Effective molar volume

In the original equation of van Laar, the effective molar volume was assumed to be independent of composition this assumption implies zero volume-change of mixing at constant temperature and pressure. While this assumption is a good one for solutions of ordinary liquids at low pressures, it is poor for high-pressure solutions of gases in liquids which expand (dilate) sharply as the critical composition is approached. The dilated van Laar model therefore assumes that... 

The volume fraction 0 = CV where V is the mean effective molar volume of the complexed forms of M (in liters) ... 

Another way to achieve uniform illumination is to lower the optical density of the sample by increasing the effective molar volume. Solid solution of an absorbing guest in a transparent host in either a normal crystal or a molecular inclusion compound can achieve this end so can using a pure crystal of a much larger molecule which includes the chromophore of interest [27]. While it is harder to reach high dilution with the latter approach, it has the virtue that the initial compound is a pure crystal and thus better suited for definitive X-ray investigation. 

If this effective molar volume is to represent the property of species f in original equimolar solution, it must be based on data for a solution of this compos However, in the process described a finite drop of water is added to the equir solution, causing a small but finite change in composition. We may, however, cons" the limiting case for which Anw - 0. Then Eq. (B) becomes... 

However, we find by experiment that the actual value of A(nV) is somewhat less than that given by this equation. Evidently, the effective molar volume of the added water in solution is less than the molar volume qf pure water at the same T and P. Designating the effective molar volume in solution by Vw, we can write... 

A number of models which can estimate density at atmospheric pressure have recently been reported. For example, Rebelo et al. [63, 64] defined the effective molar volumes of ions at 298.15 K and used the assumption of ideal behavior for the determination of the molar volume of ionic liquids. Yang et al. [65] used a theory based on the interstice model which correlated the density and the surface tension of the ionic liquid. Group contribution models have been reported by Kim et al. [66, 67] for the calculation of the density and C02 gas solubility for 1-alkyl-3-methylimidazolium based ionic liquids as a function of the temperature and C02 gas pressure with reasonable accuracy over a 50 K temperature range however, the... 

Recently Jacquemin et al. [61,71] extended the concept proposed by Rebelo et al. In this method the effective molar volume of an ionic liquid and hence density can be determined by assuming that the volumes of the ions behave as an ideal mixture. This strategy was used to calculate the effective molar volumes of a wide range of ions using a large set of previously reported data as a function of the temperature difference at 0.1 MPa and a reference temperature of 298.15 K using the following equation ... 

The coefficients (C.) were obtained for 44 anions and 102 cations which achieved a high degree of accuracy when using more than 2150 data points. This approach was further extended to include pressure [72] by applying the commonly used Tait equation. In this case the effective molar volumes are estimated using the following equation ... 

Table V was prepared from Figure 5. The critical PEG molecular weight is that molecular weight below which some penetration can occur into the substance. For untreated wood, it is about 3000. However, the meaning of this value must be interpreted with caution. According to the theory of gel permeation chromatography, the important single solute parameter is effective molar volume rather than molecular weight (9). The molar volume of a polymer depends on polymer-polymer and polymer-solvent interactions this is illustrated schematically in Figure 9 (36). The total chain lengths are the same for all the polymer systems, yet the configurations and bulkiness vary considerably.

The preceding qualitative observations about the temperature dependence of Ch and Vg — V, can be extended to a quantitative statement in cases for which the effective molecular volume of the penetrant in the sorbed state can be estimated. As a first approximation, one may assume that the effective molecular volume of a sorbed CO2 molecule is 80 A in the range of temperatures from 25 to 85 C. This molecular volume corresponds to an effective molar volume of 49 cmVmol of CO2 molecules and te similar to the partial molar volume of CO2 in various solvents, in several zeolite environments, and even as a pure subcritical liquid (see Tables 20.4-4 and 20.4-5). The implication here is not that mote than one COi molecule exists in each molecular-scale gap, but rather that the effective volume occupied by a CO2 molecule is roughly the same in the polymer sorbed state, in a saturated zeolite sorbed state, and even in a dissolved or liquidiike state since all these volume estimates tend to be similar for materials that are not too much above their critical temperatures. With the above approximation, the predictive expression given below for Cw can be compared to independently measured values for this parameter from sorption measurements ... 

The results of the gel permeation chromatography measurements by Protivova and PospisiP of the elution volumes of aromatic amines, their molecular weights, calculated molar volumes and the effective molar volumes observed and read from the calibration curves are given in Table 7.20. A comparison of the calculated and effective molar volumes revealed deviations in the behaviour of all the amines they investigated, compared to similar aliphatic hydrocarbons. 

The dependence of the effective molar volume (Veff) of transition metal oxides on their composition (on the a/b ratio ) is presented in Fig. 1. In the case of non-stoichiometric compounds of Mi-yOi-x type (oxides of V, Ti, Nb etc.) the real contents of metal and oxygen in an unit cell of these oxides were taken into account and then recalculated to the volume containing one mole of oxygen atoms. The above relations have been obtained for oxides undergoing mutual transformations. 

Fig. 1. The dependence of the effective molar volume (Veff) of the oxides Mj/iO on the composition (a/b) for the transition metal oxides and Ce oxides.

As it can be seen in Fig.l, the dependence of the effective molar volume of oxides on the a/b ratio shows linear character in the composition range 0.5ions with the same charge (2+, 3+) but also in the case of oxides showing deviation from the stoichiometry, for many oxide phases (Magneli phases) with ions in different oxidation states and, as in the case of Fei.yO or M3O4 oxides with spinel structure, when metal ions have different coordination number, 4 or 6 (tetrahedral or octahedral). 

The obtained linear character of the dependence of the effective molar volume (Veff) on the composition (4/F) in the range of composition changes of 2>q/b>0.5 (in the case of transition metal oxides l>a/b>0.5, and 0.666>( /h>0.35 in the case of/electron metal oxides), should be considered as a general relationship for the phases xmdergoing mutual transformations. The observed decrease of the molar volume of the oxides (Veff) is a result of smaller quantity of... 

Ma/bO oxides with composition in the range of 2>a/b>0.5 (in the case of actinide oxides up to 0.33) having effective molar volumes of oxides (M O) of the same metal decreasing linearly when the metal content decreases. 

All these oxide phases can be included in the oxides with framework structure. These are oxides (Ma/bO) with composition a/h<0.5 and effective molar volumes bigger than those for the oxides of compact structures (MO2) and the smallest metal content. In these oxides the coordination polyhedra can be identified, connected by corners, edges, etc. Most of these structures can be derived from the structure of the oxide ReQs where the MO octahedrons form a regular lattice. Other cations can easily be included in the structure of this type of oxides. This leads to the stabilization of the structure, and as a result, one can obtain the perovskite-type structures or their variations. 

The linear relations between the effective molar volume of the oxides of the same metal (M O) and their composition, discussed in the chapter 2 and the dependence of (V(.g) on ionic radii indicate that there should be also a simple correlation between (1/S) and the a/b ratio. The relation between (V g) and the a/b ratio for the oxides of Ti and V, and for the oxides of Ce, Pr and Tb is shown in Fig. 8. As can be seen, linear relationships have been obtained, with a high correlation coefficient, R2>0.98. Table 4 in the work (Stoklosa Laskowska, 2008b) contains parameters of the obtained linear fimctions ... 

The effective molar volume of the oxide Veff= /,.o (P r mole of oxygen atoms) is a pwameter allowing to compare oxides with different compositions, structures and with ions in different oxidation states. 

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