Ehrenfest classification

The mysteries of the helium phase diagram further deepen at the strange A-line that divides the two liquid phases. In certain respects, this coexistence curve (dashed line) exhibits characteristics of a line of critical points, with divergences of heat capacity and other properties that are normally associated with critical-point limits (so-called second-order transitions, in Ehrenfest s classification). Sidebar 7.5 explains some aspects of the Ehrenfest classification of phase transitions and the distinctive features of A-transitions (such as the characteristic lambda-shaped heat-capacity curve that gives the transition its name) that defy classification as either first-order or second-order. Such anomalies suggest that microscopic understanding of phase behavior remains woefully incomplete, even for the simplest imaginable atomic components. 

The Ehrenfest classification is not too well suited for the description of real phase transitions occurring in nature. The above remark concerns most of all the phase transitions which are not first-order. Better suited for an examination and classification of phase transitions is the Landau classification. Landau s idea is based on an assumption that in the case of many phase transitions one may always find a quantity, called the parameter of order, whose small change (with respect to the value q — 0) causes a qualitative changes in the parameters of a body (this implies that for q = 0 the system is in the sensitive state). 

The discontinuity in the first derivatives of function thus appears as the most suitable for an idealized classification of phase transitions [3,297,365]. The characteristic value of a variable, at which a phase transition occur, is termed the phase transition point fT , /cq). The changes in the derivatives can be then expressed according to Ehrenfest classification and give the limit for the first-order phase transitions... 

Though this transformation exhibits most of the typical features of a second-order phase transition according to the Ehrenfest classification, it is a true second-order phase transition due to the inherent irreversibility. 

The initial classification of phase transitions made by Ehrenfest (1933) was extended and clarified by Pippard [1], who illustrated the distmctions with schematic heat capacity curves. Pippard distinguished different kinds of second- and third-order transitions and examples of some of his second-order transitions will appear in subsequent sections some of his types are unknown experimentally. Theoretical models exist for third-order transitions, but whether tiiese have ever been found is unclear. 

Paul Ehrenfest suggested a widely used classification of thermodynamic transition phenomena according to the lowest derivative of Gibbs free energy that exhibits a mathematical discontinuity at the phase transition. 

To differentiate between the variety of phase equilibria that occur, Ehrenfest proposed a classification of phase transitions based upon the behavior of the chemical potential of the system as it passed through the phase transition. He introduced the notion of an th order transition as one in which the nth derivative of the chemical potential with respect to T or p showed a discontinuity at the transition temperature. While modern theories of phase transitions have shown that the classification scheme fails at orders higher than one, Ehrenfest s nomenclature is still widely used by many scientists. We will review it here and give a brief account of its limitations. 

Ehrenfest s classification (see [11]) into first-order and second-order transitions is based on thermodynamic criteria. First-order transitions have discontinuities in the first derivatives of the Gibbs energy with respect to temperature (= entropy) and... 

On the basis of analysis of experimental facts Ehrenfest has introduced the following classification of phase transitions a conversion is called the phase transition of nth order if successive derivatives of a thermodynamic function U up to and including (n — 1) are continuous functions, whereas the nth derivative has a step discontinuity at the transition point the... 

The initial classification of phase transitions made by Ehrenfest (1933) was extended and clarified by Pippard... 

Phase transitions can be classified according to whether they are first or second order, this classification originally being introduced by Ehrenfest. Changes in various thermodynamic properties, as well as an order parameter, (Section 1.6), for first- and second-order phase transitions as a function of temperature are illustrated in Fig 1.4. A first-order transition is defined by discontinuities in first derivatives of chemical potential. Enthalpy, entropy and volume can all be defined by appropriate first derivatives of chemical potential and all change discontinuously at a first-order phase transition. The heat capacity is defined as the derivative of enthalpy with respect to temperature. It is thus infinite for a first-order transition. The physical meaning of this is apparent when the boiling of water is considered. Any heat absorbed... 

In the modern classification scheme, phase transitions are divided into two broad categories, named similarly to the Ehrenfest classes transitions with latent heat on one hand and transitions without latent heat on the other hand. This is a thermodynamic classification. 

This type of microcanonical analysis is very similar to Ehrenfest s classification scheme for phase transitions in the thermodynamic limit. In this scheme, the order of the transition is fixed by the smallest value of n, at which the nth-order derivative of the free energy with respect to an independent thermodynamic variable, e.g., d " F T, V,N)/dI " )vj, becomes discontinuous at any point. Obviously, first-order transitions are characterized by a discontinuity in the entropy as a function of temperature, S(T) = (dF(T, V,N)/dT)yjn, at the transition temperature T x- The discontinuity at the transition point h.S corresponds to a non-vanishing of the latent heat rtrA5= Ag > 0, In a second-order phase transition, the entropy is continuous, but the second-order derivative, which is related to the heat capacity, d F T, V,N)/dT )vjs[ Cy(T), is not. The heat capacity (or better the specific heat cy= Cy/N) possesses a discontinuity (often a divergence) at the critical temperature Tct-Although higher-order phase transitions are rather rare, Ehrenfest s scheme accommodates these transitions as well. 

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