) for the reaction?

Short answer

(a) The reaction is endothermic. (b) Minimum \(\Delta H\) is \(33.61 \text{ kJ}\).

Step-by-step solution

Understanding the Reaction Basics

The reaction in question is nonspontaneous at a given temperature of \(50^{\circ} \mathrm{C}\). Nonspontaneous reactions generally have a positive Gibbs free energy change (\(\Delta G > 0\)). The problem also states that the entropy change, \(\Delta S\), is \(+104 \mathrm{~J} / \mathrm{K}\).

Identifying the Relationship Between Thermodynamic Quantities

The relation between Gibbs free energy (\(\Delta G\)), enthalpy change (\(\Delta H\)), and entropy change (\(\Delta S\)) is given by the equation: \[ \Delta G = \Delta H - T\Delta S \]Since the reaction is nonspontaneous, \(\Delta G\) must be positive.

Calculate the Minimum \(\Delta H\) Required

To find the minimum \(\Delta H\) for a nonspontaneous reaction, set \(\Delta G = 0\) for the threshold between spontaneity and nonspontaneity, then solve the equation:\[ 0 = \Delta H - T\Delta S \]\[ \Delta H = T\Delta S \]First, convert the temperature from \(^{\circ}\text{C}\) to Kelvin: \[ T = 50 + 273.15 = 323.15 \text{ K} \]Substitute into the equation: \[ \Delta H = 323.15 \text{ K} \times 104 \text{ J/K} = 33607.6 \text{ J} \]Convert \(\Delta H\) from Joules to kilojoules:\[ \Delta H = 33.61 \text{ kJ} \]

Determine if the Reaction is Endothermic or Exothermic

Since \(\Delta H = 33.61 \text{ kJ} > 0\), the reaction absorbs heat from its surroundings. Therefore, the reaction is endothermic.

Key concepts

Entropy Change

Entropy is a measure of the disorder or randomness of a system. In thermodynamic reactions, entropy change \(\Delta S\) helps us understand how the arrangement of particles in a reaction changes. It is defined as the change in entropy from the initial state to the final state of a reaction. Entropy is often expressed in units of Joules per Kelvin \(\text{J/K}\). A positive entropy change, such as \(\Delta S = 104 \, \text{J/K}\) in our reaction, suggests that the system becomes more disordered during the process.
In the context of our exercise, understanding the entropy change value aids in assessing whether a reaction increases or decreases system randomness. An increase in entropy means that the reaction creates more disorder, potentially making its products more varied or widespread than its reactants. In our example, even with an increase in disorder, the fact that the reaction is nonspontaneous indicates other energy dynamics at play, particularly involving \(\Delta H\) and \(\Delta G\).

• Positive Entropy Change: Means increased disorder in the system.
• Measurement: Expressed in \(\text{J/K}\).
• Impact on Reactions: Helps predict spontaneity along with other thermodynamic quantities.

Gibbs Free Energy

Gibbs free energy (\(\Delta G\)) provides insight into whether a reaction can occur spontaneously. It is defined as the energy available to do work when temperature and pressure are constant. The equation\[\Delta G = \Delta H - T\Delta S\]relates Gibbs free energy to two other critical thermodynamic properties: enthalpy change (\(\Delta H\)) and entropy change (\(\Delta S\)).For a reaction to be spontaneous, \(\Delta G\) must be negative. In cases where \(\Delta G\) is positive, like with our exercise, the reaction is nonspontaneous under the given conditions. By setting \(\Delta G = 0\), a threshold is defined where spontaneity shifts, allowing calculations to assess necessary conditions for either mode.

• Spontaneous Reactions: Occur when \(\Delta G < 0\).
• Nonspontaneous Reactions: Occur when \(\Delta G > 0\) (as in our example).
• Threshold Calculation: Setting \(\Delta G = 0\) helps determine the point between spontaneity and nonspontaneity.

Enthalpy Change

Enthalpy change (\(\Delta H\)) is a vital concept to understand heat exchange in reactions. It refers to the total heat content of a system and determines whether a reaction absorbs or releases heat energy. Enthalpy is usually expressed in kilojoules per mole (\(\text{kJ/mol}\)).In our exercise, the calculation of \(\Delta H\) involved determining the minimum energy needed for the reaction to stay nonspontaneous at the given temperature. By applying the equation:\[\Delta H = T\Delta S\]where temperature (\(T\)) is converted to Kelvin, we calculated:\[\Delta H = 33.61\, \text{kJ}\]This positive enthalpy change confirms the reaction is endothermic, meaning it absorbs heat from its surroundings. Understanding whether a reaction is endothermic (absorbs heat) or exothermic (releases heat) is essential, as it indicates how energy transfer occurs within the system.

• Endothermic Reactions: Have \(\Delta H > 0\) and absorb heat (as in our case).
• Exothermic Reactions: Have \(\Delta H < 0\) and release heat.
• Expression: Measured in \(\text{kJ/mol}\).