Energy vs. Entropy
Temperature, volume and pressure are far easier to control than energy and entropy, and thus one normally encounters the last two cases for the computation of equilibria. For simple one-phase systems the results are straightforward: homogeneous temperature T , and, if gravity is ignored, homogeneous pressures p. More complex systems, in particular systems in several phases, and reacting and inert mixtures of several components have additional degrees of freedom that approach equilibrium values, and it is convenient to determine these equilibrium values under the assumption that thermal and mechanical equilibrium, i.e., homogeneous temperature and pressure, are established al- ready. Then, the computation of equilibrium states typically entails to find minima of free energies, either of the Helmholtz free energy F = U − TS, or of the Gibbs free energy G = H − TS.
The free energies describe the competition between energy and entropy, with the temperature as factor to determine their relative importance. We take a look at this for the Helmholtz free energy, F = U − TS. The Helmholtz free energy can attain a minimum state either by making the energy U small, or by making the entropic term TS large. At low temperatures, the product TS is relatively small, thus the entropic term does not matter much, and energy is more important; states of low energies are assumed, for instance the liquid state, which is due to the attractive potential between molecules. For high temperatures, however, the entropic term TS dominates, and states of large entropy are assumed, e.g., the vapor state. For intermediate temperatures, energy and entropy find a compromise, e.g., the coexistence of vapor, which has large entropy, and liquid, which has low energy, in phase equilibrium.