5.1 Chemical Potential and Gibbs Free Energy

Chemical potential and Gibbs free energy are key concepts in understanding spontaneous processes and equilibrium. They help predict reaction directions and phase transitions by comparing energy states of reactants and products.

These ideas are crucial for grasping chemical equilibrium and phase changes. By linking energy differences to spontaneity, we can figure out when reactions will happen on their own and when they need a push.

Chemical potential and Gibbs free energy

Definition and relationship

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• Chemical potential is the partial molar Gibbs free energy
  • Represents the energy associated with adding or removing one mole of a substance from a system at constant temperature and pressure
• The chemical potential of a pure substance is equal to its molar Gibbs free energy
• In a mixture, the chemical potential of a component:
  • Related to its partial molar Gibbs free energy
  • Depends on its concentration or activity
• The difference in chemical potential between two states drives the direction of spontaneous processes (chemical reactions, phase transitions)

Role in driving spontaneous processes

• Spontaneous processes occur in the direction that minimizes the overall Gibbs free energy of the system
  • Examples: chemical reactions proceeding towards equilibrium, phase transitions (melting, vaporization)
• The difference in chemical potential between the initial and final states determines the direction of spontaneous change
  • If the chemical potential of the initial state is higher than the final state, the process occurs spontaneously
  • If the chemical potential of the final state is higher than the initial state, the process is non-spontaneous and requires external energy input

Chemical potential for reaction direction

Predicting reaction direction

• A chemical reaction proceeds spontaneously in the direction that minimizes the overall Gibbs free energy of the system
• The direction of a reaction can be predicted by comparing the chemical potentials of reactants and products
  • If the sum of the chemical potentials of the reactants is greater than that of the products, the reaction proceeds spontaneously in the forward direction
  • If the sum of the chemical potentials of the products is greater than that of the reactants, the reaction proceeds spontaneously in the reverse direction
• At equilibrium, the chemical potentials of reactants and products are equal, and there is no net change in the system

Examples of reaction direction prediction

• Synthesis of ammonia from nitrogen and hydrogen:
  • N2(g) + 3H2(g) ⇌ 2NH3(g)
  • At standard conditions, the chemical potentials favor the formation of ammonia, driving the reaction forward
• Decomposition of calcium carbonate:
  • CaCO3(s) ⇌ CaO(s) + CO2(g)
  • At high temperatures, the chemical potentials favor the decomposition of calcium carbonate into calcium oxide and carbon dioxide

Gibbs free energy changes

Calculating Gibbs free energy changes

• The change in Gibbs free energy (ΔG) for a process can be calculated using the equation:
  • ΔG = ΔH - TΔS
  • ΔH is the change in enthalpy, T is the absolute temperature, and ΔS is the change in entropy
• For a chemical reaction, the standard Gibbs free energy change (ΔG°) can be calculated using the standard Gibbs free energies of formation (ΔG°f) of the reactants and products:
  • ΔG° = ΣΔG°f (products) - ΣΔG°f (reactants)
• The Gibbs free energy change under non-standard conditions can be calculated using the equation:
  • ΔG = ΔG° + RT ln Q
  • R is the gas constant, T is the absolute temperature, and Q is the reaction quotient

Interpreting Gibbs free energy changes

• The sign and magnitude of ΔG determine the spontaneity and extent of a chemical process:
  • If ΔG < 0, the process is spontaneous and favored (exergonic)
  • If ΔG > 0, the process is non-spontaneous and unfavored (endergonic)
  • If ΔG = 0, the system is at equilibrium
• Examples of spontaneous processes:
  • Combustion of fuels (negative ΔG due to the release of heat and increase in entropy)
  • Dissolution of salt in water (negative ΔG due to the increase in entropy of the system)
• Examples of non-spontaneous processes:
  • Photosynthesis (positive ΔG, requires energy input from sunlight)
  • Synthesis of complex molecules (proteins, nucleic acids) from simple precursors (positive ΔG, requires energy input)

Equilibrium constant vs Gibbs free energy change

Relationship between equilibrium constant and Gibbs free energy change

• The equilibrium constant (K) of a chemical reaction is related to the standard Gibbs free energy change (ΔG°) by the equation:
  • ΔG° = -RT ln K
  • R is the gas constant and T is the absolute temperature
• A large equilibrium constant (K > 1) corresponds to a negative ΔG° and indicates that the products are favored at equilibrium
• A small equilibrium constant (K < 1) corresponds to a positive ΔG° and indicates that the reactants are favored at equilibrium
• When ΔG° = 0, the equilibrium constant is equal to 1, and the reactants and products are present in equal concentrations at equilibrium

Predicting equilibrium composition using thermodynamic data

• The relationship between ΔG° and K allows for the prediction of the direction and extent of a chemical reaction at equilibrium based on thermodynamic data
• Examples:
  • Formation of water from hydrogen and oxygen: 2H2(g) + O2(g) ⇌ 2H2O(g), K ≈ 10^48 at 298 K
    • Large equilibrium constant indicates that the reaction strongly favors the formation of water at equilibrium
  • Decomposition of dinitrogen tetroxide: N2O4(g) ⇌ 2NO2(g), K ≈ 4.6 at 298 K
    • Moderate equilibrium constant indicates that both N2O4 and NO2 are present in significant amounts at equilibrium
• Calculating equilibrium concentrations using the equilibrium constant:
  • For the reaction aA + bB ⇌ cC + dD, K = [C]^c [D]^d / [A]^a [B]^b
  • Concentrations at equilibrium can be determined by solving the equation for given initial concentrations and K value