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Anomalous Solubility of an Inert Solid in a Binary Liquid Mixture with a Critical Point of Solution James K. Baird,*,†,‡,§ Joshua R. Lang,†,§ Xingjian Wang,†,‡,§ Sijay Huang,∥ and A. Mukherjee†,§ †

Department of Chemistry, ‡Department of Physics, §Materials Science Graduate Program, ∥Department of Electrical and Computer Engineering, University of Alabama in Huntsville, Huntsville, Alabama 35899, United States ABSTRACT: We consider the dissolution of a chemically inert solid in a binary liquid mixture with a critical point of solution. When the mixture, acting as the solvent, has come to equilibrium with the solid, the state of the system is completely described by the temperature, pressure, and a concentration variable formed by dividing the molar amount of one solvent component by that of the other. Under conditions of fixed pressure, the principle of critical point isomorphism predicts that the slope of a van’t Hoff plot of the solubility of the solid should diverge toward infinity as the temperature enters the critical region. The sign of the divergence is negative when the dissolution is endothermic, whereas it is positive when the dissolution is exothermic. In experiments where excess solid phenolphthalein dissolves in a binary mixture of nitrobenzene + dodecane, we have observed exothermic dissolution concurrently with a positive divergence of the van’t Hoff slope. The data are insufficiently precise to compute an accurate numerical value for the exponent of the temperature power law expected to govern this divergence; nevertheless, on the basis of Widom scaling theory, we argue that the exponent should be equal to 0.326, which is identical to the value of the exponent that governs the temperature dependence of the shape of the liquid−liquid coexistence curve. Being entirely physical in nature, the anomalous solubility effect should be observable in the case of any chemically inert solid dissolving in any one of the more than 1000 liquid pairs known to have a critical point of solution.

1. INTRODUCTION When two chemical substances form a liquid mixture with a miscibility gap, a pair of immiscible liquid phases exists on the concave side of the phase boundary, whereas a single liquid phase exists on the convex side. When plotted in the form of temperature T versus mole fraction X, the phase boundary often ends in a critical point of solution where the two immiscible liquid phases merge. The coordinates of this point define the critical solution temperature, Toc , and the critical composition, Xoc . If the coexistence curve is concave up, then Toc is a lower critical solution temperature (LCST), whereas if it is concave down, then Toc is an upper critical solution temperature (UCST).1 More than 1000 pairs of liquids are known to exhibit this kind of behavior.2 If a small amount of a third component is added to such a binary liquid mixture, the critical temperature Toc is shifted to a new value, Tc, which can be either higher or lower than Toc , depending upon the nature and concentration of the third component.3−9 If the third component is a solid, which is allowed to come to dissolution equilibrium with the binary mixture, the mole fraction X3(T) of the third component in the liquid phase is a function of the thermostat temperature, T. This makes the critical temperature, Tc, an implicit function of T through the relation Tc = Tc(X3(T)). When T = Tc, the system is said to reach a critical endpoint.10 A pair of liquids forming a mixture with a critical point of solution can be used to advantage in solvent extraction.11,12 Early speculation suggested that critical effects in such a solvent mixture would not appear if the added solutes were inert. If so, © XXXX American Chemical Society

then the problem might be obviated if the solutes were in reaction equilibrium with one another or with one of the components of the solvent.11,13 The earliest experiment testing this idea was performed by Tveekrem, Cohn, and Greer, who studied the reaction of NO2 to form the dimer, N2O4, when the two gases were in solubility equilibrium with a mixture of the liquids, perfluoromethylcyclohexane + carbon tetrachloride (UCST).14 They observed a slight, yet detectable, effect of the critical point on the temperature derivative of the concentration of NO2.14 Subsequent experiments were carried out using isobutyric acid + water (UCST) to dissolve metal oxides by acid/base reaction.15−20 The metals were selected more or less randomly from across the periodic table. The study of the oxides was later extended to include measurements of the solubility of the metal peroxide, BaO2.21 In the case of both the oxides and the peroxide, the temperature derivative (∂s/∂T) of the solubility, s, measured in parts per million diverged toward infinity in the critical region.15−21 In our terminology, the divergence in ∂s/∂T makes the temperature dependence of solubility anomalous. The experiments mentioned above involved heterogeneous equilibria. A homogeneous equilibrium was first investigated by Shen et al., who used 2-butoxyethanol + water (LCST) as the solvent to study the position of equilibrium in the reaction of phenolphthalein with hydroxide ion.22 A very notable critical Received: November 8, 2017 Revised: February 19, 2018 Published: February 20, 2018 A

DOI: 10.1021/acs.jpcb.7b11058 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B

solubility temperature derivative effect should be completely independent of the materials involved.

effect was observed in the temperature derivative of the phenolphthalein concentration. These chemical results can be analyzed qualitatively in terms of the principle of critical point isomorphism. The formulation of this principle has its origin in the critical behavior of physical systems as diverse as superconductors, substitutional alloys, and ferromagnetic materials.23−28 The principle is based on a division of the intensive thermodynamic variables into two classes, labeled “densities” and “fields”.24 A density variable has a unique value in each coexisting phase. In the case of the chemical reaction equilibria mentioned above, the list of density variables includes the solubility, s, plus the mole fractions of the other components present in the liquid phase. By contrast, a field variable has the same value in each coexisting phase. In the experiments described above, the field variables are temperature, T, and pressure, P. The isomorphism principle states that when the experimental conditions are such that no more than one density variable is fixed, the derivative of a density with respect to a field, such as ∂s/∂T, will diverge toward infinity as T → Tc.24 The challenge in applying the isomorphism principle to chemical reactions lies in the determination of the number of intensive variables that are held fixed. The Gibbs phase rule,29 as extended to take into account chemical reactions,30 has recently been exploited for this task.20,21 In its extended form, the phase rule states that the number of independent (or free) intensive variables, F, is given by F=C−ϕ−R−I+2

2. THEORY 2.1. Expansion of the Gibbs Energy about Equilibrium. At constant pressure, the instantaneous Gibbs energy of a ternary liquid mixture consisting of n1, n2, and n3 moles of components (1), (2), and (3), respectively, is G(n3) = G(n1, n2, n3,T), where T is the Kelvin temperature. The number of moles, n1 and n2, of the two liquids forming the solvent mixture are fixed. We assume that component (3) is present in the solid form in sufficient excess to reach equilibrium at its solubility limit, where n3 = neq 3 . Evaluated in this limit, the Gibbs energy is eq G(neq 3 ) = G(n1, n2, n3 , T). We can expand G(n3) in a Taylor eq series about G(n3 ). The result is ⎛ ∂G ⎞eq G(n3) = G(n3eq ) + (n3 − n3eq )⎜ ⎟ ⎝ ∂n3 ⎠T , n , n 1

2

⎛ ∂ 2G ⎞eq 1 + (n3 − n3eq )2 ⎜ 2 ⎟ + ··· 2! ⎝ ∂n3 ⎠T , n , n 1

(2)

2

where the superscript, eq, denotes evaluation of the derivative at n3 = neq 3 . The criterion for the system to be in equilibrium is ⎛ ∂G ⎞eq =0 ⎜ ⎟ ⎝ ∂n3 ⎠T , n , n

(1)

1

(3)

2

The condition that the equilibrium is stable against fluctuations in n3 − neq 3 is

where C is the number of chemical components, ϕ is the number of coexisting phases, R is the number of linearly independent reactions involving the components, and I is the number of constraint equations derivable from the laws of conservation of mass and charge.30 Because the temperature and pressure are always controlled, any additional free variable beyond these two is necessarily density. Hence, according to the isomorphism principle, if F ≤ 3, we should expect a critical effect in ∂s/∂T.20,21 For example, in the case of the dissolution of CeO2 in a mixture of isobutryic acid + water (UCST),20 there were C = 6 components involved in R = 2 reactions. The system consisted of a solid phase and a liquid phase, so ϕ = 2. The sole constraint involving the concentrations was the electroneutrality condition, which made I = 1. When these numbers are substituted into eq 1, we obtain F = 3. Hence, only one density variable was fixed, and in agreement with the isomorphism principle, ∂s/∂T was observed to diverge in the critical region.20 Although the presence of a chemical reaction can provide the basis for observing a critical solubility effect,14−21 it is neither a necessary nor a sufficient requirement.20 Below, we find in the case of the dissolution of a chemically inert solid that the requirement F = 3 is always met. The density variable, which is fixed in this case, is the ratio of the mole fractions of the two solvent components, so in accordance with the isomorphism principle with this ratio fixed near its critical value, there should be a divergence in the temperature derivative of the solubility of the solid. We describe experiments involving the dissolution of excess solid phenolphthalein in a critical mixture of nitrobenzene + dodecane (UCST), the results of which confirm this prediction. We stress that the observance of a divergence in the temperature derivative of the solubility is not limited to this one example. Rather, as long as the solid and the liquids forming the mixture are chemically inert, the existence of the anomalous

⎛ ∂ 2G ⎞eq ⎜ 2⎟ >0 ⎝ ∂n3 ⎠T , n , n 1

(4)

2

At the critical endpoint, eq 4 becomes an equality. The resistance to fluctuations in composition disappears, and the variance in n3 − neq 3 grows without bound. The total Gibbs energy of the liquid mixture of components (1) and (2) in contact with an excess of (3) in the solid phase is G(n3) = n1μ1 + n2μ2 + n3μ3 + (n3̅ − n3)μ3o (s)

(5)

where the {μi} (i = 1, 2, 3) are the chemical potentials of the components in the liquid phase, μo3(s) is the chemical potential of (3) in the solid phase, and n3̅ is the number of moles of (3) initially present in the solid form. With n1 and n2 fixed, differentiation of eq 5 with respect to n3 yields ⎛ ∂G ⎞ ⎛ ∂μ ∂μ ⎞ ∂μ = ⎜n1 1 + n2 2 + n3 3 ⎟ ⎜ ⎟ ∂n3 ∂n3 ⎠ ⎝ ∂n3 ⎠T , n , n ⎝ ∂n3 T ,n ,n 1

2

1

+ (μ3 − μ3o (s))

2

(6)

If the temperature and pressure are constant, then by virtue of the Gibbs−Duhem equation, the sum of the three partial derivatives on the right-hand side of eq 6 is zero. The remaining term on the right-hand side of eq 6 can be recognized as the Gibbs energy of solution ΔG = μ3 − μ3o (s)

(7)

According to eqs 3 and 7, eq 6 implies ΔG = 0 B

(8) DOI: 10.1021/acs.jpcb.7b11058 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B when the dissolution is in equilibrium. The total differential of ΔG/T can be written as

To complete the conversion of eq 17 to intensive variables, we will need to use X3 to replace the remaining references to n3. To accomplish this, we start with eq 11 written for the case i = 3, that is n3 X3 = n1 + n2 + n3 (18)

⎡ ⎤ ⎛ ΔG ⎞ ⎢⎛ ∂(ΔG /T ) ⎞ ⎥d ⎟ = ⎜ ⎟ d⎜ ⎝ T ⎠ ⎢⎝ ⎠T , n , n , n ⎥⎦ ∂P ⎣ 1 2 3 ⎡⎛ ⎤ ∂(ΔG /T ) ⎞ ⎥ dT ⎟ P + ⎢⎜ ⎢⎣⎝ ⎠ P , n , n , n ⎥⎦ ∂T 1 2 3 ⎡ ⎤ ⎢⎛ ∂(ΔG /T ) ⎞ ⎥ + ∑ ⎢⎜ dni ⎟ ∂ n ⎠T , P , n , n ⎥ i i = 1 ⎣⎝ j k⎦

Differentiation of eq 18 with respect to T yields ⎛ n ⎞⎛ ∂X3 ⎞ ⎛ ∂n3 ⎞ ⎜ ⎟ =⎜ ⎟ ⎟⎜ ⎝ ∂T ⎠x ⎝ 1 − X3 ⎠⎝ ∂T ⎠x

3

where we used eq 12 to identify n. Differentiation of ΔG with respect to n3 yields

(9)

where P is the pressure, and the sum extends over i ≠ j ≠ k. Consider the system in equilibrium (ΔG = 0) with fixed pressure (dP = 0) and with fixed amounts of (1) and (2) (dn1 = dn2 = 0). Equation 9 can then be rewritten as ⎛ ∂n3 ⎞ ⎜ ⎟ ⎝ ∂T ⎠ P , n , n 1

=− 2 , ΔG = 0

⎛ ∂ΔG ⎞ ⎛ 1 − X3 ⎞⎛ ∂ΔG ⎞ ⎟⎜ ⎜ ⎟ =⎜ ⎟ ⎝ ∂n3 ⎠x ⎝ n ⎠⎝ ∂X3 ⎠x T (∂(ΔG /T )/∂T )P , x , X3 ⎛ ∂X3 ⎞ ⎜ ⎟ =− ⎝ ∂T ⎠ P , x , ΔG = 0 (∂ΔG /∂X3)P , x , T

(10)

2.2. Intensive Variables. Initially, the liquid phase consists solely of a mixture of (1) and (2). Upon addition of solid (3) in sufficient excess to dissolve up to its solubility limit, the composition of the resulting ternary liquid phase is specified by the three mole fractions n Xi = i , i = 1, 2, 3 (11) n

⎛ ∂(ΔG /T ) ⎞ ΔH ⎜ ⎟=− 2 ⎝ ⎠ ∂T T

⎛ ∂X3 ⎞ ΔH ⎜ ⎟ = ⎝ ∂T ⎠ P , x , ΔG = 0 T (∂ΔG /∂X3)P , x , T

(13)

Because the values of n1 and n2 are determined by the way the solvent is prepared, the ratio x=

n1 X = 1 n2 X2

⎛ ∂ln X3 ⎞ ⎛ ∂ln X3 ⎞ ⎟ = −T ΔH ⎜ ⎜ ⎟ ⎝ ∂ΔG ⎠ P , x . T ⎝ ∂(1/T ) ⎠ P , x , ΔG = 0

(14)

x(1 − X3) (1 + x)

(15)

(1 − X3) (1 + x)

(16)

3. EXPERIMENTAL SECTION 3.1. Materials and Methods. To confirm the inert solute solubility anomaly in the laboratory, we have carried out some preliminary experiments involving the dissolution of excess solid phenolphthalein in a mixture of nitrobenzene and dodecane. Phenolphthalein, nitrobenzene, dodecane, and NaOH were acquired from Sigma-Aldrich and used as received. Water was distilled once from a glass system. A binary liquid mixture of nitrobenzene and dodecane was prepared at the critical composition of 55.2 wt % nitrobenzene, which corresponded to a mole fraction Xoc = 0.37 of dodecane in the nitrobenzene solvent.31 The binary critical mixture was

Equations 15 and 16 express the composition of the ternary mixture in terms of the two intensive composition variables, x and X3. The remaining two intensive variables are temperature, T, and pressure, P. All three of T, P, and x are independent of one another. If we know n1, n2, and n3, then by virtue of eqs 11 and 14, we also know x and X3. These observations permit us to rewrite eq 10 in terms of fixed intensive variables T (∂(ΔG /T )/∂T )P , x , X3 ⎛ ∂n3 ⎞ ⎜ ⎟ =− ⎝ ∂T ⎠ P , x , ΔG = 0 (∂ΔG /∂n3)P , x , T

(24)

which gives the slope of a van’t Hoff, ln X3 versus 1/T, plot. With the value of x fixed in the vicinity of its critical value, xc, and T > Tc, the experimental data plotted in the form, ln X3 versus 1/T, lie on straight lines.15−22 Inside the critical region, the behavior of (∂ln X3/∂ΔG)P,xc,T and the sign of ΔH combine to determine the departure of the slope, (∂ln X3/∂(1/ T))P,xc,ΔG = 0, from a constant value.

and

X2 =

(23)

For comparison with experimental data, it is useful to reexpress eq 23 as

is a constant, independent of any changes in the position of equilibrium. By combining eqs 13 and 14, we obtain X1 =

(22)

where ΔH is the enthalpy of solution. Equation 21 then becomes

(12)

The sum of the mole fractions is

X1 + X 2 + X3 = 1

(21)

Equation 21 contains the same information as eq 10, except that it is written entirely in terms of the intensive variables. The numerator of the fraction on the right-hand side of eq 21 can be simplified by substituting the Gibbs−Helmholtz equation

where the total number of moles in the liquid phase is

n = n1 + n2 + n3

(20)

Upon substitution of eqs 19 and 20 into eq 17, we arrive at

T (∂(ΔG /T )/∂T )P , n1, n2 , n3 (∂ΔG /∂n3)P , T , n1, n2

(19)

(17) C

DOI: 10.1021/acs.jpcb.7b11058 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B introduced into a flat-bottom borosilicate glass sample vial. The vial was placed in an isothermal water bath. This thermostat for controlling and maintaining the temperature was as previously described.16 Using visual inspection to search for the formation of a meniscus, the initial critical temperature for the system was determined to be Toc = 28.12 °C. To begin a solubility experiment, 1.2 g of solid phenolphthalein was added to the critical mixture. Dissolution of the phenolphthalein caused an upward shift in the critical temperature. To find the new critical temperature, the thermostat temperature was increased by approximately 0.05 K, the mixture was stirred for 24 h to come to equilibrium, and then let to settle for an additional 24 h to clear the liquid of suspended solid. The transparent liquid was inspected for the disappearance of the meniscus. This procedure was repeated until the shifted critical temperature was located at Tc = 28.20 °C. We did not deem it necessary to search for the expected shift in the critical composition4 because experiment has shown that solubility critical effects can be observed over a surprisingly wide range of compositions to either side of the critical composition.18 This observation is a consequence of the flatness of the coexistence curve in the critical region. Once the critical temperature was located, a 5 mL sample of the clear supernatant liquid was withdrawn, placed in a vial, and set aside for subsequent analysis. The temperature was then increased by approximately 0.05 K, the mixture was stirred for 24 h, and then let to settle for 24 h before the next sample was extracted. The procedure of increasing the temperature, stirring, settling, and sampling was repeated until the thermostat temperature was far away from the critical temperature. The phenolphthalein in each sample vial was extracted from the nitrobenzene + dodecane solvent mixture by transferring the contents of the vial to a separatory funnel containing 20 mL of pH 10 sodium hydroxide. After mixing by shaking, the resulting pink, aqueous layer was transferred to a quartz cell with optical path link equal to 1 cm. The quartz cell was placed in a GENESYS 10S UV−vis spectrophotometer, the UV−vis spectrum was scanned, and the molar concentration, c3, of phenolphthalein in the cell was determined with the spectrometer set at the wavelength, λmax = 550 nm. The molar concentrations, c3, as determined by spectrophotometry, are shown as a function of temperature in tabular form in Table 1 and in graphical form in Figure 1. 3.2. Relation between X3 and c3. To compare theory with experiment, we need a relation between the mole fraction composition variable, X3, favored by theory and the molar concentration variable, c3, used in spectrophotometry. According to eq 11, the mole fraction, X3, of phenolphthalein in a sample is X3 =

n3 n

Table 1. Molar Solubility, c3, vs Temperature for the Dissolution of Phenolphthalein in a Mixture of Nitrobenzene + Dodecanea

a

(1 − X3) dn 3 n

28.20 28.27 28.34 28.45 28.48 28.58 28.65 28.72 28.78 28.92 28.99 30.19 30.67 31.65 32.47 35.66 36.87 39.48 40.00

0.205 0.181 0.223 0.213 0.165 0.165 0.225 0.185 0.172 0.164 0.174 0.145 0.133 0.134 0.134 0.131 0.131 0.124 0.122

The critical endpoint temperature was 28.20 °C.

dX3 = dln X3 = (1 − X3) dln c3 X3

(28)

Upon substitution of eq 28 into both sides of eq 24, we obtain ⎛ ∂ln c3 ⎞ ⎛ ∂ln c3 ⎞ ⎟ = −T ΔH ⎜ ⎜ ⎟ ⎝ ∂ΔG ⎠ P , x , T ⎝ ∂(1/T ) ⎠ P , x , ΔG = 0

(29)

Because c3 is a density variable, the derivation of eq 29 from eq 24 shows that if ln X3 exhibits anomalous behavior, so also should ln c3. In the critical region with x near xc, we should expect the slope of the plot of ln c3 vs 1/T to be proportional to ((T − Tc)/Tc)−z, where the exponent, z, has a universal value independent of the materials involved. Although the data in Table 1 lack sufficient precision to estimate the numerical value of the exponent, z, they clearly indicate that (∂ln c3/∂(1/T)) diverges in the critical region. In the absence of an experimental estimate of the value of z, we have provided in the Appendix arguments based on Widom23 scaling theory that indicate that z = 1 − β, where β = 0.326 is the exponent that determines the shape of the coexistence curve.28

(25)

4. DISCUSSION In the case of an inert solid dissolving in a binary liquid mixture, there are three components, so C = 3. There is a solid phase and a liquid phase, so ϕ = 2. Because, by assumption, the solid component and the two liquid components are chemically inert with respect to one another, the number of reactions is R = 0. For the same reason, the number of mass conservation equations is I = 0. According to eq 1, the number of free intensive variables is thus F = 3. Two of these are pressure and temperature, which are fields. The third is x, which is a density. In eq 24, (∂ln X3/∂ΔG)P,x,T is the derivative of a density with respect to a field with one density held fixed. Because only one

(26)

If the volume of the mixture is V, then

n3 = c3V

solubility, c3 (mol/L)

computing the differential of n3. The combination of eqs 25−27 then gives

We can use eqs 12 and 13 to compute the differential of X3. The result is dX3 =

temperature (°C)

(27)

Because the density exhibits an extremely weak critical anomaly,32 we can take the volume to be a constant when D

DOI: 10.1021/acs.jpcb.7b11058 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

Figure 1. van’t Hoff plot of the solubility of phenolphthalein in the critical mixture of nitrobenzene + dodecane. The concentration, co3 = 0.122 M, corresponding to the solubility at 40 °C, was used as the standard-state concentration. The vertical dashed line locates the reciprocal of the critical temperature, 1000/Tc, in the units of K−1.

■

density is fixed, (∂ln X3/∂ΔG)P,x,T is expected to diverge toward infinity as T → Tc for x near xc. As pointed out in Section 2, the requirement that the equilibrium be stable with respect to fluctuations in n3 about neq 3 demands (∂ΔG/∂n3)T,n1,n2 > 0. According to eq 20, this requirement is the same as (∂ΔG/ ∂X3)x > 0, or expressed in van’t Hoff form, the same as (∂ln X3/ ∂ΔG)P,x,T > 0. Because of the restriction that (∂ln X3/∂ΔG)P,x,T be positive definite, (∂ln X3/∂ΔG)P,x,T can diverge only toward positive infinity near a critical point. Hence, according to eq 24, the sign of the divergence in (∂ln X3/∂(1/T))P,x,ΔG = 0 is determined by the sign of ΔH. If ΔH > 0, (∂ln X3/∂(1/ T))P,x,ΔG = 0 will diverge toward negative infinity. If ΔH < 0, it will diverge toward positive infinity. Because in Figure 1, the enthalpy of solution outside the critical temperature appears to be negative, the positive divergence of (∂ln c3/∂(1/T))P,x,ΔG = 0 in the critical region suffices to confirm the theoretical considerations in Section 2 and also the qualitative applicability of eq 29.

APPENDIX: SOLUBILITY CRITICAL EXPONENT

A.1. Scaling in a Multicomponent Fluid Mixture

Below, we shall determine the value of the exponent for the power law governing the temperature dependence solubility in the vicinity of the critical endpoint. By invoking the principle of critical point isomorphism, we draw an analogy between variable scaling in the case of a ferromagnet and variable scaling in the case of a multicomponent fluid mixture. A.2. Scaling the Ferromagnet.23,33 The magnetization, M, of a ferromagnet depends upon the internal magnetic field, H, and the temperature, T. Within the critical region, the temperature dependence of M can be represented as a function of the ratio

ε = (T − Tc)/Tc

(A.2.1)

The critical point coincides with H = 0 and ε = 0. When the magnetization, M = M(ε, H), is projected onto the field space plane spanned by the T and H axes, the resulting magnetization coexistence curve corresponds to the T < Tc portion of the line defined by H = 0, i.e., the T < Tc portion of the temperature axis. As the coexistence curve lies along the T axis, they are by definition asymptotically parallel. According to Griffiths and Wheeler,22 we can expect that for T < Tc the magnetization should be proportional (−ε)β, as ε → 0, where β = 0.326 is the coexistence curve critical exponent.28 By extension of this argument to the perpendicular direction given by ε = 0, the magnetization should be proportional to H1/δ as H → 0, where δ = 4.8 is the exponent governing the shape of the critical isotherm.34 The thermophysical properties of the ferromagnet can be derived from a thermodynamic potential, Φ, whose differential is given by

5. CONCLUSIONS In the case of an inert solid in solubility equilibrium with a binary liquid mixture, it is clear that the condition F = 3 implies that the derivative (∂ln X3/∂(1/T))P,x,ΔG = 0 should diverge as a function of temperature in the critical region. The divergence in (∂ln X3/∂(1/T))P,x,ΔG = 0 has its origin in the breakdown of the equilibrium stability condition, (∂ln X3/∂ΔG)P,x,T > 0. The sign of the divergence in (∂ln X3/∂(1/T))P,x,ΔG = 0 is determined by the sign of enthalpy of solution, ΔH. Critical point scaling theory suggests for the case where the solubility is expressed in terms of molar concentration that within the critical region, (∂ln c3/∂(1/T))P,x,ΔG = 0 should be proportional to ((T − Tc)/ Tc)−(1 − β) in the critical region, where β = 0.326 is the coexistence curve critical exponent.28 These considerations lead us to conclude that the anomalous temperature behavior of the solubility near a critical endpoint is a general property of an inert solid dissolving in a binary liquid mixture with a critical point of solution.

dΦ = − S dT + M dH

(A.2.2)

where S is the entropy. According to eq A.2.2, the magnetization is determined by the thermodynamic derivative

M (T , H ) = E

⎛ ∂Φ ⎞ ⎜ ⎟ ⎝ ∂H ⎠T

(A.2.3) DOI: 10.1021/acs.jpcb.7b11058 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B In the critical region, Φ is assumed to be a generalized homogeneous function of its independent variables ε and H, that is λ Φ(ε , H ) = Φ(λ aεε , λ aHH )

have dP = 0 and dn1 = dn2 = dn3̅ = 0. With these simplifications, eq A.3.1 reads dG = −S dT + ΔG dn3

(A.2.4)

We next introduce the density variable, ξ, defined by

where λ is a real positive scale factor, and aε and aH are scaling powers. Upon differentiation of eq A.2.4 with respect to H, we obtain ∂Φ(ε , H ) ∂Φ(λ aεε , λ aHH ) ∂(λ aHH ) = λ −1 ∂H ∂(λ aHH ) ∂H aε aH ∂Φ(λ ε , λ H ) = λ aH − 1 ∂(λ aHH )

ξ=

(A.2.5)

Φ = G + ξ(n1 + n2)ΔG

(A.2.6)

dΦ = −S dT + (n1 + n2)ξ d(ΔG)

(A.2.7)

M(0, 1)

dϕ = −s dT + ξ d(ΔG)

dϕ = −σ dε + ξ d(ΔG)

1 β(1 + δ)

(A.2.9)

(A.2.10)

⎛ ∂ϕ ⎞ ξ=⎜ ⎟ ⎝ ∂(ΔG) ⎠

(A.2.11)

and δ aH = 1+δ

(A.3.9)

The scaling hypothesis expressed by eq A.2.4 implies that λϕ(ε , ΔG) = ϕ(λ aεε , λ aGΔG)

(A.2.12)

(A.3.10)

After substitution of eq A.3.10 into eq A.3.9 and computation of the indicated derivatives, we find

respectively. A.3. Scaling the Solubility. To draw an analogy between the ferromagnet and the solubility of an inert solid, we start with the differential of the Gibbs energy describing the excess solid plus the liquid solution

ξ(ε , ΔG) = λ aG − 1ξ(λ aεε , λ aGΔG)

(A.3.11)

Our solubility data were collected along the critical isopleth, x = xc, of the solvent mixture, which is the T > Tc extension of the coexistence curve.24 Hence, if we set λaεε = 1 and ΔG = 0 in eq A.3.11, the scaling law for ξ along the critical isopleth is found to have the form

dG = −S dT + V dP + μ1 dn1 + μ2 dn2 + μ3 dn3 + μ3o (s) d(n3̅ − n3)

(A.3.8)

For the mixture to be in equilibrium anywhere in the phase diagram, the solubility process must also be in equilibrium, which implies ΔG = 0. Thus, if the system is to approach the critical endpoint along the coexistence curve with T < Tc, ΔG = 0 is a necessary requirement. Hence, we suggest that eq A.3.8 is the analog of eq A.2.2, where ξ corresponds to M and ΔG corresponds to H. We note that the coexistence curve is located by the condition H = 0 in the case of the ferromagnet and by ΔG = 0 in the case of the solubility equilibrium. The corresponding density variables, ξ and M, are both zero at the critical point. According to eq A.3.8, the analog of eq A.2.3 is

When eqs A.2.8 and A.A.2.10 are solved simultaneously, the scaling powers are found to satisfy aε =

(A.3.7)

where we have used the definitions ϕ = Φ/(n1 + n2) and s = S/ (n1 + n2). The differential of eq A.2.1 is dT = Tc dε; hence, if we introduce this result plus σ = sTc into eq A.3.7, we have

where M(0,1) is a constant. Empirically, M(H) is proportional to H1/δ; hence, by comparison with eq A.2.9, we find for the isotherm critical exponent 1 − aH 1 = δ aH

(A.3.6)

Upon dividing eq A.3.6 through by (n1 + n2), the result is

Upon setting λaHH = 1 and ε = 0 in eq A.2.6, we arrive at the critical isotherm scaling equation M (H ) = H

(A.3.5)

After computing the total differential of eq A.3.5 and substituting the result into eq A.3.4, we have

(A.2.8)

1 − aH / aH

(A.3.4)

where we note that the mole numbers, n1 and n2, have been fixed to coincide with the critical isopleths, x = xc, of the binary mixture. We define the thermodynamic potential, Φ, as

where M(−1, 0) is a constant. Empirically, M(ε) is proportional to (−ε)β; hence, by comparison with eq A.2.7, we find for the coexistence curve critical exponent

1 − aH β= aε

(A.3.3)

dG = −S dT − ΔG(n1 + n2) dξ

Upon setting λaεε = −1 and H = 0, in eq A.2.6, we obtain for the magnetization in the critical region M(ε) = ( −ε)1 − aH / aε M( −1, 0)

n3c − n3 n1 + n2

where nc3 is the value of n3 at the critical endpoint. Upon introduction of eq A.3.3, we can write eq A.3.2 in the form

By comparison with eq A.2.3, we arrive at the magnetization scaling law in the form M(ε , H ) = λ aH − 1M(λ aεε , λ aHH )

(A.3.2)

(A.3.1)

where S is the entropy, V is the volume, and the other symbols are as defined in Section 2. Under experimental conditions, where the pressure is constant and the initial amounts of the solvent components and the solid component are fixed, we

ξ(ε) = (ε)1 − aG / aε ξ(1, 0)

(A.3.12)

where ξ(1, 0) is a constant. Assuming that aG = aH, eqs A.2.11 and A.A.2.12 combine to imply that we can write eq A.3.12 as F

DOI: 10.1021/acs.jpcb.7b11058 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B ξ(ε , 0) = ε β ξ(1, 0)

(5) Bouanz, M.; Beysens, D. Effect of ion impurities on a binary mixture of isobutyric acid and water. Chem. Phys. Lett. 1994, 231, 105− 110. (6) 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. (7) 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. (8) Venkatesu, P. Polymer modifies the critical region of the coexisting liquid phases. J. Phys. Chem. B 2006, 110, 17339. (9) 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. (10) Griffiths, R. B. Thermodynamic model for tricritical points in ternary and quaternary fluid mixtures. J. Chem. Phys. 1974, 60, 195− 206. (11) Procaccia, I.; Gitterman, M. Supercritical extraction at atmospheric pressures. J. Chem. Phys. 1983, 78, 5275−5276. (12) Ullmann, A.; Ludmer, Z.; Shinnar, R. Phase transition extraction using solvent mixtures with critical point of miscibility. AIChE J. 1995, 41, 488−500. (13) Greer, S. C. Chemical reactions and phase equilibria. Int. J. Thermophys. 1988, 9, 761−768. (14) Tveekrem, J. L.; Cohn, R. H.; Greer, S. C. The effect of critical fluctuations on chemical equilibrium. J. Chem. Phys. 1987, 86, 3602− 3606. (15) Kim, Y. W.; Baird, J. K. Chemical equilibrium and critical phenomena: The solubilities of manganese dioxide and aluminum oxide in isobutyric acid + water near its consolute point. J. Phys. Chem. B 2005, 109, 17262−17266. (16) 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. (17) 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. (18) 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, No. 154505. (19) 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. (20) 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. (21) 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. (22) Du, Z.; Yin, H.; Hao, Z.; Zheng, P.; Shen, W. Critical anomalies of alkaline fading of phenolphthalein in the critical solution of 2butoxyethanol + water. J. Chem. Phys. 2013, 139, No. 224501. (23) Widom, B. Equation of state in the neighborhood of a critical point. J. Chem. Phys. 1965, 43, 3898−3905. (24) Griffiths, R. B.; Wheeler, J. C. Critical points in multicomponent systems. Phys. Rev. A 1970, 2, 1047−1064. (25) Saam, W. F. Thermodynamics of binary Systems near the liquidgas critical point. Phys. Rev. A 1970, 2, 1461−1466. (26) Fisher, M. E. Renormalization group theory: Its basis and formulation in statistical physics. Rev. Mod. Phys. 1998, 70, 653−681. (27) Barmatz, M.; Hahn, I.; Lipa, J. A.; Duncan, R. V. Critical phenomena in microgravity: past, present, and future. Rev. Mod. Phys. 2007, 79, 1−52.

(A.3.13)

The slope of the van’t Hoff plot in the critical region depends upon the ε derivative of eq A.3.11. If we use the notations ξε = (∂ξ(ε,ΔG)/∂ε) and ξsε = (∂ξ(sε,ΔG)/∂(sε)), then we can write ξε(ε , ΔG) =

∂ξ(λ aεε , λ aGΔG) ∂(λ aεε) ∂ξ(ε , ΔG) = λ aG − 1 ∂(λ aGε) ∂ε ∂ε

= λ aG + aε − 1ξsε(λ aεε , λ aGΔG)

(A.3.14)

Upon substitution of λ ε = 1 and ΔG = 0, eq A.3.14 predicts that aε

ξε(ε) = (1/ε)aG + aε − 1/ aε ξsε(1, 0)

(A.3.15)

where ξsε(1, 0) is a constant. The exponent in eq A.3.15 can be simplified by substitution of eqs A.2.11 and A.A.2.12 with aH = aG. Upon simplification of the exponent, eq A.3.15 reads ξε(ε) = ε−(1 − β)ξsε(1, 0)

(A.3.16)

which indicates that (∂ξ(ε,ΔG)/∂ε) diverges in the critical region. Because the slope of the plot of ln c3 versus 1/T diverges in the critical region, we seek the exponent that governs (∂ln c3/ ∂(1/T)) in the critical region. We rewrite eq A.3.3 in the form

c3 = c3c −

⎛ n1 + n2 ⎞ ⎜ ⎟ξ ⎝ V ⎠

(A.3.17)

where we have introduced the molar concentration, c3 = n3/V, and its value at the critical endpoint, cc3 = nc3/V. Using eq A.3.17, we can compute ⎛ n + n ⎞ ⎛ ∂ε ⎞ ∂ln c3 1 ∂c3 2 = = −⎜ 1 ⎟ξε⎜ ⎟ ⎝ ∂T ⎠ ∂T c3 ∂T Vc ⎝ 3 ⎠

(A.3.18)

where we have ignored the temperature derivative of V, which diverges only weakly or not at all in the critical region.32 If we substitute d(1/T) = −dT/T2 and (∂ε/∂T) = 1/Tc, then sufficiently close to Tc, eq A.3.18 becomes ⎛n + n ⎞ ∂ln c3 = ⎜ 1 c 2 ⎟Tcξsε(1, 0)ε−(1 − β) ∂(1/T ) ⎝ c3 V ⎠

(A.3.19)

Equation A.3.19 indicates that (∂ln c3/∂(1/T)) should diverge as ((T − Tc)/Tc)−(1−β) in the critical region.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

James K. Baird: 0000-0002-3995-8641 Notes

The authors declare no competing financial interest.

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REFERENCES

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H

DOI: 10.1021/acs.jpcb.7b11058 J. Phys. Chem. B XXXX, XXX, XXX−XXX