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Cite This: J. Phys. Chem. B 2019, 123, 5545−5554

Phase Rule and the Universality of Critical Phenomena in Chemically Reacting Liquid Mixtures James K. Baird,*,†,‡,§ Joshua R. Lang,†,§ Xingjian Wang,‡,§ Anusree Mukherjee,†,§ and Pauline Norris§,∥ Department of Chemistry, ‡Department of Physics, and §Materials Science Graduate Program, University of Alabama in Huntsville, Huntsville, Alabama 35899, United States ∥ Advanced Materials Institute, Western Kentucky University, 2413 Nashville Road, Bowling Green, Kentucky 42101, United States Downloaded via BUFFALO STATE on July 27, 2019 at 16:04:19 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

†

ABSTRACT: Critical effects have been reported in the case of chemical equilibria, in which the solvent is a binary liquid mixture having a critical point of solution. At atmospheric pressure and for temperatures near the critical point, the critical effect manifests itself as a divergence in the temperature derivative of the extent of reaction. For a critical mixture of isobutyric acid + water (IBA/H2O) serving as the solvent, we report experimental results for three complex equilibria involving (i) parallel dissolution of aluminum oxide and manganese dioxide (involves 8 species); (ii) parallel dissolution of aluminum oxide and copper(I) oxide (involves 10 species); and (iii) dissolution of barium chromate (involves 9 species). In each case, we observe a divergence in the slope of the van’t Hoff plot of the extent of reaction in the critical region. By phase rule analysis of these and all other existing data, we find that the chemical equilibrium critical effect occurs in coincidence with three thermodynamic intensive variables being fixed, where two of these are the temperature and the pressure. The slope of the van’t Hoff plot in the critical region is observed to diverge toward negative infinity when the reaction is endothermic and toward positive infinity when it is exothermic. These two features are a characteristic of both homogeneous and heterogeneous equilibria and have been observed at both upper and lower critical solution temperatures. Taken together, these observations support the applicability of the universality concept to chemical equilibrium critical phenomena in binary liquid mixtures. respect to a field should diverge, so long as the path of approach to the critical point involves no more than one fixed density variable. If two or more density variables are fixed, there should be no critical divergence.4 As the universality concept provides a general thermodynamic framework for understanding critical effects in physical systems,2,3 it was natural to ask whether the Griffiths−Wheeler rules,4 or something related,5−7 might be used to predict the occurrence of critical effects in the position of chemical equilibrium. Binary liquid mixtures exhibiting a miscibility gap ending in a critical point of solution were an obvious choice to serve as solvents for investigating such effects. More than 1000 such pairs are known.8 Pursuing this idea, Tveekrem, Cohn, and Greer carried out an experiment in which they observed the equilibrium between nitrogen dioxide and its dimer in a critical mixture of carbon tetrachloride + perfluorocyclohexane.9 They reported that the position of chemical equilibrium exhibited only a weak critical effect. Fundamental thermodynamic reasoning based on the Griffiths−Wheeler rules was invoked to rule out the presence of critical effects in cases where more than one composition

1. INTRODUCTION The term “critical phenomenon” refers to those changes in material properties that result from the continuous transformation of one state or phase of matter into another without the immediate appearance of a phase boundary. Among these are the liquid-to-vapor transition that occurs at the critical point of pure fluids, the normal-fluid-to-superfluid transition in liquid helium, and the miscibility gap transition in binary liquid mixtures with a critical point of solution.1−3 The Ising model is used to describe these phenomena.2,3 The predictions of the Ising model form the basis for the principle of critical point universality (also called the principle of critical point isomorphism), which is thought to govern all critical phenomena.2,3 Griffiths and Wheeler4 expressed the principle of critical point universality in thermodynamic terms. They divided the intensive variables into two classes called “densities” and “fields”. A density variable is one whose value depends upon the phase in which it is measured. Density variables include the molar volume, the specific entropy, and the composition variables such as the mole fractions. By contrast, a field variable exhibits a uniform value across coexisting phases. Field variables include the temperature, the pressure, and the chemical potential. Griffiths and Wheeler predicted that as the critical point is approached, the derivative of a density with © 2019 American Chemical Society

Received: March 29, 2019 Revised: May 31, 2019 Published: June 19, 2019 5545

DOI: 10.1021/acs.jpcb.9b02978 J. Phys. Chem. B 2019, 123, 5545−5554

Article

The Journal of Physical Chemistry B variable was fixed independently of the temperature and the pressure.10 Because the fixed composition variable, or variables, prevailing in the case of the dimerization equilibrium could not be identified with any certainty, there was little incentive to investigate more complex reactions. Much later, larger chemical equilibrium critical effects were observed in the case of individual metal oxides dissolving in a critical mixture of isobutyric acid + water (IBA/H2O).11 The dissolution was assumed to be a two-step process, beginning with the dissociation of isobutyric acid to form hydronium ion plus isobutyrate ion followed by the reaction of the hydronium ion with the metal oxide to produce the dissolved metal ion and water. The slope of the van’t Hoff plot of the solubility (the derivative of a density with respect to a field) was observed to diverge toward infinity as the critical temperature was approached.11 When the dissolution was endothermic, the derivative diverged toward negative infinity, whereas when the dissolution was exothermic, it diverged toward positive infinity. This behavior with temperature was shown to be in agreement with the predictions of the Gibbs−Helmholtz equation.11 Although these metal oxide solubility experiments succeeded in demonstrating a readily observable critical effect, a method for enumerating the fixed variables was still missing. The phase rule was proposed as an answer to this problem.12−14 Although first suggested long ago,12 the connection of the phase rule to the universality concept2,3 has taken some time to develop.13,14 The phase rule provides a method for determining the total number, F, of intensive thermodynamic variables, both fields and densities, needed to describe an equilibrium. Under ordinary laboratory conditions, the two field variables, temperature and pressure, are fixed. Hence, a phase rule calculation resulting in F = 2 implies that no composition variable is fixed, whereas a calculation of F = 3 implies that one composition variable is fixed.13,14 According to the universality principle,2,3 a critical effect is anticipated in both cases.4 By contrast, if F is equal to four or greater, a critical effect is not expected.14 The dissolution of metal oxides by acid/base reaction in IBA/H2O produces mixtures having at least six chemical components. We will show that F = 3 in every case where a solubility critical effect has been reported.11,14−18 Nonetheless, we should not neglect the fact that solubility critical effect can be anticipated even in the absence of reaction,19 as has recently been demonstrated in the case of the dissolution of excess phenolphthalein in a mixture of nitrobenzene + dodecane.20 This system consists of three components, and the phase rule calculation shows F = 3.20 Below, we describe three experiments, each of which serves as a stringent test of the F = 3 concept, inasmuch as each experiment involves multiple phases and multiple reactions. None includes fewer than eight components. The experiments are (1) simultaneous dissolution of MnO2(s) and Al2O3(s) in isobutyric acid + water. This mixture consists of a liquid phase, two solid phases, and eight components supporting three parallel chemical reactions. (2) Simultaneous dissolution of Cu2O(s) and Al2O3(s) in isobutyric acid + water. This system consists of four phases and 10 components supporting a total of four reactions. One of these is the disproportionation of Cu(I) to form Cu(II) and Cu metal, which serves as an example of an oxidation/reduction reaction. (3) Dissolution of 2− BaCrO4(s) followed by dimerization of CrO2− 4 to Cr2O7 in isobutyric acid + water. This mixture involves a liquid phase, a solid phase, and nine components supporting a total of four

reactions, including the dimerization reaction. In Section 2, we show how the phase rule can be used to enumerate the fixed variables, as is required to apply the universality principle. In Section 3, we report our experimental results. We confirm that the sign of the critical effect is in agreement with the predictions of the Gibbs−Helmholtz equation. In Section 4, we use the phase rule to show that the F = 3 condition is met by all known experiments involving chemical equilibrium critical effects in binary liquid mixtures. We compare these results with a counterexample provided by an experiment illustrating the case of F = 4. In Section 5, we state our conclusions.

2. THEORY 2.1. Phase Diagram of a Binary Liquid Mixture. When the phase diagram of a binary liquid mixture exhibiting a miscibility gap is plotted in the coordinates, temperature, T, vs mole fraction, X, the boundary curve surrounding the twophase region often ends in an extremum, termed the critical point of solution. The coordinates of this point define the critical solution temperature, Toc , and the critical solution composition, Xoc . The vertical line, X = Xoc , is called the critical isopleth. If the phase boundary curve is concave up, Toc is a lower critical solution temperature (LCST), and two liquid phases coexist at temperatures, T > Toc . If the boundary curve is concave down, Toc is an upper critical solution temperature (UCST), and two liquid phases coexist at temperatures T < Toc .21 2.2. Phase Rule Used To Count the Fixed Composition Variables. Simplicity can be achieved in the description of reacting multicomponent liquid mixtures by imagining both reactants and products as vectors in a space where the chemical elements play the role of basis vectors.22,23 Any substance formed from these elements can be represented as a vector in this space. Balanced chemical reactions are represented as the sum of reactant vectors leading to a point in the space equal to the sum of the product vectors. This concept forms the foundation for the phase rule as modified to take into account chemical reactions.22,23 In this vector representation, the number of linearly independent reactions, R, linking the C components cannot exceed the difference between C and the number of chemical elements, E. The maximum value of R is thus R=C−E

(2.1)

Any additional constraint equations, I in number, linking these components can be derived by applying the law of conservation of mass,22,23 or in the case of a redox reaction, by also applying the law of conservation of charge.23 To be valid, a constraint equation must refer to a single phase and must be expressed in terms of intensive variables; moreover, it must be independent of the extent of reaction.22 The number of coexisting phases, ϕ, is determined simply by counting. The number of free, or independent intensive variables, F, is then ascertained by evaluating the equation22,23 F=C−R−ϕ−I+2

(2.2)

An example of the application of the phase rule to a reacting system with a critical point of solution can be found in the case of the binary liquid mixture of IBA/H2O (UCST). In this mixture, isobutyric acid, represented by HA(aq), reacts with water to form H3O+(aq) and the isobutyrate ion, A−(aq). The stoichiometry is 5546


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The Journal of Physical Chemistry B HA(aq) + H 2O(aq) → H3O+(aq) + A−(aq)

where ΔH is the enthalpy and ΔG is the free energy of solution. For the solubility equilibrium to be stable, the derivative (∂ln s/∂ΔG) must be positive definite.11,20 Being the derivative of the density, s, with respect to the field, ΔG, (∂ln s/ ∂ΔG) is expected to diverge toward positive infinity in the critical region when no more than one composition variable is held fixed. On the basis of eq 2.9, the van’t Hoff slope, (∂ln s/ ∂(1/T)), should diverge toward positive infinity if ΔH < 0 and toward negative infinity if ΔH > 0.

(2.3)

There are C = 4 components: HA(aq), H2O(aq), H3O+(aq), and A−(aq). Because the covalent carbon, hydrogen, and oxygen links in A−(aq) are not altered by the ionization reaction, A−(aq)can be considered as a pseudo-chemical element;23 hence, for the purposes of calculation, the four components are spanned by the three elements, which are H, A−, and O. Four components spanned by three elements can support only R = 4 − 3 = 1 linearly independent reactions, which accounts for eq 2.3. To search for constraint equations, we let noY stand for the number of moles of component Y in the mixture as originally prepared, and nY stand for the number of moles of this component present at equilibrium. The equations expressing the conservation of the chemical elements then take the form

3. EXPERIMENT 3.1. Materials. Isobutyric acid (>99% pure), manganese dioxide (>99% pure), aluminum oxide (>99% pure), copper(I) oxide (99.99% pure), and barium chromate (99.999% pure) were obtained from Sigma-Aldrich and used as received. Water was distilled once from a glass system. 3.2. Methods Shared by All Experiments. A binary liquid mixture of IBA/H2O (UCST) was prepared at the critical composition of 38.8 mass % isobutyric acid (Xoc = 0.11) by weighing and then transferred to a flat-bottomed borosilicate glass sample tube. The tube was placed in an isothermal water bath, which has been previously described.15 By visual search for the appearance of a meniscus, the initial critical temperature for the binary mixture was determined to be near Toc = 299 K. To begin a solubility experiment, excesses of the solid reactants were added to the liquid mixture. Dissolution of solutes ordinarily causes a shift in the critical temperature of a binary mixture.25−30 To find the shifted critical temperature, the thermostat temperature was altered by approximately 0.05 K, and the mixture was stirred for 24 h to come to equilibrium and then let settle for an additional 24 h to clear the liquid by gravitational sedimentation. The transparent liquid was inspected for the disappearance of the meniscus. This procedure was repeated until the shifted critical temperature was located. We did not deem it necessary to search for the collateral shift in the critical composition,25 because it has been shown that solubility critical effects can be observed over a surprisingly wide range of compositions to either side of the critical composition.17 Once the critical temperature of the mixture was located, a 5 mL aliquot of the clear supernatant liquid was withdrawn, placed in a vial, and acidified with spectroscopic grade nitric acid to prevent the precipitation of the dissolved metal ions in the form of oxides or hydroxides. The temperature was then increased by approximately 0.05 K, and the mixture was stirred for 24 h and then let settle for another 24 h before the next sample was extracted. The sequence of increasing the temperature, stirring, settling, and sampling was repeated until the thermostat temperature reached a value several degrees distant from the critical region. 3.3. Parallel Dissolutions of Aluminum Oxide and Manganese Dioxide. The simultaneous dissolutions of aluminum oxide and manganese dioxide in isobutyric acid + water combine to produce Al3+(aq) and Mn4+(aq), respectively. The reaction scheme, which begins with eq 2.3, includes the following acid/base reactions

Conservation of H: o nHA + 2n Ho 2O = nHA + 2n H2O + 3n H3O+

(2.4)

−

Conservation of A : o nHA = nHA + n A−

(2.5)

Conservation of O: n Ho 2O = n H2O + n H3O+

(2.6)

An equation independent of the extent of reaction can be found, if we combine the element conservation equations in such a way as to remove the dependence on the initial conditions, noHA and noH2O. This is achieved by forming the linear combination: eq 2.4−eq 2.5 − 2 × eq 2.6. The result is n H3O+ − n A− = 0

(2.7)

which is the electroneutrality condition. Since a constraint equation must be expressed in terms of intensive variables,22 we divide eq 2.7 through by the total number of moles, n, making up the liquid phase to obtain X H3O+ − X A− = 0

(2.8)

where XY = nY/n is the mole fraction of component, Y. As there are no other combinations of eqs 2.4−2.6 that remove reference to the initial conditions, we set I = 1. The liquid is the only phase present; hence, ϕ = 1. According to eq 2.2, with C = 4, R = 1, ϕ = 1, and I = 1, the number of independent intensive variables is then F = 4 − 1 − 1 − 1 + 2 = 3. Hence, one composition variable is fixed, a conclusion drawn previously by Wheeler without making reference to the phase rule.24 2.3. Sign of the Critical Effect. When reaction equilibrium data are collected in the form of solubility, s, as a function of temperature, T, it is useful to plot the temperature dependence of the solubility in van’t Hoff form with ln s vs 1/T. Solubility data collected outside the critical region define a straight line. If the data collected within the critical region differ significantly from the extrapolation of this straight line into the critical region, a critical effect is indicated.11 The sign of the critical effect can be accounted for by using the Gibbs−Helmholtz equation ij ∂ln s yz ij ∂ln s yz jj z j ∂(1/T ) zz = −T ΔH jj ∂ΔG zz k { k {

Al 2O3(s) + 6H3O+(aq) → 2Al3 +(aq) + 9H 2O(aq) (3.1)

MnO2 (s) + 4H3O+(aq) → Mn 4 +(aq) + 6H 2O(aq)

(2.9)

(3.2) 5547


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The Journal of Physical Chemistry B

Table 1. Concentration, s, of Dissolved Manganese in Critical 38.8 Mass % Aqueous Isobutyric Acid in Equilibrium with Excesses of Manganese Dioxide and Aluminum Oxide

According to eqs 2.3, 3.1, and 3.2, the equilibrium mixture contains the components: HA(aq), H2O(aq), H3O+(aq), A−(aq), Al2O3(s), Al3+(aq), MnO2(s), and Mn4+(aq); hence, C = 8. These eight components are spanned by the five elements: H, A−, O, Al3+, and Mn4+; thus, E = 5. The maximum possible number of linearly independent reactions is R = C − E = 8 − 5 = 3. These are the reactions summarized in eqs 2.3, 3.1, and 3.2. The mixture consists of a liquid phase and the two coexisting solid phases, Al2O3(s) and MnO2(s); hence, ϕ = 3. The equations expressing the conservation of the chemical elements include eqs 2.4 and 2.5 and the following Conservation of O: o o 3n Al + 2n MnO + n Ho 2O 2O3 2

= 3n Al 2O3 + 2n MnO2 + n H2O + n H3O+

(3.3)

3+

Conservation of Al : o 2n Al = 2n Al 2O3 + n Al3+ 2O3

temperature (K)

s (ppm)

306.84 305.81 304.80 304.02 302.79 302.27 301.48 301.04 300.60 300.39 300.25 300.11 299.92 299.86

171.1 197.3 188.0 174.9 163.1 148.7 135.6 126.3 171.1 157.2 133.1 116.4 78.99 51.97

(3.4)

4+

Conservation of Mn : o n MnO = n MnO2 + n Mn 4+ 2

(3.5)

By forming the linear combination: eq 2.4−eq 2.5 − 2 × eq 3.3 + 3 × eq 3.4 + 4 × eq 3.5, we can obtain an equation which is independent of the initial conditions, noHA, noH2O, noAl2O3, and noMnO2. After dividing this equation through by n, we obtain the constraint X H3O+ + 3X Al3+ + 4X Mn 4+ − X A− = 0

(3.6)

Equation 3.6 is the charge neutrality condition for the liquid phase. As there are no other linear combinations that lead to a different result, we conclude I = 1. Upon the substitution of C = 8, R = 3, ϕ = 3, and I = 1 into eq 2.2, we obtain F = 8 − 3 − 3 − 1 + 2 = 3. The samples extracted from the liquid phase were analyzed for their manganese content using a Thermo Scientific iCAP 6500 inductively coupled plasma optical emission spectrometer. We denote the concentration of this metal by s, which, in the Griffiths−Wheeler scheme,4 is a density variable. The values of s in ppm are listed as a function of temperature in Table 1. These data are plotted in van’t Hoff form in Figure 1. 3.4. Parallel Dissolutions of Aluminum Oxide and Copper(I) Oxide. The reaction scheme for the simultaneous dissolution of aluminum oxide and copper(I) oxide in isobutyric acid + water begins with eqs 2.3 and 3.1 and then proceeds according to the following Cu 2O(s) + 2H3O+(aq) → 2Cu+(aq) + 3H 2O(aq)

(3.7)

2Cu+(aq) → Cu 2 +(aq) + Cu(s)

(3.8)

Figure 1. van’t Hoff plot of the temperature, T, dependence of the concentration, s, of manganese dissolved in critical 38.8 mass % aqueous isobutyric acid in equilibrium with excesses of manganese(IV) oxide and aluminum oxide. The standard state of dissolved Mn4+ was taken to be so = 1 ppm. The solid line was fitted to data collected above the critical point and then extrapolated into the critical region. The vertical dashed line locates the critical end-point temperature at Tc = 299.86 K.

added to this list;23 thus, E = 6. The maximum number of linearly independent reactions that can be supported by 10 components made up from six elements is R = C − E = 10 − 6 = 4, which agrees with the scheme consisting of eqs 2.3, 3.1, 3.7, and 3.8. As the mixture consists of a liquid phase and the three coexisting solid phases, Al2O3(s), Cu2O(s), and Cu(s), we conclude that ϕ = 4. Equations expressing the conservation of the chemical elements include eqs 2.4 and 2.5 and the following Conservation of O:

11,13−18,20

None of our previous solubility experiments have included a redox couple. The disproportionation of Cu+(aq) to form Cu2+(aq) and copper metal, as shown in eq 3.8, fills this gap. According to eqs 2.3, 3.1, 3.7, and 3.8, the reaction mixture consists of the components: HA(aq), H2O(aq), H3O+(aq), A−(aq), Al2O3(s), Al3+(aq), Cu2O(s), Cu+(aq), Cu2+(aq), and Cu(s); hence, C = 10. The stoichiometric formulae for these 10 components include the five “elements”: H, A−, O, Al3+, and Cu. Because eq 3.8 involves redox, however, the electron serves as a pseudo-element and must be

o o 3n Al + nCu + n Ho 2O 2O3 2O

= 3n Al 2O3 + nCu 2O + n H2O + n H3O+

(3.9)

Conservation of Al: o 2n Al = 2n Al 2O3 + n Al3+ 2O3

5548

(3.10) DOI: 10.1021/acs.jpcb.9b02978 J. Phys. Chem. B 2019, 123, 5545−5554

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The Journal of Physical Chemistry B Conservation of Cu: o 2nCu = 2nCu 2O + nCu+ + nCu2+ + nCu 2O

(3.11)

Conservation of charge: n H3O+ + nCu+ + 2nCu 2+ + 3n Al3+ − n A− = 0

(3.12)

We can remove the reference to initial conditions by forming the linear combination: eq 2.4−eq 2.5 − 2 × eq 3.9 + 3 × eq 3.10 + eq 3.11. The result is n H3O+ + nCu+ + nCu 2+ + 3n Al3+ − n A− + nCu = 0

(3.13)

Although free of initial conditions, eq 3.13 fails as a constraint equation because it mixes the variable, nCu, that describes the number of moles of copper in the solid metal phase with other variables, all of which refer to the composition of the liquid phase. We notice, however, that eq 3.12 by itself satisfies the single-phase condition. After the division of eq 3.12 through by n, we obtain X H3O+ + XCu+ + 2XCu 2+ + 3X Al3+ − X A− = 0

Figure 2. van’t Hoff plot of the temperature, T, dependence of the concentration, s, of aluminum dissolved in critical 38.8 mass % aqueous isobutyric acid in equilibrium with excesses of aluminum oxide and copper(I) oxide. The standard state of dissolved Al3+ was taken to be so = 0.1 ppm. The solid line was fitted to data collected above the critical point and then extrapolated into the critical region. The vertical dashed line locates the critical end-point temperature at Tc = 301.22 K.

(3.14)

Equation 3.14 states that the liquid phase must be electrically neutral. As no other constraint equations are possible, we conclude that I = 1. Upon substitution of C = 10, R = 4, ϕ = 4, and I = 1 into eq 2.2, the number of independent intensive variables is found to be F = 10 − 4 − 4 − 1 + 2 = 3. The liquid samples collected from this experiment were analyzed for their aluminum content using a Thermo Scientific iCAP 6500 Inductively Coupled Plasma Optical Emission Spectrometer. The concentrations, s, of the dissolved aluminum are listed as a function of temperature in Table 2. These data are plotted in van’t Hoff form in Figure 2.

H3O+(aq) + CrO24 −(aq) → HCrO−4 (aq) + H 2O 2H3O+(aq) + 2CrO24 −(aq) → Cr2O27 −(aq) + 3H 2O

(3.17)

According to eqs 2.3 and 3.15−3.17, the components are HA(aq), H2O(aq), H3O+(aq), A−(aq), BaCrO4(s), Ba2+(aq), − 2− CrO2− 4 (aq), HCrO4 (aq), and Cr2O7 (aq); hence, C = 9. These components are spanned by the chemical elements, H, A−, O, Ba2+, and Cr; thus, E = 5. The maximum number of linearly independent reactions is R = 9 − 5 = 4, which is consistent with the scheme summarized by eqs 2.3 and 3.15−3.17. The coexisting phases are the liquid phase and BaCrO4(s); hence, ϕ = 2. The equations of conservation of the chemical elements include eq 2.5 and Conservation of H:

Table 2. Concentration, s, of Dissolved Aluminum in Critical 38.8 Mass % Aqueous Isobutyric Acid in Equilibrium with Excesses of Aluminum Oxide and Copper(I) Oxide temperature (K)

s (ppm)

306.14 305.30 305.05 304.43 303.56 303.35 303.00 302.84 302.66 302.29 302.12 301.74 301.41 301.22

115.6 111.5 103.7 96.00 88.57 26.52 12.26 10.07 5.877 5.877 3.863 1.903 0.9447 0.9447

o nHA + 2n Ho 2O = nHA + 2n H2O + 3n H3O+ + n HCrO−4

(3.18)

Conservation of O: o n Ho 2O + 4n BaCrO = n H2O + n H3O+ + 4n BaCrO4 + 4nCrO24− 4

+ 4n HCrO−4 + 7nCr2O27−

(3.19)

Conservation of Ba2+: o n BaCrO = n BaCrO4 + n Ba 2+ 4

(3.20)

Conservation of Cr: o n BaCrO = n BaCrO4 + nCrO24− + n HCrO−4 + 2nCr2O27− 4

(3.21)

o A relation, which is independent of the initial conditions, nH , 2O o o nHA, and nBaCrO4, can be derived by forming the linear combination: −eq 2.5 + eq 3.18 − 2 × eq 3.19 + 8 × eq 3.20. The result is

3.5. Dissolution of Barium Chromate. The dissolution of BaCrO4(s) in isobutyric acid + water produces Ba2+(aq) and + CrO2− 4 (aq). The latter can react with H3O (aq) in two ways. In the first reaction, the product is HCrO2− 4 (aq), whereas in the second, the “dimer”, Cr2O2− 7 (aq) is formed. The overall reaction scheme includes eq 2.3 and the following BaCrO4 (s) → Ba 2 +(aq) + CrO24 −(aq)

(3.16)

n H3O+ + 8n Ba 2+ − 8nCrO24− − 7n HCrO24− − 14nCr2O27− − n A−

(3.15)

=0 5549


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The Journal of Physical Chemistry B A second equation can be derived by subtracting eq 3.21 from eq 3.20. The result is n Ba 2+ − nCrO24− − n HCrO24− − 2nCr2O27− = 0

(3.23)

Although eq 3.23 states the stoichiometric requirement that the number of barium atoms in the liquid phase must equal the number of chromium atoms, the physical meaning of eq 3.22 is a bit obscure. Its significance is clarified, however, if we subtract six times eq 3.23 from eq 3.22. The result is 2n Ba 2+ + n H3O+ − 2nCrO24− − n HCrO−4 − 2nCr2O27− − n A− = 0 (3.24)

Equation 3.24 is easily recognized as the liquid-phase charge neutrality condition. If we divide eqs 3.23 and 3.24 by the total number of moles in the liquid phase, n, we obtain X Ba 2+ = XCrO24− + X HCrO24− + 2XCr2O27−

Figure 3. van’t Hoff plot of the temperature, T, dependence of the concentration, s, of total chromium dissolved in critical 38.8 mass % aqueous isobutyric acid in equilibrium with excess barium chromate. The standard state of dissolved chromium was taken to be so = 0.00275 g/L, which is the aqueous solubility of barium chromate at 293.15 K. The solid line was fitted to data collected above the critical point and then extrapolated into the critical region. The vertical dashed line locates the critical temperature at Tc = 299.07 K.

(3.25)

and 2X Ba 2+ + X H3O+ − 2XCrO24− − X HCrO−4 − 2XCr2O27− − X A− (3.26)

=0

respectively, which qualify as constraint equations. We thus have I = 2. Upon substitution of C = 9, R = 4, ϕ = 2, and I = 2 into eq 2.2, the number of independent intensive variables is found to be F = 9 − 4 − 2 − 2 + 2 = 3. The concentration of dissolved chromium, s, in the liquid phase was determined by optical absorption spectrophotometry using a Genesys 10S UV−vis spectrophotometer. Our experimental data for s are listed as a function of the temperature in Table 3. These data are plotted in van’t Hoff form in Figure 3.

The results obtained from a number of other experiments provide support to this hypothesis. Table 4 summarizes these examples. The first column of the table lists the reactant or reactants, or in the case of an inert substance, the solute that was added to the liquid mixture. The chemical element symbols, which appear adjacent to the designation, MxOy(s), refer to the metals whose oxides were individually tested for their solubility. Where two oxides appear together, for example, in the case of Al2O3(s) + MnO2(s) and Al2O3(s) + Cu2O(s), the oxide pairs were in simultaneous equilibrium with the liquid. These metal oxide solubility mixtures, along with the entry for BaCrO4(s), correspond to the three experiments, which we have described above. The second column identifies the binary liquid mixture used as the solvent, whereas the third column lists the critical solution temperature as an upper (UCST) or a lower (LCST) critical solution temperature. The columns labeled C, E, R, Φ, I, and F give the numerical values of these variables, as defined by eqs 2.1 and 2.2. The sign (either + or −) in the next to the last column refers to the sign of the enthalpy, ΔH, of the reaction/dissolution. The last column, labeled “slope”, gives the sign of the divergence exhibited by the slope of the van’t Hoff plot in the critical region. Below, we support our case for F = 3 by providing or citing a proof for each of the numerical values listed in the table for the variables C, E, R, Φ, I, and F. In considering the examples in the table, we start with the top line and proceed to the bottom line through increasing values for C. The first example involves the extraction of solid phenolphthalein into a critical mixture of nitrobenzene + dodecane (UCST).20 All three components are mutually inert, so C = 3, R = 0, and the number of constraints is I = 0. Excess solid phenolphthalein is in equilibrium with the liquid phase, so ϕ = 2. Evaluation of eq 2.2 gives F = 3 − 0 − 2 − 0 + 2 = 3. In agreement with the F = 3 hypothesis, the phenolphthalein solubility, s, in the critical region has been observed to depart from the van’t Hoff background.20 The next level of complexity in Table 4 is presented by the reaction of NO2 to form N2O4 in the critical mixture of carbon tetrachloride + perfluoromethylcyclohexane (UCST).9 At

4. DISCUSSION The three experiments described above suggest a correlation between F = 3 and the occurrence of a critical solubility effect. Table 3. Concentration, s, of Total Chromium Dissolved in Critical 38.8 Mass % Aqueous Isobutyric Acid in Equilibrium with an Excess of Barium Chromate temperature (K)

s (g/L)

299.07 299.1 299.18 299.21 299.26 299.32 299.41 299.47 299.58 299.67 299.79 299.89 300.01 300.59 303.71 306.15 309.29 312.58 314.44 319.23

0.065 0.061 0.057 0.047 0.052 0.047 0.048 0.044 0.044 0.045 0.040 0.039 0.038 0.036 0.036 0.035 0.033 0.033 0.032 0.033 5550


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Table 4. Summary of Critical Effects in Equilibrium Measurements of Solutes and Reactants in Binary Liquid Mixtures with a Critical Point of Solution reactant/solute

solvent pair

crit pt

C

E

R

Φ

I

F

ΔH

slopea

phenolphthalein NO2(g)c MxOy(s): Cu(II)e, Co(II)e, Ni(II)f, Fe(III)g, Mn(IV)i, Ce(IV)j, In(VI)k MxOy(s): Sn(II)e, Al(III)i Co3O4(s)g BaO2(s)l phenolphthalein + NaOHm Al2O3(s) + MnO2(s) BaCrO4(s) Al2O3(s) + Cu2O(s) PbSO4(s) + KI(aq)q

C6H5NO2/C12H26 CCl4/PFd IBAh/H2O IBA/H2O IBA/H2O IBA/H2O 2BEn/H2O IBA/H2O IBA/H2O IBA/H2O IBA/H2O

UCST UCST UCST UCST UCST UCST LCST UCST UCST UCST UCST

3 4 6 6 7 7 8 8 9 10 11

N/A 3 4 4 5 5 5 5 5 6 7

0 1 2 2 2 2 3 3 4 4 4

2 2 2 2 2 2 1 3 2 4 3

0 0 1 1 2 2 3 1 2 1 2

3 3 3 3 3 3 3 3 3 3 4

− N/A + − + + + + − + +

+ N/A − + − − − −o + −p none

b

a

Slope refers to the sign of the critical divergence in the van’t Hoff slope of the measured solubility. bSee ref 20. cSee ref 9. dPF refers to perfluoromethylcyclohexane. eSee ref 18. fSee ref 17. gSee ref 16. hIBA refers to isobutyric acid. iSee ref 11. jSee ref 14. kSee ref 15. lSee ref 13. m See ref 31. n2BE refers to 2-butoxyethanol. oThe sign refers to the slope of the divergence as determined by the measured concentration of Mn4+ and not Al3+. pThe sign refers to the slope of the divergence as measured by the concentration of Al3+ and not Cu+. qSee ref 14.

oxide,14 and indium(VI) oxide15 in a critical mixture of IBA/ H2O (UCST). The reaction scheme consists of eq 2.3 and the generic metal oxide dissolution equation

equilibrium, the system consists of NO2, N2O4, CCl4, and C7F13, so C = 4. In this experiment, which was carried out at constant pressure,9 a gas phase was in equilibrium with the liquid phase; hence, ϕ = 2. For the purpose of enumerating the chemical elements, carbon tetrachloride will be abbreviated as CT, while perfluoromethylcyclohexane will be abbreviated as PF. Both CT and PF are inert with respect to one another and also with respect to NO2; hence, CT and PF can be considered as pseudo-elements. The NO2 molecule functions as an “element” when dimerizing to form N2O4; hence E = 3. The maximum number of linearly independent reactions involving four components spanned by three elements is R = 4 − 3 = 1; hence, we need to consider only 2NO2 (l) → N2O4 (l)

MxOy (s) + xZ H3O+(aq) → x MZ +(aq) + 3y H 2O(aq) (4.5)

where the metal, MZ+, exhibits the oxidation state, Z+. According to eqs 2.3 and 4.5, the mixture consists of the six components: HA(aq), H2O(aq), H3O+(aq), A−(aq), MxOy(s), and MZ+(aq). The elements are H, O, MZ+, and A−, so E = 4. The maximum number of linearly independent reactions is R = C − E = 6 − 4 = 2, which accounts for eqs 2.3 and 4.5. There are two phases, the solid metal oxide, MxOy(s), and the liquid phase, so ϕ = 2. For ease in keeping track of the algebra associated with the metal oxide stoichiometry, where (2y/x) = Z, our list of the elemental conservation relations repeats the equations for hydrogen (previously eq 2.4) and isobutyrate (previously eq 2.5). The full list is

(4.1)

noY

To search for equations of constraint, we let be the initial (l) total number of moles of component, Y, nY be the equilibrium number of moles of Y in the liquid phase, and n(g) Y be the equilibrium number of moles of Y in the gas phase. The laws of conservation of elemental masses can then be expressed in the form Conservation of NO2: (g) o (l) n NO = n NO + n NO + 2n N(l)2O4 + 2n N(g)2O4 2 2 2

Conservation of H: o nHA + 2n Ho 2O = nHA + 2n H2O + 3n H3O+

Conservation of O: o yn M + n Ho 2O = yn M xOy + n H2O + n H3O+ xOy

(4.2)

o xn M = xn MxOy + n MZ+ xOy

(4.3)

=

(l) nPF

+

(g) nPF

(4.8)

Conservation of A−:

Conservation of PF: o nPF

(4.7)

Conservation of M:

Conservation of CT: (g) o (l) nCT = nCT + nCT

(4.6)

o nHA = nHA + n A−

(4.4)

(4.9)

If we form the linear combination: eq 4.6 − 2 × eq 4.7 + Z× eq 4.8−eq 4.9, and divide by the total number of moles in the liquid phase, we obtain

Because eqs 4.2−4.4 have no quantities in common, it is impossible to construct a constraint equation, which is free of the initial conditions. We thus conclude that I = 0. If we substitute the values C = 4, R = 1, ϕ = 2, and I = 0 into eq 2.2, we obtain F = 4 − 1 − 2 − 0 + 2 = 3. Tveekrem, Cohn, and Greer reported a critical effect in this system.9 Their observation is consistent with our calculation of F = 3. The next case, C = 6, is encountered upon the dissolution of the individual metal oxides, copper(II) oxide,18 cobalt(II) oxide,18 nickel(II) oxide,17 tin(II) oxide,18 iron(III) oxide,16 aluminum oxide,11 manganese(IV) oxide,11 cerium(IV)

X H3O+ + ZX MZ+ − X A− = 0

(4.10)

which is the electroneutrality condition. We conclude I = 1. Upon substitution of C = 6, R = 2, ϕ = 2, and I = 1 into eq 2.2, we calculate F = 6 − 2 − 2 − 1 + 2 = 3. In all cases where the metal exhibits the single oxidation state, Z+, the corresponding oxide, MxOy(s), exhibits a critical solubility effect when reacting with IBA/H2O.11,14−18 As the metals, MZ+, have included copper, cobalt, nickel, tin, iron, aluminum, 5551


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phenolphthalein in 2-butoxyethanol + water (LCST). This experiment, in which hydroxide ion reacts with the pink dianion form of phenolphthalein to produce the colorless trivalent anion form, was carried out by Shen et al.31 To simplify the phase rule analysis, we shall denote the colorless neutral lactone32 form of phenolphthalein by PH2, the pink divalent quinonoid32 form by P2−, and the colorless trivalent carbinol32 form by POH3−. The inert solvent component, 2butoxyethanol, will be represented by the symbol, 2BE. In the experimental arrangement of Shen et al.,31 the starting materials consisted of PH2, NaOH, 2BE, and H2O. The four starting materials and their reaction products occupied a single liquid phase, so ϕ = 1. As phenolphthalein is pH-sensitive, we include the autoionization of water in our proposed reaction scheme, which is

manganese, cerium, and indium, the existence of the metal oxide solubility critical effect would seem to be independent of the position of the metal in the periodic table. We consider next in Table 4 the dissolution of the mixed valence metal oxide, Co3O4, in IBA/H2O (UCST),16 a case for which C = 7. The reaction scheme consists of eq 2.3 and Co3O4 (s) + 8H3O+(aq) → Co2 +(aq) + 2Co3 +(aq) + 12H 2O(aq) +

(4.11) −

The components are HA(aq), H2O(aq), H3O (aq), A (aq), Co3O4(s), Co2+(aq), and Co3+(aq). These seven components are spanned by the five elements, which are H, O, A−, Co2+, and Co3+. Because the two cobalt oxidation states are not linked by a redox reaction, they can be treated as separate chemical elements; hence, E = 5. The number of linearly independent reactions involving seven components spanned by five elements is R = 7 − 5 = 2. These two reactions are shown in eqs 2.3 and 4.11. There are two coexisting phases, the liquid and Co3O4(s), so ϕ = 2. The element mass conservation equations are eqs 2.4 and 2.5 and Conservation of O: n Ho 2O

+

o 4nCo 3O4

= n H2O + n H3O+ + 4nCo3O4

(4.12)

(4.13)

3+

Conservation of Co : o 2nCo = 2nCo3O4 + nCo3+ 3O4

(4.14)

P2 −(aq) + OH−(aq) → POH3 −(aq)

(4.20)

(4.21)

−

Conservation of OH : o o nHOH + nNaOH = nHOH + nOH− + n H3O+ + nPOH3−

(4.15)

(4.22)

Conservation of P: o nPH = nPH2 + nP2− + nPOH3− 2

(4.16)

(4.23)

Conservation of Na+:

Although eq 4.16 is clearly a constraint equation based on stoichiometry, the meaning of eq 4.15 is not obvious. A more transparent equation is obtained by forming the linear combination of eq 4.15 − 3 × eq 4.16, which gives X H3O+ + 2XCo2+ + 3XCo3+ − X A− = 0

(4.19)

o o nHOH + 2nPH = nHOH + 2nPH2 + 2n H3O+ 2

A second linear combination consisting of 2 × eq 4.13−eq 4.14 gives 2XCo2+ = XCo3+

PH 2(aq) + 2OH−(aq) → P2 −(aq) + 2H 2O(aq)

Conservation of H:

If we form the linear combination: eq 2.4−eq 2.5 − 2 × eq 4.12 + 8 × eq 4.13, followed by the division of the total number of moles in the liquid phase, we obtain X H3O+ + 8XCo2+ = X A−

(4.18)

Given the identity of the starting materials and eqs 4.18−4.20, the equilibrium composition comprises H2O(aq), H3O+(aq), OH−(aq), PH2(aq), P2−(aq), POH3−(aq), Na+(aq), and 2BE(aq). Hence, C = 8. These eight components are spanned by the elements, which we choose to be H, OH−, P, Na+, and 2BE; hence, E = 5. The maximum number of linearly independent reactions is R = 8 − 5 = 3, which is consistent with eqs 4.18−4.20. The elemental conservation equations are

Conservation of Co2+: o nCo = nCo3O4 + nCo2+ 3O4

H 2O(aq) + H 2O(aq) → H3O+(aq) + OH−(aq)

o nNaOH = n Na+

(4.24)

Conservation of 2BE: o n2BE = n2BE

(4.17)

(4.25)

If we form the linear combination: eq 4.21−eq 4.22 − 2 × eq 4.23 + eq 4.24, and divide by the total number of moles, n, we arrive at

Equation 4.17 can be recognized as the electroneutrality condition. Eqs 4.16 and 4.17 serve as constraints; hence I = 2. Since C = 7, R = 2, ϕ = 2, and I = 2, the evaluation of eq 2.2 gives F = 7 − 2 − 2 − 2 + 2 = 3. We note that a critical effect in the total dissolved cobalt concentration has been reported.16 The dissolution of BaO2 in IBA/H2O (UCST)13 is also described by C = 7. Like the dissolution of Co3O4 in isobutyric acid + water, this system includes an element exhibiting more than one oxidation state. Specifically, the oxygen in BaO2 is in the −1 state, whereas the oxygen in the water solvent is in the −2 state. Evaluation of F is based upon C = 7, R = 2, ϕ = 2, and I = 2. According to eq 2.2, F = 7 − 2 − 2 − 2 + 2 = 3. A critical effect in the concentration of dissolved Ba2+ has been observed.13 In Table 4, the next level of complexity, C = 8, is encountered in the case of the alkaline fading reaction of

X Na+ + X H3O+ − XOH− − 2X P2− − 3X POH3− = 0

(4.26)

which we recognize as the charge neutrality condition. No use has yet been made of eq 4.25, so we divide eqs 4.23 and 4.24 through by eq 4.25 to obtain o nPH 2 o n2BE

=

nPH2 + nP2− + nPOH3− n2BE

(4.27)

and o nNaOH n + = Na o n2BE n2BE

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4.30 as constraint equations;22 hence, I = 3. With C = 8, R = 3, ϕ = 1, and I = 3, we use eq 2.2 to compute F = 8 − 3 − 1 − 3 + 2 = 3. The slope of the van’t Hoff plot of the concentration of POH3− vs 1/T was negative outside the critical region;31 hence, ΔH > 0. Inside the critical region, the slope diverged in the negative sense in agreement with eq 2.9. One should note that 2-butoxyethanol + water (LCST) is characterized by a lower critical solution temperature, as opposed to the other three solvent pairs mentioned above, all of which have upper critical solution temperatures. We thus conclude that both kinds of critical points are capable of supporting a critical reaction equilibrium effect. Moreover, the phenolphthalein fading reaction is a strictly homogeneous equilibrium, which distinguishes it from the other reaction schemes discussed above, all of which include at least one heterogeneous equilibrium.9,11,13−18,20 To complete our discussion of critical effects in reaction systems, we refer to Figures 1−3. A van’t Hoff slope critical effect is apparent in each of the three IBA/H2O (UCST)-based systems: Al2O3(s) + MnO2(s) for which C = 8 (Figure 1), BaCrO4(s) for which C = 9 (Figure 3), and Al2O3(s) + Cu2O(s) for which C = 10 (Figure 2). In each case, we calculated F = 3, and a critical effect was observed. In Figures 1 and 2, the slope of the van’t Hoff plot outside the critical region is negative, indicating that ΔH > 0. In both cases, the slope in the critical region diverges in the negative sense in agreement with eq 2.9. By contrast, in Figure 3, everything is reversed. The slope outside the critical region is positive, indicating that ΔH < 0, and the slope in the critical region diverges in a positive sense. This agrees with eq 2.9. Given the close association of an equilibrium critical effect with F = 3, it is important to examine what happens when F = 4. The reaction of KI with PbSO4(s) to produce PbI2(s) in IBA/H2O (UCST) serves this purpose.14 The results are tabulated in the last line of Table 4. For this system, C = 11, R = 4, ϕ = 3, and I = 2.14 Upon substitution of these results into eq 2.2, we obtain F = 11 − 4 − 3 − 2 + 2 = 4. Measurements of the temperature dependence of the concentration of dissolved lead revealed no critical effect.14 Finally, we should add that the association of F = 3 with the observation of a critical point divergence in a thermodynamic derivative is not restricted to chemical effects but includes physical effects as well.33 The isobaric heat capacity CP,X, for example, is given by CP,X = T(∂S/∂T)P,X, where S is the specific entropy (a density), T is the temperature (a field), and X is the mole fraction of one of the components. In the case of a mixture of carbon disulfide + methanol, CP,X diverges as T → Tc.34 In this mixture, the components are inert, and there is but one phase, so, C = 2, R = 0, ϕ = 1, and I = 0. Equation 2.2 gives F = 2 − 0 − 1 − 0 + 2 = 3. By contrast, the mixture of IBA/ H2O sustains an ionization reaction and contains four components, being described by C = 4, R = 1, ϕ = 1, and I = 1; nonetheless, F = 3. In agreement with the F = 3 rule, the value of CP,X is observed to diverge for this mixture as T → Tc.34

respectively. After the definition of the mole fraction, X2BE = n2BE/n, and division of the numerators and denominators of the right-hand sides of eqs 4.27 and 4.28 through by n, we arrive at o ij nPH yz X PH2 + X P2− + X POH3− − jjjj o 2 zzzzX 2BE = 0 k n2BE { and

(4.29)

ij n o yz zzX X Na+ − jjj NaOH zz 2BE = 0 o j n2BE (4.30) k { respectively. Each of eqs 4.26, 4.29, and 4.30 is a candidate constraint equation. Eq 4.26, which is free of any explicit reference to the initial conditions, qualifies because it is necessarily independent of the extent of reaction. By contrast, eqs 4.29 and 4.30 involve the initial conditions, noPH2, noNa+, and no2BE, and must be proved to be independent of the extent of reaction. This can be demonstrated by showing that they are orthogonal to the reaction coefficient matrix, i.e., Jouguet matrix.14,22 For the sake of completeness, we shall subject all three of eqs 4.26, 4.29, and 4.30 to this test. The process begins by using the coefficients in eqs 4.26, 4.29, and 4.30 to construct the matrix shown in Table 5. The rows are identified by the

Table 5. Jouguet Matrix of Coefficients of the Mole Fractions in Equations 4.14, 4.17, and 4.18 H2O

H3O+

OH−

PH2

P2−

POH3−

Na+

eq 4.26 eq 4.29

0 0

1 0

−1 0

0 1

−2 1

−3 1

1 0

eq 4.30

0

0

0

0

0

0

1

2BE 0 −(noPH2/ no2BE) −(noNa+/ no2BE)

equation numbers and the columns are identified by the chemical species. Each entry in the matrix is identical to the coefficient of the corresponding mole fraction, as found in the numbered equation. The reaction coefficient matrix, i.e., Jouguet matrix, is shown in Table 6. The magnitude of an Table 6. Matrix of Stoichiometric Coefficients in Equations 4.6−4.8 a eq 4.18

eq 4.19

eq 4.20

species

−2 1 1 0 0 0 0 0

2 0 −2 −1 1 0 0 0

0 0 −1 0 −1 1 0 0

H2O H3O+ OH− PH2 P2− POH3− Na+ 2BE

a

Negative signs indicate reactants. Positive signs indicate products.

entry in this matrix is identical with the stoichiometric coefficient of the corresponding species as it appears in eqs 4.18−4.20. The sign of the entry is negative if the species is a reactant and positive if the species is a product. To test for orthogonality, the columns of the Jouguet matrix in Table 6 are left multiplied by the rows of the matrix in Table 5. The result is seen to be the null matrix, which establishes the orthogonality requirement and qualifies eqs 4.26, 4.29, and

5. CONCLUSIONS In Table 4, we have summarized the results for all reported systems exhibiting an equilibrium critical effect where the solvent was a binary liquid mixture with a critical point of solution. The number of chemical components involved in a given system ranges from 3 to 11. In all cases where the phase rule calculation resulted in F = 3, a critical effect in the position 5553


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(12) Gitterman, M. Singularity in the degree of dissociation near a critical point. Phys. Rev. A 1983, 28, 358−360. (13) Baird, J. K.; Hu, B.; Lang, J. R.; Richey, R. D. Phase rule method for applying the principle of critical point universality to chemically reacting open systems. J. Mol. Liq. 2015, 204, 222−226. (14) Baird, J. K.; Baker, J. D.; Hu, B.; Lang, J. R.; Joyce, K. E.; Sides, A. K.; Richey, R. D. A chemical test of the principle of critical point isomorphism: Reactive dissolution of ionic solids in isobutyric acid + water near the consolute point. J. Phys. Chem. B 2015, 119, 4041− 4047. (15) Hu, B.; Richey, R. D.; Baird, J. K. Chemical equilibrium and critical phenomena: solubility of indium oxide in isobutyric acid + water near the consolute point. J. Chem. Eng. Data 2009, 54, 1537− 1540. (16) Hu, B.; Baird, J. K. Chemical Equilibrium and Critical Phenomena: The solubilities of iron(III) oxide and cobalt(II,III) oxide in isobutyric acid + water near the consolute point. Int. J. Thermophys. 2010, 31, 717−726. (17) Hu, B.; Baird, J. K.; Richey, R. D.; Reddy, R. G. A chemical test of the principle of critical point universality: The solubility of nickel (II) oxide in isobutyric acid + water near the consolute point. J. Chem. Phys. 2011, 134, 154505−154512. (18) Hu, B.; Baird, J. K.; Alvarez, P. K.; Melton, K. C.; Barlow, D. A.; Richey, R. D. Diverging thermodynamic derivatives associated with heterogeneous chemical equilibrium in a binary liquid mixture with a consolute point. Int. J. Thermophys. 2014, 35, 841−852. (19) Procaccia, I.; Gitterman, M. Supercritical extraction at atmospheric pressures. J. Chem. Phys. 1983, 78, 5275−5276. (20) Baird, J. K.; Lang, J. R.; Wang, X.; Huang, S.; Mukherjee, A. Anomalous solubility of an inert solid in a binary liquid mixture with a critical point of solution. J. Phys. Chem. B 2018, 122, 2949−2956. (21) Savoy, J. D.; Baird, J. K.; Lang, J. R. Ion exchange at the critical point of solution. J. Chromatogr. A 2016, 1437, 58−66. (22) Lee, V. J. Generalization of the Gibbs phase rule for heterogeneous chemical equilibrium. J. Chem. Educ. 1967, 44, 164− 166. (23) Kyle, B. G. Chemical and Process Thermodynamics, 3rd ed.; Prentice Hall PTR: Upper Saddle River, NJ, 1999; Chapter 13. (24) Wheeler, J. C. Singularity in the degree of dissociation of isobutyric acid, water solutions at the critical point of solution. Phys. Rev. A 1984, 30, 648−649. (25) Tveekrem, J. L.; Jacobs, D. T. Impurity effects in a near critical binary-fluid mixture. Phys. Rev. A 1983, 27, 2773−2776. (26) Bouanz, M.; Beysens, D. Effect of ion impurities on a binary mixture of isobutyric acid and water. Chem. Phys. Lett. 1994, 231, 105−110. (27) Kim, Y. W.; Baird, J. K. Kinetics of SN1 reactions in binary liquid mixtures near the critical point of solution. J. Phys. Chem. A 2003, 107, 8435−8443. (28) Toumi, A.; Bouanz, M. Effect of the (K+, Cl−) Ions on the order parameters and on the Lorentz−Lorenz relation in the isobutyric acid−water critical mixture. J. Mol. Liq. 2005, 122, 74−83. (29) Venkatesu, P. Polymer modifies the critical region of the coexisting liquid phases. J. Phys. Chem. B 2006, 110, 17339−17346. (30) Reddy, P. M.; Venkatesu, P.; Bohidar, H. B. Influence of polymer molecular weight and concentration on coexistence curve of isobutyric acid + water. J. Phys. Chem. B 2011, 115, 12065−12075. (31) Du, Z.; Yin, H.; Hao, Z.; Zeng, P.; Shen, W. Critical anomalies of alkaline fading of phenolphthalein in the critical solution of 2butoxyethanol + water. J. Chem. Phys. 2013, 139, 224501−224508. (32) Kunimoto, K. K.; Sugiura, H.; Kato, T.; Senda, H.; Kuwae, A.; Hanai, K. Molecular structure and vibrational spectra of phenolphthalein and its dianion. Spectrochim. Acta, Part A 2001, 57, 265−271. (33) Baird, J. K.; Wang, X.; Lang, J. R.; Norris, P. Phase rule classification of physical and chemical critical effects in liquid mixtures. Chem. Phys. Lett. 2019, 729, 73−78. (34) Kumar, A.; Krishnamurthy, H. R.; Gopal, E. S. R. Equilibrium critical phenomena in binary liquid mixtures. Phys. Rep. 1983, 98, 57− 143.

of equilibrium was reported. The effect could be observed at both upper and lower critical solution temperatures. As predicted by eq 2.9 and summarized in Table 4, the sign of the divergence of the van’t Hoff slope in the critical region was opposite to the sign of the enthalpy of reaction. Whereas the enthalpy of reaction is negative11 when Al2O3 dissolves alone in a mixture of IBA/H2O, it changes to slightly positive, as can be seen in Figure 2, when Al2O3 dissolves together with Cu2O in this solvent. Although this change in the sign is not understood, the sign of ΔH in both cases is still opposite to that of the corresponding divergence in the van’t Hoff slope in the critical region. In the case of the reaction of KI with PbSO4 in IBA/H2O (UCST),14 the phase rule predicts F = 4. The absence of a divergence in the slope of the van’t Hoff plot for this reaction confirms the Griffiths−Wheeler rule that if two density variables are fixed, a critical effect is forbidden. In the case of chemical reactions at a fixed temperature and pressure in binary liquid mixtures with a critical point of solution, it would thus seem that a phase rule calculation resulting in F = 3 has the same force in predicting a critical effect as the principle of critical point universality.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

James K. Baird: 0000-0002-3995-8641 Anusree Mukherjee: 0000-0001-8311-7738 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

This research did not benefit from financial support from any governmental or philanthropic agency.

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