Partial molar volumes from density measurements

PARTIAL MOLAR VOLUMES FROM DENSITY MEASUREMENTS 

Based on compilations of values of A it can be seen that typical values are within the range of +30 cm3 mol although some organic reactions extend that range to 

However, actual precision can depend on the reaction, the particular method of high pressure kinetics measurement and the quality of the instrument. As noted earlier, measurement of pressure at a precision or accuracy of 1-2% is acceptable, as reaction rate sensitivity to pressure would not be detectable within this range. 

For many inorganic and bioinorganic reactions the volume of activation has been obtained from measurements at 25 °C. It prevails that the volume of activation is not particularly sensitive to temperature change. This is convenient in that values obtained for different reactions at different temperatures in most cases can be compared directly. Differences at various temperatures for a given reaction can be expected to be within the error of each measurement. Clearly this does not apply to any temperature, and a useful guideline could be that + 20 °C from the standard 25 °C of measurement should yield no significant difference in the value of AV. This can be seen from Equation (9). 

When it is experimentally feasible, the temperature dependence of AV can be compared with the pressure dependence of the entropy of activation. The equality of these terms is defined from extending the Maxwell thermodynamic equality  

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It is clear that from the integrated form of Equation (4) the volume of reaction can be obtained if the equilibrium constant can be determined over a range of pressure. If the volume of activation is not experimentally accessible for one of the directions of the reaction, A V can be used to calculate its value. Under certain conditions and with suitable properties of reactants and/or products it may be possible to determine their partial molar volumes, hence allowing development of a volume profile on an absolute volume basis, as noted above. Even if A V can be determined either from the pressure dependence of the equilibrium constant and/or from use of Equation (5), it may be possible to confirm its value by determination of the partial molar volumes from density measurements. The conditions for conducting successful determinations of partial molar volumes are rather stringent and will be described in Section 2. The method depends on measuring the density, d, of several solutions of different concentrations of the reactant or product. The following equation is used to obtain... 

PARTIAL MOLAR VOLUMES FROM DENSITY MEASUREMENTS... 

For aqueous electrolytes the ionic association become important when b is higher than 5, a value typical of a 2 2 electrolyte at room temperature or a 1 1 electrolyte above 300 °C. Thus, the extrapolation of the apparent partial volume of these electrolytes at infinite dilution to obtain the standard partial molar volume is uncertain, because the free ions concentration depends on the stoichiometric electrolyte concentration. For a 2 2 electrolyte, as MgS04, at 25 °C the apparent partial molar volume approach the DHLL value at concentrations bellow 0.01 mol kg (Franks and Smith, 1967) and, at least the density could be measured with a precision of 1 ppm, V° for MgS04 can not be obtained by extrapolation. In this case one can calculate the standard partial molar volume from the known values of standard partial volume of 1 2 and 1 1 electrolytes by using the additivity rule (Lo Surdo et al, 1982) ... 

A similar procedure may be used to obtain the partial molar volumes from measurements of the volume change on mixing, i.e. from the density of the mixture. In this instance the absolute values of the volumes, and V2, of the pure components are known, and it is therefore possible to obtain the absolute values of the partial molar volumes. In some mixtures it may occur that the partial molar volume of a particular component is negative, implying that an... 

The apparent and partial molar volumes of aggregated sodium octyl, decyl, dodecyl, and tetradecyl sulfate molecules have been studied in detail by Vass et al. [144] from densities measured by a vibrating capillary densitometer in normal and 99.85% heavy water at 25°C and by Vass [130] from density, small-angle scattering, and positron annihilation measurements. 

The effect of pressure on chemical equilibria and rates of reactions can be described by the well-known equations resulting from the pressure dependence of the Gibbs enthalpy of reaction and activation, respectively, shown in Scheme 1. The volume of reaction (AV) corresponds to the difference between the partial molar volumes of reactants and products. Within the scope of transition state theory the volume of activation can be, accordingly, considered to be a measure of the partial molar volume of the transition state (TS) with respect to the partial molar volumes of the reactants. Volumes of reaction can be determined in three ways (a) from the pressure dependence of the equilibrium constant (from the plot of In K vs p) (b) from the measurement of partial molar volumes of all reactants and products derived from the densities, d, of the solution of each individual component measured at various concentrations, c, and extrapolation of the apparent molar volume 4>... 

For the mechanistic interpretation of activation volume data for nonsymmetrical electron-transfer reactions, it is essential to have information on the overall volume change that can occur during such a process. This can be calculated from the partial molar volumes of reactant and product species, when these are available, or can be determined from density measurements. Efforts have in recent years focused on the electrochemical determination of reaction volume data from the pressure dependence of the redox potential. Tregloan and coworkers (139, 140) have demonstrated how such techniques can reveal information on the magnitude of intrinsic and solvational volume changes associated with electron-transfer reactions of transition... 

Fig. 2.14 The volume profile for reaction (2.173) at 25 °C, Ref. 170. The partial molar volumes of COj, Co(NHj)5H20 + and CofNHjljOH were measured with a digital density apparatus. Reprinted with permission from U. Spitzer, R. van Eldik and H. Kelm, Inorganic Chemistry, 21, 2821 (1982). (1982) American Chemical Society.

The reaction volume may be of interest in itself, and furthermore its determination can provide a route to the volume of activation in the reverse direction if that parameter is not experimentally accessible and when AV for the reaction in the forward direction is known. As indicated above, AV may be determined from the dependence upon pressure of the equilibrium constant. It may also be obtained under certain circumstances from the partial molar volumes of the reactants and products. Density measurements d are made on several solutions of different concentrations of the reactant(s) and the product(s). The following equation is used to obtain the apparent molar volume of each species, tp, at each molar concentration c. 

In this experiment the partial molar volumes of sodium chloride solutions will be calculated as a function of concentration from densities measured with a pycnometer. 

The composition dependence of the total volume of a solution at constant temperature and pressure is expressed in terms of the partial molar volumes of the solute and the solvent. Since we are concerned with solvation properties, the quantities which we need to discuss are the partial molar volumes in infinite dilution of the solute so that solute-solute interactions make no contribution. In practice, partial molar volumes are obtained indirectly from precise density measurements. The partial molar volumes at infinite dilution of the amino acids are compiled in Table 2 [7]. It is apparent from these data that an approximately linear correlation exists between the partial molar volume and the number of carbon atoms in the backbone. The data indicate volume contributions from the polar head group (NH, COj) and from the CH2 group and to be about... 

Thus, if the dependency of the molar volume of the system on composition is known from the density measurement, it is possible to calculate the partial molar volumes of both the components according to Eqs. (5.13) and (5.14). In the graphic representation they are the intercepts, which for a given composition, cut the tangent of the molar volume versus composition plot on the y axes at X2 = 0 and X2 = 1. 

Experimental values for some of the partial molar quantities can be obtained from laboratory measurements on mixtures. In particular, mixture density measurements can be used to obtain partial molar volumes, and heat-of-mixing data yield information on partial molar enthalpies. Both of these measurements are considered here. In Chapter 10 phase equilibrium measurements that provide information on the partial molar Gibbs energy of a component in a mixture are discussed. Once the partial molar enthalpy and partial molar Gibbs energy are known at the anie temperature, the partial molar entropy can be computed from the relation S-, = (G — H-,)/T. 

So far we have considered only the volume as a partial molar quantity. But calculations involving solutes will require knowledge of all the thermodynamic properties of dissolved substances, such as H, S, Cp, and of course G, as well as the pressure and temperature derivatives of these. These quantities are for the most part derived from calorimetric measurements, that is, of the amount of heat released or absorbed during the dissolution process, whereas V is the result of volume or density measurements. 

The partial molar volume of Ge(CgH5)4 has been determined by measuring the densities of solvents and solutions in dimethylformamide and hexamethylphosphoric trIamide V° = 292.2 + 2.0 and 298.3 + 1.2 cm /mol, respectively. These volumes are nearly equal to the crystal volume, = 300.7+ 3.1 cm /mol, determined by density measurements of freshly prepared crystals in water. From the experimental data and the calculated van der Waals ... 

Figure 6.10 shows activity coefficient derivatives over the whole composition range for experiment from three correlations and the Verlet method. A procedure for experimental data analysis was described by Wooley and O Connell (1991), in which one extracts the isothermal compressibility, partial molar volumes, and activity coefficient derivatives from experimental data. The activity coefficient derivatives are obtained by fitting mixture vapor-liquid equilibrium data to obtain parameters for at least two different models. Wooley and O Connell employed the Wilson, non-random, two liquid (NRTL) and modified Margules (mM) models. Partial molar volumes are obtained from correlations of mixture densities (Handa and Benson 1979). Isothermal compressibilities are either taken from measurements or estimated with... 

In Figure 2.13, the partial molar volumes of n-hexane and toluene derived from thepVTx measurements (Abdulagatov et al, 2001, 2005 Rabezkii et al, 2001 Degrange, 1998) and the values calculated with semiempirical equation developed by Majer et al (1999) and crossover model (Kiselev et al, 2002 and Abdulagatov et al, 2005) are shown as a function of pure solvent (water) density along the various near-critical and supercritical isotherms. 

A V determined from dilatometric or partial molar volume (density) measurements. Theoretical predicted value. 

To evaluate molar masses from equilibrium runs it is necessary to know (very accurately) the value of the partial specific volume of the species. The partial spedfic volumes of proteins and nucleic acids are not dependent on conformation so they can be evaluated from the amino acid or nucleotide composition (Laue et al. 1992). A satisfactory working value for proteins is v = 0.735 10-3 m3 kg-1. Partial specific volumes can also be measured by comparison of equilibrium runs carried out in H20, D20 and H280, using the different densities of the three isotopic forms of water. 

Equation 9.8.22 is needed for sedimentation equilibrium, a method of determining the molar mass of a macromolecule. A dilute solution of the macromoleeule is placed in the cell of an analytical ultracentrifuge, and the angular velocity is selected to produce a measurable solute concentration gradient at equilibrium. The solute concentration is measured optically as a function of r. The equation predicts that a plot of In (cb/c°) versus will be linear, with a slope equal to Mb (l — E p) jlRT. The partial specific volume is found from measurements of solution density as a function of solute mass fraction (page 234). By this means, the molar mass Mb of the macromolecule is evaluated. 

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