Thermodynamic Functions of State

The following four thermodynamic functions of state can be derived from the first law of thermodynamics and the second law of thermodynamics, assuming the equilibrium state (i.e., the driving force is equal to zero):

dG = -SdT + VdP + μ1dN1 + μ2dN2 + ... + μidNi

dA = -SdT - PdV + μ1dN1 + μ2dN2 + ... + μidNi

dU = TdS - PdV + μ1dN1 + μ2dN2 + ... + μidNi

dH = TdS + VdP + μ1dN1 + μ2dN2 + ... + μidNi

Depending on how the system under consideration is controlled, one of the four thermodynamic functions of state, as opposed to any other, is advantageous to use:

The Gibbs energy (G) is the most useful thermodynamic function of state (from a practical point of view) and is used when the temperature, the pressure, and the composition of a system are controlled. The system is in equilibrium when its content of Gibbs energy reaches the minimum value.

The Helmholtz energy (A) is used when the temperature, the volume, and the composition of a system are controlled. The system is in equilibrium when its content of Helmholtz energy reaches the minimum value.

The internal energy (U) is used when the entropy, the volume, and the composition of a system are controlled. The system is in equilibrium when its content of internal energy reaches the minimum value.

The enthalpy (H) is used when the entropy, the pressure, and the composition of a system are controlled. The system is in equilibrium when its content of enthalpy reaches the minimum value. The enthalpy is related the heat content of the system.