Equilibrium first order transition

In these equations T, T2. .. represent equilibrium first order transition temperatures and AH, AH-. .. the corresponding equilibrium transition enthalpies. is the enthalpy of the material analyzed at 0 K. 

Thus if one starts with one pure isomer of a substance, this isomer can undergo first-order transitions to other forms, and in turn these other forms can undergo transitions among themselves, and eventually an equilibrium mixture of different isomers will be generated. The transitions between atomic and molecular excited states and their ground states are also mostly first-order processes. This holds both for radiative decays, such as fluorescence and phosphorescence, and for nonradiative processes, such as internal conversions and intersystem crossings. We shall look at an example of this later in Chapter 9. 

Returning to the general case in which kb is permitted to be nonzero, we comment on one more feature of this stochastic model, namely that the equilibrium distribution, considered as a function of kfjkb, displays a first order transition in the limit K = oo. With dNi(oo)/dt = 0 the equilibrium solution to Eqs. (1)—(3) is seen to be of the form Af(oo) = A(kfjkby. The conservation condition, Eq. (6), gives the value of A, and we have... 

Complete neglect of the Coulomb interaction between charges fixed on the network is also a problem. It has been argued [33] that the contribution from fixed charges to the osmotic pressure plays an important role in determining the swelling equilibrium of gels. However, we will not discuss these theoretical problems further because the main concern of the present article is to clarify experimentally the nature of the first-order transition rather than to improve theoretical treatment. 

Another unique feature of ionized NIPA gel has been found recently both of the equilibrium swelling ratio a and the first-order transition temperature T0 depend strongly on the shape of samples [31]. The measurement of equilibrium a has been made on ionized NIPA gel rods of various diameters, and also on plates and cubes. The gel contained 680 mM NIPA, 20 mM acrylic acid (AA), and 8.6 mM BIS. All samples were prepared from the same pregel solution at the same time so as to guarantee that the composition and the structure of all samples were the same. 

The phase coexistence observed around the first-order transition in NIPA gels cannot be interpreted by the Flory-Rehner theory because this theory tacitly assumes that the equilibrium state of a gel is always a homogeneous one. Heterogeneous structures such as two-phase coexistence are ruled out from the outset in this theory. Of course, if the observed phase coexistence is a transient phenomenon, it is beyond the thermodynamical theory. However, as will be described below, the result of the detailed experiment strongly indicates that the coexistence of phases is not a transient but rather a stable or metastable equilibrium phenomenon. At any rate, we will focus our attention in this article only on static equilibrium phenomena. 

The phase coexistence of gels at the first-order transition is accompanied by a number of unusual features, of which a few will be mentioned below. First, the fact that the triphasic equilibrium persists over a wide temperature range is an apparent contradiction to the Gibbs phase rule. This rule predicts that the... 

We note here that gel is a coherent solid because its structure is characterized by a polymer network, and hence, the above theoretical considerations on crystalline alloys should be applicable to gels without essential alteration. It is expected that the curious features of the first-order transition of NIPA gels will be explained within the concept of the coherent phase equilibrium if the proper calculation of the coherent energy and the elastic energy of the gel network is made. This may be one of the most interesting unsolved problems related to the phase transitions of gels. 

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). 

As with liquid crystals (Sect. 5.1.2), only the equilibrium transition to the fully ordered crystal and the transition to the isotropic melt have been studied extensively16c 102). Both transitions are of the first order transition type, similar to that shown in Fig. 1. A series of transition data are collected in Table 7. Plastic crystals with motifs as small as... 

In a first order transition, two phases co-exist at the transition temperature. From the Gibbs phase rule, if thermodynamic equilibrium is maintained, these phases can only co-exist at a single temperature (for a fixed pressure). At that... 

Other features of interest in the phase diagram of 4He include triple points between various liquid and solid phases of the element. At point c in Figure 13.11, liquid I, liquid II and a body-centered cubic (bcc) solid phase are in equilibrium. The bcc solid exists over a narrow range of pressure and temperature. It converts by way of a first-order transition to a hexagonal close packed (hep) solid, or to liquid I or liquid II. At point d, liquid I and the two solids (bcc and hep) are in equilibrium liquid II and the two solids are in equilibrium at point e. 

In their statistical model for microphase separation of block copolymers, Leary and Williams (43) proposed the concept of a separation temperature Ts. It is defined as the temperature at which a first-order transition occurs when the domain structure is at equilibrium with a homogeneous melt, i.e.,... 

Simple molecules may occur in three states, the solid, the liquid and the gaseous state. The transitions between these phases are sharp and associated with a thermodynamic equilibrium. Under these conditions, phase changes are typical first-order transitions, in which a primary thermodynamic function, such as volume or enthalpy, shows a sudden jump. 

There is a fundamental difference between Tg and Tm however, in that the melting point is a real first-order transition point, at which the free energies of both phases in equilibrium are equal. Thus ... 

The use of the continuous quasi-equilibrium technique (see Chapter 3) made it possible to determine the corresponding adsorption isotherm in sufficient detail to reveal the sub-step shown in Figure 9.5 and located at the same 6 as the sharp calorimetric peak. The isotherm sub-step and associated calorimetric peak are evidently associated with an increase in packing density of the adsorbate. Rouquerol el al. (1977) and Grillet etal. (1979) concluded that these changes were due to a degenerated first-order transition from a hypercritical two-dimensional (2-D) fluid state to a 2-D localized state. 

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