8.1 Helmholtz and Gibbs free energies

Helmholtz and Gibbs free energies are key thermodynamic potentials that help us understand how systems behave. They show us the energy available for useful work under different conditions, like constant temperature and volume or pressure.

These concepts are crucial for predicting spontaneous processes, determining equilibrium states, and calculating maximum work output. They're used in everything from battery design to chemical reaction analysis, making them essential tools in thermodynamics.

Helmholtz Free Energy

Helmholtz free energy definition

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• Thermodynamic potential measuring useful work obtainable from a closed system at constant temperature and volume
• Defined as [A = U - TS](https://www.fiveableKeyTerm:a_=_u_-_ts) where:
  • $U$ internal energy
  • $T$ absolute temperature (K)
  • $S$ entropy (J/K)
• Change in Helmholtz free energy ($dA$) equals maximum work extractable from closed system during isothermal process (constant temperature)

Physical significance of Helmholtz energy

• Represents portion of internal energy available for useful work at constant temperature and volume
  • Remaining portion $TS$ represents unavailable energy due to system's entropy
• In isothermal process, change in Helmholtz free energy equals maximum work system can perform on surroundings
  • Decrease in Helmholtz free energy during isothermal process indicates system can perform work on surroundings (battery discharging, gas expansion)
• Minimized at equilibrium for system at constant temperature and volume
  • Helps determine spontaneous direction of process and equilibrium state (chemical reactions, phase transitions)

Differential forms of free energies

• Differential form of Helmholtz free energy derived from definition and first law of thermodynamics:
  1. Start with $dA = dU - TdS - SdT$
  2. Substitute $dU = TdS - PdV$ (first law of thermodynamics)
  3. $dA = TdS - PdV - TdS - SdT$
  4. Simplify to $dA = -SdT - PdV$
• Differential form of Gibbs free energy derived similarly:
  1. Start with $dG = dH - TdS - SdT$
  2. Substitute $dH = TdS + VdP$ (enthalpy differential form)
  3. $dG = TdS + VdP - TdS - SdT$
  4. Simplify to $dG = -SdT + VdP$

Gibbs Free Energy

Gibbs free energy definition

• Thermodynamic potential measuring maximum reversible work performable by a system at constant temperature and pressure
• Defined as $G = H - TS$ where:
  • $H$ enthalpy (J)
  • $T$ absolute temperature (K)
  • $S$ entropy (J/K)
• Change in Gibbs free energy ($dG$) equals maximum non-expansion work extractable from system during isobaric (constant pressure) and isothermal process

Physical significance of Gibbs energy

• Represents portion of enthalpy available for useful work at constant temperature and pressure
  • Remaining portion $TS$ represents unavailable energy due to system's entropy
• In isobaric process, change in Gibbs free energy equals maximum non-expansion work system can perform on surroundings
  • Decrease in Gibbs free energy during isobaric process indicates system can perform non-expansion work on surroundings (electrochemical cells, pumps)
• Minimized at equilibrium for system at constant temperature and pressure
  • Helps determine spontaneous direction of process, equilibrium state, and stability of chemical reactions at constant temperature and pressure (fuel cells, batteries)