Problem 61 - Chemistry An Atoms First Approach

Question

Consider the reaction $$\mathrm{Fe}_{2} \mathrm{O}_{3}(s)+3 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{Fe}(s)+3 \mathrm{H}_{2} \mathrm{O}(g)$$ a. Use \(\Delta G_{\mathrm{f}}^{\circ}\) values in Appendix 4 to calculate \(\Delta G^{\circ}\) for this reaction. b. Is this reaction spontaneous under standard conditions at \(298 \mathrm{K} ?\) c. The value of \(\Delta H^{\circ}\) for this reaction is \(100 .\) kJ. At what temperatures is this reaction spontaneous at standard conditions? Assume that \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) do not depend on temperature.

Step-by-Step Solution

Answer

We first calculate the standard Gibbs free energy change (\(\Delta G^{\circ}\)) for the reaction using the given standard Gibbs free energy of formation values \(\Delta Gf^{\circ}\) and the formula: \(\Delta G^{\circ} = \sum n \Delta G_{f}^{\circ}(\text{products}) - \sum n \Delta G_{f}^{\circ}(\text{reactants})\) Then, we determine the spontaneity of the reaction at 298 K by comparing the calculated \(\Delta G^{\circ}\) value with zero. If \(\Delta G^{\circ} < 0\), the reaction is spontaneous under standard conditions at 298 K. Lastly, to find the temperature range in which the reaction is spontaneous under standard conditions, we use the relationship between Gibbs free energy, enthalpy, and entropy: \(\Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ}\) We determine \(\Delta S^{\circ}\) using the given \(\Delta H^{\circ}\) value, and then find the temperature range in which \(\Delta G^{\circ} < 0\).

Step 1: Write the balanced equation for the reaction

Given the balanced equation: \(\mathrm{Fe}_{2} \mathrm{O}_{3}(s)+3 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{Fe}(s)+3 \mathrm{H}_{2} \mathrm{O}(g)\) Now, find the standard Gibbs free energy change for this reaction using the standard Gibbs free energy of formation values.

Step 2: Calculate the ΔG° using the ΔGf° values

Use the formula: \(\Delta G^{\circ} = \sum n \Delta G_{f}^{\circ}(\text{products}) - \sum n \Delta G_{f}^{\circ}(\text{reactants})\) Where n is the stoichiometric coefficient, and \(\Delta G_{f}^{\circ}\) is the standard Gibbs free energy of formation. The values for ΔGf° can be found in Appendix 4. For this reaction: \(\Delta G^{\circ} = (2\Delta G_{f}^{\circ}(\mathrm{Fe}) + 3\Delta G_{f}^{\circ}(\mathrm{H}_{2}\mathrm{O}) - (\Delta G_{f}^{\circ}(\mathrm{Fe}_{2} \mathrm{O}_{3}) + 3\Delta G_{f}^{\circ}(\mathrm{H}_{2}))\)

Step 3: Determine if the reaction is spontaneous at 298 K

The reaction is spontaneous under standard conditions at 298 K if \(\Delta G^{\circ} < 0\). Compare the calculated \(\Delta G^{\circ}\) value with zero to check the spontaneity of the reaction.

Step 4: Find the temperature range for spontaneous reaction

We are given that the standard enthalpy change for the reaction is 100 kJ, and we need to find the temperature range for which the reaction is spontaneous. We can use the relationship between Gibbs free energy, enthalpy, and entropy: \(\Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ}\) where T is the temperature in Kelvin, and ΔS° is the standard entropy change. First, determine \(\Delta S^{\circ}\) using the relationship: \(\Delta S^{\circ} = \frac{\Delta H^{\circ} - \Delta G^{\circ}}{T}\) Then, find the temperature range in which \(\Delta G^{\circ} < 0\) under standard conditions (\(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) do not depend on temperature). In other words, find the temperature range in which: \(\Delta H^{\circ} - T\Delta S^{\circ} < 0\)

Key Concepts

Spontaneous Reaction

In chemistry, a spontaneous reaction is one that occurs without the need for external energy input. It happens on its own, naturally, under a given set of conditions. The spontaneity of a reaction is determined by the Gibbs Free Energy change (\( \Delta G \)), which combines the effects of enthalpy (\( \Delta H \)) and entropy (\( \Delta S \)). For a reaction to be spontaneous, the Gibbs Free Energy change needs to be negative (\( \Delta G < 0 \)).

Spontaneity does not indicate the speed of the reaction; it only tells us whether the process will ultimately occur without external influence.

• If \( \Delta G < 0 \), the reaction is spontaneous.
• If \( \Delta G = 0 \), the reaction is at equilibrium.
• If \( \Delta G > 0 \), the reaction is non-spontaneous.

Understanding the concept of spontaneous reactions helps predict whether a chemical process can occur naturally under specific conditions.

Standard Conditions

Standard conditions, often referred to as standard state conditions, include a defined set of criteria that ensure consistency and comparability in chemical datasets. These conditions are typically:

• Temperature: 298 K (25°C)
• Pressure: 1 atmosphere (atm)
• Concentration for solutions: 1 molarity (1 M)

Under these conditions, chemical reactions are assessed for their thermodynamic properties, such as \( \Delta H^{\circ} \), standard enthalpy change, \( \Delta S^{\circ} \), standard entropy change, and \( \Delta G^{\circ} \), standard Gibbs Free Energy change.

Different elements and compounds have their properties tabulated per these conditions to provide a benchmark for scientists and students. For example, in the context of the original exercise, the \( \Delta G^{\circ} \) values used in calculations are standardized to ensure resultant data is reliable and coherent across different reactions.

Entropy Change

Entropy change (\( \Delta S \)) is a measure of the disorder or randomness in a system during a chemical process. When a reaction occurs, the arrangement and energy distribution of molecules can change, affecting the overall entropy of the system.

• A positive \( \Delta S \) indicates an increase in disorder, typically favorable in natural processes.
• A negative \( \Delta S \) reflects a decrease in disorder, which might require external work to be done.

In thermodynamics, entropy is always a key factor when predicting the feasibility of a reaction through the Gibbs Free Energy equation:

\[ \Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ} \]
Here, the balance between enthalpy and entropy helps decide if a reaction is spontaneous. For example, a negative \( \Delta G^{\circ} \) might be achieved with either a large positive \( \Delta S^{\circ} \), a negative \( \Delta H^{\circ} \), or a combination thereof at a given temperature \( T \). Understanding entropy change alongside \( \Delta H \) and \( \Delta G \) is crucial for grasping why and when reactions occur.