Entropy first-order transitions

Phase transitions at which the entropy and enthalpy are discontinuous are called first-order transitions because it is the first derivatives of the free energy that are disconthuious. (The molar volume V= (d(i/d p) j is also discontinuous.) Phase transitions at which these derivatives are continuous but second derivatives of G... 

As a result, there will be a continuous change in G as the transition of one phase into another takes place. However, for some phase transitions (known as first-order transitions), it is found that there is a discontinuity in the first derivative of G with respect to pressure or temperature. It can be shown that the partial derivative of G with pressure is the equal to volume, and the derivative with respect to temperature is equal to entropy. Therefore, we can express these relationships as follows ... 

The electronic heat capacity naturally has a pronounced effect on the energetics of insulator-metal transitions and the entropy of a first-order transition between an insulating phase with y = 0 and a metallic phase with y= ymet at Ttrs is in the first approximation Ains met5m = 7met7trs. 

We now distinguish solid state transformations as first-order transitions or lambda transitions. The latter class groups all high-order solid state transformations (second-, third-, and fourth-order transformations see Denbigh, 1971 for exhaustive treatment). We define first-order transitions as all solid state transformations that involve discontinuities in enthalpy, entropy, volume, heat capacity, compressibility, and thermal expansion at the transition point. These transitions require substantial modifications in atomic bonding. An example of first-order transition is the solid state transformation (see also figure 2.6)... 

Here Cp, a and are the heat capacity, volume thermal expansivity and compressibility respectively. First-order transitions involving discontinuous changes in entropy and volume are depicted in Fig. 4.1. In this figure curves G Gu represent variations in free energies of phases I and II respectively, while // Hu and F, represent variations in... 

Both kinetic and thermodynamic approaches have been used to measure and explain the abrupt change in properties as a polymer changes from a glassy to a leathery state. These involve the coefficient of expansion, the compressibility, the index of refraction, and the specific heat values. In the thermodynamic approach used by Gibbs and DiMarzio, the process is considered to be related to conformational entropy changes with temperature and is related to a second-order transition. There is also an abrupt change from the solid crystalline to the liquid state at the first-order transition or melting point Tm. 

Ramirez et al (1970) discussed a metal-insulator transition as the temperature rises, which is first order with no crystal distortion. The essence of the model is—in our terminology—that a lower Hubbard band (or localized states) lies just below a conduction band. Then, as electrons are excited into the conduction band, their coupling with the moments lowers the Neel temperature. Thus the disordering of the spins with consequent increase of entropy is accelerated. Ramirez et al showed that a first-order transition to a degenerate gas in the conduction band, together with disordering of the moments, is possible. The entropy comes from the random direction of the moments, and the random positions of such atoms as have lost an electron. The results of Menth et al (1969) on the conductivity of SmB6 are discussed in these terms. 

Whether the phase transition is first- or second-order depends on the relative magnitudes of the coefficients in the Landau expansion, Eq. 17.2. For a first-order transition, the free energy has a discontinuity in its first derivative, as at the temperature Tm in Fig. 17.1a, and higher-order derivative quantities, such as heat capacity, are unbounded. In second-order transitions, the discontinuity occurs in the second-order derivatives of the free energy, while first derivatives such as entropy and volume are continuous at the transition. 

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). 

Figure 20 shows the phase diagram of polyethylene119). The existence range of the condis crystals increases with pressure and temperature. The enthalpy of the reasonably reversible, first order transition from the orthorhombic to the hexagonal condis phase of polyethylene is 3.71 kJ/mol at about 500 MPa pressure 121) which is about 80 % of the total heat of fusion. The entropy of disordering is 7.2 J/(K mol), which is more than the typical transition entropy of paraffins to their high temperature... 

Poly (diethyl siloxane) was suggested by Beatty et al. 1651 based on DSC, dielectric, NMR, and X-ray measurements to possess liquid crystalline type order between about 270 and 300 K. The macromolecule shows two large lower temperature first order transitions, one at about 200 K, the other at about 270 K166 ll,7). The transition of the possible mesophase to the isotropic liquid at 300 K is quite small and irre-producible, so that variable, partial crystallinity was proposed 165) [measured heat of transition about 150 J/mole1S8)], Very little can be said about this state which may even consist of residual crystals. It is of interest, however, to further analyze the high temperature crystal phase between 200 and 270 K. It is produced from the, most likely, fully ordered crystal with an estimated heat and entropy of transition of 5.62kJ/mol and 28J/(Kmol), respectively [calculated from calorimetric data 1S6)... 

These conclusions are summarized graphically in Figure 2.34. The first-order transition of importance in polymer chemistry is the melting point Tm, and the top two sketches show the discontinuities AVm and A,Sm in the volume and entropy, respectively. 

So-called first-order transitions include evaporation or fusion, where volume (V), entropy (S), and enthalpy (H) all exhibit a discontinuity upon differentiation of Eq. (18.5) with respect to state variables pressure (p), or temperature (7). 

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. 

There has been a wealth of activity based on the idea that glassy dynamics is due to some underlying thermodynamic transition [1-25], If a glass former shows a jump in some an appropriate order parameter without the evolution of latent heat, then such a system is said to exhibit a random first-order transition [94,95]. Models of this kind, which include the p-spin glasses [110], and the random energy model [111], do not have symmetry between states but do have quenched random long-range interactions and exhibit the so-called Kauzmann entropy crisis. 

An alternative way to clarify the nature of this state is to test its stability with respect to a metal-insulator transition. This has received a lot of theoretical attention recently. The JT singlet ground state makes these compounds free from the tendency towards a magnetic instability observed in so many Mott insulators. In fact, their ground state does not break any symmetry and Capone et al. explained [43] that it then has a zero entropy, which makes a direct connection with a metal impossible (it would violate the Luttinger theorem). These authors predict that the only way to go from the insulator to the metal would be through an exotic superconducting phase or a first-order transition. 

Fig. 2.4 Free-energy changes at transitions (a) first-order transition (b) change in S at constant T and, consequently, latent heat (c) second-order transition (d) continuous change in entropy and so no latent heat (discontinuity in d2G/dT2).

Continuous phase transitions show anomalies in the specific heat and magnetic susceptibility (or magnetization) at or very near Tg. There is, however, no latent heat as in so-called first-order transitions but of course there is a decrease in spin entropy. 

The evaluation of solid-state transitions involves first the recognition of the type of transition, which may not always be obvious. A first-order transition such as fusion involves a discontinuous change of enthalpy and entropy at the transition point, whereas second-order transitions involve only discontinuities in heat capacity. Because of impurities and other factors, first-order transitions often do not occur sharply at one temperature instead, they spread a little on either side and are sometimes difficult to distinguish fromA-type second-order transitions. 

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... 

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