Gibbs plot

FIG. 6 Gibbs plot CMC values for LAS homologs measured by surface tension at 38°C in 0.01 M Na2S04. 

FIG. 8 Surface tension vs. log concentration (Gibbs plot) for C12 average DATS at 25°C in 0.01 M Na2S04. 

Fig. 12. Adam-Gibbs plots of the dielectric relaxation time of 2-methyltetrahydrofuran (2-MTHF) and 3-bromopentane (3-BP) versus (Tsconi) . The lines are VTF fits, 7 fus is the fusion temperature, and Tb is the temperature below which the VTF equation applies. /I ag and Avf are prefactors in the Adam-Gibbs and VTF equations, respectively. Tk is the calorimetri-cally determined Kauzmann temperature, and To is the VTF singular temperature, which were set equal in the VTF (line) fits. (Reprinted with permission from R. Richer and C. A. Angell. Dynamics of glass-forming liquids. V. On the link between molecular dynamics and configurational entropy. J. Chem. Phys. (1998) 108 9016. Copyright 1998, American Institute of Physics.)...

Note The equilibrium melting temperature (tJJ,) of copolymers depends on the molecular weight, sequence distribution and counit content. The T, value is determined by two commonly used techniques the Hoffman-Weeks plot and the Thompson-Gibbs plot. Tire application of the Hoffman-Weeks method to determine the tJ, of a copolymer is unreliable (see reference 43). The more reliable method is to use the Tliompson-Gibbs relationship of Tm as a function of lamellar thickness, provided a large range of lamella thickness can be obtained. Considerable disagreement exists between different authors on the exact value of transition that can be identified for fhe copolymers. Consequently, values tabulated in this table must be used cautiously. See references (39, 43, and 44) for detailed discussions. 

Fig. 1, Gibbs Plots of Surface Tension vs. Log Molarity in Solutions of Oligomer Salt Fractions...

Figure 10. The Thomson-Gibbs plot (T vs I/lc ) of samples with various degrees of branching...

Langevin and coworkers have studied the spreading behavior of PDMS on only one non-ionic surfactant—CiqEOs. The effect of PDMS on the relevant Gibbs plot revealed a small reduction in the surfactant surface excess, which does imply some partial penetration of the surfactant monolayer by the polymer [83]. Binks and... 

Methyl ester ethoxylates and their alcohol ethoxylate counterparts have similar surface properties. Gibbs plot for pure C14 7-mol (no other ethox-ymers except the 7-mol homolog) methyl ester ethoxylate is compared to its pure Ci4 7-mol alcohol ethoxylate counterpart in Fig. 11. The methyl ethoxylate shows a higher CMC and a lower surface tension at the CMC than its alcohol ethoxylate equivalent. This increase in CMC is presumably due to the presence of the ester moiety which adds rigidity and steric constraint to the methyl ester ethoxylate molecule. This would likely reduce the tendency of a molecule to micellize, leading to a slightly higher CMC. 

FIG. 11 Gibbs plot for C14 pure (ethoxylates are pure 7-mol ethoxylates there are no other homologs) 7-mol methyl ester and alcohol ethoxylates. 

The impact of unsaturation stems from the increase in rigidity caused by the presence of one or more double bonds to the alkyl chain. Although studies so far show that unsaturation has a relatively low impact on water solubility and viscosity, it has been reported to lower melting points by about 5-10°C [16]. Studies also show (Fig. 20) that unsaturation affects surface properties. Gibbs plots of 14.5-mole ethoxylates produced from hydrogenated and nonhydrogenated Cig is methyl ester show that unsaturation appears to increase surface tension below the CMC. This suggests that unsaturation reduces the hydrophobic character of the methyl ester chain. 

InFig. 9, aplotofEq. (114) results in predicted adsorption as a function of pH with total cation and surface-site mole fraction held constant. For the Gibbs plot in Fig. 10 of equation (115), adsorption is plotted as the areal surface concentration, Tg, as a function of increasing total cation mole fraction [or equilibrium solution mole fraction with Eq. (102)] at constant pH and total surface-site concentration. 

The type of behavior shown by the ethanol-water system reaches an extreme in the case of higher-molecular-weight solutes of the polar-nonpolar type, such as, soaps and detergents [91]. As illustrated in Fig. Ul-9e, the decrease in surface tension now takes place at very low concentrations sometimes showing a point of abrupt change in slope in a y/C plot [92]. The surface tension becomes essentially constant beyond a certain concentration identified with micelle formation (see Section XIII-5). The lines in Fig. III-9e are fits to Eq. III-57. The authors combined this analysis with the Gibbs equation (Section III-SB) to obtain the surface excess of surfactant and an alcohol cosurfactant. 

The deviation of Gibbs monolayers from the ideal two-dimensional gas law may be treated by plotting xA// 7 versus x, as shown in Fig. III-15c. Here, for a series of straight-chain alcohols, one finds deviations from ideality increasing with increasing film pressure at low x values, however, the limiting value of unity for irAfRT is approached. 

The data could be expressed equally well in terms of F versus P, or in the form of the conventional adsorption isotherm plot, as shown in Fig. Ill-18. The appearance of these isotherms is discussed in Section X-6A. The Gibbs equation thus provides a connection between adsorption isotherms and two-dimensional equations of state. For example, Eq. III-57 corresponds to the adsorption isotherm... 

Equation 9 states that the surface excess of solute, F, is proportional to the concentration of solute, C, multipHed by the rate of change of surface tension, with respect to solute concentration, d /dC. The concentration of a surfactant ia a G—L iaterface can be calculated from the linear segment of a plot of surface tension versus concentration and similarly for the concentration ia an L—L iaterface from a plot of iaterfacial teasioa. la typical appHcatioas, the approximate form of the Gibbs equatioa was employed to calculate the area occupied by a series of sulfosucciaic ester molecules at the air—water iaterface (8) and the energies of adsorption at the air-water iaterface for a series of commercial aonionic surfactants (9). 

Fig. 5.6. Plot of the Gibbs functions for ice and water os functions of temperature. Below the melting point Tn, > Gjjj and ice is the stable state of HjO above T , G...

Fig. 16. A. Plot of log iNa as a function of T 1 (°K) using the experimental values of the rate constants and the location of the binding sites in Eq. 4. The Gibbs free energy of activation is calculated from Eq. 3 the AS are taken to be zero, and the current is calculated by means of Eq. 4. The purpose is to demonstrate that multibarrier channel transport can be seen as single rate process with average values for the enthalpies of activation. Non-linearity of such a plot is then taken to arise form the dynamic nature of the channel.

It was shown some time ago that one can also use a similar thermodynamic approach to explain and/or predict the composition dependence of the potential of electrodes in ternary systems [22-25], This followed from the development of the analysis methodology for the determination of the stability windows of electrolyte phases in ternary systems [26]. In these cases, one uses isothermal sections of ternary phase diagrams, the so-called Gibbs triangles, upon which to plot compositions. In ternary systems, the Gibbs Phase Rule tells us... 

Thus, values for C°p m T, S°m T, (H°m T - H°m 0) and (G°mT H°m0) can be obtained as a function of temperature and tabulated. Figure 4.16 summarizes values for these four quantities as a function of temperature for glucose, obtained from the low-temperature heat capacity data described earlier. Note that the enthalpy and Gibbs free energy functions are graphed as (// , T - H°m 0)/T and (G T — H q)/T. This allows all four functions to be plotted on the same scale. Figure 4.16 demonstrates the almost linear nature of the (G°m T H°m 0)/T function. This linearity allows one to easily interpolate between tabulated values of this function to obtain the value at the temperature of choice. 

If the property evaluated, for instance, the critical micelle concentration, can be approximated by a suitable plot, it is depicted in the ternary system as a concave area (e.g., cM area) located in the space above the Gibbs triangle as the basis for the distinct concentrations. The property axis describing the cM data stands vertically on the base triangle. 

Figure 26. Plot of the Gibbs energy of adsorption of organic substances at a = 0 vs. the interfacial parameter, AX. (1) 1-Hexanol, (2) 1-pentanol, and (3) acetonitrile. From Ref. 32, updated. Additional points (1) Au(l 11),910 Bi(l 11),152 and (2) Ga916...

BrCl(g), K = 0.2. Construct a plot of the Gibbs free energy of this system as a function of partial pressure of BrCl as the reaction approaches equilibrium. 

Use the Living Graph Variation of Equilibrium Constant on the Web site for this book to construct a. if plot from 250 K to 350 K for reactions with standard g reaction Gibbs free energies of + 11 kj-mol 1 to 4 15 kj-mol 1 in increments of 1 kj-mol. Which equilibrium constant is most sensitive to changes in temperature ... 

Figure 13. The Gibbs energy available from a reaction, A B, depends on its displacement from equilibrium when IB)/IA) = K. The AC value is plotted against the mass-action ratio, and this is the value when B1/ A] is maintained constant in the steady state if the rate of substrate supply and substrate removal is constant.

Fig. 1.1 (a) The ionization enthalpies of dipositive lanthanide ions with configurations of the type [Xe]4f" (upper plot left-hand axis), (b) The standard Gibbs energy change of reaction 1 (lower plot right-hand axis estimated value ... 

Avdeef, A., pH-metric solubility. 1. Solubility-pH profiles from Bjerrum plots. Gibbs buffer and pfCa in the solid state. Pharm. Pharmacol. Commun. 1998, 4,165-178. 

---