Equilibrium of phases

This is Van der Waals equation. We shall later come to the question of how far it can he justified theoretically by statistical mechanics. First, however, we shall study its properties as an equation of state and see how useful it is in describing the equilibrium of phases. 

One of the most fundamental problems of chemical physics is the study of the forces between atoms and molecules. We have seen in many preceding chapters that these forces are essential to the explanation of equations of state, specific heats, the equilibrium of phases, chemical equilibrium, and in fact all the problems we have taken up. The exact evaluation of these forces from atomic theory is one of the most difficult branches of quantum theory and wave mechanics. The general principles on which the evaluation is based, however, are relatively simple, and in this chapter we shall learn what these general principles are, and see at least qualitatively the sort of results they lead to. 

As we have seen the anisotropy parameter a in the anisotropic phase II near the high temperature corridor is of order unity (in the sense of the independence of p, although a is numerically large). Hence, in order to estimate the relative importance of the repulsive and attractive interactions for the equilibrium of phases I and II one can simply average Br(y) and B.,(y) over all possible angles y. It turns out that even at the triple point temperature... 

On a cost-benefit basis, it is interesting to outline that the best mechanical performances of the PE-PS polyblends can be reached using a minimum amount (ca. 2wt%) of an appropriate diblock copolymer (19). Furthermore, not only reproducible samples can be prepared under processing conditions, but an apparent equilibrium of phase morphology and mechanical properties is obtained within half to a few minutes depending on the melt viscosity of the blend and especially the microstructure of the diblock copolymer. 

The condition of equilibrium of phases is the equality of chemical potentials of components in both phases ... 

The conditions of thermodynamic equilibrium of phases for a multi-component system can written as a system of equations... 

If the adsorption equilibrium of phases leaving the stage is just reached, the result of stage 1 is characterized by the loading differences and X — in the op-... 

Similarly, we could also study the equilibria of the crystal with its vapor by adding an equilibrium of phase change between the stmcture elements of the solid and the vapor, such as, for instance ... 

As in section 2.5.1, this representation is incorrect because of the intervention of variable x. Also, it is more accurate to write the double equilibrium of phase transformation and transfer of water in the form ... 

The most frequent application of phase-equilibrium calculations in chemical process design and analysis is probably in treatment of equilibrium separations. In these operations, often called flash processes, a feed stream (or several feed streams) enters a separation stage where it is split into two streams of different composition that are in equilibrium with each other. 

It is important to stress that unnecessary thermodynamic function evaluations must be avoided in equilibrium separation calculations. Thus, for example, in an adiabatic vapor-liquid flash, no attempt should be made iteratively to correct compositions (and K s) at current estimates of T and a before proceeding with the Newton-Raphson iteration. Similarly, in liquid-liquid separations, iterations on phase compositions at the current estimate of phase ratio (a)r or at some estimate of the conjugate phase composition, are almost always counterproductive. Each thermodynamic function evaluation (set of K ) should be used to improve estimates of all variables in the system. 

In the highly nonlinear equilibrium situations characteristic of liquid separations, the use of priori initial estimates of phase compositions that are not very close to the true compositions of these phases can lead to divergence of iterative computations or to spurious convergence upon feed composition. 

The amounts of each phase and their compositions are calculated by resolving the equations of phase equilibrium and material balance for each component. For this, the partial fugacities of each constituent are determined ... 

Hamiltonian) trajectory in the phase space of the model from which infonnation about the equilibrium dyuamics cau readily be extracted. The application to uou-equilibrium pheuomeua (e.g., the kinetics of phase separation) is, in principle, straightforward. 

In this chapter we shall consider some thermodynamic properties of solutions in which a polymer is the solute and some low molecular weight species is the solvent. Our special interest is in the application of solution thermodynamics to problems of phase equilibrium. 

Reaction 1 is highly exothermic. The heat of reaction at 25°C and 101.3 kPa (1 atm) is ia the range of 159 kj/mol (38 kcal/mol) of soHd carbamate (9). The excess heat must be removed from the reaction. The rate and the equilibrium of reaction 1 depend gready upon pressure and temperature, because large volume changes take place. This reaction may only occur at a pressure that is below the pressure of ammonium carbamate at which dissociation begias or, conversely, the operating pressure of the reactor must be maintained above the vapor pressure of ammonium carbamate. Reaction 2 is endothermic by ca 31.4 kJ / mol (7.5 kcal/mol) of urea formed. It takes place mainly ia the Hquid phase the rate ia the soHd phase is much slower with minor variations ia volume. 

One of the simplest cases of phase behavior modeling is that of soHd—fluid equilibria for crystalline soHds, in which the solubility of the fluid in the sohd phase is negligible. Thermodynamic models are based on the principle that the fugacities (escaping tendencies) of component are equal for all phases at equilibrium under constant temperature and pressure (51). The soHd-phase fugacity,, can be represented by the following expression at temperature T ... 

These are the criteria of phase equilibrium apphed in the solution of practical problems. 

Vapor/liquid equilibrium (XT E) relationships (as well as other interphase equihbrium relationships) are needed in the solution of many engineering problems. The required data can be found by experiment, but such measurements are seldom easy, even for binaiy systems, and they become rapidly more difficult as the number of constituent species increases. This is the incentive for application of thermodynamics to the calculation of phase-equilibrium relationships. 

Q Residual of phase-equilibrium X Feedback-reset time s t... 

This subsec tion summarizes and presents examples of phase equilibrium data currently available to the designer. The thermodynamic concepts utilized are presented in the subsection Thermodynamics of Sec. 4. 

The phase-distribution restrictions reflect the requirement that ff =ff at equilibrium where/is the fugacity. This may be expressed by Eq. (13-1). In vapor-hquid systems, it should always be recognized that all components appear in both phases to some extent and there will be such a restriction for each component in the system. In vapor-liquid-hquid systems, each component will have three such restrictions, but only two are independent. In general, when all components exist in all phases, the uumDer of restricting relationships due to the distribution phenomenon will be C(Np — 1), where Np is the number of phases present. 

All these processes are, in common, liquid-gas mass-transfer operations and thus require similar treatment from the aspects of phase equilibrium and kinetics of mass transfer. The fluid-dynamic analysis ofthe eqmpment utihzed for the transfer also is similar for many types of liquid-gas process systems. 

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