Molar volume characteristic

The molar volume characteristics, V, for petroleum fractions and hydrocarbons can be obtained from the known density at temperature T ... 

The transformation between water and steam can be depicted in the manner shown in Figure 2.3.2, which represents a plot of P vs the molar volume V at various temperatures close to the critical point. The shaded region indicates an excluded domain for example, at F = 15.3 bar and T = 473.2 K water with molar volume Va coexists with steam whose molar volume is V, . No intermediate molar volumes characteristic of a single phase can exist under these conditions. As the pressure is raised beyond 15.3 bar at 473.2 K steam gas at molar volume Vb condenses to form liquid at molar volume Va- The diagram also shows that as the temperature is raised the two volumes approach each other and ultimately coalesce at 647.2 K to a common value Vc- Above that temperature the two phases merge into one fluid phase. 

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. 

The adjustable parameters are related to pure component molar volumes and to characteristic energy differences as following... 

Correctness of the sixth parameter equation (7) and its simplified forms for the generalization of the swelling data was proved for other coals including the Donbas coal [32] at the parameters B and VM- If to apply the equation (7) to the coal extraction data, then the factor of molar volume VM is insignificant, and the connection between quantities of extracted substance (in g/mole of the solvent) and physical-chemical characteristics can be satisfactorily described by fifth parameter equation (6) or by its simplified forms in this case possible acid-base interaction is the decisive factor, that is factor B [33 - 35], This confirmation is in good agreement with the above-said bigger molecules harder introduce... 

Using PCA, Cramer found that more than 95% of the variances in six physical properties (activity coefficient, partition coefficient, boiling point, molar refractivity, molar volume, and molar vaporization enthalpy) of 114 pure liquids can be explained in terms of only two parameters which are characteristic of the solvent molecule (Cramer 111, 1980). These two factors are correlated to the molecular bulk and cohesiveness of the individual solvent molecules, the interaction of which depends mainly upon nonspecific, weak intermolecular forces. 

Note that for the derivation of Eq. 5-22 we have adopted a very simple characteristic atomic volume contribution method estimating Vt (see Box 5.1), which we denote as Vix. Since the various methods commonly used to assess molar volumes yield quite different absolute values (see e.g., Mackay et al., 1992-1997), Vix values in cm3 mol-1 calculated by this method should be used when applying Eq. 5-22. Hence, if, in addition to Vix, p L, nDi, p a and b, are known or can be estimated for a given... 

The constant b is regarded as the correction to the molar volume due to the volume occupied by the molecules. Constants a and b are characteristic of the particular gas, whereas R is the universal gas constant. 

To obtain the relationship between the particle size d and the molar surface s, we consider 1 mole of a finely powdered solid consisting of N uniform particles of equal size. The surface of a single particle is S = fed2, where d denotes an arbitrarily chosen characteristic dimension of the particle. The volume is V = id3. Hence, the molar volume is... 

The anisotropic interaction, characteristic of the intermediate phase i, is most likely to occur when the bulk molar volume is consistent with a preferential volume Vy. In fact, each particle has an optimal local volume for the formation of anisotropic bonds to its neighbours. Changing the particle density, when V 4 Vyt, the anisotropic interactions are only a fraction f of the total, since V is no longer consistent with the possibility that all anisotropic interactions are saturated at the optimal volume. The remaining fraction of bonds, 1-f, occurs in an unfavourable local volume and therefore, they cannot form the anisotropic bonds of the i phase. 

The common characteristics of phase transitions are that the Gibbs energy is continuous. Although the conditions of equilibrium and the continuity of the Gibbs energy demand that the chemical potential must be the same in the two phases at a transition point, the molar entropies and the molar volumes are not. If, then, we have two such phases in equilibrium, we have a set of two Gibbs-Duhem equations, the solution of which gives the Clapeyron equation (Eq. (5.73))... 

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