Single-component systems equilibria

Having considered single component systems, multicomponent systems need to be addressed now. If a closed system contains more than one phase, the equilibrium condition can be written as ... 

A phase boundary for a single-component system shows the conditions at which two phases coexist in equilibrium. Recall the equilibrium condition for the phase equilibrium (eq. 2.2). Letp and Tchange infinitesimally but in a way that leaves the two phases a and /3 in equilibrium. The changes in chemical potential must be identical, and hence... 

In this first example, a single-component system consisting of a liquid and a gas phase is considered. If the surface between the two phases is curved, the equilibrium conditions will depart from the situation for a flat surface used in most equilibrium calculations. At equilibrium the chemical potentials in both phases are equal ... 

Figure 2.31. Schematic representation of the P/T equilibria in a simple two-component system (forming continuous solid and liquid solutions). In (a) a perspective view of the P-T-X diagram is shown in (b) its projection on the P/T plane. Notice the two single-component systems represented, for instance, for the component B by the three lines SB/G (sublimation line of B representing the gas/so lid equilibrium), SB/LB (melting equilibrium of B) and the boiling line LB/G. The solid solution is indicated by a. Notice in (a) the isobaric and isothermal sections of the diagrams (compare with Fig. 2.1).

Single component system (SCS) adsorption models actually mean one pollutant component in aqueous system or in a SWM leachate [34]. Since water is simply assumed to be inert, and the pollutant/leachate adsorption is assumed to be unaffected by water, the system is treated as an SCS. To represent the equilibrium relation for SCS adsorption, a number of isotherm models reported in the literature are reviewed in the following. 

Since TOC, for some solid wastes, was used as a criterion to measure leachate sorption for organic compounds, TOC by itself is considered as a single component system (i.e., SCS, see Sect. 2.1). To represent the SCS equilibrium system for various waste materials, the sorption characteristics of different soils and sediments were analyzed and evaluated using three different sorption iso-... 

Gas and Liquid Phases. Equilibrium data (P-V-T) and thermodynamic properties for the single-component systems water (steam) and ammonia are complete and apparently of the best accuracy because of the extensive use of these substances in cyclic systems 14,20). 

Next consider the triple point of the single-component system at which the solid, liquid, and vapor phases are at equilibrium. The description of the surfaces and tangent planes at this point are applicable to any triple point of the system. At the triple point we have three surfaces, one for each phase. For each surface there is a plane tangent to the surface at the point where the entire system exists in that phase but at the temperature and pressure of the triple point. There would thus seem to be three tangent planes. The principal slopes of these planes are identical, because the temperatures of the three phases and the pressures of the three phases must be the same at equilibrium. The three planes are then parallel. The last condition of equilibrium requires that the chemical potential of the component must be the same in all three phases. At each point of tangency all of the component must be in that phase. Consequently, the condition... 

In the three-dimensional diagram, the curve formed by the intersection of the two surfaces represents all of the equilibrium points of the two-phase system. Such a curve is obtained for each type of a two-phase equilibrium existing in a single-component system. At any triple point of the system three such curves meet at a point, giving the temperature and pressure of the triple point. 

In the case of a single-component system, the adsorption isotherm gives the concentration in the stationary phase C (in moles per liter of bead or grams per liter of bead) versus the mobile phase concentration C when equilibrium is reached (in the same units as for C), at a given temperature. 

In our previous works [1-3], we reported experimental and theoretical equilibrium isotherms for adsorption of L-glutamic acid in the single component system on polyaminated highly porous chitosan (hereafter called PEl-CH), weakly basic ion exchanger, and crosslinked chitosan fiber. We found that the adsorption of L-glutamic acid, which is a kind of acidic amino acid, was controlled by the acid/base neutralization reaction between neutral L-glutamic acid (zwitterion, A and those adsorbents. 

A straightforward determination of is obtained by centrifuging until an equilibrium distribution of the molecules of a protein or other macromolecular material is obtained and by recording the variation in concentration from the center to the periphery of the centrifuge celF 483,i85,i87-i90 ggg Section A,2). Using short cells, this sedimentation equilibrium can be attained in 1-5 hours instead of the 1-2 days needed with older instruments. For a single component system the concentration distribution at equilibrium is given by Eq. 3-12. 

Having obtained the adsorption equilibrium and mass transfer parameters of single component systems (Tables 1 to 2), we are ready to examine the predictability of the model in simulating the sorption kinetics of multicomponent systems on Norit activated carbon. 

The minor disturbance or perturbation method relies on equilibrium theory too and was first suggested by Reilley et al. (1962). As known from linear chromatography, the retention time of a small pulse injected into a column filled with pure eluent can be used to obtain the initial slope of the isotherm. This approach is expanded to cover the whole isotherm range. For the example of a single-component system (Fig. 6.24) the procedure is as follows The column is equilibrated with a concentration ca and, once the plateau is established, a small pulse is injected at a time fstart a and a pulse of a different concentration is detected at the corresponding retention time tR a. 

Piqtkowski el al. measured the single-component and the competitive equilibrium isotherms of phenetole (ethoxy-benzene) and n-propyl benzoate on a 150 x 3.9 mm S3onmetry -Cig (endcapped) column (Waters), using a methanol/water (65 35, v/v) as the mobile phase [26]. The adsorption equilibrium data of the single-component systems were acquired by frontal analysis. For both compoimds. 

The factors that determine the crystal structure of particles formed in aerosol reactors have not been studied systematically. In this section, we identify key theoretical concepts and review relevant experimental observations. Consideration is limited to single-component systems. Panicle crystal structure depends on a combination of thermodynamic (equilibrium) factors and rate processes. The equilibrium shape of a particle is detennined by the surface energies of its crystal face.s according to the Wulff construction (Chapter 8). Another factor that inay enter into the process is the excess pressure inside small particles according to the Laplace formula (Chapter 9). Thus the equilibrium form may vary with panicle size depending on the phase diagram,... 

The phase behavior of single-component systems has been discussed as part of thepVT relationship presented in Section 4.2.1. Examiifing the phase behavior of mixtures, we observe that, with mixtures, phase behavior remains one facet of the pVT relationship. But a new phenomenon is encountered with mixtures phases at equilibrium are generally of different compositions. These mixtures show a great variety of phase behavior that can often be exploited to make separations. We examine in broad terms the qualitative features of the phase behavior of binary mixtures of various types. Experience has shown a wealth of phenomena displayed by binary mixtures. 

To find the vapor pressure using an equation of state (eos), we begin with vapor-liquid equilibrium for a single-component system ... 

Gas-solid equilibrium for a single-component system is commonly referred to as sublimation equilibrium. Sublimation pressure, the vapor pressure of a solid, is basic to the modeling of solid-gas equilibrium. Sublimation pressure changes with temperature by an equation similar to that of the vapor pressure of a liquid. Equation (4.453),... 

The ground-state anion of the proton emitter is the straightforward detector for the ejected protons and represents the simplest system for analysis (Forster and Volker, 1975, Gutman etal., 1981). In the absence of other acceptors, the transient increment of H+ concentration (AH+), is identical with the increment of < >0 above its prepulse (equilibrium) concentration (Achemical relaxation, the concentration of the reactants is given as a sum of the equilibrium concentration plus the incremental deviation from equilibrium, which is the time-dependent variable. For the single-component system... 

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