Molar volume corrections

Although ignoring the oxygen molar volume correction allows more rapid calculation of estimated dielectric constants. 

However the agreement of the predicted solubility, X by Eq. (46), with the corresponding experimental data for naphthalene and phenanthrene in toluene (56) was found to be inadequate. The deviation is significant at high values of both X and JTi. To overcome this limitation, a new model has been developed. For this purpose, first a molar volume correction is incorporated (57), as shown in Eq. (49), to predict the binary solid solubility Z3, and then Xi is replaced with Xi, leading to Eq. (50), to predict X3 from binary data, since Xi = Xi/(1 — V3) and Z3 = x jil — Xi). 

T were eliminated beyond that point, the vapor-phase correction, as calculated here, is inadequate and the liquid molar volume is no longer constant with pressure. 

Standard-state fugacities at zero pressure are evaluated using the Equation (A-2) for both condensable and noncondensable components. The Rackett Equation (B-2) is evaluated to determine the liquid molar volumes as a function of temperature. Standard-state fugacities at system temperature and pressure are given by the product of the standard-state fugacity at zero pressure and the Poynting correction shown in Equation (4-1). Double precision is advisable. 

The rate of chemical diffusion in a nonfiowing medium can be predicted. This is usually done with an equation, derived from the diffusion equation, that incorporates an empirical correction parameter. These correction factors are often based on molar volume. Molecular dynamics simulations can also be used. 

The solvent triangle classification method of Snyder Is the most cosDBon approach to solvent characterization used by chromatographers (510,517). The solvent polarity index, P, and solvent selectivity factors, X), which characterize the relative importemce of orientation and proton donor/acceptor interactions to the total polarity, were based on Rohrscbneider s compilation of experimental gas-liquid distribution constants for a number of test solutes in 75 common, volatile solvents. Snyder chose the solutes nitromethane, ethanol and dloxane as probes for a solvent s capacity for orientation, proton acceptor and proton donor capacity, respectively. The influence of solute molecular size, solute/solvent dispersion interactions, and solute/solvent induction interactions as a result of solvent polarizability were subtracted from the experimental distribution constants first multiplying the experimental distribution constant by the solvent molar volume and thm referencing this quantity to the value calculated for a hypothetical n-alkane with a molar volume identical to the test solute. Each value was then corrected empirically to give a value of zero for the polar distribution constant of the test solutes for saturated hydrocarbon solvents. These residual, values were supposed to arise from inductive and... 

The molar volumes are in some cases at the stated temperature and in other cases at the normal boiling point. Certain calculated molecular volumes are also used thus the reader is cautioned to ensure that when using a molar volume in any correlation, it is correctly selected. In the case of polynuclear aromatic hydrocarbons, the Le Bas molar volume is regarded as suspect because of the compact nature of the multi-ring compounds. It should thus be regarded as merely an indication of relative volume, not an absolute volume. 

First-order estimates of entropy are often based on the observation that heat capacities and thereby entropies of complex compounds often are well represented by summing in stoichiometric proportions the heat capacities or entropies of simpler chemical entities. Latimer [12] used entropies of elements and molecular groups to estimate the entropy of more complex compounds see Spencer for revised tabulated values [13]. Fyfe et al. [14] pointed out a correlation between entropy and molar volume and introduced a simple volume correction factor in their scheme for estimation of the entropy of complex oxides based on the entropy of binary oxides. The latter approach was further developed by Holland [15], who looked into the effect of volume on the vibrational entropy derived from the Einstein and Debye models. 

Correctness of the sixth parameter equation (7) and its simplified forms for the generalization of the swelling data was proved for other coals including the Donbas coal [32] at the parameters B and VM- If to apply the equation (7) to the coal extraction data, then the factor of molar volume VM is insignificant, and the connection between quantities of extracted substance (in g/mole of the solvent) and physical-chemical characteristics can be satisfactorily described by fifth parameter equation (6) or by its simplified forms in this case possible acid-base interaction is the decisive factor, that is factor B [33 - 35], This confirmation is in good agreement with the above-said bigger molecules harder introduce... 

To apply the correlation of Figure 10 for prediction of the salting out of carbon dioxide by ammonium hydrosulfide and bicarbonate solutions we need to correct for the differences of their partial molar volumes from that of sodium chloride. Partial molar volumes were obtained from Ellis and McFadden(53). Volume change of the hydrosulfide and bicarbonate are equal within 0.2 cm3/mol at temperatures up to 100 C and differ very little at still higher temperatures thus, we assume that the changes with temperature of the salting-out coefficients of the two salts are equal up to... 

Partial molar volumes at infinite dilution were adopted from the correlation of Brelvi and O Connell (20). (In the pressure range regarded here (p 100 atm) Poynting corrections are very small and can be neglected for all electrolytes as well as for water (eqs. E1, E2) ). ... 

An important attribute of Equation 5.16 is that the pressure exerted on both phases, Ptot, is common to both isotopomers. The important difference between Equations 5.16 and 5.9 is that the isotopic vapor pressure difference (P/ — P) does not enter the last two terms of Equation 5.16 as it does in Equation 5.9. Also isotope effects on the second virial coefficient AB/B = (B — B)/B and the condensed phase molar volume AV/V are significantly smaller than those on AP/P ln(P7P). Consequently the corrections in Equation 5.16 are considerably smaller than those in Equations 5.9 and 5.10, and can sooner be neglected. Thus to good approximation ln(a") is a direct measure of the logarithmic partition function ratio ln(Qv Q7QvQcO> provided the pressure is not too high, and assuming ideality for the condensed phase isotopomer solution. For nonideal solutions a modification to Equation 5.16 is necessary. 

AV is then the excess molar volume of products over that of reactants, in their standard states. For dilute solutions, where activity corrections may be neglected, and where Kx is expressed in mole fraction units... 

Although Equations (8.28) and (8.32) are formally alike, they refer to different types of processes. The former is strictly true for a process that occurs at a constant pressure throughout a temperature range. Vaporization or sublimation does not fulfill this restriction, but nevertheless. Equation (8.32) is approximately correct because the molar volume of the condensed phase is small compared with that of the gas, and the vapor pressure is small enough that the vapor behaves as an ideal gas. 

As shown by Helgeson et al. (1978), satisfactory estimates of standard state molar entropy for crystalline solids can be obtained through reversible exchange reactions involving the compound of interest and an isostructural solid (as for heat capacity, but with a volume correction). Consider the generalized exchange reaction... 

A direct quantitative comparison between AG theory and measurements requires the resolution of two issues. First, the excess entropy Sexc must be normahzed by the molar volume. We suggest that the lack of this normalization is partly responsible for previous claims [15, 49] that AG theory breaks down for small molecule fluids. Second, the vibrational contribution to which is absent in s, must be subtracted reliably. While the first correction can readily be introduced, the inclusion of the second correction requires further investigation [63, 240]. 

The above procedure is now applied to two ethanol-water (8, 9) and five 1-propanol-water systems (9) which have been saturated with an inorganic salt and which show partial miscibility. The vapor pressures and molar volumes (10), and second virial coefficients of water (11), ethanol (12), and 1-propanol (IS) were obtained by interpolation of literature data. The vapor pressures of water saturated with salts over a temperature range are available for all salts (14) except lead nitrate. Such data are unavailable for both alcohols saturated with salt. Hence a correction to the saturation vapor pressure is made by multiplying by the ratio of the vapor pressure of alcohol saturated with salts to the vapor pressure... 

It only remains to replace the uncorrected equation (2.14) by substituting the corrections expressed by (2.19) and (2.25). When these substitution are made, we obtain the Van der Waals equation (2.13), which can also be expressed in terms of molar volume Vm=V/n as... 

Peneloux et al. [35] have introduced a clever method of improving the saturated liquid molar volume predictions of a cubic equation of state, by translating the calculated volumes without efffecting the prediction of phase equilibrium. The volume-translation parameter is chosen to give the correct saturated liquid volume at some temperature, usually at a reduced temperature Tr = T/Tc = 0.7, which is near the normal boiling point. It is possible to improve the liquid density predictions further by making the translation parameter temperature dependent. 

The constant b is regarded as the correction to the molar volume due to the volume occupied by the molecules. Constants a and b are characteristic of the particular gas, whereas R is the universal gas constant. 

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