Molar critical volume

Included is the density of water as a function of temperature, the vapor pressure, specific heat, partial molar volume, critical properties, and viscosity of water from 0 to 500 °C and to 8 kbar. Also see items [43] and [134]. 

Chemical name (lUPAC) Critical temperature Critical pressure Critical molar volume Critical compress. Molar heat capacity of vapor (Cp/JK mor ) c =a+bT+cT +dT Cp (298K) Antoine s vapor pressure equation with P in mmHg and T in K ln z =A-B/(r-fC) Vapor pressure (293.15K)... 

Chemical name (lUPAC) Critical temperature Critical pressure Critical molar volume Critical compress. Molar heat capacit c =a+bT+cT +dT ... 

Other physical and thermodynamic properties of ethyl lactate, such as molar volume, critical temperature, critical pressure, critical volume, heat of vaporization, and liquid heat capacity, among others, can be found in the work reported by Pereira and coworkers. 20.4.1.2.1 Ethyl lactate vapor pressure... 

The critical molar volume is defined using the acentric factor by the following, relations . 

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,... 

Unlike the pressure where p = 0 has physical meaning, the zero of free energy is arbitrary, so, instead of the ideal gas volume, we can use as a reference the molar volume of the real fluid at its critical point. A reduced Helmlioltz free energy in tenns of the reduced variables and F can be obtained by replacing a and b by their values m tenns of the critical constants... 

The critical pressure, critical molar volume, and critical temperature are the values of the pressure, molar volume, and thermodynamic temperature at which the densities of coexisting liquid and gaseous phases just become identical. At this critical point, the critical compressibility factor, Z, is ... 

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. 

Deterioration of electrode performance due to corrosion of electrode components is a critical problem. The susceptibility of MHt electrodes to corrosion is essentially determined by two factors surface passivation due to the presence of surface oxides or hydroxides, and the molar volume of hydrogen, VH, in the hydride phase. As pointed out by Willems and Buschow [40], VH is important since it governs alloy expansion and contraction during the charge-discharge cycle. Large volume changes... 

Chueh s method for calculating partial molar volumes is readily generalized to liquid mixtures containing more than two components. Required parameters are and flb (see Table II), the acentric factor, the critical temperature and critical pressure for each component, and a characteristic binary constant ktj (see Table I) for each possible unlike pair in the mixture. At present, this method is restricted to saturated liquid solutions for very precise work in high-pressure thermodynamics, it is also necessary to know how partial molar volumes vary with pressure at constant temperature and composition. An extension of Chueh s treatment may eventually provide estimates of partial compressibilities, but in view of the many uncertainties in our present knowledge of high-pressure phase equilibria, such an extension is not likely to be of major importance for some time. 

In the past, it has been customary to assume that partial molar volumes depend only on temperature and are independent of composition and pressure (Cl, P13). This assumption is very poor in the critical region. Primarily... 

For components near or above their critical temperatures, the liquid volume t>j was evaluated by extrapolation with respect to temperature. For supercritical components, the fugacity f° was also evaluated by extrapolation the effect of pressure was found from the Poynting relation using the previously extrapolated liquid molar volumes. 

Chao and Seader assume that the partial molar volumes are independent of composition this assumption is equivalent to saying that at constant temperature and pressure there is no volume change upon mixing the pure liquid components, be they real or hypothetical. The term on the right-hand side of Eq. (46) is assumed to be zero for all temperatures, pressures, and compositions. This assumption is very poor near critical conditions, and is undoubtedly the main reason for the poor performance of the Chao-Seader correlation in the critical region. 

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. 

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... 

Figure 8.4 Graph of temperature against molar volume (a), and density (b). for CO (gas) and C02 (liquid) in the temperature range from the triple point to the critical point. The dashed line in (b) is the average density. The area enclosed within the curves is a two-phase region, with the molar volume or the density of the gas and liquid at a particular temperature given by the horizontal (dotted) tie-lines connecting the gas and liquid sides of the curve.

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