Van der Waals molar volume

It is interesting to note that such molar volumes can be calculated also by group increments using the van der Waals molar volume for a carbon atom, which is 6.95 ml/mol (Van Krevelen, 1990). 

The Arw and AVW are the van der Waals radius of the atom of substituents directly bound to the ring and the van der Waals molar volume of substituents, respectively, those of the chlorine substituent being taken as reference. 

Aroma compound Molecular weight (g/mol) Experimental van der Waals molar volume molar volume 25 °C (Bondi, 1968) (mL/mol) (mL/mol) Saturated Vapor Pressure 25 °C (Pa) Structure... 

Tc, numbers of adsorbent sites occupied by the solute and solvent, respectively, and V and van der Waals molar volumes of solute and solvent, respectively Kf is the ratio of the equilibrium concentrations of the solute in the stationary and mobile phases at infinite dilution of the solute ... 

The openness of a liquid measures its free volume, which is the difference between its bulk molar volume and its intrinsic molar volume. The latter can be expressed as the van der Waals volume of its molecules per mole. A liquid cannot be compressed by an external pressure to have a lower molar volume. The van der Waals molar volume of water is Vydw = 12.4 cm mol at 25 °C. Another measure of the intrinsic volume of a liquid takes into account the exclusion volume adjacent to a given molecule, where another particle cannot penetrate because the crevices between the peripheral atoms are smaller than any atomic size. This is the McGowan intrinsic volume, which for water is Vx = 16.7 cm mol (Abraham and McGowan 1987). Both of these measures of the intrinsic molar volume of water are considerably smaller than its molar volume at 25 °C, V = 18.07cm moK (Marcus 1998)leaving some free volume. The fractions of free volume according to these two measures are (1 - Kdw/V) = 0.314 and (1 — Vx/10 = 0.076. The former is less relevant for the discussion of solutions, because it does not allow for the exclusion volume. 

The molar volume P(T) is inter-and extrapolated between the molar volume v, of the adsorptive at the normal boiling point and the van der Waals molar volume... 

FIGURE 1.3 Comparison of ideal gas and van der Waals molar volume. 

EVtype descriptors -> van der Waals excluded volume method EVwhole descriptors - van der Waals excluded volume method excess electron polarizability -> electric polarization descriptors excess molar refractivity -> molar refractivity expanded distance Cluj matrices -> expanded distance matrices expanded distance indices -> expanded distance matrices expanded distance matrices... 

Observed stability constants of metal ion-acetato complexes Calculated constant of metal ion complexes with NH3 Enthalpy of formation of sulfides Stability constant of metal ion complexes with sulfate Average molar metal concentration in soil at 10% moisture Medians of elemental composition of soils Calculated mean of the elemental content in land plants Intrinsic (van der Waals) molecular volume... 

In 1873, van der Waals [2] first used these ideas to account for the deviation of real gases from the ideal gas law P V= RT in which P, Tand T are the pressure, molar volume and temperature of the gas and R is the gas constant. Fie argried that the incompressible molecules occupied a volume b leaving only the volume V- b free for the molecules to move in. Fie further argried that the attractive forces between the molecules reduced the pressure they exerted on the container by a/V thus the pressure appropriate for the gas law isP + a/V rather than P. These ideas led him to the van der Waals equation of state ... 

Although later models for other kinds of systems are syimnetrical and thus easier to deal with, the first analytic treatment of critical phenomena is that of van der Waals (1873) for coexisting liquid and gas [. The familiar van der Waals equation gives the pressure p as a fiinction of temperature T and molar volume F,... 

The problem with figure A2.5.6 and figure A2.5.7 is that, because it extends to infinity, volume is not a convenient variable for a graph. A more usefiil variable is the molar density p = 1 / V or the reduced density p. = 1 / Fj. which have finite ranges, and the familiar van der Waals equation can be transfonned into an alternative although relatively unfamiliar fonn by choosing as independent variables the chemical potential p and the density p. 

The molar volume is usually larger than the van der Waals volume because two additional influences must be added. The first is the amount of empty space in the bulk material due to constraints on how tightly together the chains can pack. The second is the additional space needed to accommodate the vibrational motion of the atoms at a given temperature. 

The virial equations are unsuitable forhquids and dense gases. The simplest expressions appropriate (in principle) for such fluids are equations cubic in molar volume. These equations, inspired by the van der Waals equation of state, may be represented by the following general formula, where parameters b, 9 5, S, and Tj each can depend on temperature and composition ... 

In this approach, connectivity indices were used as the principle descriptor of the topology of the repeat unit of a polymer. The connectivity indices of various polymers were first correlated directly with the experimental data for six different physical properties. The six properties were Van der Waals volume (Vw), molar volume (V), heat capacity (Cp), solubility parameter (5), glass transition temperature Tfj, and cohesive energies ( coh) for the 45 different polymers. Available data were used to establish the dependence of these properties on the topological indices. All the experimental data for these properties were trained simultaneously in the proposed neural network model in order to develop an overall cause-effect relationship for all six properties. 

Experimental data for Van der Waals volumes Molar volumes Heat capacities Solubility parameter and glass transition temperature... 

Figure 25 ANN model (5-8-6) training and testing results for van der Waals volume, molar volume, heat capacity, solubility parameter, and glass transition temperature of 45 different polymers.

Since Eqs. (5) and (6) are not restricted to the vapor phase, they can, in principle, be used to calculate fugacities of components in the liquid phase as well. Such calculations can be performed provided we assume the validity of an equation of state for a density range starting at zero density and terminating at the liquid density of interest. That is, if we have a pressure-explicit equation of state which holds for mixtures in both vapor and liquid phases, then we can use Eq. (6) to solve completely the equations of equilibrium without explicitly resorting to the auxiliary-functions activity, standard-state fugacity, and partial molar volume. Such a procedure was discussed many years ago by van der Waals and, more recently, it has been reduced to practice by Benedict and co-workers (B4). 

SCHEME 2. Van der Waals volumes, partial molar volumes and packing coefficients... 

TABLE 6. Comparison between molar volumes V, van der Waals volumes Vw (cm3 mol 1) and packing coefficients t) for selected examples of acyclic and cyclic ground and transition states... 

All volumes are given in cm3 mol-1. The structural parameters necessary for the calculation of the van der Waals volume for the transition state (TS) were taken from ab initio calculations159,160. The partial molar volume for the TS was calculated from the equation ... 

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