Problem 52 - Chemistry: An Atoms-Focused Approach

Question

The values of \(\Delta H_{\mathrm{rxn}}^{\circ}\) and \(\Delta S_{\mathrm{rxn}}^{\circ}\) for the reaction $$2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) \rightarrow 2 \mathrm{NO}_{2}(g)$$ are \(-12 \mathrm{kJ}\) and \(-146 \mathrm{J} / \mathrm{K}\) a. Use these values to calculate \(\Delta G_{\text {rxn }}^{\circ}\) at \(298 \mathrm{K}\) b. Explain why the value of \(\Delta S_{\text {rxn }}^{\circ}\) is negative.

Step-by-Step Solution

Answer

Question: Calculate the standard Gibbs free energy change at 298 K for the given reaction and explain why the standard entropy change is negative. Answer: The standard Gibbs free energy change (ΔG°rxn) at 298 K is 31508 J/mol. The standard entropy change (ΔS°rxn) is negative (-146 J/mol K) because the reaction results in fewer gas molecules being produced than consumed, leading to a decrease in the overall system's disorder or randomness.

Step 1: Determine the given values

We are given the values for the reaction: ΔH°rxn = -12 kJ/mol ΔS°rxn = -146 J/mol K T = 298 K

Step 2: Convert units if necessary

To make the units consistent, we need to convert the ΔH°rxn value from kJ/mol to J/mol: ΔH°rxn = -12 kJ/mol * (1000 J/1 kJ) = -12000 J/mol

Step 3: Calculate the standard Gibbs free energy change

Now, we can use the formula to calculate the standard Gibbs free energy change (ΔG°rxn) at 298 K: ΔG°rxn = ΔH°rxn - TΔS°rxn ΔG°rxn = -12000 J/mol - (298 K * (-146 J/mol K)) ΔG°rxn = -12000 J/mol + 43508 J/mol ΔG°rxn = 31508 J/mol a. The standard Gibbs free energy change (ΔG°rxn) at 298 K is 31508 J/mol.

Step 4: Explain the negative value of the standard entropy change

b. The standard entropy change (ΔS°rxn) is negative (-146 J/mol K) because the reaction results in fewer gas molecules being produced than consumed, which leads to a decrease in the overall system's disorder or randomness. In this case, 2 moles of NO (g) and 1 mole of O₂ (g) combine to form only 2 moles of NO₂ (g). As a result, the total number of gas molecules decreases, which causes a decrease in entropy.

Key Concepts

Enthalpy Change

Enthalpy (\( \Delta H \)) is a measure of the thermal energy released or absorbed during a chemical reaction at constant pressure. It fundamentally represents the difference in energy content between reactants and products.
An enthalpy change value can be either negative or positive:

• **Exothermic Reaction**: If \( \Delta H \) is negative, the reaction releases energy, making the surrounding environment warmer.
• **Endothermic Reaction**: If \( \Delta H \) is positive, the reaction absorbs energy from the surroundings, resulting in a cooler environment.

In the context of the provided reaction involving nitrogen monoxide (NO) and oxygen (O₂), the enthalpy change (\( \Delta H_{\mathrm{rxn}}^{\circ} = -12 \text{kJ/mol} \)) indicates the reaction releases heat, qualifying it as exothermic.
This concept helps us understand whether a reaction requires energy input or supplies energy to the surroundings.

Entropy Change

Entropy (\( \Delta S \)) quantifies the degree of disorder or randomness in a system. In thermodynamics, it's crucial because it indicates how energy disperses in a process.
For chemical reactions, entropy change can guide us on the system's energy distribution:

• **Positive Entropy Change**: Implies increased randomness, with more freedom in movement and distribution of particles.
• **Negative Entropy Change**: Implies decreased randomness, usually because particles become more ordered.

In the problem, the standard entropy change (\( \Delta S_{\mathrm{rxn}}^{\circ} = -146 \text{ J/mol K} \)) is negative. This negative value stems from the fact that the reaction produces fewer gas molecules than it consumes. Two moles of NO and one mole of O₂ combine to form only two moles of NO₂, reducing the total motion and number of molecules in the system, thus decreasing disorder.

Thermodynamics

Thermodynamics is the scientific discipline that assesses heat and energy changes within a physical or chemical system. It provides laws that govern these transformations.Thermodynamic properties help predict if a reaction will occur spontaneously. Gibbs free energy (\( \Delta G \)) is particularly critical in this evaluation.

• If \( \Delta G < 0 \), the reaction is spontaneous. It can proceed on its own without energy input.
• If \( \Delta G > 0 \), the reaction is non-spontaneous, needing energy to occur.
• If \( \Delta G = 0 \), the system is at equilibrium, and no net reaction occurs at this stage.

In the given task, the calculated Gibbs free energy change (\( \Delta G_{\mathrm{rxn}}^{\circ} = 31508 \text{ J/mol} \)) helps determine if our reaction from NO and O₂ to NO₂ happens spontaneously. The positive value suggests that we need an energy input for this reaction to proceed under standard conditions, meaning it's non-spontaneous.

Chemical Reaction

Chemical reactions involve the transformation of reactants into products through the breaking and forming of chemical bonds. Each reaction follows the law of conservation of mass, meaning the same amount of each element should be present on both sides of the equation.
These reactions can be characterized by:

• **Changes in energy**, involving enthalpy (\( \Delta H \)), where energy is absorbed or released.
• **Entropy Shift**, reflecting changes in the system's disorder as reactants convert to products.
• **Reaction Dynamics**, wherein thermodynamic functions like Gibbs free energy (\( \Delta G \)) predict spontaneity.

In our example reaction, NO and O₂ react to form NO₂. Here, the enthalpy indicates energy release, the entropy suggests a decrease in randomness, and the calculated Gibbs free energy talks about the reaction's spontaneity under defined conditions. Understanding these parameters gives a clear picture of what happens during and after the reaction.