Exhaustive Review of Thermodynamic Potentials H and G

Comprehensive Analysis of Enthalpy ($H$) and Gibbs Free Energy ($G$)

Fundamental Concepts and Definition of Enthalpy ($H$)

Enthalpy, represented by the symbol $H$, is a fundamental thermodynamic property defined as the total heat content of a system. It is mathematically expressed as the sum of the system's internal energy ($U$) and the product of its pressure ($P$) and volume ($V$), yielding the formula $H = U + PV$. As a state function, enthalpy depends only on the current state of the system, not on the path taken to arrive at that state. Enthalpy is also an extensive property, meaning its value scales with the mass or amount of matter present in the system.

In chemical reactions, we primarily focus on the change in enthalpy ($\Delta H$). Under conditions of constant pressure, the change in enthalpy is equal to the heat ($q$) exchanged with the surroundings. This relationship is crucial for categorizing reactions. An exothermic reaction releases energy, resulting in a negative enthalpy change ($\Delta H < 0$), while an endothermic reaction absorbs energy, resulting in a positive enthalpy change ($\Delta H > 0$). These measurements are typically expressed in units of Kilojoules per mole ($kJ\,mol^{-1}$).

Introduction to Gibbs Free Energy ($G$)

Gibbs Free Energy, denoted by the symbol $G$, is a thermodynamic potential that quantifies the maximum amount of reversible work a system can perform at constant temperature and pressure. Named after the scientist Josiah Willard Gibbs, it serves as the definitive criterion for predicting the spontaneity of a process. The definitive equation for Gibbs Free Energy is $G = H - TS$, where $H$ represents enthalpy, $T$ is the absolute temperature in Kelvin ($K$), and $S$ is the entropy, or the degree of disorder, of the system.

The significance of Gibbs Free Energy lies in its ability to account for the competing influences of energy (enthalpy) and randomness (entropy). A process is considered thermodynamically favorable, or spontaneous, if it leads to a decrease in the system's free energy ($\Delta G < 0$). If $\Delta G > 0$, the process is non-spontaneous and requires an external input of energy to proceed. When $\Delta G = 0$, the system has reached chemical equilibrium, where the rates of the forward and reverse processes are equal.

The Gibbs-Helmholtz Relationship and Spontaneity

The interplay between enthalpy ($H$) and Gibbs Free Energy ($G$) is most effectively captured by the Gibbs-Helmholtz equation for changes in a system: $\Delta G = \Delta H - T\Delta S$. This equation allows scientists to determine how temperature affects the spontaneity of a chemical reaction. Based on the signs of $\Delta H$ and $\Delta S$, four distinct scenarios for spontaneity emerge:

1. If $\Delta H < 0$ (exothermic) and $\Delta S > 0$ (increased disorder), then $\Delta G$ is always negative, and the reaction is spontaneous at all temperatures.
2. If \Delta H > 0 (endothermic) and \Delta S < 0 (decreased disorder), then $\Delta G$ is always positive, and the reaction is non-spontaneous at all temperatures.
3. If \Delta H > 0 and \Delta S > 0, the reaction is spontaneous only at high temperatures, where the $-T\Delta S$ term becomes large enough to outweigh the positive $\Delta H$.
4. If \Delta H < 0 and \Delta S < 0, the reaction is spontaneous only at low temperatures, where the $-T\Delta S$ term is small enough that the negative $\Delta H$ dominates.

Standard States and Thermodynamic Calculations

To facilitate the comparison of different substances, thermodynamic values are standardized at specific conditions, usually identified by a superscript circle ($\Delta H^\circ$, $\Delta G^\circ$). These standard state conditions typically refer to a pressure of $1\,atm$ (or $10^5\,Pa$), a temperature of $298.15\,K$ ($25\,^\circ C$), and a concentration of $1\,mol\,dm^{-3}$ for solutes. The standard Gibbs free energy of formation ($\Delta G_f^\circ$) is the change in free energy when one mole of a compound is formed from its constituent elements in their most stable states.

For any given chemical reaction, the total change in Gibbs Free Energy can be calculated using the following summation formula: $\Delta G_{rxn}^\circ = \sum n\Delta G_f^\circ(\text{products}) - \sum m\Delta G_f^\circ(\text{reactants})$. Similarly, the enthalpy of a reaction is calculated as $\Delta H_{rxn}^\circ = \sum n\Delta H_f^\circ(\text{products}) - \sum m\Delta H_f^\circ(\text{reactants})$. These calculations are essential for designing industrial chemical processes, understanding biological metabolism, and predicting the stability of materials.