Molar volume: defined, 93: equation

Equation 14.10 is the defining equation for molar volume. Defining equations and their corresponding Per expressions are summarized in Section 3.8. 

This equation is now in a very interesting form. By defining new variables Pr = P/PC (the reduced pressure), Tr = T/Tc (the reduced temperature), and Vmr= Vm/Vmr (the reduced molar volume), the equation takes the form... 

Quantitatively, the number of degrees the freezing point is lowered is directly proportional to the concentration of solute particles when that concentration is expressed in molality. The molality (m) of a solution is not the same as its molarity (M) because molality is based on the mass of solvent (kg) and not the volume of solution (L) as is used in molarity. The defining equation for molality is... 

Herein Pa and Pb are the micelle - water partition coefficients of A and B, respectively, defined as ratios of the concentrations in the micellar and aqueous phase [S] is the concentration of surfactant V. ai,s is fhe molar volume of the micellised surfactant and k and k , are the second-order rate constants for the reaction in the micellar pseudophase and in the aqueous phase, respectively. The appearance of the molar volume of the surfactant in this equation is somewhat alarming. It is difficult to identify the volume of the micellar pseudophase that can be regarded as the potential reaction volume. Moreover, the reactants are often not homogeneously distributed throughout the micelle and... 

In these equations, is the molar volume and 5 is the solubiUty parameter of the pure Hquid k. The solubiUty parameter is the square root of an energy density, defined as... 

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. 

Density is defined as the concentration of matter, measured by the mass per unit volume [1]. The molar volume, Vm, is defined as the volume occupied by 1 mol of a substance. The molar volume of an ideal gas is 22.4140dm3mol-1 (22.4140liter mol-1) at 1 atm pressure and 0°C. Vapor densities pv are derived through rearrangement of the ideal gas law equation as... 

This equation acknowledges that real molecules have size. They have an exclusion volume, defined as the region around the molecule from which the centre of any other molecule is excluded. This is allowed for by the constant b, which is usually taken as equal to half the molar exclusion volume. The equation also recognizes the existence of a sphere of influence around each molecule, an interaction volume within which any other molecule will experience a force of attraction. This force is usually represented by a Lennard-Jones 6-12 potential. The derivation below follows a simpler treatment (Flowers Mendoza 1970) in which the potential is taken as a square-well function as deep as the Lennard-Jones minimum (figure 2a). Its width x is chosen to give the same volume-integral, and defines an interaction volume Vx around the molecule, which will contain the centre of any molecule in the square well. This form of molecular pair potential then appears in the Van der Waals equation as the constant a, equal to half the product of the molar interaction volume and the molar interaction energy. 

For a given set of values for H, V, and n, values of n, n", and n " can be calculated if the molar enthalpies and molar volumes of the three phases can be determined. The molar volumes can be obtained experimentally, but the absolute values of the molar enthalpies are not known. In order to solve this problem, we make use of the concept of standard states. We choose one of the three phases and define the standard state to be the state of the system when all of the component exists in that phase at the temperature and pressure of the triple point. If we choose the triple-primed phase as the standard phase, we subtract nH " from each side of Equation (8.50) and obtain... 

Partial molar quantities can be defined as the change of an extensive variable with respect to the mole number of one component at constant temperature, pressure, electric field, and mole numbers of all other components. Then, with Equations (14.73) and (14.74), the change of the partial molar entropy and partial molar volume with the electric field is given by... 

The volumetric constitutive equations for a chemoporoelastic material can be formulated in terms of the stress S = a,p, it and the strain 8 = e, (, 9, i.e., in terms of the mean Cauchy stress a, pore pressure p, osmotic pressure it, volumetric strain e, variation of fluid content (, and relative increment of salt content 9. Note that the stress and strain are measured from a reference initial state where all the stress fields are equilibrated. The osmotic pressure it is related to the change in the solute molar fraction x according to 7r = N Ax where N = RT/v is a parameter with dimension of a stress, which is typically of 0( 102) MPa (with R = 8.31 J/K mol denoting the gas constant, T the absolute temperature, and v the molar volume of the fluid). The solute molar fraction x is defined as ms/m with m = ms + mw and ms (mw) denoting the moles of solute (solvent) per unit volume of the porous solid. The quantities ( and 9 are defined in terms of the increment Ams and Amw according to... 

Several empirical equations of state (EOS), representing correlations between pressure and molar volume data have been defined, one of which is the Birch-Mumaghan EOS,... 

Liquids and solids are in the condensed state in which chemical substances are very dense and hardly undergo any volume change with changing pressure in the range of ordinary pressures. Let us now consider a condensed system of a pure substance. The coefficient of thermal expansion a and the compressibility (rare defined in terms of the molar volume v by the following two equations, respectively ... 

The aim of this Chapter is the development of an uniform model for predicting diffusion coefficients in gases and condensed phases, including plastic materials. The starting point is a macroscopic system of identical particles (molecules or atoms) in the critical state. At and above the critical temperature, Tc, the system has a single phase which is, by definition, a gas or supercritical fluid. The critical temperature is a measure of the intensity of interactions between the particles of the system and consequently is a function of the mass and structure of a particle. The derivation of equations for self-diffusion coefficients begins with the gaseous state at pressures p below the critical pressure pc. A reference state of a hypothetical gas will be defined, for which the unit value D = 1 m2/s is obtained at p = 1 Pa and a reference temperature, Tr. Only two specific parameters, Tc, and the critical molar volume, VL, of the mono-... 

This equation defines the partial molar volume of species i in solution. It is simply the volumetric response of the system to the addition at constant T and P of a differential amount of species i A partial molar property may be defined in like fashion for each extensive thermodynamic property. Letting M represent the molar value of such a property, we write the general defining equation for a partial molar property as... 

An adsorption step, during which an amount of liquid na is vaporized from the liquid phase (with a molar energy of vaporization Avap ) and adsorbed on the solid surface (with an integral molar adsorption energy A k, precisely defined for an adsorption process at constant volume (cf. Equation (2.59)), at temperature T and for the surface excess concentration r=n°/A. 

To derive these equations, log P (hydrophobic parameter), MR (molar refrac-tivity index), and MV (molar volume) were calculated using software freely available on the internet (wwwlogP.com, www.daylight.com). The first-order valence molecular connectivity index of substituents was calculated as suggested by Kier and Hall [46,47]. In these equations, is cross-vahdated obtained by the leave-one-out jackknife procedure. Its value higher than 0.6 defines the good predictive ability of the equation. The different indicator variables in these equations were defined as follows. 

Partial Molar Quantities Equation (1.16) may be simplified by introducing the quantity Vi called the partial molar volume of the component i defined by ... 

The coefficients of thermal expansion (a) and of compressibility k) are defined in terms of the molar volume v T, p) by the equations... 

The compressibility factor Z can be calculated by using the defining equation Z = PV/ RT), where F is pressure, V is molar volume, R is the gas constant, and T is absolute temperature. Values of F, V, and T for substances listed in Table 2-184 are given in tables in the Thermodynamic Properties section. For the units used in these tables, R is 0.008314472 MPa dmV mol-K). Values at temperatures and pressures other than those in the tables can be generated for many of the substances in Table 2-184 by going to http //webbook. nist.gov and selecting NIST Chemistry WebBook, then Thermo-... 

Equation (46) implies that the fraction of solute crystallized is proportional to the reduction in the partial molar volume of the solvent. Accordingly, the characteristic parameter for the antisolvent crystallization is the relative partial molar volume reduction (RPMVR), which is defined as... 

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