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 \(\Delta G_{f}^{\circ}\) 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.

Short answer

ΔG° for the reaction is -685.74 kJ/mol, making it spontaneous at 298 K. The reaction is spontaneous at temperatures above 37.92 K under standard conditions, assuming ΔH° and ΔS° don't depend on temperature.

Step-by-step solution

Calculate ΔG° using ΔGf° values

To calculate ΔG° for this reaction, use the formula: ΔG° = Σ ΔGf°(products) - Σ ΔGf°(reactants) From Appendix 4, we get the following ΔGf° values: ΔGf°(Fe₂O₃) = -740 kJ/mol ΔGf°(H₂) = 0 kJ/mol (since it's a diatomic gas) ΔGf°(Fe) = 0 kJ/mol (since it's in its standard state) ΔGf°(H₂O) = -228.58 kJ/mol Now we can calculate ΔG°: ΔG° = (2 x 0 + 3 x (-228.58)) - (1 x (-740) + 3 x 0) ΔG° = -685.74 kJ/mol

Determine if the reaction is spontaneous at 298 K

To check if the reaction is spontaneous, compare the calculated ΔG° value with zero: If ΔG° < 0, then the reaction is spontaneous. If ΔG° > 0, then the reaction is non-spontaneous. In our case, since ΔG° = -685.74 kJ/mol, which is less than 0, the reaction is spontaneous at 298 K.

Find the temperature range for spontaneous reaction

To determine the temperature range for which the reaction is spontaneous, we can use the Gibbs-Helmholtz equation: ΔG° = ΔH° - TΔS° We need to find ΔS° using the given ΔH° value: ΔH° = 100 kJ/mol Rearrange the equation to find ΔS°: ΔS° = (ΔH° - ΔG°) / T Using the given temperature of 298 K: ΔS° = (100 - (-685.74)) / 298 ΔS° = 2.637 kJ/mol K Now, we need to find the temperature at which ΔG° = 0: 0 = ΔH° - TΔS° Rearrange to find T: T = ΔH° / ΔS° T = 100 / 2.637 T = 37.92 K Since the problem asks for the temperature range where the reaction is spontaneous, we can conclude that this reaction is spontaneous for all temperatures above 37.92 K under standard conditions, given that ΔH° and ΔS° do not depend on temperature.

Key concepts

Reaction Spontaneity

In the world of chemistry, a reaction is considered spontaneous if it can occur without the need for external energy. To figure out if a reaction happens spontaneously, we look at the change in Gibbs Free Energy (ΔG). If ΔG is less than zero, the reaction is spontaneous and can occur naturally.
For the reaction of Fe₂O₃ with H₂, calculated ΔG is -685.74 kJ/mol, indicating spontaneity. Simply put, because this number is below zero, the reaction can happen on its own without extra input. This key concept helps chemists understand which reactions are favorable under certain conditions.

Thermodynamics

Thermodynamics is the study of energy, heat, and their transformations. In chemical reactions, it helps us understand how energy changes affect reactions. One key concept is the Gibbs Free Energy equation:

• ΔG = ΔH - TΔS

Where ΔG is Gibbs Free Energy, ΔH is enthalpy change, and ΔS is entropy change.
It's all about balance. Entropy, or disorder, and enthalpy, or heat content, influence a reaction’s spontaneity.
Understanding these terms helps chemists predict how reactions behave, making thermodynamics essential for studying chemical processes.

Standard Conditions

Standard conditions are essential for consistent comparison of chemical reactions. They refer to a common set of conditions: a pressure of 1 bar and a temperature of 298 K (25°C).
These conditions allow scientists to predict how reactions will behave in a controlled environment.

• Under these conditions, substances are in their most stable forms.

For example, the Gibbs Free Energy values used for calculations are standardized, providing reliable data across studies.
Knowing these conditions helps compare different reactions and predict their outcomes.

Gibbs-Helmholtz Equation

The Gibbs-Helmholtz Equation is crucial for understanding how temperature affects spontaneity:

• ΔG = ΔH - TΔS

This equation allows us to calculate the temperature range for spontaneous reactions. For instance, if we know ΔH and ΔS, we can solve for temperature (T) where ΔG changes.
For the given reaction, this equation showed that spontaneity occurs above 37.92 K under standard conditions.
Using this equation enables chemists to find out at what temperatures reactions become energetically favorable, aiding in the design and control of chemical processes.