Free energy and equilibrium Knowledge-Based Study

Study notes

1. Non-Standard Conditions and the Reaction Quotient

   In chemical thermodynamics, most tabulated thermodynamic values are provided under standard-state conditions, denoted by the standard Gibbs free energy change, $\Delta G^\circ$. Standard state refers to pure substances, gases at a partial pressure of 1 bar (or 1 atm), and solutes at a concentration of 1 M, usually at 298.15 K.

   However, actual chemical processes rarely occur under standard conditions throughout their entire course. The actual Gibbs free energy change, $\Delta G$, depends on the instantaneous composition of the reaction mixture. This relationship is quantified using the reaction quotient, $Q$:

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

   In this equation:

   • $\Delta G$ represents the instantaneous change in Gibbs free energy (in J/mol).
   • $\Delta G^\circ$ represents the standard change in Gibbs free energy (in J/mol).
   • $R$ is the universal gas constant (8.314 J/(mol·K)).
   • $T$ is the absolute temperature in kelvins (K).
   • $Q$ is the reaction quotient, expressed in terms of partial pressures ($Q_p$) for gas-phase reactions or molar concentrations ($Q_c$) for aqueous solutions.

   The sign of $\Delta G$ determines the direction of spontaneity:

   • If $\Delta G < 0$, the forward reaction is spontaneous under the given conditions.
   • If $\Delta G > 0$, the forward reaction is non-spontaneous, meaning the reverse reaction is spontaneous.
   • If $\Delta G = 0$, the system has reached chemical equilibrium.

   The Condition for Dynamic Equilibrium

   As a chemical reaction proceeds, reactants are converted into products or vice versa, causing the value of $Q$ to change continuously until the system reaches chemical equilibrium. At equilibrium, the rates of the forward and reverse reactions are equal, and the macroscopic concentrations of all species remain constant.

   Thermodynamically, dynamic equilibrium corresponds to the minimum possible Gibbs free energy for the reacting system. At this state:

   $$\Delta G = 0$$

   Furthermore, when a system is at equilibrium, the reaction quotient $Q$ is equal to the equilibrium constant, $K$:

   $$Q = K$$

   Substituting $\Delta G = 0$ and $Q = K$ into the non-standard free energy equation gives:

   $$0 = \Delta G^\circ + RT \ln K$$

   Rearranging this yields the fundamental thermodynamic relationship connecting standard free energy to the equilibrium constant:

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

   Alternatively, solving for $K$ gives:

   $$K = e^{-\frac{\Delta G^\circ}{RT}}$$

   Predicting Reaction Extent from $\Delta G^\circ$

   The relationship $\Delta G^\circ = -RT \ln K$ allows chemists to predict the extent of a reaction at equilibrium directly from standard thermodynamic data without running the reaction.

   Consider the three fundamental cases:

   • $\Delta G^\circ < 0$: When the standard free energy change is negative, $\ln K > 0$, which means $K > 1$. At equilibrium, products dominate over reactants. The reaction is product-favored under standard conditions.
   • $\Delta G^\circ > 0$: When the standard free energy change is positive, $\ln K < 0$, which means $K < 1$. At equilibrium, reactants dominate over products. The reaction is reactant-favored under standard conditions.
   • $\Delta G^\circ = 0$: When $\Delta G^\circ$ equals zero, $\ln K = 0$, which means $K = 1$. At equilibrium, reactants and products are present in comparable thermodynamic activities.

   It is crucial to distinguish between $\Delta G$ and $\Delta G^\circ$:

   • $\Delta G$ changes continuously as a reaction proceeds, driving the mixture toward equilibrium until $\Delta G = 0$.
   • $\Delta G^\circ$ is a fixed constant for a given reaction at a specific temperature and directly indicates the numerical value of $K$.

   Connecting Enthalpy, Entropy, and Equilibrium

   The standard Gibbs free energy change can also be expressed in terms of standard enthalpy change ($\Delta H^\circ$) and standard entropy change ($\Delta S^\circ$):

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

   Setting this equal to the equilibrium relationship gives:

   $$-RT \ln K = \Delta H^\circ - T\Delta S^\circ$$

   Dividing both sides by $-RT$ yields the van 't Hoff equation in linear form:

   $$\ln K = -\frac{\Delta H^\circ}{R}\left(\frac{1}{T}\right) + \frac{\Delta S^\circ}{R}$$

   This equation explains how temperature influences the equilibrium position according to Le Chatelier's principle:

   • Endothermic reactions ($\Delta H^\circ > 0$): An increase in temperature increases $K$, shifting the equilibrium toward products.
   • Exothermic reactions ($\Delta H^\circ < 0$): An increase in temperature decreases $K$, shifting the equilibrium toward reactants.

   Plotting $\ln K$ versus $1/T$ produces a straight line with a slope equal to $-\Delta H^\circ / R$ and a y-intercept equal to $\Delta S^\circ / R$, enabling experimental determination of standard enthalpy and entropy changes.