Gibbs free energy Knowledge-Based Study

Study notes

1. Defining Gibbs Free Energy

   In chemical thermodynamics, the spontaneity of a process cannot be evaluated solely by considering the enthalpy change of the system. According to the Second Law of Thermodynamics, a spontaneous process must increase the total entropy of the universe: $\Delta S_{\text{univ}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}} > 0$.

   To predict spontaneity using exclusively system-defined state functions at constant temperature and pressure, Josiah Willard Gibbs introduced the concept of Gibbs free energy ($G$). It represents the maximum reversible work that can be extracted from a closed thermodynamic system at constant temperature and pressure.

   • Spontaneous process (exergonic): $\Delta G < 0$
   • Non-spontaneous process (endergonic): $\Delta G > 0$ (the reverse process is spontaneous)
   • System at dynamic equilibrium: $\Delta G = 0$

   The Gibbs-Helmholtz Relation

   At constant temperature ($T$, in kelvins) and constant pressure, the change in Gibbs free energy for a system is given by the foundational thermodynamic equation:

   $$\Delta G = \Delta H - T\Delta S$$

   This equation demonstrates that chemical spontaneity is governed by the competition between two distinct thermodynamic driving forces:

   • Enthalpic factor ($\Delta H$): Exothermic processes ($\Delta H < 0$) release heat to the surroundings and favor spontaneity.
   • Entropic factor ($-T\Delta S$): An increase in system dispersal or positional disorder ($\Delta S > 0$) creates a favorable negative contribution to $\Delta G$.

   Caution: Absolute temperature $T$ must always be expressed in kelvins ($\text{K}$). Because enthalpy values are typically reported in kilojoules per mole ($\text{kJ}\cdot\text{mol}^{-1}$) and entropy values in joules per mole-kelvin ($\text{J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}$), unit conversion is critical when calculating $\Delta G$.

   Predicting Spontaneity Under Varying Conditions

   Because the entropic term is scaled by absolute temperature ($T$), temperature plays a decisive role when enthalpy and entropy provide opposing thermodynamic tendencies:

   • $\Delta H < 0$ and $\Delta S > 0$: Both driving forces are favorable. $\Delta G < 0$ at all temperatures; the process is unconditionally spontaneous.
   • $\Delta H > 0$ and $\Delta S < 0$: Both driving forces are unfavorable. $\Delta G > 0$ at all temperatures; the process is non-spontaneous at all temperatures.
   • $\Delta H < 0$ and $\Delta S < 0$: Enthalpy favors spontaneity, but entropy opposes it. $\Delta G < 0$ only at low temperatures where $|\Delta H| > |T\Delta S|$.
   • $\Delta H > 0$ and $\Delta S > 0$: Entropy favors spontaneity, but enthalpy opposes it. $\Delta G < 0$ only at high temperatures where $T\Delta S > \Delta H$.

   The crossover temperature at which a system shifts between spontaneity and non-spontaneity occurs when $\Delta G = 0$, defined as: $$T = \frac{\Delta H}{\Delta S}$$

   Connecting $\Delta G^\circ$ to the Equilibrium Constant

   Under standard thermodynamic conditions ($1\text{ bar}$, solute concentrations of $1\text{ M}$, and a specified temperature usually of $298.15\text{ K}$), the standard free energy change ($\Delta G^\circ$) reflects the intrinsic thermodynamic drive from pure reactants to pure products:

   $$\Delta G^\circ = \sum n\Delta G_{\text{f}}^\circ(\text{products}) - \sum m\Delta G_{\text{f}}^\circ(\text{reactants})$$

   For any arbitrary composition of a reaction mixture, the actual free energy change $\Delta G$ relates to the reaction quotient $Q$ and the equilibrium constant $K$:

   $$\Delta G = \Delta G^\circ + RT\ln Q$$

   At dynamic chemical equilibrium, $\Delta G = 0$ and $Q = K$, yielding the central thermodynamic link:

   $$\Delta G^\circ = -RT\ln K$$

   • If $\Delta G^\circ < 0$, then $K > 1$, and products predominate at equilibrium.
   • If $\Delta G^\circ > 0$, then $K < 1$, and reactants predominate at equilibrium.
   • If $\Delta G^\circ = 0$, then $K = 1$, indicating balanced equilibrium concentrations under standard states.