Question

Given the values of \(\Delta H_{\mathrm{rxn}},\) and \(T,\) determine \(\Delta S_{\mathrm{rxn}},\) and predict whether or not each reaction is spontaneous. (Assume that all reactants and products are in their standard states.) a. \(\Delta H_{\mathrm{rxn}}^{\circ}=-95 \mathrm{~kJ} ; \Delta S_{\mathrm{rxn}}^{\circ}=-157 \mathrm{~J} / \mathrm{K} ; T=298 \mathrm{~K}\) b. \(\Delta H_{\mathrm{rxn}}^{\circ}=-95 \mathrm{~kJ} ; \Delta S_{\mathrm{rxn}}^{\circ}=-157 \mathrm{~J} / \mathrm{K} ; T=855 \mathrm{~K}\) c. \(\Delta H_{\mathrm{rxn}}^{\circ}=+95 \mathrm{~kJ} ; \Delta S_{\mathrm{rxn}}^{\circ}=-157 \mathrm{~J} / \mathrm{K} ; T=298 \mathrm{~K}\) d. \(\Delta H_{\mathrm{rxn}}^{\circ}=-95 \mathrm{~kJ} ; \Delta S_{\mathrm{rxn}}^{\circ}=+157 \mathrm{~J} / \mathrm{K} ; T=398 \mathrm{~K}\)

Short answer

a. Non-spontaneous; b. Spontaneous; c. Non-spontaneous; d. Spontaneous. Reactions are spontaneous if \(\Delta G_{\mathrm{rxn}}\) is negative based on the calculated values from the Gibbs free energy equation for each reaction.

Step-by-step solution

Understanding the Gibbs Free Energy Equation

To determine if a reaction is spontaneous, we can use the Gibbs free energy equation: \(\Delta G_{\mathrm{rxn}} = \Delta H_{\mathrm{rxn}} - T \times \Delta S_{\mathrm{rxn}}\). A reaction is considered spontaneous if \(\Delta G_{\mathrm{rxn}}\) is negative.

Calculate \(\Delta G_{\mathrm{rxn}}\) for Reaction a

For reaction a, plug the given values into the Gibbs free energy equation:\[\Delta G_{\mathrm{rxn}} = (-95 \mathrm{~kJ}) - (298 \mathrm{~K})(-157 \mathrm{~J} / \mathrm{K} \times 10^{-3} \mathrm{~kJ}/\mathrm{J})\] then calculate the value of \(\Delta G_{\mathrm{rxn}}\).

Calculate \(\Delta G_{\mathrm{rxn}}\) for Reaction b

Repeat the calculation from Step 2 for reaction b, using the new temperature of 855 K.

Calculate \(\Delta G_{\mathrm{rxn}}\) for Reaction c

Repeat the calculation from Step 2 for reaction c, but remember that \(\Delta H_{\mathrm{rxn}}\) is now positive while \(\Delta S_{\mathrm{rxn}}\) is still negative.

Calculate \(\Delta G_{\mathrm{rxn}}\) for Reaction d

Repeat the calculation from Step 2 for reaction d, noting that both \(\Delta H_{\mathrm{rxn}}\) and \(\Delta S_{\mathrm{rxn}}\) are negative.

Predict Reaction Spontaneity

Compare each value of \(\Delta G_{\mathrm{rxn}}\) calculated in previous steps to determine if the reactions are spontaneous (\(\Delta G_{\mathrm{rxn}} < 0\)) or non-spontaneous (\(\Delta G_{\mathrm{rxn}} \geq 0\)).

Key concepts

Chemical Thermodynamics

Chemical thermodynamics is a branch of physical chemistry that deals with the relationship between heat, work, and energy within chemical reactions. It rests on two fundamental concepts: the conservation of energy, which states that energy cannot be created or destroyed, only transferred or transformed, and thermodynamic equilibrium, a condition in which there is no net change in the distribution of energy among components of the system.

Within this context, the First Law of Thermodynamics assures that the total energy of an isolated system remains constant, while the Second Law introduces the concept of entropy, revealing that energy spontaneously spreads out if not hindered. These principles help us understand why some reactions occur spontaneously and others do not, leading us to the Gibbs free energy concept, which combines enthalpy, entropy, and temperature to predict the spontaneity of reactions.

Spontaneous Reactions

Spontaneous reactions are processes that occur without the need for continuous input of energy from an external source. They are favored by conditions that result in a decrease in the system's overall energy, leading to an increase in disorder or entropy, or both. To predict whether a reaction will occur spontaneously, we calculate the Gibbs free energy change (\(Delta G_{\text{rxn}}\)), which incorporates both the entropic and enthalpic aspects of the system.

When \(Delta G_{\text{rxn}} < 0\), the reaction tends to be spontaneous, indicating that the process can proceed under the given conditions. If \(Delta G_{\text{rxn}}\) is greater than zero, the reaction is non-spontaneous, requiring additional energy to proceed. Importantly, a reaction with a \(Delta G_{\text{rxn}}\) of zero is in equilibrium, meaning the rate of the forward reaction equals that of the reverse reaction.

Entropy in Chemistry

Entropy, symbolized as \(S\), is a central concept in chemistry that describes the amount of disorder or randomness in a system. The Second Law of Thermodynamics states that the total entropy of an isolated system can never decrease over time, which means that the universe tends toward increased entropy. In the context of chemical reactions, entropy refers to the distribution of energy among the particles involved.

When a reaction proceeds to a state where the product has higher entropy, it suggests a more disordered system. Changes in state, such as solid to liquid or liquid to gas, typically lead to an increase in entropy. Additionally, reactions that produce a greater number of gas molecules from fewer ones often show an increase in entropy. Considering entropy helps us understand how chemical reactions contribute to the universal trend towards disorder and how it impacts reaction spontaneity.

Enthalpy Change

Enthalpy change, denoted as \(Delta H_{\text{rxn}}\), refers to the heat absorbed or released by a system during a chemical reaction at constant pressure. It is a measure of the total energy of a thermodynamic system, including both internal energy and the energy associated with pressure and volume.

A negative \(Delta H_{\text{rxn}}\) indicates that the reaction is exothermic, releasing heat to the surroundings, which can contribute to the spontaneity of a reaction, as per the Gibbs free energy equation. Conversely, a positive \(Delta H_{\text{rxn}}\) signifies an endothermic reaction, whereby the system absorbs heat from the surroundings. Understanding enthalpy change is crucial in predicting not only the energy requirements of a reaction but also its potential to occur spontaneously.