Principle of Minimum Energy - Thermodynamic Potentials

Thermodynamic Potentials

The principle of minimum energy can be generalized to apply to constraints other than fixed entropy. For other constraints, other state functions with dimensions of energy will be minimized. These state functions are known as thermodynamic potentials. Thermodynamic potentials are at first glance just simple algebraic combinations of the energy terms in the expression for the internal energy. For a simple, multicomponent system, the internal energy may be written:

where the intensive parameters (T, P, μ_j) are functions of the internal energy's natural variables via the equations of state. As an example of another thermodynamic potential, the Helmholtz free energy is written:

where temperature has replaced entropy as a natural variable. In order to understand the value of the thermodynamic potentials, it is necessary to view them in a different light. They may in fact be seen as (negative) Legendre transforms of the internal energy, in which certain of the extensive parameters are replaced by the derivative of internal energy with respect to that variable (i.e. the conjugate to that variable). For example, the Helmholtz free energy may be written:

and the maximum will occur when the variable T becomes equal to the temperature since

The Helmholtz free energy is a useful quantity when studying thermodynamic transformations in which the temperature is held constant. Although the reduction in the number of variables is a useful simplification, the main advantage comes from the fact that the Helmholtz free energy is minimized at equilibrium with respect to any unconstrained internal variables for a closed system at constant temperature and volume. This follows directly from the principle of minimum energy which states that at constant entropy, the internal energy is minimized. This can be stated as:

where and are the value of the internal energy and the (fixed) entropy at equilibrium. The volume and particle number variables have been replaced by x which stands for any internal unconstrained variables.

As a concrete example of unconstrained internal variables, we might have a chemical reaction in which there are two types of particle, an A atom and an A₂ molecule. If and are the respective particle numbers for these particles, then the internal constraint is that the total number of A atoms is conserved:

we may then replace the and variables with a single variable and minimize with respect to this unconstrained variable. There may be any number of unconstrained variables depending on the number of atoms in the mixture. For systems with multiple sub-volumes, there may be additional volume constraints as well.

The minimization is with respect to the unconstrained variables. In the case of chemical reactions this is usually the number of particles or mole fractions, subject to the conservation of elements. At equilibrium, these will take on their equilibrium values, and the internal energy will be a function only of the chosen value of entropy . By the definition of the Legendre transform, the Helmholtz free energy will be:

The Helmholtz free energy at equilibrium will be:

where is the (unknown) temperature at equilibrium. Substituting the expression for :

Assuming the order of the extrema can be exchanged:

showing that the Helmholtz free energy is minimized at equilibrium.

The Enthalpy and Gibbs free energy, are similarly derived.