Critical Isopleth Research Articles

Phase rule method for applying the principle of critical point universality to chemically reacting open systems

We have measured the solubility of barium peroxide in the liquid mixture, isobutyric acid+water, in the single phase region along the critical isopleth at temperatures just above the critical solution temperature at 299K. When the concentration, s, of Ba2+ is plotted in van't Hoff form with ln s vs. 1/T, where T is the Kelvin temperature, a straight line having negative slope is obtained for values of the temperature which are sufficiently in excess of the critical temperature Tc. Because the liquid phase is in equilibrium with a non-critical solid phase consisting of BaO2, the liquid constitutes an open system. A phase rule analysis taking this into account shows that one mole fraction describing the liquid phase can be fixed. Under such conditions, the principle of critical point universality predicts that the magnitude of the slope of the van't Hoff plot, (∂lns/∂(1/T)), should become infinite as T→Tc. Moreover, according to the Gibbs–Helmholtz equation, the sign of (∂lns/∂(1/T)) in the critical region should be negative. In agreement with both of these predictions, we find that when T is within 1.9K of Tc, (∂lns/∂(1/T)) approaches negative infinity.

Diverging Thermodynamic Derivatives Associated with Heterogeneous Chemical Equilibrium in a Binary Liquid Mixture with a Consolute Point

The solubilities of tin(II) oxide, copper(II) oxide, and cobalt(II) oxide have been determined in the liquid mixture, isobutyric acid + water, along the critical isopleth. When plotted in van’t Hoff form with \(\ln s\) versus \(1/T\), the solubility measurements, \(s\), lie on a straight line for values of the temperature, \(T\), which are sufficiently in excess of the critical solution temperature, \(T_\mathrm{c}.\) In the case of SnO, the dissolution reaction is exothermic, and the slope of the van’t Hoff plot diverges toward positive infinity as \(T\rightarrow T_\mathrm{c} .\) In the case of both CuO and CoO, the dissolution reaction is endothermic, and the slope of the van’t Hoff plot diverges toward negative infinity as \(T\rightarrow T_\mathrm{c} .\) Analysis of these ternary, heterogeneous equilibria using finite dimensional vector space stoichiometry theory shows that each contains two linearly independent components. According to the Gibbs phase rule, two-phase equilibria of this type can be described by two fixed, intensive variables, which are accounted for by the temperature and the pressure, respectively. The Gibbs–Helmholtz equation and the principle of critical-point universality can be combined to predict under conditions of fixed temperature and pressure that when dissolution is exothermic, \((\partial \ln s/\partial (1/T))\) should diverge toward positive infinity in the critical region, while when dissolution is endothermic, \((\partial \ln s/\partial (1/T))\) should diverge toward negative infinity. Our experiments include examples confirming both these predictions.

The critical behavior of the refractive index near liquid-liquid critical points

The nature of the critical behavior in the refractive index n is revisited in the framework of the complete scaling formulation. A comparison is made with the critical behavior of n as derived from the Lorentz-Lorenz equation. Analogue anomalies to those predicted for the dielectric constant ε, namely, a leading |t|(2β) singularity in the coexistence-curve diameter in the two-phase region and a |t|(1-α) along the critical isopleth in the one phase region, are expected in both cases. However, significant differences as regards the amplitudes of both singularities are obtained from the two approaches. Analysis of some literature data along coexistence in the two-phase region and along the critical isopleth in the one-phase region provide evidence of an intrinsic effect, independent of the density, in the critical anomalies of n. This effect is governed by the shift of the critical temperature with an electric field, which is supposed to take smaller values at optical frequencies than at low frequencies in the Hz to MHz range.

A chemical test of the principle of critical point universality: The solubility of nickel (II) oxide in isobutyric acid + water near the consolute point

A mixture of isobutyric acid + water has an upper consolute point at 38.8 mass % isobutyric acid and temperature near 26 °C. Nickel (II) oxide dissolves in this mixture by reacting with the acid to produce water and nickel isobutyrate. The solubility of nickel (II) oxide in isobutyric acid + water has been measured as a function of temperature at compositions, 25, 38.8, and 60 mass % isobutyric acid. For values of the temperature, T, which were at least 2 K in excess of the liquid-liquid phase transition temperature, the measured values of the solubility, s, lie on a straight line when plotted in van't Hoff form with ln s versus 1∕T. The slope, (∂ln s∕∂(1∕T)), of the line is negative indicating that the dissolution reaction is endothermic. When the temperature was within 2 K of the phase transition temperature, however, (∂ln s∕∂(1∕T)) diverged toward negative infinity. The principle of critical point universality predicts that when excess solid nickel (II) oxide is in dissolution equilibrium with liquid isobutyric acid + water, (∂ln s∕∂(1∕T)) should diverge upon approaching the consolute point along the critical isopleth at 38.8 mass % isobutyric acid. As determined by the sign of the enthalpy of solution, the sign of this divergence is expected to be negative. Not only do our experiments confirm these predictions, but they also show that identical behavior can be observed at both 25 and 60 mass % isobustyric acid, compositions which lie substantially to either side of the critical composition.

Dielectric constant of fluids and fluid mixtures at criticality

The behavior of the dielectric constant epsilon of pure fluids and binary mixtures near liquid-gas and liquid-liquid critical points is studied within the concept of complete scaling of asymmetric fluid-fluid criticality. While mixing of the electric field into the scaling fields plays a role, pressure mixing is crucial as the asymptotic behavior of the coexistence-curve diameter in the epsilon-T plane is concerned. Specifically, it is found that the diameters, characterized by a |T-Tc|1-alpha singularity in the previous scaling formulation [J. V. Sengers, D. Bedeaux, P. Mazur, and S. C. Greer, Physica A 104, 573 (1980)], gain a more dominant |T-Tc|2beta term, whose existence is shown to be supported by literature experimental data. The widely known |T-Tc|1-alpha singularity of epsilon along the critical isopleth in the one-phase region is found to provide information on the effect of electric fields on the liquid-liquid critical temperature: from experimental data it is inferred that Tc usually decreases as the magnitude of the electric field is enhanced. Furthermore, the behavior of mixtures along an isothermal path of approach to criticality is also analyzed: theory explains why the observed anomalies are remarkably higher than those associated to the usual isobaric path.

Temperature, concentration, and frequency dependence of the dielectric constant near the critical point of the binary liquid mixture nitrobenzene-tetradecane

Detailed results are reported for the dielectric constant epsilon as a function of temperature, concentration, and frequency near the upper critical point of the binary liquid mixture nitrobenzene-tetradecane. The data have been analyzed in the context of the recently developed concept of complete scaling. It is shown that the amplitude of the low frequency critical Maxwell-Wagner relaxation (with a relaxation frequency around 10 kHz) along the critical isopleth is consistent with the predictions of a droplet model for the critical fluctuations. The temperature dependence of epsilon in the homogeneous phase can be well described with a combination of a (1-alpha) power law term (with alpha the heat capacity critical exponent) and a linear term in reduced temperature with the Ising value for alpha. For the proper description of the temperature dependence of the difference Deltaepsilon between the two coexisting phases below the critical temperature, it turned out that good fits with the Ising value for the order parameter exponent beta required the addition of a corrections-to-scaling contribution or a linear term in reduced temperature. Good fits to the dielectric diameter epsilon(d) require a (1-alpha) power law term, a 2beta power law term (in the past considered as spurious), and a linear term in reduced temperature, consistent with complete scaling.

Chemical Equilibrium and Critical Phenomena: The Solubilities of Iron(III) Oxide and Cobalt(II,III) Oxide in Isobutyric Acid + Water Near the Consolute Point

The solubilities of iron(III) oxide, formula Fe2O3, and cobalt(II,III) oxide, formula Co3O4, have been determined in the liquid mixture, isobutyric acid + water, along the critical isopleth at temperatures above the upper critical solution temperature near 299 K. When plotted in van’t Hoff form with ln s versus 1/T, the measurements of solubility, s, lie on a straight line for values of the temperature, T, in kelvin, which are sufficiently in excess of the critical solution temperature, Tc. The sign of the slope, (∂ ln s/∂(1/T)), indicates that in the case of both oxides, the dissolution reaction is endothermic. When the temperature is within 1K of Tc, however, the slope departs from its constant value and appears to diverge toward negative infinity. The principle of critical-point universality predicts that a divergence in (∂ ln s/∂(1/T)) is to be expected for T near Tc in those cases where the stoichiometry of the dissolution reaction involves both components of the solvent; moreover, the Gibbs–Helmholtz equation predicts that, if the heat of solution is endothermic, the sign of the divergence should be negative. Both of these predictions are confirmed by the solubilities of Fe2O3 and Co3O4 measured as a function of temperature along the critical isopleth of isobutyric acid + water.

Chemical Equilibrium and Critical Phenomena: Solubility of Indium Oxide in Isobutyric Acid + Water Near the Consolute Point

We have determined the solubility, s, of indium oxide in the liquid mixture isobutyric acid + water along the critical isopleth at temperatures above the upper critical solution temperature near 299 K. When plotted in van’t Hoff form with ln s vs 1/T, the measurements of solubility lie on a straight line for values of the absolute temperature, T, which are sufficiently in excess of the critical solution temperature, Tc. The sign of the slope, (∂ ln s/∂(1/T)), indicates that the enthalpy of dissolution is endothermic. When the temperature is within 1 K of Tc, however, the slope departs from its constant value and appears to diverge toward negative infinity. The principle of critical point universality predicts that a divergence in (∂ ln s/∂(1/T)) is to be expected for T near Tc in those cases where the stoichiometry of the dissolution reaction involves both components of the solvent; moreover, the Gibbs−Helmholtz equation predicts that if the enthalpy of solution is endothermic, the sign of the divergence should be negative. Both of these predictions are confirmed by the experimental data.

Shear Viscosities of Methycyclohexane, Perfluoromethylcyclohexane, and Their Mixtures in the Vicinity of the Upper Critical Mixing Temperature. 1. Critical Isopleth and Coexistence Curve

Shear viscosities η are reported for pure liquid methylcyclohexane (MCH) from 298.610 K to 333.694 K, for perfluoromethylcyclohexane (PFMCH) from 319.196 K to 333.114 K and for a MCH + PFMCH mixture of overall PFMCH near-critical mole fraction, xc = 0.3640, from (TUCS/K − 7) in the region of biphase liquid coexistence to (TUCS/K + 20) in the uniphase region, where TUCS = 320.13 K is the air-saturated upper liquid−liquid critical solution temperature. The measurements were made using a capillary rheometer that permits the measurement of the viscosity of thermally equilibrated coexisting-liquid phases. The results confirm that the near-critical viscosity exhibits a weak enhancement that strictly speaking becomes a divergence when account is taken of the finite shear gradients in the capillary during measurements. The viscosity of the uniphase mixture of critical composition is well-described after shear gradient correction by a multiplicative combination of an Arrhenius background and a critical power expression with an index close to the now-accepted universal value y = 0.0435. The chief objective of the work, in addition to contributing to knowledge of this aspect of near-critical rheology, is the development for the biphase of a simple expression for the temperature dependence of the viscosities of the coexisting phases, η+ and η-, that combines (a) an expression for the viscosity diameter <η> = 1/2(η+ + η-) similar to that for the viscosity of the critical mixture in the uniphase region, with a similar best critical index y‘ between 0.041 and 0.0435, and (b) an expression for Δη = (η+ − η-) that behaves like an order parameter, with an index β very close to the normal value β = 0.325 and as many Wegner correction terms as the data require. The best two-phase fit emerges from a freely fitted exponent y‘ ≈ 0.037 with one Wegner-extended scaling term, but we believe that were shear gradient correction to be applied, the best y‘ would be the consensus value y‘ = 0.0435. The magnitude of y‘ notwithstanding, we believe that our primary objective has been satisfied, namely, the formulation of an expression that affords a good description of the shear and background viscosities of near-critical mixtures in the one- and two-liquid phases in relation to our estimates of the nano- or molecular-viscosity derived from measurements of fluorescence polarization decay rates.

Chemical Equilibrium and Critical Phenomena: The Solubilities of Manganese Dioxide and Aluminum Oxide in Isobutyric Acid + Water near Its Consolute Point

We have measured the solubilities of manganese dioxide and aluminum oxide in isobutyric acid + water along its critical isopleth at temperatures above the upper critical solution temperature at 26.2 degrees C. Both oxides are basic anhydrides that react with isobutyric acid to produce the metal isobutyrate and water. In both cases, the measured solubility values, s, were in the parts per million range. When the solubility data for either of these oxides were plotted in van't Hoff form with ln s vs 1/T, a straight line resulted for absolute temperatures, T, sufficiently in excess of the critical solution temperature, T(c). The sign of the slope of the line indicated that manganese dioxide dissolved endothermically, while aluminum oxide dissolved exothermically. When the temperature was within 1 K of T(c), however, the local van't Hoff slope, ( partial differential ln s/ partial differential(1/T)), appeared to diverge toward negative infinity in the case of manganese dioxide, while it appeared to diverge toward positive infinity in the case of aluminum oxide. By applying the principle of critical point universality to the critical behavior of thermodynamic derivatives, we have shown that the existence of a divergence in ( partial differential ln s/ partial differential(1/T)) as T approaches T(c) is to be expected when the dissolution reaction of the oxide involves both components of the solvent. Application of the Gibbs-Helmholtz equation showed that the sign of the divergence must be opposite to the sign of the heat of solution. The experimental solubility data sets for both manganese dioxide and aluminum oxide are in good agreement with these theoretical assertions.

Interdiffusion Coefficient and Dynamic Viscosity for the Mixture 2,6-Lutidine + Water near the Lower Consolute Point

The mixture, 2,6-lutidine + water has a lower consolute point at a temperature of 33.9°C and a lutidine concentration of 2.78 M. The interdiffusion coefficient of this mixture was measured over a wide range of concentrations along the isotherms at 25.00°C and 33.00°C, and over a wide range of temperatures along the critical isopleth. Critical slowing down of diffusion was evident along all three paths. The dynamic viscosity was measured along the isotherm at 25.00°C and along the critical isopleth at temperatures below critical. Along the critical isopleth, the viscosity increased rapidly as the critical temperature was approached. These data are discussed in terms of the cross over function theory, which is used to connect divergent data near the critical point to smoothly varying data away from the critical point.

A forward approach to the problem of nonlinear viscoelasticity and the thickness of the mechanical lithosphere

The physical significance of the mechanical lithosphere is clarified using a nonlinear Maxwell flexure model. This model reconciles high‐temperature creep data for rock with the observed long‐term strength of the mechanical lithosphere. An approximate correspondence is established between the nonisothermal Maxwell model and an elastic flexure model to define an effective lithosphere thickness he. The model calculations are essentially forward, since he does not depend in a significant way upon the nominal or prescribed value of beam thickness, depending rather upon such variables as temperature and creep activation energy. The effective thickness of the Maxwell model, though a function of time, is stable enough over appropriate time scales to explain the long‐term integrated strength of the lithosphere. The strength index Σ is defined as the ratio of axial stress to the axial stress that would be present in a purely elastic fiber suffering the same strain. The base of the effective lithosphere is marked by a critical isopleth of strength, Σc ≅ 0.3. This critical level (though not its position) is independent of time since loading. The strength index also maps out zones of predominant rheology. The narrowness of the transition zone between the elastic lithosphere and the viscous asthenosphere explains the success of the elastic plate in modeling many examples of lithosphere flexure. The elastic plate approximation may apply best where the lithosphere is relatively young and hot. Isolated, weak zones in the continental crust may or may not weaken the lithosphere as a whole, depending upon the distribution of these zones and the manner in which they deform. In general, such weakening is likely to be most effective when the base of the crust occurs at midlithospheric depths.