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Heterogeneous (Multiphase) Mixtures
Published in R. Ravi, Chemical Engineering Thermodynamics, 2020
Thus the degree of freedom is (number of variables – number of equations) 3, the same as that given by the phase rule: = 3 − 2 + 2 = 3. A typical phase diagram would have a plot of xI vs xII (Fig. 5.4) at a given temperature and pressure. Both liquid phases can be modeled using activity coefficient models. From eq. (4.136)2, we may write eq. (44) as xIfA(T,p)γAI(T,p,xI)=xIIfA(T,p)γAII(T,p,xII).
Phase equilibria: non-reactive systems
Published in W. John Rankin, Chemical Thermodynamics, 2019
The phase rule is a tool which enables us to examine and greatly simplify the complex nature of multi-component, multi-phase systems. In examining such systems we are faced with the question: How many variables must be specified so that all the other variables of the system have unique values? The relevant variables are the state functions for the system (temperature, total pressure, partial pressures of the gaseous species) and the concentrations (or, alternatively, the activities) of the solid and liquid species. The answer to this question is important for at least two reasons. Firstly, in order to obtain relations (experimental or theoretical) between variables it is necessary to know how many variables must be specified (that is, measured or controlled) so that the relation is unique. Secondly, for controlling a process it is necessary to know how many variables must be controlled so that all other variables have unique values.
Vapour–liquid phase stratifying
Published in Yu. K. Tovbin, The Molecular Theory of Adsorption in Porous Solids, 2017
where r > 2, then f = k - n + r. The phase rule is used in the study of complex systems, since it allows to calculate the possible number of phases n and the degrees of freedom f in equilibrium systems for any number of components k, [1,7 ].
Abe Clearfield and Inorganic Ion Exchange: A Crystallographer’s Approach to Understanding Reactions and Mechanisms
Published in Solvent Extraction and Ion Exchange, 2020
The field of inorganic ion exchange comprises a wide variety of materials, including zeolites, clays, and silicates and extends to layered materials like layered double hydroxides, layered metal oxides, and layer metal sulfides, to name a few. To predict the kinetic and thermodynamic properties of the ion exchange material a priori requires a thorough understanding of its structure. This was not lost on Abe Clearfield, who first crystallized alpha-zirconium phosphate (α-ZrP) in 1964, solving its structure in 1969 and further refining it in 1977. From a structural basis, the mechanism of exchange was determined and was found to be dependent on the crystallinity of the material. Highly crystalline samples lead to distinct exchange phases, where the exchange occurs systematically at two sites, first at the crystallographically unique exchange site and then at the second. Additionally, the high crystallinity also results in specificity during ion exchange, preferring smaller ions, which are more easily accommodated by the limited interlay space. In contrast, less crystalline samples form solid solutions, not being restrained by Gibbs’ Phase Rule. The discovery of α-ZrP and its interesting properties led to the development of a whole new field of study centering on zirconium phosphate and its derivatives. Since then, a large number of inorganic ion exchange materials have been synthesized and tuned for specific separations by modifying their structure with a desired outcome in mind.
Dissolution, nucleation, and crystal growth mechanism of calcium aluminate cement
Published in Journal of Sustainable Cement-Based Materials, 2019
Lapyote Prasittisopin, Issara Sereewatthanawut
Figure 4(a,b) shows the plots of ion concentration of calcium and aluminate ions of CAC paste versus the tind, respectively. The plots indicate that there is a good agreement between the experimental data and theoretical principle of hydration process of ions shown in Figure 1. Figure 5 shows the binary solubility diagram of calcium-aluminate phases cured at 20 °C, plotted on orthogonal axes. Points A, B, and C represent invariant points at 20 °C. The invariant point is a unique condition of temperature, pressure, and concentration in the liquid system, resulting in a number of phases to coexist in equilibrium stage, based on Gibbs phase rule [26]. For the binary diagram, two phases (calcium and aluminate) exist in the hydrating CAC system. Point A is the invariant point of the AH3, CAH10, and water. Point B is the invariant point of the CAH10, C2AH8, and water. Point C is the invariant point of the C2AH8, Ca4AlO7•13H2O (C4AH13), and water. These points (A, B, and C) were obtained from several empirical experiments from Jones [27] and Brown [28]. A ‘tie line’ is the line of particular hydrated product at a constant pressure and temperature condition representing a phase composition beyond the solubility limit. Its slope is calculated by the ratio between calcium and aluminate (C/A) for each hydrated phase.
Modelling of liquid phases and metal distributions in copper converters: transferring process fundamentals to plant practice
Published in Mineral Processing and Extractive Metallurgy, 2019
E. Jak, T. Hidayat, D. Shishin, P. J. Mackey, P. C. Hayes
The thermodynamic equilibria between gas, slag and matte in the Cu–Fe–O–S–Si system have been characterised by (Yazawa and Kameda 1955; Kameda and Yazawa 1961; Korakas 1964; Kuxmann and Bor 1965; Bar and Tarassaff 1971; Geveci and Rosenqvist 1973; Nagamori 1974; Tavera and Davenport 1979; Kaiura et al. 1980; Jalkanen 1981; Yazawa et al. 1983a; Shimpo et al. 1986; Tavera and Bedolla 1990; Li and Rankin 1994; Takeda 1997a, 1997b; Font et al. 1998; Chen et al. 2017a, 2017b; Fallah-Mehrjardi et al. 2017a, 2017b, 2017c ; Hidayat et al. 2018a, 2018b). According to the Gibbs phase rule, in order to study the Cu–Fe–O–S–Si system experimentally or to perform a thermodynamic calculation, 5 + 2 = 7 degrees of freedom must be fixed. (1) Temperature kept constant. (2) Total pressure of 1 atm. (3) Presence of the slag phase. (4) Presence of the matte phase. (5) Fixed Fe/SiO2 in slag. This is usually achieved either by tridymite saturation, giving lowest possible Fe/SiO2, or by spinel saturation, corresponding to the highest Fe/SiO2. (6) Fixed P(SO2). (7) The last degree of freedom can be set by changing Cu2S/FeS in the initial mixture, or varying the P(O2) in the controlled gas flow. If all degrees of freedom are properly fixed, the slag and matte in equilibrium will have a certain, or invariant, composition.