Molar compressibility

Note that the constants in Equation (2.4) are distinguishable from those in Equation (2.3) because they lack the prime symbol. For both Equations (2.3) and (2.4), the terms in brackets represents the molar compressibility Z. Table 2.5 lists a few virial coefficients. 

Equation 163, written as liu = (f- / RT, clearly shows that In (j) is a partial molar property with respect to GR/ RT. MultipHcation of equation 175 by n and differentiation with respect to at constant 7, P, and in accord with equation 116 yields, after reduction, equation 179 (constant 7, .x), where Z, is the partial molar compressibility factor. This equation is the partial-property analogue of equation 178. 

Further evidence for the reality of a structural transition in the vicinity of Ira may be implied from a study by Satyanarayanamurty and Krishnamurty on ultrasonic velocities and compressibilities of aqueous solutions (131). They found that plots of the apparent molar compressibility (for metal nitrates) vs. square root concentration become linear only at concentrations above Ira. 

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.

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. 

Molar compressibility (6) = Mlp)P where /and M are the molar volume and molecular weight, respectively. 

Further deviatives yield the partial molar compressibility, Iq, and the partial molar heat capacity, These quantities are required to calculate chemical potentials at higher pressures and over a wider range of temperatures, respectively. 

If instead AV° is a function of pressure, we may define the standard partial molar compressibility which for single species i equals... 

Naumov et al. (1974), Helgeson et al. (1981), Millero(1982), andTanger and Helgeson (1988) report molar volume data for aqueous species at 25°C. The last two references also give V° data for aqueous species from 0°C up to 50°C and 350°C, respectively. Molar compressibilities of minerals are available in Birch (1966) Mathieson and Conway (1974), and Millero (1982). Millero (1982) also lists values of AV° and AK° for aqueous species from 0 to 50 C. Molar volumes and compressibilities vary as a function of temperature and ionic strength (Millero 1982). Such effects may be important for aqueous species and especially for gases. 

Table 6.10 shows that the partial molar volumes and particularly the isentropic partial molar compressibilities at infinite dilution are significantly higher in the micellar environment than in water. As far as the partial molar volume is concerned, the increment per CH2 group is reasonably constant. The values are 15.9 cmVmol in water and 16.8 cmVmol in the micelles. This can be compared to a CH2 group value of 16.1 in octane and 16.2 in heptane and 16.8 for the molar volume of pure alcohols. It means that as far as the volume of a CH2 group in the micelle is concerned, pure alcohol is the best comparison. 

Partial Molar Volumes and Isentropic Partial Molar Compressibilities of Alcohols at Infinite Dilution in Aqueous Surfactant Solutions at 298.15 K (volumes in cm /mol, compressibilities in cm /mol/bar)... 

If we look at the absolute values of partial molar volumes and compressibilites of alcohols solubilized in micelles, it appears that for the lower alcohols, up to heptanol, volumes and compressibilities exhibit lower values in SDS than in carboxylates. It also appears that the partial molar compressibilities of these alcohols are higher than the molar compressibility of the pure alcohols when solubilized in the carboxylate micelles and lower when solubilized in SDS. 

In cationic micelles the partial molar compressibility is generally slightly higher than for the anionic ones. There are two possible explanations for the differences in partial molar compressibility for an alcohol in the various micellar systems. It is possible that the micellar structure is slightly different and that it is easier to compress a carboxylate micelle around an alcohol molecule than it is for a dodecylsulfate micelle. However, the partial molar compressibility of a surfactant molecule... 

Here, quantity 2, is a partial molar compressibility factor evaluated at constant temperature and total volume ... 

Here, pis molar density, and is a partial molar compressibility factor given by... 

This letter to the editor provides no thermodynamic information. The author has used experimental determination of the molar compressibility to calculate hydration numbers of different metal ions, including Th". Data of this type give no information on the number of water molecules in the first coordination sphere, only an operational coordination number, in this case 14 for Th". Operational coordination numbers of this type are always dependent on the method used and are therefore of limited chemical interest. 

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