Phase Equilibrium State

In practice it is the International Practical Temperature Scale of1968 (IPTS-68) which is used for calibration of scientific and industrial instruments-t This scale has been so chosen that temperatures measured on it closely approximate ideal-gas temperatures the differences are within the limits of present accuracy of measurement. The IPTS-68 is based on assigned values of temperature for a number of reproducible equilibrium states (defining fixed points) and on standard instruments calibrated at these temperatures. Interpolation between the fixed-point temperatures is provided by formulas that establish the relation between readings of the standard instruments and values of the international practical temperature. The defining fixed points are specified phase-equilibrium states of pure substances, t a given in Table 1.2. 

The formulas presented here are valid for fluids at a homogeneous one-phase equilibrium state. They are not to be directly apphed to a fluid at an unstable state, for ordinary interest is not on the unstable fluid as a homogeneous phase, but on the saturated phases that separate from the unstable fluid. Separate calculations on the separated phases need to be performed with the eos-derived formulas for the individual saturated phases, and summed if desired. The calculations to find the saturated equilibrium phases are the subject of Section 4.4. 

The subscripts L and V denote the saturated liquid and the saturated vapor, respectively. An isotherm is an S-shaped curve, while an isobar is a horizontal line. At a phase-equilibrium state, f =fy and Pl = Pv = P > where p stands for the vapor pressnre. Eqnation (4.465) rearranges to... 

Another type of vapor-liquid equilibrium problem, and one that is more important for designing separation equipment, is computing the two-phase equilibrium state when either a liquid of known composition is partially vaporized or a vapor is partially condensed as a result of a change in temperature and/or pressure. Such a problem is gener-... 

Although by starting with Eq. 11.2-2 one can proceed directly to the calculation of the liquid-liquid phase equilibrium state, this equation does not provide in.sight into the reason that phase separation and critical solution temperature behavior occur. To obtain this insight it is necessary to study the Gibbs energy versus composition diagram for various mixtures. For an ideal binary mixture, we have (Table 9.3-1)... 

It should be evident from the examples in Chapters 10, 11, and 12 that the evaluation of species fugacities or partial molar Gibbs energies (or chemical potentials) is central to any phase equilibrium calculation. Two different fugacity descriptions have been used, equations of. state and activity coefficient models. Both have adjustable parameters. If the values of these adjustable parameters are known or can be estimated, the phase equilibrium state may be predicted. Equally important, however, is the observation that measured phase equilibria can be used to obtain these parameters. For example, in Sec. 10.2 we demonstrated how activity coefficients could be computed directly from P-T-x-y data and how activity coefficient models could be fit to such data. Similarly, in Sec. 10.3 we pointed out how fitting equation-of-state predictions to experimental high-pressure phase equilibrium data could be used to obtain a best-fit value of the binary interaction parameter.. /"... 

Unstable states defined by negative curvature of A-V isotherms, between the spinodal points. In this region, dp/dV)T,ai Ni > 0. Single-phase equilibrium states of this nature are completely disallowed. Small fluctuations in system properties grow in unrestricted fashion until the system splits into two phases. 

Given the estimate of the reactor effluent in Example 4.2 for fraction of methane in the purge of 0.4, calculate the.actual separation in the phase split assuming a temperature in the phase separator of 40°C. Phase equilibrium for this mixture can be represented by the Soave-Redlich-Kwong equation of state. Many computer programs are available commercially to carry out such calculations. 

Clapeyron-Clausius equation A thermodynamic equation applying to any two-phase equilibrium for a pure substance. The equation states ... 

In the preceding derivation, the repulsion between overlapping double layers has been described by an increase in the osmotic pressure between the two planes. A closely related but more general concept of the disjoining pressure was introduced by Deijaguin [30]. This is defined as the difference between the thermodynamic equilibrium state pressure applied to surfaces separated by a film and the pressure in the bulk phase with which the film is equilibrated (see section VI-5). 

Radiation probes such as neutrons, x-rays and visible light are used to see the structure of physical systems tlirough elastic scattering experunents. Inelastic scattering experiments measure both the structural and dynamical correlations that exist in a physical system. For a system which is in thennodynamic equilibrium, the molecular dynamics create spatio-temporal correlations which are the manifestation of themial fluctuations around the equilibrium state. For a condensed phase system, dynamical correlations are intimately linked to its structure. For systems in equilibrium, linear response tiieory is an appropriate framework to use to inquire on the spatio-temporal correlations resulting from thennodynamic fluctuations. Appropriate response and correlation functions emerge naturally in this framework, and the role of theory is to understand these correlation fiinctions from first principles. This is the subject of section A3.3.2. 

A homogeneous metastable phase is always stable with respect to the fonnation of infinitesimal droplets, provided the surface tension a is positive. Between this extreme and the other thennodynamic equilibrium state, which is inhomogeneous and consists of two coexisting phases, a critical size droplet state exists, which is in unstable equilibrium. In the classical theory, one makes the capillarity approxunation the critical droplet is assumed homogeneous up to the boundary separating it from the metastable background and is assumed to be the same as the new phase in the bulk. Then the work of fonnation W R) of such a droplet of arbitrary radius R is the sum of the... 

The strong dependence of the PES on molecular orientation also leads to strong coupling between rotational states, and hence rotational excitation/de-excitation in the scattering. This has been observed experimentally for H2 scattering from Cu surfaces. Recent work has shown that for H2 the changes m rotational state occur almost exclusively when the molecular bond is extended, that is, longer than the gas-phase equilibrium value [ ]. 

In practice, colloidal systems do not always reach tlie predicted equilibrium state, which is observed here for tlie case of narrow attractions. On increasing tlie polymer concentration, a fluid-crystal phase separation may be induced, but at higher concentration crystallization is arrested and amorjihous gels have been found to fonn instead [101, 102]. Close to the phase boundary, transient gels were observed, in which phase separation proceeded after a lag time. 

Because the macroscopic-intensive properties of homogeneous fluids in equilibrium states ate functions of T, P, and composition, it follows that the total property of a phase fiM can be expressed functionally as in equation 113 ... 

Critical Temperature The critical temperature of a compound is the temperature above which a hquid phase cannot be formed, no matter what the pressure on the system. The critical temperature is important in determining the phase boundaries of any compound and is a required input parameter for most phase equilibrium thermal property or volumetric property calculations using analytic equations of state or the theorem of corresponding states. Critical temperatures are predicted by various empirical methods according to the type of compound or mixture being considered. 

Since the phase rule treats only the intensive state of a system, it apphes to both closed and open systems. Duhem s theorem, on the other hand, is a nJe relating to closed systems only For any closed system formed initially from given masses of preseribed ehemieal speeies, the equilibrium state is completely determined by any two propeities of the system, provided only that the two propeities are independently variable at the equilibrium state The meaning of eom-pletely determined is that both the intensive and extensive states of the system are fixed not only are T, P, and the phase compositions established, but so also are the masses of the phases. 

The mechanism of ion polymerization in formaldehyde crystals proposed by Basilevskii et al. [1982] rests on Semenov s [1960] assumption that solid-phase chain reactions are possible when the arrangement of the reactants in the crystal prepares the configuration of the future chain. The monomer crystals capable of low-temperature polymerization fulfill this condition. In the initial equilibrium state the monomer molecules are located in the lattice sites and the creation of a chemical bond requires surmounting a high barrier. However, upon creation of the primary dimer cation, the active center shifts to the intersite, and the barrier for the addition of the next link... 

Concentration-time curves. Much of Sections 3.1 and 3.2 was devoted to mathematical techniques for describing or simulating concentration as a function of time. Experimental concentration-time curves for reactants, intermediates, and products can be compared with computed curves for reasonable kinetic schemes. Absolute concentrations are most useful, but even instrument responses (such as absorbances) are very helpful. One hopes to identify characteristic features such as the formation and decay of intermediates, approach to an equilibrium state, induction periods, an autocatalytic growth phase, or simple kinetic behavior of certain phases of the reaction. Recall, for example, that for a series first-order reaction scheme, the loss of the initial reactant is simple first-order. Approximations to simple behavior may suggest justifiable mathematical assumptions that can simplify the quantitative description. 

The a — 0 transformation has a large hysteresis in hydrogenated titanium alloys, and different thermal treatments change their phase content. Various degrees of metastability due to hysteresis are implicit for the alloys after different thermal treatments. Metastable phases undergo transformation to a more equilibrium state during deformation, which can effect the flow of the alloy. Below we consider the effect of the thermal pre-strain treatment on ductility on the strength of the Ti-6A1-2Zr-1.5V-lMo-rH alloys. ... 

Consider the equilibrium state of the system assuming that m grams of n-heptane are dissolved in the benzene phase. Then the mass fraction of n-heptanc in this phase is to. 2 = m/(400 + m). 

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