Gibbs free energy is a crucial concept in chemistry, helping us predict whether reactions will happen spontaneously. It combines enthalpy and entropy, giving us a powerful tool to understand chemical processes and their likelihood of occurring under specific conditions.

Calculating Gibbs free energy allows us to determine reaction spontaneity, equilibrium constants, and the stability of compounds. This knowledge is vital for understanding real-world applications, from industrial processes to biological systems, and helps us manipulate reactions to achieve desired outcomes.

Gibbs Free Energy and Spontaneity

Gibbs free energy fundamentals

• Gibbs free energy ($G$) thermodynamic quantity determines spontaneity of process at constant temperature and pressure
  • Process spontaneous if $\Delta G < 0$ (diamond formation from graphite)
  • Process non-spontaneous if $\Delta G > 0$ (water flowing uphill)
  • Process at equilibrium if $\Delta G = 0$ (saturated solution of sugar in water)
• $G$ defined as: $G = H - TS$
  • $H$ enthalpy, measure of total heat content of system (energy released or absorbed during chemical reaction)
  • $S$ entropy, measure of disorder or randomness of system (gas has higher entropy than solid)
  • $T$ absolute temperature in Kelvin (K) (room temperature ~298 K)
• Change in Gibbs free energy ($\Delta G$) given by: $\Delta G = \Delta H - T\Delta S$
  • $\Delta H$ change in enthalpy (heat released or absorbed)
  • $\Delta S$ change in entropy (increase or decrease in disorder)

Calculation of Gibbs free energy

• Standard Gibbs free energy of formation ($\Delta G_f^\circ$) change in Gibbs free energy when one mole of compound formed from constituent elements in standard states at 1 atm pressure and specified temperature (usually 298 K)
  • Table of standard Gibbs free energies of formation available for many compounds (CRC Handbook of Chemistry and Physics)
• Standard Gibbs free energy change ($\Delta G^\circ$) for reaction calculated using standard Gibbs free energies of formation:
  • $\Delta G^\circ = \sum \nu_p \Delta G_{f,p}^\circ - \sum \nu_r \Delta G_{f,r}^\circ$
    1. Identify the chemical reaction and write a balanced equation
    2. Look up the $\Delta G_f^\circ$ values for each reactant and product
    3. Multiply each $\Delta G_f^\circ$ value by its stoichiometric coefficient ($\nu$)
    4. Sum the products and subtract the reactants to obtain $\Delta G^\circ$

Spontaneity prediction using Gibbs energy

• Sign of $\Delta G$ determines spontaneity of reaction:
  • $\Delta G < 0$, reaction spontaneous (product-favored) (rusting of iron)
  • $\Delta G > 0$, reaction non-spontaneous (reactant-favored) (electrolysis of water)
  • $\Delta G = 0$, reaction at equilibrium (vapor pressure of liquid)
• Magnitude of $\Delta G$ indicates driving force for reaction:
  • Large negative $\Delta G$ indicates strongly spontaneous reaction (combustion of methane)
  • Small negative $\Delta G$ indicates weakly spontaneous reaction (dissolution of salt in water)
  • Large positive $\Delta G$ indicates strongly non-spontaneous reaction (decomposition of water into hydrogen and oxygen)
  • Small positive $\Delta G$ indicates weakly non-spontaneous reaction (melting of ice)

Interpretation of Gibbs energy diagrams

• Gibbs free energy diagrams plot $G$ as function of reaction coordinate (progress of reaction)
  • Reactants and products represented as local minima on diagram (stable states)
  • Transition state represented as local maximum between reactants and products (highest energy state)
• Equilibrium constant ($K$) determined from Gibbs free energy change:
  • $\Delta G^\circ = -RT \ln K$
    • $R$ ideal gas constant (8.314 J/mol·K)
    • $T$ absolute temperature in Kelvin (K)
    • Larger $K$ values indicate equilibrium favors products (reaction spontaneous)
    • Smaller $K$ values indicate equilibrium favors reactants (reaction non-spontaneous)
• Direction of spontaneous change determined from Gibbs free energy diagram:
  • System will spontaneously move from higher $G$ state to lower $G$ state (downhill on diagram)
  • At equilibrium, reactants and products have same $G$ value (no net change)

Applications of Gibbs free energy

• Feasibility of chemical processes assessed using Gibbs free energy:
  • Processes with $\Delta G < 0$ feasible and occur spontaneously (synthesis of ammonia from nitrogen and hydrogen)
  • Processes with $\Delta G > 0$ not feasible and require input of energy (decomposition of water into hydrogen and oxygen)
• Stability of compounds compared using standard Gibbs free energies of formation:
  • Compounds with lower (more negative) $\Delta G_f^\circ$ values more stable (diamond vs graphite)
  • Compounds with higher (less negative or positive) $\Delta G_f^\circ$ values less stable (ozone vs oxygen)
• Relationship between $\Delta G^\circ$ and equilibrium constant ($K$) given by:
  • $\Delta G^\circ = -RT \ln K$
    • $\Delta G^\circ < 0$, $K > 1$ (products favored at equilibrium) (formation of water from hydrogen and oxygen)
    • $\Delta G^\circ > 0$, $K < 1$ (reactants favored at equilibrium) (decomposition of calcium carbonate into calcium oxide and carbon dioxide)
    • $\Delta G^\circ = 0$, $K = 1$ (reactants and products equally favored at equilibrium) (vaporization of water at its boiling point)
• Equilibrium position shifted by changing conditions (temperature, pressure, or concentration) to alter value of $\Delta G$
  • Conditions changed to make $\Delta G$ more negative, equilibrium shifts towards products (increasing temperature for endothermic reaction)
  • Conditions changed to make $\Delta G$ more positive, equilibrium shifts towards reactants (decreasing pressure for reaction with fewer moles of gas)