Molar volume definition

Solubility parameters are generally tabulated, together with the corresponding liquid molar volumes, only at 25°C. Although solubility parameters are themselves temperature-dependent, the combination of quantities in Eq. 70 is not. Differentiating Eq. 70 with respect to temperature gives — the excess entropy, a quantity which has been assumed to be zero in accord with the definition of a regular solution. Thus only data at 25°C are needed. Solubility parameters may be... 

Volume is an extensive property. Usually, we will be working with Vm, the molar volume. In solution, we will work with the partial molar volume V, which is the contribution per mole of component i in the mixture to the total volume. We will give the mathematical definition of partial molar quantities later when we describe how to measure them and use them. Volume is a property of the state of the system, and hence is a state function.1 That is... 

A comparahve analysis of coefficients and descriptors clarifies the relationship between lipophilicity and hydrophobicity (Y in Eq. 4 is the molar volume which assesses the solute s capacity to elicit nonpolar interactions (i.e. hydrophobic forces) which, as also clearly stated in the International Union of Pure and Applied Chemistry definitions [3] are not synonyms but, when only neutral species are concerned, may be considered as interchangeable. In the majority of partitioning systems, the lipophilicity is chiefly due to the hydrophobicity, as is clearly indicated by the finding that the product of numerical values of the descriptors V and of the coefficient v is larger in absolute value than the corresponding product of other couples of descriptors/coefficients [9]. This explains the very common linear rela-... 

As an alternative to the use of Fiery s interaction parameter, we intend to express AH based on the solubiHty parameter, 6, being slightly hmited in accuracy but superior in simpHcity, needing only pen and paper. According to its definition, 6 is equal to the square root of the molar energy of vaporization, AE per unit molar volume, thus [79]... 

By definition, molar volume V, liquid density plL, and molar mass Mi are related by... 

Another way to achieve uniform illumination is to lower the optical density of the sample by increasing the effective molar volume. Solid solution of an absorbing guest in a transparent host in either a normal crystal or a molecular inclusion compound can achieve this end so can using a pure crystal of a much larger molecule which includes the chromophore of interest [27]. While it is harder to reach high dilution with the latter approach, it has the virtue that the initial compound is a pure crystal and thus better suited for definitive X-ray investigation. 

It follows from the definition of the molar volume that... 

It is, therefore, not surprising that there exists a definite relationship between Aand the enthalpy of vaporization, Av H, the former constituting a fraction between 0.2 and 0.3 of the latter, as is readily obtained from the data in Tables 3.1 and 3.9. The pressure dependence of the viscosity is also closely related to the free volume of the solvent. The fluidity (O = l/r ) is proportional to the ratio between the free and the occupied volume, the former, as mentioned above, being the difference between the actual molar volume and the intrinsic molar volume (Tables 3.1 and 3. 4) (Hildebrand 1978). In fact, the logarithm of the viscosity of liquids was found (Marcus 1998) to be described well for some 300 liquids by the empirical relationship ... 

The partial molar properties are not measured directly per se, but are readily derivable from experimental measurements. For example, the volumes or heat capacities of definite quantities of solution of known composition are measured. These data are then expressed in terms of an intensive quantity—such as the specific volume or heat capacity, or the molar volume or heat capacity—as a function of some composition variable. The problem then arises of determining the partial molar quantity from these functions. The intensive quantity must first be converted to an extensive quantity, then the differentiation must be performed. Two general methods are possible (1) the composition variables may be expressed in terms of the mole numbers before the differentiation and reintroduced after the differentiation or (2) expressions for the partial molar quantities may be obtained in terms of the derivatives of the intensive quantity with respect to the composition variables. In the remainder of this section several examples are given with emphasis on the second method. Multicomponent systems are used throughout the section in order to obtain general relations. 

Apparent molar quantities for multicomponent systems, other than binary systems, have been used to only a limited extent because of the difficulty of dealing with more than two mole numbers. Actually, there is no single definition of an apparent molar quantity for such systems. If we limit the discussion to the volume and refer to the definition of the apparent molar volume of the binary system (Eq. 6.53)), then we see that for multicomponent systems we must consider the difference between the volume of the solution and some other volume, and the meaning of the molar quantity. There appear to be two possible cases for these definitions. In the first case we could choose... 

The measurement of partial molar quantities will be illustrated with reference to partial molar volumes. We can measure absolute volumes of solution and, thus, can determine partial molar volumes directly from its definition ... 

The concept of free volume varies on how it is defined and used, but is generally acknowledged to be related to the degree of thermal expansion of the molecules. When liquids with different free volumes are mixed, that difference contributes to the excess functions (Prausnitz et al., 1986). The definition of free volume used by Bondi (1968) is the difference between the hard sphere or hard core volume of the molecule (Vw= van der Waals volume) and the molar volume, V ... 

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-... 

For the formulation of a dimensionless adsorption equilibrium constant the definition of a standard state is crucial. The standard state is freely selectable, regardless of the possibility of its physical realization. It is defined according to its expediency. The standard state of adsorption is assumed to be the ratio of a standard molar volume to the standard molar surface [4],... 

Partial molar entropies of ions can, for example, be calculated assuming S (H+) = 0. Alternatively, because K+ and Cl ions are isoelectronic and have similar radii, the ionic properties of these ions in solution can be equated, e.g. analysis of B-viscosity coefficients (Gurney, 1953). In other cases, a particular theoretical treatment which relates solvation parameters to ionic radii indicates how the subdivision could be made. For example, the Bom equation requires that AGf (ion) be proportional to the reciprocal of the ionic radius (Friedman and Krishnan, 1973b). However, this approach involves new problems associated with the definition of ionic radius (Stem and Amis, 1959). In another approach to this problem, the properties of a series of salts in solution are plotted in such a way that the value for a common ion is obtained as the intercept. For example, when the partial molar volumes of some alkylammonium iodides, V (R4N+I ) in water (Millero, 1971) are plotted against the relative molecular mass of the cation, M+, the intercept at M + = 0 is equated to Ve (I-) (Conway et al., 1966). This procedure has been used to... 

For an ideal mixture, the enthalpy of mixing is zero and so a measured molar enthalpy of mixing is the excess value, HE. The literature concerning HE -values is more extensive than for GE-values because calorimetric measurements are more readily made. The dependence of HE on temperature yields the excess molar heat capacity, while combination of HE and GE values yields SE, the molar excess entropy of mixing. The dependences of GE, HE and T- SE on composition are conveniently summarized in the same diagram. The definition of an ideal mixture also requires that the molar volume is given by the sum, Xj V + x2 V2, so that the molar volume of a real mixture can be expressed in terms of an excess molar volume VE (Battino, 1971). 

When we refer to the volume of a gas, we are in fact talking about the volume of the container. The definition of the volume of a gas is the space available for gas molecules to move around in. The kinetic molecular theory of gases assumes that the volume of each gas molecule is essentially zero. Thus, the amount of space for them to move around in is the volume of the container. For all gases, Vgas = the volume of the container holding the gas. (Do not confuse this with the molar volume of gases. You will learn about molar volume in Chapter 12.)... 

Chen-Chen [7] Their correlation was based on diffusion measurements of 50 combinations of conditions with 3 to 4 replicates each and exhibited an average error of 6 percent. In this correlation, Vr = VB/[0.9724 (Vrf, + 0.04765)] and V B = the liquid molar volume at the melting point, as discussed previously. Their association parameter P [which is different from the definition of that symbol in Eq. (5-229)] accounts for hydrogen bonding of the solvent. Values for acetonitrile and methanol are P = 1.58 ana 2.31, respectively. 

This equation has been obtained by combining Eq. (1-8) with a similar equation in which T is replaced by Tg, which is the temperature at the ice point (273.15 K). For definitions of the quantities in this expression, see Exp. 1 quantities with subscript zero are those corresponding to Tq. The volume V of the thermometer bulb is not to be confused with V == RTo po, which is the molar volume of the gas employed. The direct partial... 

PARTIAL MOLAR VOLUMES OF IONS IN SOLUTION 2.6.1. Definition... 

By the general definition (99) of statistical averaging, in dealing with dense systems we have to include the perturbed potential energy for the system as a whole, i.e. to perform a summation of the energies (121) over all the molecules in the molar volume... 

We begin with the definition of the molar volume of a mixture by writing V = V/J2ini- On introducing mole fractions we then convert (1.19.2) to the form... 

Let us consider an ideal ternary mixture. According to the definition of an ideal mixture (18), the activities of the components ( i) are equal to their mol fractions (xi) and their partial molar volumes are equal to those of the pure components (V = Tf). 

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