Mole fractions in solutions

Equation (4-49) is merely a special case of Eq. (4-48) however, Eq. (4-50) is a vital new relation. Known as the summahility equation, it provides for the calculation of solution properties from partial properties. Thus, a solution property apportioned according to the recipe of Eq. (4-47) may be recovered simply by adding the properties attributed to the individual species, each weighted oy its mole fraction in solution. The equations for partial molar properties are also valid for partial specific properties, in which case m replaces n and the x, are mass fractions. Equation (4-47) applied to the definitions of Eqs. (4-11) through (4-13) yields the partial-property relations ... 

Standard state unit mole fraction in solution and unit coverage in monolayer. r=-15 °C. 

This is true in terms of mole fractions in solution but, for... 

Titration results for the mixed erne s of the SDS/CgE4 and C12E2S/C8E4 systems as a function of their relative mole fraction in solution are shown in Figures 2 and 3, respectively. Here, the experimentally determined points are compared with calculated results from the nonideal mixed micelle model (solid line) and the ideal mixed micelle model (dashed line). Good agreement with the nonideal model is seen in each case. 

The preceding discussion has led us to the conclusion that the surface is the only locus of polymerization which needs to be considered in the heterogeneous polymerization of acrylonitrile. Radicals arrive at the surface at a rate determined by the decomposition of the initiator and efficiency of initiation. Propagation occurs on the surface at a rate determined by the activity of monomer at the surface. By analogy with emulsion polymerization, where monomer diffuses into the particles rapidly enough to maintain near equilibrium activity (14), we assume that the activity of the monomer adsorbed on the particle surface is approximately equal to the mole fraction in solution. The propagation rate constant is presumably influenced somewhat by the presence of the solid surface. 

The effect of injection of large samples on the retention behavior of minor sample components can be either to decrease or, more likely, to increase retention volumes, even though their mole fraction in solution may approach zero under ail conditions, because at least at the column inlet the major sample component can act in part as the stationary phase. For example, Deans observed substantially increased retention times for octane, nonane, and decane injected as minor components in large samples of hexane. 

The chemical potential does not need a superscript because it is the same everywhere it C in now be written as p° + RT In where J. and are the activity coefficient and mole fraction in solution. 

The Henry constant is defined as the limiting value of the ratio of the gas partial pressure to its mole fraction in solution as the latter tends to zero [8]. 

The preceding section described the properties of solutions of nonvolatile solutes in liquid solvents. The concept of an ideal solution can be extended to mixtures of two or more components, each of which is volatile. In this case, an ideal solution is one in which the vapor pressure of each species present is proportional to its mole fraction in solution over the whole range of mole fraction ... 

For an ideal solution or a sufficiently dilute real solution, the vapor pressure of any volatile component is proportional to its mole fraction in solution. 

The vapor pressure T of a solvent is equal to the product of its mole fraction in solution,... 

Fig. 1.2 Plots of the molar volume of aqueous solutions of acetonitrile (AcN) and methanol (MeOH) against their mole fraction in solution.

In summary, there are three important characteristics of ideal solutions that one should remember in assessing the properties of any non-ideal system (i) the vapor pressure of each component is proportional to its mole fraction in solution over the whole composition range (Raoult s law) (ii) the enthalpy of mixing is zero (iii) the volume change associated with mixing is zero. The sections which follow deal with non-ideal solutions. 

This equation states that chemical potentials of component A in the liquid solution and vapor are equal and that each relates to the vapor pressure of A. However, one would like to have a way of relating the chemical potential of A to its mole fraction in solution. This is achieved by relating the vapor pressure of A to its mole fraction in the liquid solution using a correction factor to make the value of Pa predicted by Raoult s law equal to the true value. Thus, one writes... 

The equilibrium pressures (0.5—760Torr) of hydrogen existing above mixtures of lithium with lithium hydride (0.5—99 mol% LiH) sealed in iron capsules have been measured from 983 to 1176 K. The isotherms confirm the phase diagram to consist of two immiscible liquid phases with boundaries 25.2 and 98.4 mol% LiH at 983 K and 45.4 and 85.8 mol% LiH at 1176 K. For dilute solutions of lithium hydride in liquid lithium, the relationship between the mole fraction in solution, Xi.iH, and the equilibrium pressure, (phj)S at T(K) is given by... 

Temperature (K) Mole Fraction in Solution Temperature (K) Mole Fraction in Solution ... 

Note that the partial vapor pressure of X above an ideal solution depends only on its mole fraction in solution and it is completely independent of the vapor pressures of the other volatile components of the solution. If all components other than X are nonvolatile, the total vapor pressure of the mixture will be equal to the partial pressure of X, since the vapor pressure of nonvolatile compounds may be taken as zero. Accordingly, the distillate from such a mixture will always be pure X. This is the case for the distillation of a solution of sugar and water, as discussed earlier. 

Figure 5.8. Isotherm of excess adsorption on activated carbon of (f) n-butylamine and (2) methyl acetate, from fheir respecfive solutions in benzene (Blackburn et al., f 957), where x is mole fraction in solution and the excess adsorption is given by Eq. 5.1.

Figure 4 4-C12-DTPA mole fraction in micelles, as a function of 4-C12-DTPA mole fraction in solution, xj, for the different surfactant systems at pH 5.0...

Therefore the difficulty in the study of solutions arises when we wish to relate the composition of the gas phase with the composition of the solution with which it is in contact. In general, the partial pressure of any component of the vapour may be expected to be a function of temperature, together with its mole fraction in solution and also the mole fractions of all other species (or rather a function of N — 1 mole fractions if there are N substances in solution). Thus... 

Xa is the mole fraction in solution of component a Xa is the molar heat of fusion of component a R is the gas constant... 

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