Transition first-order chemical phase

As one would expect, developments in the theory of such phenomena have employed chemical models chosen more for analytical simplicity than for any connection to actual chemical reactions. Due to the mechanistic complexity of even the simplest laboratory systems of interest in this study, moreover, application of even approximate methods to more realistic situations is a formidable task. At the same time a detailed microscopic approach to any of the simple chemical models, in terms of nonequilibrium statistical mechanics, for example, is also not feasible. As is well known, the method of molecular dynamics discussed in detail already had its origin in a similar situation in the study of classical fluids. Quite recently, the basic MD computer model has been modified to include inelastic or reactive scattering as well as the elastic processes of interest at equilibrium phase transitions (18), and several applications of this "reactive" molecular dynamicriRMD) method to simple chemical models involving chemical instabilities have been reported (L8j , 22J. A variation of the RMD method will be discussed here in an application to a first-order chemical phase transition with many features analogous to those of the vapor-liquid transition treated earlier. 

First Order Chemical Phase Transition in a Cooperative Isomerization Reaction. S convenient model of a first order transition is provided by a reversible isomerization reaction in a macroscopically homogeneous system... 

Matheson, I., Walls, D., and Gardiner, C., Stochastic models of first-order nonequilibrium phase transitions in chemical reactions, Journal of Statistical Physics, Vol. 12, 1975, pp. 21-34. 

The Ehrenfest17 classification of phase transitions (first-order, second-order, and lambda point) assumes that at a first-order phase transition temperature there are finite changes AV 0, Aft 0, AS VO, and ACp VO, but hi,lower t = hi,higher t and changes in slope of the chemical potential /i, with respect to temperature (in other words (d ijdT)lowerT V ((9/i,7i9T)higherT). At a second-order phase transition AV = 0, Aft = 0, AS = 0, and ACp = 0, but there are discontinuous slopes in (dV/dT), (dH/<)T), (OS / <)T), a saddle point in and a discontinuity in Cp. A lambda point exhibits a delta-function discontinuity in Cp. 

First, a strong volume change can be excited by a large spectrum of different physical and chemical factors such as temperature, electrical voltage, pH, concentration of organic compounds in water, and salt concentrations. The possibility of a first-order volume phase transition in gels was suggested by K. Dusek and... 

Certain polyelectrolyte hydrogels display first-order volume phase transitions, with hysteresis, in response to external stimuli 2,7). Hydrophobic polyelectrolyte hydrogels, in particular, may undergo discrete transitions in response to changes in external pH. Such hydrogels have been considered as chemically-sensitive mechanical switches 8). 

The SmS semiconductor to metal transition was later verified by the direct observation of a discontinuous change in the optical reflectivity at 6 kbar (Kirk et al., 1972). This is consistent with a first order magnetic phase transition which was directly verified by magnetic susceptibility measurements under pressure by Maple and Wohlleben (1971). In the collapsed phase the susceptibility of SmS showed no magnetic order down to 0.35 iC and was almost identical to the susceptibility of SmBa (see fig. 20.10 of volume 2). Bader et al. (1973) measured the heat capacity (fig. 11.16) and electrical resistivity (fig. 11.17) of SmS under pressure. They found a large electronic contribution to the heat capacity ( y = 145 mJ/mole-K ) and a resistivity reminiscent of SmB. Mossbauer isomer shift measurements of SmS under pressure by Coey et al. (1976) reveal the transition from a Sm isomer shift at zero pressure to an intermediate value at pressures above 6 kbar (fig. 11.18). The isomer shift of SmS above 6 kbar was found to be about the same as the isomer shifts for chemically collapsed Smo.77Yo.23S and SmBo at zero pressure. 

For both first-order and continuous phase transitions, finite size shifts the transition and rounds it in some way. The shift for first-order transitions arises, crudely, because the chemical potential, like most other properties, has a finite-size correction p(A)-p(oo) C (l/A). An approximate expression for this was derived by Siepmann et al [134]. Therefore, the line of intersection of two chemical potential surfaces Pj(T,P) and pjj T,P) will shift, in general, by an amount 0 IN). The rounding is expected because the partition fiinction only has singularities (and hence produces discontinuous or divergent properties) in tlie limit i—>oo otherwise, it is analytic, so for finite Vthe discontinuities must be smoothed out in some way. The shift for continuous transitions arises because the transition happens when L for the finite system, but when i oo m the infinite system. The rounding happens for the same reason as it does for first-order phase transitions whatever the nature of the divergence in thennodynamic properties (described, typically, by critical exponents) it will be limited by the finite size of the system. 

Reconstructive phase transitions Chemical bonds are broken and rejoined the reconstruction involves considerable atomic movements. Such conversions are always first-order transitions. 

Fig. 35 Phase diagrams AB miktoarm-star copolymers for n = 2, n = 3, n = 4 and n = 5. mean-field critical point through which system can transition from disordered state to Lam phase via continuous, second-order phase transition. All other phase transitions are first-order. From [112]. Copyright 2004 American Chemical Society...

Fig. 68 Comparison of temperature-dependent intensity of first-order Bragg peak for bare matrix copolymer (A) containing 0.5 wt% nanocomposites with plate-like (V), spherical (o) and rod-like ( ) geometry. Data are vertically shifted for clarity. Inset dependence of ODT temperature on dimensionality of fillers (spherical 0, rod-like 1, plate-like 2). Vertical bars width of phase transition region. Pure block copolymer is denoted matrix . From [215]. Copyright 2003 American Chemical Society...

In [25, 26] it is shown that at given pq the diquark gap is independent of the isospin chemical potential for Pi ) < Pic(Pq), otherwise vanishes. Increase of isospin asymmetry forces the system to pass a first order phase transition by tunneling through a barrier in the thermodynamic potential (2). Using this property we choose the absolute minimum of the thermodynamic potential (2) between two /3-equilibrium states, one with and one without condensate for the given baryochemical potential Pb = Pu + 2pd-... 

Fig. 27. Phase diagram of an adsorbed film in- the simple cubic lattice from mean-fleld calculations (full curves - flrst-order transitions, broken curves -second-order transitions) and from a Monte Carlo calculation (dash-dotted curve - only the transition of the first layer is shown). Phases shown are the lattice gas (G), the ordered (2x1) phase in the first layer, lattice fluid in the first layer F(l) and in the bulk F(a>). For the sake of clarity, layering transitions in layers higher than the second layer (which nearly coincide with the layering of the second layer and merge at 7 (2), are not shown. The chemical potential at gas-liquid coexistence is denoted as ttg, and 7 / is the mean-field bulk critical temperature. While the layering transition of the second layer ends in a critical point Tj(2), mean-field theory predicts two tricritical points 7 (1), 7 (1) in the first layer. Parameters of this calculation are R = —0.75, e = 2.5p, 112 = Mi/ = d/2, D = 20, and L varied from 6 to 24. (From Wagner and Binder .)...

Figures 15 and 16 show the temperature dependence of the chemical shift for 80% deuterated KD2PO4 and RbH2P04. Obviously, iso varies significantly with temperature in the paraelectric phase and shows a clear break at Tc of 202 K for DKDP and 147 K for RbH2P04, respectively. The shift exhibits a distinct discontinuity at Tc while the line width shows an abrupt increase below Tc, in agreement with the (close to) first-order nature of the phase transition, and an anticipated pronounced distortion in the PO4 moiety. The cause of the line width increase below Tc has yet to be explained, but is at least partly due to a lack of the increased spinning speed to average out the enhanced chemical shift anisotropy below Tc.

A characteristic manifestation of the coexistence of two gel phases and hence of the first-order phase transition in a swollen network consists of the van der Waals loop which appears in the dependence of the swelling pressure P (or of the chemical potential of the solvent plf see Eq. (1)) on 0. The composition of coexisting gel phases at the collapse (values

2) is given by the condition of equality of the chemical potentials of the solvent px and polymer p2 in both phases... 

If the ranges of homogeneity of the phases taking part in the transformation are wider than those of line compounds, the kinetic coefficients in Eqns. (12.22) and (12.23), that is v jf, yb, and A b, are certainly not composition independent. It may then be questionable if transport across the boundary (Eqn. (12.22)) and the simultaneous structure change (Eqn. (12.23)) are independent processes as was tacitly assumed by formulating the kinetic relations in Eqns. (12.22) and (12.23). Let us emphasize that the foregoing analysis is meant to clarify the physico-chemical conceptual frame in which first-order transitions which include matter transport should be discussed. Pertinent experiments are still rare. 

What is described above is an idea of the so-called chemical clock, that is a reaction with periodic (oscillating) change of reactant concentrations its period could be estimated as 5t > nc/p. In the condensed matter theory a leap in densities is interpreted as phase transitions of the first order. From this point of view, the oscillations correspond to a sequence in time of phase transitions where the two phases (i.e., big clusters of A s containing inside rare and small clusters of B s and vice versa) differ greatly in their structures. 

In this volume, we will apply the principles developed in Principles and Applications to the description of topics of interest to chemists, such as effects of surfaces and gravitational and centrifugal fields phase equilibria of pure substances (first order and continuous transitions) (vapor + liquid), (liquid 4-liquid), (solid + liquid), and (fluid -f fluid) phase equilibria of mixtures chemical equilibria and properties of both nonelectrolyte and electrolyte mixtures. But do not expect a detailed survey of these topics. This, of course, would require a volume of immense breadth and depth. Instead, representative examples are presented to develop general principles that can then be applied to a wide variety of systems. 

The equality of chemical potentials in a first-order phase transition leads to two important relationships. The first is the Clapeyron equation1... 

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