Normalized molar volume

If a Type IV isotherm has a distinctive plateau, which cuts the p° axis at an angle 90°, we may generally arrive at an acceptable assessment of the total mesopore volume vp. The amount adsorbed, n(sat), at the plateau is a measure of the adsorption capacity and to obtain vp it is assumed that die adsorbate has the normal molar volume, Vlm, of the liquid at die operational temperature. This simple method for the determination of the pore volume is based on a general principle, which was put forward 80 years ago by Gurvich (1915) and is still known as the Gurvich rule. 

An overview of some basic mathematical techniques for data correlation is to be found herein together with background on several types of physical property correlating techniques and a road map for the use of selected methods. Methods are presented for the correlation of observed experimental data to physical properties such as critical properties, normal boiling point, molar volume, vapor pressure, heats of vaporization and fusion, heat capacity, surface tension, viscosity, thermal conductivity, acentric factor, flammability limits, enthalpy of formation, Gibbs energy, entropy, activity coefficients, Henry s constant, octanol—water partition coefficients, diffusion coefficients, virial coefficients, chemical reactivity, and toxicological parameters. 

For both polar and nonpolar nonhydrocaihon gaseous mixtui es at low pressui es, the most accurate viscosity prediction method is the method of Brokaw. The method is quite accurate but requires the dipole moment and the Stockmayer energy parameter (e/A ) for polar components as well as pure component viscosities, molecular weights, the normal boding point, and the hq-uid molar volume at the normal boding point. The Technical Data Manual should be consulted for the fidl method. 

Component 1 is the solute, while component 2 is the solvent. Units of T, P, and V are °R, psia, and cmVgmole, respectively. Diffusivity is then in ftVhr. The molar volumes Vi and Vo at the normal boiling point... 

Component 1 is the diffusing gas, while component 2 is the solvent. The solvent viscosity l2 in Pa sec, the solute molar volume at the normal boihng point Vi in mVkmole, and the solvent association parameter Xo multiplied by the solvent... 

For predic ting diffiisivities in binary polar or associating liquid systems at liign solute dilution, the method of Wilke and Chang " defined in Eq. (2-156) can be utilized. The Tyn and Cains equation (2-152) can be used to determine the molar volume of the solute at the normal boihng point. Errors average 20 percent, with occasional errors of 35 percent. The method is not considered to be accurate above a solute concentration of 5 mole percent. 

Component 1 is the solute, while component 2 is water. The molar volume of the solute in mVkmole is at the solute normal boiling point, while the viscosity of water in Pa sec is at the temperature of the system resulting in a diffusivity in mVsec. The average error is about 9 percent when tested on 36 experimental systems. 

V Molar volume of i at its normal boiling point mVkmol or cmVmol ftVlbmol... 

In addition to deciding on the method of normalization of activity coefficients, it is necessary to undertake two additional tasks first, a method is required for estimating partial molar volumes in the liquid phase, and second, a model must be chosen for the liquid mixture in order to relate y to x. Partial molar volumes were discussed in Section IV. This section gives brief attention to two models which give the effect of composition on liquid-phase thermodynamic properties. 

In their correlation, Chao and Seader use the original Redlich-Kwong equation of state for vapor-phase fugacities. For the liquid phase, they use the symmetric convention of normalization for y and partial molar volumes which are independent of composition, depending only on temperature. For the variation of y with temperature and composition, Chao and Seader use the equation of Scatchard and Hildebrand for a multicomponent solution ... 

The difficulties encountered in the Chao-Seader correlation can, at least in part, be overcome by the somewhat different formulation recently developed by Chueh (C2, C3). In Chueh s equations, the partial molar volumes in the liquid phase are functions of composition and temperature, as indicated in Section IV further, the unsymmetric convention is used for the normalization of activity coefficients, thereby avoiding all arbitrary extrapolations to find the properties of hypothetical states finally, a flexible two-parameter model is used for describing the effect of composition and temperature on liquid-phase activity coefficients. The flexibility of the model necessarily requires some binary data over a range of composition and temperature to obtain the desired accuracy, especially in the critical region, more binary data are required for Chueh s method than for that of Chao and Seader (Cl). Fortunately, reliable data for high-pressure equilibria are now available for a variety of binary mixtures of nonpolar fluids, mostly hydrocarbons. Chueh s method, therefore, is primarily applicable to equilibrium problems encountered in the petroleum, natural-gas, and related industries. 

The apparent and partial molar volumes of aggregated sodium octyl, decyl, dodecyl, and tetradecyl sulfate molecules have been studied in detail by Vass et al. [144] from densities measured by a vibrating capillary densitometer in normal and 99.85% heavy water at 25°C and by Vass [130] from density, small-angle scattering, and positron annihilation measurements. 

There may be more than a thousandfold increase in volume when liquids or solids react to form a gas. The molar volumes of gases are close to 25 L-mol 1 under normal conditions (room temperature and pressure), whereas liquids and solids occupy only about a few tens of milliliters per mole. The molar volume of liquid water, for instance, is only 18 mL-mol 1. In other words, 1 mol of gas molecules at 25°C and 1 atm occupies as much as 1000 times the volume of 1 mol of molecules in a typical liquid or solid. 

An approximate estimate of the density at the normal boiling point can be obtained from the molar volume (see Table 8.6)... 

From the viewpoint of developing quantitative correlations it is desirable to seek a linear relationship between descriptor and property, but a nonlinear or curvilinear relationship is adequate for illustrating relationships and interpolating purposes. In this handbook we have elected to use the simple descriptor of molar volume at the normal boiling point as estimated by the Le Bas method (Reid et al. 1987). This parameter is very easily calculated and proves to be adequate for the present purposes of plotting property versus relationship without seeking linearity. 

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. 

Most substances have a slightly smaller molar volume as a solid than as a liquid substances generally decrease in volume when they freeze. Water is the notable exception it expands (increases in molar volume) upon freezing. That is why closed containers holding water and other aqueous solutions will break in freezing weather. Although iron is much more dense overall than the compounds in Table 12-1, and its molar volumes are much less than those of benzene and carbon tetrachloride, iron still behaves normally. The molar volume of liquid iron is still greater than that of solid iron. 

Figure 6.8 Surface tension of fused salts as a function of melting temperature normalized with the molar volume TvapVm2n [14, 15].

Va Molar volume of dissolved gas A at its normal boiling temperature m3 mol 1... 

Vb = liquid molar volume at the normal boiling point, can be found fr molecular contributions. 

Compressing ammonia gas under high pressure forces the molecules into close proximity. In a normal gas, the separation between each molecule is generally large - approximately 1000 molecular diameters is a good generalization. By contrast, the separation between the molecules in a condensed phase (solid or liquid) is more likely to be one to two molecular diameters, thereby explaining why the molar volume of a solid or liquid is so much smaller than the molar volume of a gas. 

Table 12.5 Molar volume isotope effects for some liquids, mostly at or below their normal boiling...

To model the solubility of a solute in an SCF using an EOS, it is necessary to have critical properties and acentric factors of all components as well as molar volumes and sublimation pressures in the case of solid components. When some of these values are not available, as is often the case, estimation techniques must be employed. When neither critical properties nor acentric factors are available, it is desirable to have the normal boiling point of the compound, since some estimation techniques only require the boiling point together with the molecular structure. A customary approach to describing high-pressure phenomena like the solubility in SCFs is based on the Peng-Robinson EOS [48,49], but there are also several other EOS s [50]. 

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