Question

Using data from Appendix \(\mathrm{C}\) , calculate \(\Delta G^{\circ}\) for the following reactions. Indicate whether each reaction is spontaneous at 298 \(\mathrm{K}\) under standard conditions. (a) \(2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{SO}_{3}(g)\) (b) \(\mathrm{NO}_{2}(g)+\mathrm{N}_{2} \mathrm{O}(g) \longrightarrow 3 \mathrm{NO}(g)\) (c) \(6 \mathrm{Cl}_{2}(g)+2 \mathrm{Fe}_{2} \mathrm{O}_{3}(s) \rightarrow 4 \mathrm{FeCl}_{3}(s)+3 \mathrm{O}_{2}(g)\) (d) \(\mathrm{SO}_{2}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{S}(s)+2 \mathrm{H}_{2} \mathrm{O}(g)\)

Short answer

(a) \( \Delta G^\circ = -140.0 \, kJ/mol \): The reaction is spontaneous. (b) \( \Delta G^\circ = 46.9 \, kJ/mol \): The reaction is non-spontaneous. (c) \( \Delta G^\circ = -784.0 \, kJ/mol \): The reaction is spontaneous. (d) \( \Delta G^\circ = -157.0 \, kJ/mol \): The reaction is spontaneous.

Step-by-step solution

Gather Gibbs free energy values

Use Appendix C to find the Gibbs free energy values for each species in the given reactions.

Calculate \(\Delta G^\circ\) for each reaction

Use the equation \(\Delta G^\circ(products) - \Delta G^\circ(reactants)\) to calculate the Gibbs free energy change for each reaction. (a) $2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{SO}_{3}(g)$ \(\Delta G^\circ(2 SO_3) - \Delta G^\circ(2 SO_2 + O_2) = (-740.4) - (2(-300.2) + 0) = -140.0 \, kJ/mol \) (b) $\mathrm{NO}_{2}(g)+\mathrm{N}_{2} \mathrm{O}(g) \longrightarrow 3 \mathrm{NO}(g)$ \(\Delta G^\circ(3 NO) - \Delta G^\circ(NO_2 + N_2O) = (3(-86.6)) - (-51.3 - 103.6) = 46.9 \, kJ/mol \) (c) $6 \mathrm{Cl}_{2}(g)+2 \mathrm{Fe}_{2} \mathrm{O}_{3}(s) \rightarrow 4 \mathrm{FeCl}_{3}(s)+3 \mathrm{O}_{2}(g)$ \(\Delta G^\circ(4 FeCl_3 + 3 O_2) - \Delta G^\circ(6 Cl_2 + 2 Fe_2O_3) = (4(-399.5) + 3(0)) - (6(0) + 2(-824.2)) = -784.0 \, kJ/mol \) (d) $\mathrm{SO}_{2}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{S}(s)+2 \mathrm{H}_{2} \mathrm{O}(g)$ \(\Delta G^\circ(S + 2 H_2O) - \Delta G^\circ(SO_2 + 2 H_2) = (0 + 2(-228.6)) - (-300.2 + 2(0)) = -157.0 \, kJ/mol \)

Determine spontaneity of each reaction

If \(\Delta G^\circ < 0\), then the reaction is spontaneous. If \(\Delta G^\circ > 0\), the reaction is non-spontaneous. If \(\Delta G^\circ = 0\), the system is in equilibrium. (a) \(\Delta G^\circ = -140.0 \, kJ/mol < 0\): The reaction is spontaneous. (b) \(\Delta G^\circ = 46.9 \, kJ/mol > 0\): The reaction is non-spontaneous. (c) \(\Delta G^\circ = -784.0 \, kJ/mol < 0\): The reaction is spontaneous. (d) \(\Delta G^\circ = -157.0 \, kJ/mol < 0\): The reaction is spontaneous.

Key concepts

Thermodynamics

Thermodynamics is the study of energy transformations in chemical processes. It helps us understand how energy is transferred and transformed during a reaction. One of the core concepts in thermodynamics is Gibbs free energy, denoted as \( \Delta G \).
This value helps predict whether a reaction will occur on its own, without needing external energy, by giving insights on the energy availability.
In a chemical reaction, various energies like enthalpy and entropy play key roles.
Gibbs free energy combines these aspects to provide a comprehensive view of the thermodynamic potential of a reaction.
To predict how a reaction will proceed, you calculate \( \Delta G \) using the formula:
\[\Delta G = \Delta H - T \Delta S\]
where:

• \( \Delta H \) is the change in enthalpy
• \( T \) is the temperature in Kelvin
• \( \Delta S \) is the change in entropy

When \( \Delta G \) is negative, it indicates that the reaction releases free energy and tends to be spontaneous under the given conditions.

Chemical Spontaneity

Chemical spontaneity determines if a reaction can proceed on its own without external input. A spontaneous chemical reaction will occur naturally once initiated, contributing to the transformation of reactants to products.
This spontaneity can be gauged through the sign of \( \Delta G \).
A few key points to remember about spontaneity include:

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

These rules help us quickly determine the likelihood of a reaction happening naturally at standard conditions.

Standard Conditions

Standard conditions are a set of predefined conditions used to provide consistency in thermodynamic measurements.
They allow chemists to compare results from different experiments uniformly.
In most cases, standard conditions refer to:

• Temperature: 298 K (25°C)
• Pressure: 1 atm
• Each species present at 1 M concentration, if in solution

Gibbs free energy changes calculated under these conditions are denoted with a superscript circle, \( \Delta G^\circ \), to differentiate them from non-standard conditions.
These values are universal and used for comparing the fundamental thermodynamic properties of various reactions.

Reaction Equilibrium

Reaction equilibrium occurs when the rate of the forward reaction equals the rate of the reverse reaction.
At this point, the concentrations of reactants and products remain constant over time.
No net change is observed in the system.
The Gibbs free energy change at equilibrium is zero, \( \Delta G = 0 \).
This means neither the forward nor the backward reaction is favored under the prevailing conditions.
Equilibrium is crucial in studying the conditions that favor the formation of either reactants or products.
It provides valuable information about the potential amounts of substances present when a reaction reaches a stable point.