The thermodynamics of phase equilibrium

Abstract Thermodynamics is usually subdivided into a theory dealing with equilibrium and into one concerned with irreversible processes. In the present paper this subdibvision is carried further and the Gibbsian thermodynamics of phase equilibrium is distinguished from the thermodynamics of Clausius and Kelvin. The latter was put into an axiomatic form by Caratheodory; the present paper attempts a similar task for the Gibbs theory. The formulation of this theory as an autonomous logical structure reveals characteristic aspects that were not evident until the two logical structures were differentiated. The analysis of the basic assumptions of the Gibbs theory allows the identification and removal of defects that marred the classical formulation. In the new theory thermodynamic systems are defined as conjunctions of spatially disjoint volume elements (subsystems), each of which is characterized by a set of additive conserved quantities (invariants): the internal energy, and the mole numbers of the independent chemical components. For the basic theory, it is convenient to assume the absence of elastic, electric, and magnetic effects. This restriction enables us to define thermodynamic processes as transfers of additive invariants between subsystems. Following Gibbs, we postulate that all thermostatic properties of system are contained in a fundamental equation representing the entropy as a function of the additive invariants. Geometrically, this equation is represented as a surface in a space to which we refer as Gibbs space. In order to make the information contained in the fundamental equation complete, we have to use, in many cases, additional quasi-thermodynamic variables to specify the intrinsic symmetry properties of the system. The walls, or boundaries limiting thermodynamic systems are assumed to be restrictive or nonrestrictive with respect to the transfer of the various invariants. The manipulations of the boundary conditions (imposition and relaxation of constraints) are called thermodynamic operations. In systems with nonrestrictive internal boundaries, the constraints are consistent with infinitely many distributions of the invariants over the subsystems. These virtual states serve as comparison states for the entropy maximum principle. This principle allows us to identify the state of thermodynamic equilibrium, attained asymptotically by real systems. The thermostatic extremum principle is the basis of a theory of stability. Stability may be normal or critical. In the latter case, the compliance coefficients (specific heat c p , expansion coefficient, and isothermal compressibility) tend, in general, to infinity. The phases of thermodynamic systems are each represented by a primitive surface in Gibbs space, the points of these surfaces correspond to modifications of the phase. The actual distribution of the invariants of a system over phases is determined by the entropy maximum principle. Among the fundamental theorems are the two phase rules. The first rule specifies the dimension of the set of points in the space of intensities, in which a given number of modifications can coexist. The second rule specifies the dimensionality of the set of critical points. The phase rules are somewhat more general than those of Gibbs because of our use of symmetry considerations. At a critical point two modifications become identical, and we obtain critical points of two kinds: (i) the modifications differ in densities, as for liquid and vapor; (ii) the modifications differ in symmetry, as the two directions of ordering in the Ising model. The second case relates to the well-known λ-points and λ-lines in the p - T diagram, hence these phenomena fit into the framework of the theory without ad hoc assumptions. For the case of liquid helium, this interpretation requires that the superfluid ground stage be degenerate. This conclusion is not inconsistent with the third law, but it requires substantiation by quantum-mechanical methods. Another fundamental theorem is the principle of thermostatic determinism: a reservoir of given intensities determines the densities (energy and components per unit volume) of a system which is in equilibrium with it and, conversely, the densities of the system determine the intensities. The mutual determination is unique, except at critical points and at absolute zero. A more satisfactory description of these singular situations calls for the use of statistical methods. The present approach leads to several statistical theories, the simplest of which is developed in the second paper of this series.