Volume apparent molar

Figure 5.4 Significance of the apparent molar volume 6v usj aqueous sodium acetate solutions as an example. The dots represent the ex erimental results at T = 298.15 K with m the molality of the sodium acetate.

The apparent molar volume of interfacial water in AOT-reversed micelles is lower and its refractive index is greater than that of pure water. These findings, together with other experimental evidence, emphasize that these water molecnles are destructured, immobilized, and polarized by the ionic head of AOT [2,84,89]. In particular, it has been reported that the... 

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>... 

Apparent molar volume. No trend with pressure was observed. 

Here, V refers to the molar volume of the nonelectrolyte, V and Vs° are the intrinsic3 and apparent molar volume of the salt, respectively, and /3 is the isothermic compressibility coefficient of the solution. Although this equation is strictly valid in the limit of Vi - 0 and ca - 0, it works quite satisfactorily for small nonpolar solutes. The equation shows that the effect is greatest for nonelectrolytes of large molar volume Vi, and for salts that cause the largest electrostriction, Vs - Vs°. A difficulty with this expression is that it is not always easy to evaluate the intrinsic volume of a salt (the mere volume, without... 

According to Macinnes and Dayhoff [7], the apparent molar volume of K1 in CH3OH can be expressed by the equation... 

When a solute particle is introduced into a liquid, it interacts with the solvent particles in its environment. The totality of these interactions is called the solvation of the solute in the particular solvent. When the solvent happens to be water, the term used is hydration. The solvation process has certain consequences pertaining to the energy, the volume, the fluidity, the electrical conductivity, and the spectroscopic properties of the solute-solvent system. The apparent molar properties of the solute ascribe to the solute itself the entire change in the properties of the system that occur when 1 mol of solute is added to an infinite amount of solution of specified composition. The solvent is treated in the calculation of the apparent molar quantities of the solute as if it had the properties of the pure solvent, present at its nominal amount in the solution. The magnirndes of quantities, such as the apparent molar volume or heat content, do convey some information on the system. However, it must be realized that both the solute and the solvent are affected by the solvation process, and more useful information is gained when the changes occurring in both are taken into account. 

The apparent molar volumes and heat capacities, V2. and Cp 2 of NaDec In water and In 0.05 mol kg l BE are shown In figure 1. As expected from the formation of mixed micelles, BE shifts the CMC of NaDec to lower values. 

Figure 1 Apparent molar volumes and heat capacities of sodium deca noate In water (ref. 11) and In 0.05 mol kg of 2-butoxy-ethanol at 25°C.

The enthalpies of solution and apparent molar volumes in various solvents have been determined for methyltriphenylphosphonium iodide (10) rate constants, activation enthalpies, activation volumes and reaction enthalpies were also determined for its synthesis from triphenylphosphine and methyl iodide156. (Equation 3). 

It is observed from Figure 12 that at infinite dilution the apparent molar volume of all alkali metals in ammonia approach roughly the same value. This suggests complete dissociation of metal into metal ions and solvated electrons as has been demonstrated by conductivity and other measurements. As we continue to increase the concentration from infinite dilution, a rather peculiar phenomenon has been shown to occur... 

Figure 12. The apparent molar volumes of alkali metals in liquid ammonia...

PI 1.7 Start with equation (11.96) applied to the apparent molar volume V, solve for V, and differentiate to show that V is related to the partial molar volume of the solute by... 

Volumetric Properties Equations for V, the apparent molar volume, can also be obtained. They are as follows... 

Figure 18.7 Volumetric properties of aqueous NaCl solutions as a function of temperature, pressure, and concentration, (a) apparent molar volume (b) apparent molar compressibility and (c) apparent molar expansibility. The effect of pressure is shown as alternating grey and white surfaces of V, K, and E at p = 0.1 or saturation, 20, 30,40, 50, 70, and 100 MPa. In all three instances, the surfaces increase with increasing pressure.

Mixtures of these surfactants with water result in solutions with unique properties that we want to consider. We will use the alkylpyridinium chlorides as examples. Figure 18.11 compares the osmotic coefficient 0, apparent relative molar enthalpy 4>L, apparent molar heat capacity Cp, and apparent molar volumes V as a function of molality for two alkylpyridinium chlorides in water.w19... 

Figure 18.11 (a) Osmotic coefficient (b) apparent relative molar enthalpy (c) apparent molar volume and (d) apparent molar heat capacity, at T = 298.15 K and p = 0.1 MPa, for (1) n-decylpyridinium chloride and (2) n-dodecylpyridinium chloride. 

Figure 18.13 Effect of temperature on (a) apparent relative molar enthalpies (b) apparent molar volumes and (c) apparent molar heat capacities, for n-dodecylpyridinium chloride. The temperatures are (1) 298.15 K (2) 313.15 K and (3) 328.15 K.

Figure 18.15 Surfactant pseudo-phase model prediction for (a) the apparent molar volume and (b) the apparent molar heat capacity. Drawing courtesy of K. Ballerat-Busserolles from the Institut de Chemie des Surfaces et Interfaces, Mulhouse, France.

Figure 18.16 (a) Apparent molar volume and (b) apparent molar heat capacities for aqueous sodium dodecylsulfate at T = 298.15 K and /> = 0.1 MPa, graphed as a function of 1 /m. The insets give the values at low m where a second transition occurs in the micelle. 

The thermodynamic properties at T = 298.15 K shown in Figure 18.11 come from S. Causi, R. De Lisi, and S. Milioto, Thermodynamic properties of N-octyl-, N-decyl- and N-dodecylpyridinium chlorides in water , J. Solution Chem., 20, 1031-1058 (1991). Results at the other two temperatures are courtesy of K. Ballerat-Busserolles, C. Bizzo, L. Pezzimi, K. Sullivan, and E. M. Woolley, Apparent molar volumes and heat capacities at aqueous n-dodecyclpyridium chloride at molalities from 0.003 molkg-1 to 0.15 molkg-1, at temperatures from 283.15 K. to 393.15 K, and at the pressure 0.35 MPa , J. Chem. Thermodyn., 30, 971-983 (1998). 

In this approach the molecule is treated as a sphere of radius a in a uniform cavity characterized by a microviscosity ij and an apparent molar volume Vm. Over the years many studies correlating T, with solution viscosity have been carried out. The record of linear correlations is patchy, partly because the microviscosity is not an experimentally available number. However, in large, relatively flat... 

The apparent molar volume is given as a function of one of the composition variables. The methods of evaluating the partial derivatives in Equations (6.55) and (6.56) are identical to those discussed in Section 6.3. The final expressions for the partial molar volumes of the two components in terms of the molality are... 

Figure 6.1 illustrates the concept of the apparent molar volume. AB is the portion of the total volume AC for n2 moles of solute that is attributed to the pure solvent. Then, the volume BC is apparently due to the solute. The slope of the line passing through point C and V is the apparent molar volume. The slope of the curve of the total volume at point C is the partial molar volume of component 2. Indeed, the slope of the total volume curve at any point is the partial molar volume of component 2 at that concentration. It is obvious that partial molar properties and apparent molar properties are both functions of concentration. 

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... 

In the second case we might choose a solution of fixed mole ratios of the components and consider the change in the volume of the solution as another solute is added to it. The apparent molar volume of the added solute would then be defined by... 

From studies of the concentration dependence of density and isentropic compressibility coefficients, the apparent molar volume and the isentropic apparent molar compressibility may be obtained above and below the CMC. Such studies have recently been performed for several systems by Brun, Holland and Vikingstad32,39-41 who deduced the change in partial molar volume and compressibility on micelle formation. This gives information on the counterion hydration and the packing of the hydrocarbon chains in the micelles. 

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. 

The density of the solvent is d0 and MW is the molar mass of the solute. The values of the apparent molar volume are plotted against concentration and the partial molar volume of a species is the value obtained by extrapolation to zero concentration. 

An example not previously discussed is the Pitzer-Debye-Hiickel slope for apparent molar volume (Av) that is required in Eqs. 2.76, 2.80, and 2.81. A numerical equation for Ay as a function of temperature and pressure was derived from the database of Ananthaswamy and Atkinson (1984) over a temperature range of 273 to 298 K and over a pressure range of 1 to 1000 bars ... 

Fig. 3.25. The Pitzer-Debye-Hiickel slope for apparent molar volume (Av) as a function of temperature and pressure. Symbols are from Ananthaswamy and Atkinson (1984) lines represent model estimates...

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