Comprehensive Notes on Gibbs Free Energy

Gibbs Free Energy

Definition

  • Gibbs free energy (G) combines enthalpy (H), temperature (T), and entropy (S).

  • Change in Gibbs free energy (\Delta G) indicates spontaneity of a reaction.

  • It measures the maximum energy released to perform useful work at constant temperature and pressure.

  • Defined as: \Delta G = \Delta H - T\Delta S

    • T is temperature in Kelvins.
    • T\Delta S represents energy absorbed during reversible entropy increase.

Conceptual Overview

  • Visualized as a valley between hills; systems move to reduce free energy, like a ball rolling downhill.
  • Bottom of the valley: Equilibrium.
  • Sides of the hill: Points toward or away from equilibrium.
  • Movement towards equilibrium: \Delta G < 0 (spontaneous, exergonic).
  • Movement away from equilibrium: \Delta G > 0 (non-spontaneous, endergonic).
  • At equilibrium: \Delta G = 0. System resists change.

Spontaneity Criteria

  • \Delta G < 0: Spontaneous reaction.
  • \Delta G > 0: Non-spontaneous reaction.
  • \Delta G = 0: System at equilibrium (\Delta H = T\Delta S).

Phase Equilibria

  • Phase equilibria: States where multiple phases coexist.
  • At equilibrium, \Delta G = 0.
  • For gas-solid equilibrium: \Delta G = G{gas} - G{solid} = 0, so G{gas} = G{solid}.
  • Temperature in Gibbs free energy calculation is always in Kelvins (positive).

Temperature Dependence and Phase Changes

  • Phase changes are temperature-dependent.
  • Example: Water boiling.
    • Boiling is endothermic (\Delta H > 0) as hydrogen bonds break.
    • Entropy increases (\Delta S > 0) as water becomes gas.
    • Reaction is spontaneous only if T\Delta S > \Delta H (resulting in negative \Delta G).
    • For water, this occurs above 373 K (100°C).
    • Below 100°C, \Delta G > 0 and boiling is non-spontaneous.
    • At 100°C, equilibrium exists between liquid and gas (\Delta H - T\Delta S = 0), and vapor pressure equals ambient pressure (boiling point).

Reaction Rate vs. Spontaneity

  • Reaction rate depends on activation energy (E_a), not \Delta G.
  • Spontaneous reactions can be fast or slow.

Kinetic vs. Thermodynamic Control

  • Reversible reactions may yield products differing in stability and kinetics.
  • Thermodynamically stable product may have slower kinetics due to higher E_a.
  • Kinetic control: Initial major product is formed faster (lower E_a).
  • Thermodynamic control: Given sufficient time, the dominant product is more stable (lower free energy).
  • Eventually, the reaction reaches equilibrium, defined by Keq.

Standard Gibbs Free Energy

  • Measured under standard conditions to yield \Delta G^{\circ}_{rxn}.
  • Standard conditions: 1 M solutions.
  • Standard free energy of formation (\Delta G^{\circ}_f) refers to the free energy when one mole of a compound is formed from its elements in their standard states.
  • \Delta G^{\circ}_f = 0 for elements in standard state.
  • Standard free energy of a reaction (\Delta G^{\circ}_{rxn}) is measured when reactants convert to products under standard conditions (298 K, 1 atm).
  • Calculated as: \Delta G^{\circ}{rxn} = \Sigma \Delta G^{\circ}{f(products)} - \Sigma \Delta G^{\circ}_{f(reactants)}.

Free Energy, Keq, and Q

  • Relationship: \Delta G^{\circ}_{rxn} = -RTln(Keq)

    • R: Ideal gas constant.
    • T: Temperature in Kelvins.
    • Keq: Equilibrium constant.

  • Higher Keq means more negative \Delta G^{\circ}_{rxn}, indicating greater spontaneity.

  • For reactions in progress (non-standard conditions), use the reaction quotient Q.
    \Delta G{rxn} = \Delta G^{\circ}{rxn} + RTln(Q) = RTln(\frac{Q}{Keq})

  • If \frac{Q}{Keq} < 1 (Q < Keq): \Delta G_{rxn} < 0, reaction proceeds forward spontaneously.

  • If \frac{Q}{Keq} > 1 (Q > Keq): \Delta G_{rxn} > 0, reaction proceeds in reverse spontaneously.

  • If \frac{Q}{Keq} = 1 (Q = Keq): \Delta G_{rxn} = 0, reaction is at equilibrium.

Catalysts

  • Catalysts reduce activation energy (E_a) without altering overall \Delta G.

Conclusion of Thermochemistry

  • Systems: Open, closed, isolated.
  • Processes: Isothermal, adiabatic, isobaric, isovolumetric.
  • State functions: Pressure, density, temperature, volume, enthalpy, internal energy, Gibbs free energy, entropy.
  • Phase change equilibria: \Delta G = 0.
  • Enthalpy: Heat content of a system related to intermolecular interactions and bonds.
  • Hess's law: Calculates total enthalpy change for a series of reactions.
  • Entropy: Measure of energy dispersal; not always disorder.
  • Gibbs free energy: Combines temperature, enthalpy, and entropy to determine spontaneity.
  • Spontaneous: \Delta G < 0.
  • Non-spontaneous: \Delta G > 0.
  • Coupling reactions: Non-spontaneous reactions can occur by coupling them with spontaneous reactions in biological systems.