Temperature 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. 

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 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... 

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.,... 

Another problem which obscures the analogy between different phase transitions is the fact that one does not always wish to work with the corresponding statistical ensembles. Consider, for example, a first-order transition where from a disordered lattice gas islands of ordered c(2x2) structure form. If we consider a physisorbed layer in full thermal equilibrium with the surrounding gas, then the chemical potential of the gas and the temperature would be the independent control variables. In equilibrium, of course, the chemical potential jx of subsystems is the same, and so the chemical potential of the lattice gas and that of the ordered islands would be the same, while the surface density (or coverage 9) in the islands will differ from that of the lattice gas. The three-dimensional gas acts as a reservoir which supplies adsorbate atoms to maintain the equilibrium value of the coverage in the ordered islands when one cools the adsorbed layer through the order-disorder transition. However, one often considers such a transition at... 

The liquid-vapour phase transition is a first-order transition. For example, the molar entropy S, being a derivative of te Gibbs potential G, S = —(8G/dT)p, where T is the temperature and p the pressure, has a discontinuity at the boiling point T = Tb. For a description of the liquid-vapour equilibrium, various approximate state equations are employed. Let us consider one of the most common equations of this type, the semiempirical van der Waals equation, written in terms of temperature T, volume V and pressure p or a gas... 

Polyethylene data are shown in Fig. 2.23. At the equilibrium melting temperature of 416.4 K, the heat of fusion and entropy of fusion are indicated as a step increase. The free enthalpy shows only a change in slopes, characteristic of a first-order transition. Actual measurements are available to 600 K. The further data are extrapolated. This summary allows a close connection between quantitative DSC measurement and the derivation of thermodynamic data for the limiting phases, as well as a connection to the molecular motion. In Chaps. 5 to 7 it will be shown that this information is basic to undertake the final quantitative step, the analysis of nonequilibrium states as are common in polymeric systems. 

A simple analysis of an irreversible first-order transition is the cold crystallization, defined in Sect 3.5.5. For polymers, crystallization on heating from the glassy state may be so far from equilibrium that the temperature modulation will have little effect on its rate, as seen in Fig. 4.122. The modeling of the measurement of heat capacity in the presence of large, irreversible heat flows in Fig. 4.102, and irreversible melting in Figs. 3.89 and 4.123, document this capability of TMDSC to separate irreversible and reversible effects. Little needs to be added to this important application. 

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