Molar volume basic value

Here we will use a simplified example to illustrate some basic aspects of the mass transport process for carbonates that avoids most of the more complex relationships. In this example, the calcium and carbonate ion concentrations are set equal, and values of the activity coefficients, temperature, and pressure are held constant. The carbonate ion concentration is considered to be independent of the carbonic acid system. The resulting simple (and approximate) relation between the change in saturation state of a solution and volume of calcite that can be dissolved or precipitated (Vc) is given by equation 7.4, where v is the molar volume of calcite. 

In this equation, R, is the excess molar refraction iTs is the dipolarity/polarizability 2 and E jSp. are the summation of hydrogen bond acidity and basicity values, respectively and Vx is McClowan s volume. 

We use the value of at infinite dilution (c —> 0) as the basic value of the molar volume of a dissolved substance, meaning the volume demand of the substance in the practically pure solvent. The basic value at standard temperature... 

Often the interactions between small spherical atoms and some (rotationaUy-averaged) molecules similarly depend on the relative interparticle separation the Lennard-Jones interaction [12] reasonably describes their pair interachon (see [13] for an in-depth critical discussion) for quite a number of substances. For these systems the phase diagrams scaled by the critical values of temperature, pressure and molar volume appear similar as well. The fact that the thermodynamic properties of all simple gases exhibit basic similarities is expressed by the law of corresponding states of Van der Waals. A statistical mechanical derivation of this law was provided by Pitzer [14]. 

The heat capacities that have been discussed previously refer to closed, single-phase systems. In such cases the variables that define the state of the system are either the temperature and pressure or the temperature and volume, and we are concerned with the heat capacities at constant pressure or constant volume. In this section and Section 9.3 we are concerned with a more general concept of heat capacity, particularly the molar heat capacity of a phase that is in equilibrium with other phases and the heat capacity of a thermodynamic system as a whole. Equation (2.5), C = dQ/dT, is the basic equation for the definition of the heat capacity which, when combined with Equation (9.1) or (9.2), gives the relations by which the more general heat capacities can be calculated. Actually dQ/dT is a ratio of differentials and has no value until a path is defined. The general problem becomes the determination of the variables to be used in each case and of the restrictions that must be placed on these variables so that only the temperature is independent. 

The volume is rather a basic concept of a region, as it appears in the derivation of all kinds of equations of continuity. For this reason, we introduce quantities that refer to the unit of volume. We will use the superscript X in order to indicate that this is X per volume. Remember that we are using C for the molar heat capacity (at constant volume), Cv for the specific heat capacity, C for a reduced heat capacity, i.e., divided by a reference value, and now C for the heat capacity density. The quantities X are basically X densities, for example, the mass density or the molar density ... 

Conductance measurements on dilute solutions are of special interest for electrolyte theory. These measurements can be carried out at high precision for almost all electrolytes in almost all solvents at various temperatures and pressures and thus provide an efficient method for determining the basic data of electrolyte solutions, i.e. A , and R, under various conditions. Values of and R are found to be compatible with the values obtained from thermodynamic methods. The enthalpies and volumes of ion-pair formation, AH and AV, as determined from temperature- and pressure-dependence of conductance, are compatible with the corresponding relative apparent molar quantities, ii (IP) and Ov (IP), from thermodynamic measurements, cf. Section 5.2. R-vahies are found to be almost independent of temperature. 

As a state property, the molar (or specific) volume can be determined once as a function of pressure and temperature, and tabulated for future use. Tabulations have been compiled for a large number of pure fluids. In very common use are the steam tables, which contain tabulations of the properties of water. Steam is a basic utility in chemical plants as a heat transfer fluid for cooling or heating, as well as for power generation (pressurized steam), and its properties are needed in many routine calculations. Thermodynamic tables for water are published by the American Society of Mechanical Engineers (ASME) and are available in various forms, printed and electronic. A copy is included in the appendix. We will use them not only because water is involved in many industrial processes but also as a demonstration of how to work with tabulated values in general. 

Volume fractions imply a temperature dependence and, as they are defined in equation (38), neglect excess volumes of mixing and, very often, the densities of the copolymer in the slate of the solution are not known correctly. However, volume fractions can he calculated without the exact knowledge of the copolymer molar mass (or its averages). Base mole fractions are seldom applied for copolymer systems. The value for A o, the molar mass of a basic unit of the copolymer, has to he determined according to the corresponding average chemical composition. Sometimes it is chosen arbitrarily, however, and has to he specified. 

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