For a mixture

In vapor-liquid equilibria, if one phase composition is given, there are basically four types of problems, characterized by those variables which are specified and those which are to be calculated. Let T stand for temperature, P for total pressure, for the mole fraction of component i in the liquid phase, and y for the mole fraction of component i in the vapor phase. For a mixture containing m components, the four types can be organized in this way ... 

The fugacity coefficient is a function of temperature, total pressure, and composition of the vapor phase it can be calculated from volumetric data for the vapor mixture. For a mixture containing m components, such data are often expressed in the form of an equation of state explicit in the pressure... 

For a pure vapor the virial coefficients are functions only of temperature for a mixture they are also functions of composition. An important advantage of the virial equation is that there are theoretically valid relations between the virial coefficients of a mixture and its composition. These relations are ... 

ENTH calculates vapor or liquid enthalpies, J/mole, (reference, ideal gas at 300 K) for a mixture of N components (N <... 

FLASH determines the equilibrium vapor and liquid compositions resultinq from either an isothermal or adiabatic equilibrium flash vaporization for a mixture of N components (N 20). The subroutine allows for presence of separate vapor and liquid feed streams for adaption to countercurrent staged processes. 

BUDEP calculates the bubble-point pressure or the dew-point pressure for a mixture of N components (N j< 20) at specified temperature and liquid or vapor composition. The subroutine also furnishes the composition of the incipient vapor or liquid and the vaporization equilibrium ratios. 

Molecular weight for a mixture of components such as one would encounter in petroleum cuts, the molecular weight is ... 

It is difficult to judge the accuracy of these methods because data are scarce. Table 4.9 compares the values obtained by different weighting methods with experimental values for a mixture of n-hexane-n-hexadecane at 25°C. The ASTM method shows results very close to those obtained experimentally. 

The specific heat of gases at constant pressure is calculated using the principle of corresponding states. The for a mixture in the gaseous state is equal to the sum of the C g of the ideal gas and a pressure correction term ... 

In 1972, Soave published a method of calculating fugacities based on a modification of the Redlich and Kwong equation of state which completely changed the customary habits and became the industry standard. In spite of numerous attempts to improve it, the original method is the most widespread. For hydrocarbon mixtures, its accuracy is remarkable. For a mixture, the equation of state is ... 

If an ideal solution is formed, then the actual molecular A is just Aav (and Aex = 0). The same result obtains if the components are completely immiscible as illustrated in Fig. IV-21 for a mixture of arachidic acid and a merocyanine dye [116]. These systems are usually distinguished through the mosaic structure seen in microscopic evaluation. 

Lebowitz J L 1964 Exact solution of the generalized Percus-Yevick equation for a mixture of hard spheres Phys. Rev. 133 A895... 

A water-soluble mixture may be in the form of a mixture of water-soluble solids or in the form of a liquid. The liquid mixtimes are frequently aqueous solutions. The prelirninary examination of a liquid mixture (see 1) will indicate whether a volatile solvent (i.e., removable on a boiling water bath) is present. If a volatile solvent is present, distil 20 g. of the mixtime from a water bath until no more hquid passes over set aside the volatile solvent for identification. Dissolve the residue (B) in water as detailed below for a mixture of solids. 

Figure 9.8b shows a titration curve for a mixture consisting of two weak acids HA and HB. Again, there are two equivalence points. In this case, however, the equivalence points do not require the same volume of titrant because the concentration of HA is greater than that for HB. Since HA is the stronger of the two weak acids, it reacts first thus, the pH before the first equivalence point is controlled by the HA/A buffer. Between the two equivalence points the pH reflects the titration of HB and is determined by the HB/B buffer. Finally, after the second equivalence point, the excess strong base titrant is responsible for the pH. 

The scale of operations, accuracy, precision, sensitivity, time, and cost of methods involving redox titrations are similar to those described earlier in the chapter for acid-base and complexometric titrimetric methods. As with acid-base titrations, redox titrations can be extended to the analysis of mixtures if there is a significant difference in the ease with which the analytes can be oxidized or reduced. Figure 9.40 shows an example of the titration curve for a mixture of Fe + and Sn +, using Ce + as the titrant. The titration of a mixture of analytes whose standard-state potentials or formal potentials differ by at least 200 mV will result in a separate equivalence point for each analyte. 

Titration curve for a mixture of 1 and Cl using AgNOa as a titrant. 

The acidity of a water sample is determined by titrating to fixed end points of 3.7 and 8.3, with the former providing a measure of the concentration of strong acid, and the latter a measure of the combined concentrations of strong acid and weak acid. Sketch a titration curve for a mixture of 0.10 M HCl and 0.10 M H2CO3 with 0.20 M strong base, and use it to justify the choice of these end points. 

A linear-potential scan hydrodynamic voltammogram for a mixture of Le + and Le + is shown in the figure, where and... 

Mag nesia. ndAlumina. Suspension. A mixture of salts, available as Maalox, Mylanta, Gelusil, and Aludrox, contains magnesium hydroxide [1309-42-8] Mg(OH)2, and variable amounts of aluminum oxide in the form of aluminum hydroxide and hydrated aluminum oxide, ie, 2.9—4.2% magnesium hydroxide and 2.0—2.4% aluminum oxide, Al O, for a mixture of 4.9—6.6% combined magnesium hydroxide and aluminum oxide. This mixture may contain a flavoring and antimicrobial agents in a total amount not to exceed 0.5% (see Aluminum compounds, aluminum oxide). 

Liquid crystals may be divided into two broad categories, thermotropic and lyotropic, according to the principal means of breaking down the complete order of the soHd state. Thermotropic Hquid crystals result from the melting of mesogenic soHds due to an increase in temperature. Both pure substances and mixtures form thermotropic Hquid crystals. In order for a mixture to be a thermotropic Hquid crystal, the different components must be completely miscible. Table 1 contains a few examples of the many Hquid crystal forming compounds (2). Much more is known about calamitic (rod-Hke) Hquid crystals then discotic (disk-like) Hquid crystals, since the latter were discovered only recendy. Therefore, most of this section deals exclusively with calamities, with brief coverage of discotics at the end. 

Example 42 Estimate surface tension of a mixture. At 298.15 K, Daiibert et al. " report the hqiiid density of n-pentane to be 8.617 kmol/nd and its surface tension to be 15.47 mN/m. From the same source, the corresponding values for dichloromethane are 15.52 kmol/m and 27.22 mN/m. Using Eqs. (2-170) and (2-169) for a mixture of 0.1606 mole fraction n-pentane and 0.8394 mole fraction dichloromethane ... 

The ideal-gas-state heat capacity Cf is a function of T but not of T. For a mixture, the heat capacity is simply the molar average X, Xi Cf. Empirical equations giving the temperature dependence of Cf are available for many pure gases, often taking the form... 

Although the Pitzer correlations are based on data for pure materials, they may also be used for the calculation of mixture properties. A set of recipes is required relating the parameters T, Pc, and (0 for a mixture to the pure-species values and to composition. One such set is given by Eqs. (2-80) through (2-82) in Sec. 2, which define pseudopa-rameters, so called because the defined values of T, Pc, and (0 have no physical significance for the mixture. 

When i = J, all equations reduce to the appropriate values for a pure species. When i j, these equations define a set of interaction parameters having no physical significance. For a mixture, values of By and dBjj/dT from Eqs. (4-212) and (4-213) are substituted into Eqs. (4-183) and (4-185) to provide values of the mixture second virial coefficient B and its temperature derivative. Values of and for the mixture are then given by Eqs. (4-193) and (4-194), and values of In i for the component fugacity coefficients are given by Eq. (4-196). 

When multicomponent mixtures are to be separated into three or more products, sequences of simple distillation columns of the type shown in Fig. 13-1 are commonly used. For example, if aternaiy mixture is to be separated into three relatively pure products, either of the two sequences in Fig. 13-4 can be used. In the direct sequence, shown in Fig. 13-4, all products but the heaviest are removed one by one as distillates. The reverse is true for the indirect sequence, shown in Fig. 13-4 7. The number of possible sequences of simple distillation columns increases rapidly with the number of products. Thus, although only the 2 sequences shown in Fig. 13-4 are possible for a mixture separated into 3 products, 14 different sequences, one of which is shown in Fig. 13-5, can be synthesized when 5 products are to be obtained. 

When only a few solutes are separated, they may occupy only a small portion of the total column volume at any given instant. In such cases, the productivity is improved by cyclic feed injections, timed so that the most strongly retained component from an injection elutes just before the least strongly retained component from the following injection (see Fig. 16-57). For a mixture of two components with k > 1, when the same resolution is maintained between bands of the same injections and bands of successive injections, the cycle time tc and the plate number requirement are ... 

The simplest and most common method of establishing pseudocriticais for a mixture is Kay s Rule. 

Since the boiling point properties of the components in the mixture being separated are so critical to the distillation process, the vapor-liquid equilibrium (VLE) relationship is of importance. Specifically, it is the VLE data for a mixture which establishes the required height of a column for a desired degree of separation. Constant pressure VLE data is derived from boiling point diagrams, from which a VLE curve can be constructed like the one illustrated in Figure 9 for a binary mixture. The VLE plot shown expresses the bubble-point and the dew-point of a binary mixture at constant pressure. The curve is called the equilibrium line, and it describes the compositions of the liquid and vapor in equilibrium at a constant pressure condition. 

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