Problem 70 - Chemistry: An Atoms-Focused Approach

Question

How do you calculate the value of \(\Delta G^{\circ}\) for a reaction that is the result of coupling a spontaneous reaction \(\left(\Delta G_{\text {spon }}^{\circ}<0\right)\).and a nonspontaneous reaction $$\left(\Delta G_{\text {nonspon }}^{\circ}>0\right) ?$$.

Step-by-Step Solution

Answer

Answer: The overall reaction will be spontaneous if the absolute value of the Gibbs free energy change of the spontaneous reaction is greater than the absolute value of the Gibbs free energy change of the nonspontaneous reaction.

Step 1: Understand the concept of Gibbs free energy

The Gibbs free energy (G) is a thermodynamic potential that measures the maximum reversible work that may be performed by a system at constant temperature and pressure. The change in the Gibbs free energy of the system, \(\Delta G\), is determined from the changes in enthalpy (\(\Delta H\)) and entropy (\(\Delta S\)), as well as the temperature (T) of the system: \(\Delta G = \Delta H - T \Delta S\). For a reaction at standard conditions, the Gibbs free energy change is denoted as \(\Delta G^{\circ}\). In general, a spontaneous reaction occurs when \(\Delta G^{\circ} < 0\), whereas a nonspontaneous reaction occurs when \(\Delta G^{\circ} > 0\).

Step 2: Find the relation between the Gibbs free energy of the overall reaction and the individual reactions

When two reactions are coupled, the overall reaction is the sum of the individual reactions. Consequently, the Gibbs free energy change of the overall reaction (\(\Delta G_{\text{total}}^{\circ}\)) is the sum of the Gibbs free energy changes for the spontaneous (\(\Delta G_{\text{spon}}^{\circ}\)) and the nonspontaneous (\(\Delta G_{\text{nonspon}}^{\circ}\)) reactions: $$\Delta G_{\text{total}}^{\circ} = \Delta G_{\text{spon}}^{\circ} + \Delta G_{\text{nonspon}}^{\circ}$$

Step 3: Compute the value of \(\Delta G^{\circ}\) for the coupled reaction

As we don't have concrete values of \(\Delta G_{\text{spon}}^{\circ}\) and \(\Delta G_{\text{nonspon}}^{\circ}\), we can't give a specific numerical value for \(\Delta G_{\text{total}}^{\circ}\). However, we can give a general conclusion based on the signs of the spontaneous and nonspontaneous reactions' Gibbs free energy values: Since \(\Delta G_{\text{spon}}^{\circ} < 0\) and \(\Delta G_{\text{nonspon}}^{\circ} > 0\), if the absolute value of \(\Delta G_{\text{spon}}^{\circ}\) is greater than the absolute value of \(\Delta G_{\text{nonspon}}^{\circ}\), the overall reaction will be spontaneous since the total Gibbs free energy change will be negative. On the other hand, if the absolute value of \(\Delta G_{\text{nonspon}}^{\circ}\) is greater than the absolute value of \(\Delta G_{\text{spon}}^{\circ}\), the overall reaction will be nonspontaneous since the total Gibbs free energy change will be positive. In the case where the absolute values of the Gibbs free energy changes for the two coupled reactions are equal, the overall reaction will be at equilibrium, and \(\Delta G_{\text{total}}^{\circ} = 0\).

Key Concepts

Spontaneous Reaction

A spontaneous reaction is one that occurs naturally, without any external input of energy, under given conditions. The driving force behind such reactions is the tendency of a system to move towards a state of lower energy. This is quantified by the Gibbs free energy change, \(\Delta G^{\circ}\).
For a reaction to be spontaneous, the system must release free energy, making \(\Delta G^{\circ} < 0\). Here, the negative sign of \(\Delta G^{\circ}\) signifies that the process will occur on its own over time.
The spontaneity of a reaction is dictated by:

• The enthalpy change (\(\Delta H\)), which should be negative (exothermic reaction), releasing heat to the surroundings.
• The entropy change (\(\Delta S\)), which should be positive, increasing randomness or disorder in the system.
• The temperature (\(T\)), which influences the impact of \(\Delta S\) on the reaction's spontaneity.

Thus, spontaneous reactions result when the increase in entropy or the release of heat lowers the Gibbs free energy below zero.

Nonspontaneous Reaction

Nonspontaneous reactions are the opposite of spontaneous reactions. They require an input of energy to proceed. In terms of Gibbs free energy, this means \(\Delta G^{\circ} > 0\). Such reactions do not happen on their own; they need continuous energy supply.
A nonspontaneous reaction may happen under certain conditions if external energy or intervention is provided, such as:

• Providing heat to increase the system's temperature.
• Applying mechanical work or electrical energy to drive the process.
• Changing the pressure or concentration to favor the reaction's progress.

In essence, nonspontaneous reactions are processes that move toward a higher energy state, requiring more energy than they release.

Thermodynamic Potential

Thermodynamic potential refers to the potential energy available in a system to perform work under specified conditions. Gibbs free energy is one of these important thermodynamic potentials, especially relevant for reactions at constant temperature and pressure.
By definition, Gibbs free energy gives an idea of the capacity of a system to perform non-mechanical work and the tendency of the system to change towards equilibrium. The main equation: \(\Delta G = \Delta H - T\Delta S\), connects enthalpy (\(\Delta H\)), temperature (\(T\)), and entropy (\(\Delta S\)) to determine if a process is energetically favorable.
Thermodynamic potentials help predict:

• The direction of chemical processes.
• The equilibrium states of reactions.
• The feasibility of reactions under given conditions.

Understanding these potentials gives insight into how reactions can be coupled, where a spontaneous reaction can drive a nonspontaneous one if the overall Gibbs free energy remains negative.