Ehrenfest first-order

For a second-order transition Eq. (4.53) is the analog of Eq. (4.5), which is useful for first-order transitions. Equation (4.53) and the first- and second-order terminology are due to Ehrenfest. 

If one of these quantities experiences a discontinuous change, i.e. if AS 0 or AV 0, then the phase transition is called a first-order transition according to Ehrenfest. It is accompanied by the exchange of conversion enthalpy AH = TAS with the surroundings. 

When the free energies F of the two crystal structures are identical, the system is at a critical point. The identity of F does not imply identical fimctions (otherwise the two phases would be indistinguishable). Therefore, at the critical point first derivatives of F might differ and therefore enthalpy, volume, and entropy of the two phases would be different. These transformations are first-order phase transitions, according to Ehrenfest [105]. A discontinuous enthalpy imphes heat exchange at the transition temperature, which can easily be measured with DSC experiments. A discontinuous volume is evident under the microscope or, more precisely, with diffraction experiments on single crystals or powders. Some phase transitions are however characterized by continuous first derivatives of the free energy, whereas the second derivatives (specific heat, compressibility, or thermal expansivity, etc.) are discontinuous. These transformations are second-order transitions and are clearly softer. 

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 transitions between the bottom five phases of Fig. 2 may occur close to equilibrium and can be described as thermodynamic first order transitions (Ehrenfest definition 17)). The transitions to and from the glassy states are limited to the corresponding pairs of mobile and solid phases. In a given time frame, they approach a second order transition (no heat or entropy of transition, but a jump in heat capacity, see Fig. 1). 

The transitions between phases discussed in Section 10.1 are classed as first-order transitions. Ehrenfest [25] pointed out the possibility of higher-order transitions, so that second-order transitions would be those transitions for which both the Gibbs energy and its first partial derivatives would be continuous at a transition point, but the second partial derivatives would be discontinuous. Under such conditions the entropy and volume would be continuous. However, the heat capacity at constant pressure, the coefficient of expansion, and the coefficient of compressibility would be discontinuous. If we consider two systems, on either side of the transition point but infinitesimally close to it, then the molar entropies of the two systems must be equal. Also, the change of the molar entropies must be the same for a change of temperature or pressure. If we designate the two systems by a prime and a double prime, we have... 

Ehrenfest s concept of the discontinuities at the transition point was that the discontinuities were finite, similar to the discontinuities in the entropy and volume for first-order transitions. Only one second-order transition, that of superconductors in zero magnetic field, has been found which is of this type. The others, such as the transition between liquid helium-I and liquid helium-II, the Curie point, the order-disorder transition in some alloys, and transition in certain crystals due to rotational phenomena all have discontinuities that are large and may be infinite. Such discontinuities are particularly evident in the behavior of the heat capacity at constant pressure in the region of the transition temperature. The curve of the heat capacity as a function of the temperature has the general form of the Greek letter lambda and, hence, the points are called lambda points. Except for liquid helium, the effect of pressure on the transition temperature is very small. The behavior of systems at these second-order transitions is not completely known, and further thermodynamic treatment must be based on molecular and statistical concepts. These concepts are beyond the scope of this book, and no further discussion of second-order transitions is given. 

Collecting all first-order terms obtained through the BCH expansion of the Ehrenfest theorem given in Eq. (74) leads to a system of differential equations... 

If we consider an equilibrium transformation of the system, then if this is an ordinary phase change, at least one of these first order derivatives of 0 must exhibit a discontinuity. For this reason ordinary phase changes are called transformations of the first order (Ehrenfest). 

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... 

Apart from temperature, hydrostatic pressure is the other intensive thermodynamical parameter that can be modified with high-pressure cells to build (T, P) phase diagrams. By changing the relative distances between the atoms and molecules, the strength of the interactions are modified, thereby modifying the transition temperature or even inducing new phases. The change of the transition temperature as a function of pressure depends whether the transition is continuous (second order) or discontinuous (first order). The Clausius-Clapeyron (dTc/dP) and Ehrenfest (dTc/dP)2 relationships apply to first- and second-order phase transitions, respectively,... 

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 thermodynamics of the I-N phase transition has been extensively investigated for resolving the issue concerning the order of the transition. Following the Ehrenfest scheme, a phase transition is classified into a first-order transition or a second-order one, depending upon the observation of finite discontinuities in the first or the second derivatives of the relevant thermodynamic potential at the transition point. An experimental assessment of the order of the I-N transition has turned out to be not a simple task because of the presence of only small discontinuities in enthalpy and specific volume. It follows from high-resolution measurements that I-N transition is weakly first order in nature [85]. 

The derivatives of AG with respect to temperature, on the other hand, do not have to be zero. This suggests a way to characterize the transitions thermodynamically. Ehrenfest suggested that a transition for which (5AG/5T) (= - AS, see Fig. 2.19) is not equal to zero be called a first-order transition [25]. A second-order transition would analogously have the first derivative (8AG/5T) equal to zero, but the second derivative (8 AG/8T ) would not equal zero. The second derivative of AG is equal... 

There are two types of phase transitions in solids first-order and continuous-phase. According to Ehrenfest the order of the phase transition is equal to the lowest order of the derivative of the Gibbs energy, which is discontinuous at the transition. 

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... 

Exercise 11.5 Derive the first-order equation Eq. (11.41) from the Ehrenfest theorem... 

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... 

Phase transitions are generally classified by the criteria developed by Ehrenfest. It is considered that first-order transitions are those at which the free energy as a function of a given state variable (volume (V), pressure (p), and temperature (7)) is continuous, while the first derivative is discontinuous or steplike. First-order thermodynamic quantities are those that can be expressed as the first derivative of Gibbs free energy, such as... 

According to Ehrenfest S a first-order transition is one for which the free energy as a function of any given state variable (F, P, T) is continuous, but the first partial derivatives of the free energy with respect to the relevant state variables are discontinuous. Thus, if the Gibbs free energy G at the transition temperature is continuous, but dG/dT)p and (dG/dP) are discontinuous, we have a first-order transition. At a typical first-order transition point, such as fusion or vaporization, there is a discontinuity in entropy S, volume V and enthalpy ff, since ... 

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. 

According to Ehrenfest, a second-order transition occurs when p and its first derivatives are continuous across the transition region, but the second derivatives, Cp and compressibility, k are discontinuous. This behavior is illustrated in the second column of Figure 13.1. For these transitions, the enthalpy, in addition to the entropy and volume, is continuous across the transition. 

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