Total solution properties

Vapor-liquid distribution coefficients (/ -values) may be calculated from equations of state using Equations 1.21, 1.23, and 1.25. These calculations require the evaluation of partial properties of individual components, defined as the change in the total solution property resulting from the addition of a differential amount of the component in question to the solution, while holding constant the remaining component amounts and the temperature and pressure. Mathematically, the partial property fl of component i is given by... 

Apply thermodynamics to mixtures. Write the differential for any extensive property, dK, in terms of m + 2 independent variables, where m is the number of species in the mixture. Define and find values for pure species properties, total solution properties, partial molar properties, and property changes of mixing. 

When a species becomes part of a mixture, it loses its identity yet it still contributes to the properties of the mixture, since the total solution properties of the mixture depend on the amount present of each species and its resultant interactions. We wish to develop a way to account for how much of a solution property (V, H, U, S, Q.. .) can be assigned to each species. We do this through a new formalism the partial molar property. 

Mathematically, we can write the extensive total solution property K in terms of T, P, and the number of moles of m different species ... 

Equation (6.17) indicates that the extensive total solution property K is equal to the sum of the partial molar properties of its constituent species, each adjusted in proportion to the quantity of that species present. Similarly, Equation (6.18) shows that the intensive solution property k is simply the weighted average of the partial molar properties of each of the species present. The partial molar property, Kj, can then be thought of as species is contribution to the total solution property, K. 

There are many different types of properties of which to keep track in mixtures. In this section, we review our nomenclature and see how we keep track of the different types of properties. We consider total solution properties, pure species properties, and partial molar properties. 

The total solution properties are the properties of the entire mixture. They are written as ... 

Partial molar properties can be viewed as the specific contribution of species i to the total solution property, as discussed in the previous section. They are written ... 

We have introduced a new type of property, the partial molar property. This property tells us about the contribution of a given species to the properties of a mixture. Our next question is How do we obtain values for these partial molar properties There are several ways in which to accomplish this task. In this section, we consider two examples of how we might calculate a partial molar property by analytical means when we have an equation that describes the total solution property or by graphical means from plots of total solution data. 

Often an analytical expression for the total solution property, k, is known as a function of composition. In that case, the partial molar property, Kj, can be found by differentiation of the extensive expression for K with respect to Uj, holding T, P, and the number of moles of the otherj species constant, as prescribed by Equation (6.15). 

Thermodynamic properties of a mixture are affected both by the like (i-i) interactions and by the unlike i-j) interactions, that is, how each of the species in the mixture interacts with all of the other species it encounters. The total solution property fC, (fC = V, ff, 17, S, G,.. . ), represents a given property of the entire mixture. It can be written as the sum of the partial molar properties of its constituent species, each adjusted in proportion to how much is present ... 

The ideal gas provides the reference state for the vapor phase. To set up our discussion of reference states for the liquid phase, we will review the property changes of mixing for an ideal gas as discussed in Section 6.3. Recall that a property change of mixing is defined as the difference between the total solution property and the sum of the pure species properties apportioned by the amount each species present in the mixture ... 

Partial molar property Total solution property... 

Orthophosphate salts are generally prepared by the partial or total neutralization of orthophosphoric acid. Phase equiUbrium diagrams are particularly usehil in identifying conditions for the preparation of particular phosphate salts. The solution properties of orthophosphate salts of monovalent cations are distincdy different from those of the polyvalent cations, the latter exhibiting incongment solubiUty in most cases. The commercial phosphates include alkah metal, alkaline-earth, heavy metal, mixed metal, and ammonium salts of phosphoric acid. Sodium phosphates are the most important, followed by calcium, ammonium, and potassium salts. 

Physical properties of some commercially available polyamines appear in Table 1. Generally, they are slightly to moderately viscous, water-soluble Hquids with mild to strong ammoniacal odors. Although completely soluble in water initially, hydrates may form with time, particularly with the heavy ethyleneamines (TETA, TEPA, PEHA, and higher polyamines), to the point that gels may form or the total solution may soHdify under ambient conditions. The amines are also completely miscible with alcohols, acetone, benzene, toluene and ethyl ether, but only slightly soluble in heptane. 

Partial Molar Properties Consider a homogeneous fluid solution comprised of any number of chemical species. For such a PVT system let the symbol M represent the molar (or unit-mass) value of any extensive thermodynamic property of the solution, where M may stand in turn for U, H, S, and so on. A total-system property is then nM, where n = Xi/i, and i is the index identifying chemical species. One might expect the solution propei fy M to be related solely to the properties M, of the pure chemical species which comprise the solution. However, no such generally vahd relation is known, and the connection must be establi ed experimentally for eveiy specific system. 

If the validity of Eq. (1.3.31) is assumed for the mean activity coefficient of a given electrolyte even in a mixture of electrolytes, and quantity a is calculated for the same measured electrolyte in various mixtures, then different values are, in fact, obtained which differ for a single total solution molality depending on the relative representation and individual properties of the ionic components. 

The variable is called the particular solution. It is the function that satisfies the original ODE with a specified One of the most useful properties of linear ODEs is that the total solution is the sum of the complementary solution and the particular solution. 

Table IV shows a comparison of some aqueous solution properties of nonionics having the same total number (ten) of carbon atoms in the hydrophobe and a comparable POE chain length, but different hydrophobe structure. It can be seen that multi-chain hydrophobes bring about a striking decrease in the cloud point and in the surface tension at the cmc, and an increase in the cmc, while cyclic fixation of the alkyl chain causes a large increase in the cloud point, the cmc and in the surface tension at the cmc.

Kbp is the molal boiling-point constant, a property of the solvent and independent of the nature of the solutes msolutes refers to the total concentration of independent solute particles whether they are neutral molecules or ions. Thus a 0.01 molal NaCl solution has a total solute con-... 

In the ideally dilute limit, colligative properties depend on the total solute mole fraction or molality. If there are a number of different solutes present, this can be expressed as (for the case of freezing-point depression)... 

Table 1. Relationship between X and the physical solute properties using different FFF techniques [27,109] with R=gas constant, p=solvent density, ps=solute density, co2r=centrifugal acceleration, V0=volume of the fractionation channel, Vc=cross-flow rate, E=electrical field strength, dT/dx=temperature gradient, M=molecular mass, dH=hydrodynamic diameter, DT=thermal diffusion coefficient, pe=electrophoretic mobility, %M=molar magnetic susceptibility, Hm=intensity of magnetic field, AHm=gradient of the intensity of the magnetic field, Ap = total increment of the chemical potential across the channel...

In certain cases we can adequately describe the chemical properties of species / by using the concentration of that solute, Cj. Owing to molecular interactions, however, this usually requires that the total solute concentration be low. Molecules of solute species j interact with each other as well as with other solutes in the solution, and this influences the behavior of species /. Such intermolecular interactions increase as the solution becomes more concentrated. The use of concentrations for describing the thermodynamic properties of some solute thus indicates an approximation, except in the limiting case of infinite dilution for which interactions between solute molecules are negligible. Where precision is required, activities—which may be regarded as corrected concentrations—are used. Consequently, for general thermodynamic considerations, as in Equation 2.4, the influence of the amount of a particular species / on its chemical potential is handled not by its concentration but by its activity, aj. The activity of solute j is related to its concentration by means of an activity coefficient, y ... 

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