Systems with Macroscopically Relevant Internal Degrees of Freedom

Abstract

Equilibrium thermodynamics only retains its validity as long as the molecular internal degrees of freedom of the considered system do not attain a primary macroscopic relevance. This condition is fulfilled if the macroscopic perturbation of the system occurs so slowly (quasi-static) that the statistical equilibrium of the internal degrees of freedom is not disturbed during the perturbation. The system is then in a so-called internal equilibrium. In this case, the internal degrees of freedom are not explicitly noticeable, but only become globally apparent, for example, by a change dQ of the heat content of the system. This condition, however, is also fulfilled if the macroscopic perturbation occurs so quickly that one or more of the molecular internal degrees of freedom do not have time to react to the perturbation. The degrees of freedom concerned then appear frozen with respect to the macroscopic perturbation. Provided that the other part of the internal degrees of freedom is still able to restore equilibrium practically instantaneously during the perturbation (so that we can still talk about a thermodynamic equilibrium), the system is in a so-called arrested equilibrium with respect to the frozen degrees of freedom for the duration of the perturbation. Arrested (frozen) internal degrees of freedom do not exhibit any macroscopic effects. They can be ignored as long as different arrested states do not have to be compared with each other or with other states. How fast a system must be perturbed so that an internal degree of freedom appears arrested with respect to the perturbation naturally depends on the degree of freedom itself, but also on the physical environment in which it is embedded. Some internal degrees of freedom can, in certain temperature and pressure ranges, still be regarded as being at least approximately arrested, even in the case of a quasistatic perturbation. Two typical examples are the degrees of freedom of the diffusive translational motion of the molecules in a glass and the degree of advancement of the reaction in an oxygen and hydrogen gas mixture.