Volume, excess partial molar

The new reference state reduces to GV in the limiting case of an ideal mixture, but also satisfies the volume conservation condition. The following differences exist between the ML and SR excesses the ML excesses have non-zero values if either the partial molar volumes differ from the ideal ones or D = + Xi d a.yi/dxi)pj 7 1, where P represents the pressure and y- is the activity coefficient of component v, the SR excesses have non-zero values only if D 1. The present reference state is a hypothetical one similar to the ideal state, in which the molar volume, the partial molar volumes and the isothermal compressibility are the real ones. 

In obtaining the last expressionln Eq. 9.3-20 we have assumed that the excess partial molar volume is independent of pressure. (Note that although Eqs. 9.3-18 and 9.3-19 are correct, they are difficult to use in practice since the activity coefficient description is applied to fluid mixtures not well described by an equation of state.) Next, taking the, temperature derivative of Eq. 9.3-18 at constant pressure and composition, we obtain... 

The excess partial molar entropy, enthalpy, and volume can now be obtained from familiar partial derivatives of free energy as follows ... 

These results are plotted in Figure 12-2. The excess partial molar volumes are indicated by the arrows and may be obtained by the graphical method. For a more accurate determination, the experimental excess volume is fitted to a polynomial, and the excess partial molar volumes are obtained by application of eqs. ri2.8 ) and fi2.8 ). The fitted polynomial for is... 

Fig. 1.6 Excess partial molar volumes of water in water-rich mixtures at 25 °C with dimethylsulfoxide,...

The subsequent step is the calculation of the quantities that appear in the matrix elements A-. By means of Equation 4.15 through Equation 4.18, all chemical potential derivatives are calculated. The partial molar volumes are obtained by adding to the molar volume of the pure components the excess partial molar volumes, obtained through Equation 4.21 with = V, by applying the same procedure used to calculate In y. The mixture molar volume is then obtained as = x V + X2V2 + 3 3 and Kj. from Equation 4.5. The values of are finally obtained from Equation 4.12, for which the calculations of the concentrations, the determinant, and the cofactors are all straightforward. 

The scope of the present work remains essentially that of Special Publication 454, The general aim 1 to assist the reader in locating those publications which contain thermochemical data which can best serve his needs. Equilibrium data Is taken in its most general sense and includes equilibrium constants, enthalpies, entropies, heat capacities, volumes, and partial molar and excess property data. To a much lesser extent, transport and other properties have been included. Unfortunately, much of the data on biochemical systems is scattered throughout much of the literature and there is a need for... 

Many of the theories and models described in this section were developed for the excess thermodynamic properties of solutions, including not only the excess partial molar volume, but also other excess properties. In the following subsections we have restricted the discussion to the volumetric properties of aqueous systems. 

Misawa, M., Inamura, Y., Hosaka, D., and Yamamuro, O. (2006). Hydration of alcohol clusters in 1-propanol-water mixture studied by quasielastic neutron scattering and an interpretation of anomalous excess partial molar volume J. Chem. Phys., 125, 6. 

Other measures of the effect of cosolvents on the structure of water do not confirm this general conclusion. The excess partial molar volume of water in water-rich mixtures with cosolvents is a clear indication, if positive, that the bulky, low-density structure of the water is enhanced as argned by Marcus [35], When the excess molar volume of the binary aqueons mixtures with the cosolvents are expressed as a third-order polynomial in the water-rich region, (x <03) = b + + b xl + b xl, the... 

Thermodynamics gives limited information on each of the three coefficients which appear on the right-hand side of Eq. (1). The first term can be related to the partial molar enthalpy and the second to the partial molar volume the third term cannot be expressed in terms of any fundamental thermodynamic property, but it can be conveniently related to the excess Gibbs energy which, in turn, can be described by a solution model. For a complete description of phase behavior we must say something about each of these three coefficients for each component, in every phase. In high-pressure work, it is important to give particular attention to the second coefficient, which tells us how phase behavior is affected by pressure. 

Therefore, if we have information on the partial molar volumes and the excess Gibbs energy of the ternary system, we can use Eqs. (119)—(122) to find the ends of the tie lines which comprise the coexistence curve. 

Here it i s assumed that only excess water causes swelling. The parameter p = y /V is the ratio of partial molar volumes of ionomer molecules and water and v is the number of polar head groups (SO3) per ionomer molecule. 

CALCULATION OF PARTIAL MOLAR QUANTITIES AND EXCESS MOLAR QUANTITIES FROM EXPERIMENTAL DATA VOLUME AND ENTHALPY... 

In this chapter, we shall consider the methods by which values of partial molar quantities and excess molar quantities can be obtained from experimental data. Most of the methods are applicable to any thermodynamic property J, but special emphasis will be placed on the partial molar volume and the partial molar enthalpy, which are needed to determine the pressure and temperature coefficients of the chemical potential, and on the excess molar volume and the excess molar enthalpy, which are needed to determine the pressure and temperature coefficients of the excess Gibbs function. Furthermore, the volume is tangible and easy to visualize hence, it serves well in an initial exposition of partial molar quantities and excess molar quantities. 

The excess molar volumes of 10-40 mol % methanol/C02 mixtures at 26°C as a function of pressure has been determined. The excess molar volumes varied with composition and pressure significant interaction between CO2 and methanol was noted from the observed excess molar volumes. To better characterize the interaction and its effect on analyte solubility, the partial molar volume of naphthalene at infinite dilution in liquid 10 and 40 mol % methanol/C02 mixtures was determined. The variation of the partial molar volume at infinite dilution with pressure correlated well with isothermal compressibility of the methanol/C02 mixtures (Souvignet and Olesik, 1995). 

Here, Av is the activation volume. It is the excess of the partial molar volume of the transition state over the partial molar volume of the initial species, at the composition of the mixture. 

Figure 17.5 Derived thermodynamic properties at T — 298.15 K and p = 0.1 MPa for (2Cic-CfiHi2 + X2n-CjHi4) (a) excess molar heat capacities obtained from the excess molar enthalpies (b) relative partial molar heat capacities obtained from the excess molar heat capacities (c) change of the excess molar volume with temperature obtained from the excess molar volumes and (d) change of the excess molar enthalpies with pressure obtained from the excess molar volumes.

In the grand equilibrium method, a simulation of the condensed phase is done to calculate the excess chemical potentials, /x,ex, and the partial molar volumes, V,-, of all components. One may use the test-particle insertion method [59] to calculate the excess chemical potentials and the partial molar volumes as... 

Equations (XV.5.8) and (XV.5.9) predict a change of rate constant with pressure which will depend logarithmically on the partial molar volume change for the transition-state reaction. An exactly similar equation, the Kelvin equation, can be written for the change of equilibrium constant with pressure. Since AY /RT is of the order of magnitude of 10 atm " for solution reactions, it is evident that the effect of these pressure changes will be of importance only at pressures in excess of 10 atm, and indeed this is verified experimentally. 

The quantities pj are the densities of the pure phases at the specified pressure p, that is, with attractive forces operating to achieve experimental densities the partial molar volumes of these pure phases are i)i = 1/pi. Similarly, the densities of the mixture correspond to the same pressure p. Rememhering Eq. (4.39), a way for this mixing free energy, Eq. (4.41), to achieve the form of the initial two terms of Eq. (4.38) is that the excess quantities within the brackets of Eq. (4.41) vanish ... 

They are used as industrial solvents for small- and large-scale separation processes, and they have unusual thermodynamic properties, which depend in a complicated manner on composition, pressure, and temperature for example, the excess molar enthalpy (fp-) of ethanol + water mixture against concentration exhibits three extrema in its dependence on composition at 333.15 K and 0.4 MPa. The thermodynamic behavior of these systems is particularly intricate in the water-rich region, as illustrated by the dependencies of the molar heat capacity and partial molar volume on composition. This sensitivity of the partial molar properties indicates that structural changes occur in the water-rich region of these mixtures. Of course, the unique structural properties of water are responsible for this behavior. ... 

Ci being parameters provided by [33]. The derivatives of the excess molar volumes in binary and ternary mixtures can be calculated using Eqs. (24)-(26) and thus the partial molar volumes can be obtained. 

The present paper is devoted to the local composition of liquid mixtures calculated in the framework of the Kirkwood—Buff theory of solutions. A new method is suggested to calculate the excess (or deficit) number of various molecules around a selected (central) molecule in binary and multicomponent liquid mixtures in terms of measurable macroscopic thermodynamic quantities, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility, and the partial molar volumes. This method accounts for an inaccessible volume due to the presence of a central molecule and is applied to binary and ternary mixtures. For the ideal binary mixture it is shown that because of the difference in the volumes of the pure components there is an excess (or deficit) number of different molecules around a central molecule. The excess (or deficit) becomes zero when the components of the ideal binary mixture have the same volume. The new method is also applied to methanol + water and 2-propanol -I- water mixtures. In the case of the 2-propanol + water mixture, the new method, in contrast to the other ones, indicates that clusters dominated by 2-propanol disappear at high alcohol mole fractions, in agreement with experimental observations. Finally, it is shown that the application of the new procedure to the ternary mixture water/protein/cosolvent at infinite dilution of the protein led to almost the same results as the methods involving a reference state. 

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