Gibbs Free Energy and Reaction Spontaneity

1. From Universal Entropy to Gibbs Energy

In the previous lesson, we established that for any spontaneous process, the total entropy of the universe must increase ($\Delta S_{universe}>0$). While fundamentally correct, this isn't practical for a chemist, as measuring the entropy change of the entire universe is impossible. We need a criterion that depends only on the properties of the system (our reaction).

This is where Gibbs free energy originates. By considering a system in thermal and pressure equilibrium with its surroundings (a constant T and P environment, typical for lab chemistry), it can be rigorously shown that the condition $\Delta S_{universe}>0$ is mathematically equivalent to $\Delta G_{system}<0$.

Let's briefly trace this crucial derivation. The total entropy change is the sum of the system's and surroundings' entropy changes:
$\Delta S_{universe}=\Delta S_{system}+\Delta S_{surroundings}>0$
The heat released by the system ($-\Delta H_{system}$) is absorbed by the surroundings, so the entropy change of the surroundings is $\Delta S_{surroundings}=\frac{\Delta H_{surroundings}}{T}=\frac{-\Delta H_{system}}{T}$. Substituting this in gives:
$\Delta S_{universe}=\Delta S_{system}-\frac{\Delta H_{system}}{T}>0$
Multiplying by $-T$ (and reversing the inequality) yields:
$-T\Delta S_{system}+\Delta H_{system}<0$
This expression, $\Delta H_{system}-T\Delta S_{system}$, is defined as the change in Gibbs Free Energy, $\Delta G_{system}$. Thus, we arrive at the second law of thermodynamics, restated in its most useful form for chemists:

At constant temperature and pressure, a process is spontaneous if and only if $\Delta G<0$.

To begin, let's watch a brief overview of this concept.

This leads to the following clear criteria:

• $\Delta G<0$: The reaction is spontaneous in the forward direction. These are also called exergonic reactions.
• $\Delta G>0$: The reaction is non-spontaneous in the forward direction. It is, however, spontaneous in the reverse direction. These are called endergonic reactions.
• $\Delta G=0$: The system is at equilibrium, with the forward and reverse reaction rates being equal. There is no net change.

2. The Interplay of Enthalpy, Entropy, and Temperature

The equation $\Delta G=\Delta H-T\Delta S$ reveals a fascinating competition between enthalpy and entropy, with temperature acting as the weighting factor for the entropy term. By analyzing the signs of $\Delta H$ and $\Delta S$, we can predict the conditions under which a reaction will be spontaneous.

Here is a summary of the four scenarios:

Case 3 ($\Delta H<0,\Delta S<0$): This is an enthalpically-driven reaction. It is favorable from an energy standpoint but unfavorable from a disorder standpoint. At low temperatures, the favorable $\Delta H$ term dominates, making $\Delta G$ negative. At high temperatures, the unfavorable $-T\Delta S$ term becomes large and positive, making $\Delta G$ positive. Many organic addition reactions fall into this category.

Case 4 ($\Delta H>0,\Delta S>0$): This is an entropically-driven reaction. It is unfavorable energetically but favorable entropically. At low temperatures, the unfavorable $\Delta H$ term dominates. At high temperatures, the favorable $-T\Delta S$ term becomes large and negative, driving the reaction forward. Many organic elimination or decomposition reactions fit this profile.

3. Gibbs Free Energy and the Equilibrium Constant ($K_{eq}$)

Spontaneity tells us the direction a reaction will proceed from a given state, but it doesn't tell us the extent. To understand the final composition of a reaction mixture at equilibrium, we must relate $\Delta G$ to the equilibrium constant, $K_{eq}$.

This is achieved through the equation:

$\Delta G^{\circ }=-RT\ln K_{eq}$

Here, $\Delta G^{\circ }$ is the standard Gibbs free energy change, which is $\Delta G$ under standard conditions (1 atm pressure, 1 M concentration, 298 K). $R$ is the gas constant (8.314 J/mol·K).

This simple equation has profound implications:

The key relationships are:

• If $\Delta G^{\circ }<0$, then $\ln K_{eq}>0$, so $K_{eq}>1$. The reaction is product-favored at equilibrium.
• If $\Delta G^{\circ }>0$, then $\ln K_{eq}<0$, so $K_{eq}<1$. The reaction is reactant-favored at equilibrium.
• If $\Delta G^{\circ }=0$, then $\ln K_{eq}=0$, so $K_{eq}=1$. Reactants and products are present in comparable amounts at equilibrium.

Let's watch a short segment to solidify this connection.

The magnitude of $\Delta G^{\circ }$ is a powerful indicator of how complete a reaction will be. Even small values have a dramatic effect on the equilibrium position.

4. A Practical Calculation

Let's walk through a calculation to see how these principles are applied. It is crucial to be mindful of units, as $\Delta H$ is typically given in kJ/mol while $\Delta S$ is in J/mol·K. You must convert one to match the other.

This example highlights that a reaction can be spontaneous ($\Delta G<0$) even if it is endothermic ($\Delta H>0$), as long as the temperature is high enough to make the entropy term ($-T\Delta S$) sufficiently negative to overcome the positive enthalpy.

Conclusion

Today, we have unified the concepts of enthalpy and entropy into the single, decisive function of Gibbs free energy. You are now equipped to analyze the thermodynamic viability of any organic reaction.

Key Takeaways:

• Gibbs free energy ($\Delta G$) is the ultimate criterion for spontaneity at constant temperature and pressure. A negative $\Delta G$ signifies a spontaneous (exergonic) process.
• The spontaneity of a reaction is governed by the interplay between enthalpy, entropy, and temperature, as described by the equation $\Delta G=\Delta H-T\Delta S$.
• Reactions can be either enthalpically-driven (spontaneous at low T) or entropically-driven (spontaneous at high T).
• The standard Gibbs free energy change ($\Delta G^{\circ }$) is directly related to the equilibrium constant ($K_{eq}$) by the equation $\Delta G^{\circ }=-RT\ln K_{eq}$. This allows you to predict the position of equilibrium from thermodynamic data.

Preview of the Next Lesson:
Thermodynamics tells us where a reaction is going (the equilibrium state), but it tells us nothing about how fast it will get there. A reaction can be highly exergonic ($\Delta G\ll 0$) but proceed at an imperceptibly slow rate. In our next lesson, we will shift our focus from thermodynamics to kinetics. We will analyze reaction coordinate diagrams to understand the energy barriers (activation energy) that control the speed of a reaction, and we will identify key features like transition states and intermediates.