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Kinetics of the Reaction of Crystal Violet with Hydroxide Ion in the Critical Solution of 2‑Butoxyethanol + Water Zhongyu Du,† Shiyan Mao,† Zhiyun Chen,‡ and Weiguo Shen*,†,‡ †

Department of Chemistry, Lanzhou University, Lanzhou 730000, China School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai 200237, China

‡

ABSTRACT: The kinetics of alkaline fading of crystal violet (CV) has been studied by UV spectrophotometry and microcalorimetry in the critical binary solution of 2-butoxyethanol + water at the initial reaction stage and various temperatures. It was found that the first-order rate constants obtained from these two methods are well accorded with each other, and the temperature dependence of the rate constant obeyed the Arrhenius equation in a temperature region far from the critical point. The critical slowing down was detected by both methods near the critical point. A simple empirical crossover model was proposed and used to analyze the experimental data to obtain the critical exponents, which were 0.158 ± 0.013 and 0.133 ± 0.012 from UV spectrophotometry and microcalorimetry, respectively, and the former was in good agreement with the theoretical prediction of 0.151. The slight lower value derived from microcalorimetry was attributed to the stirring in the microcalorimeter, which weakened the critical reduction of the diffusion coefficient.

1. INTRODUCTION It is well-known that some partially miscible binary solutions exhibit interesting critical behaviors resulting from the critical fluctuation near the critical solution point. For example, in a mixture with the critical composition, some thermodynamic properties such as the heat capacity, the osmotic compressibility, and the correlation length diverge as the reduced temperature τ (τ = |(T − Tc)/(Tc)|, Tc is the critical solution temperature) approaches zero, while the derivative of the Gibbs free energy change of the reaction with respect to the extent of the reaction at the equilibrium (∂ΔG/∂ξ)ξe vanishes. Moreover, some dynamic properties such as the viscosity and the mutual diffusion coefficient exhibit divergence or vanish, respectively, near the critical solution point. The influence of the critical phenomena on chemical reactions has been studied both theoretically and experimentally for several decades.1−5 Although speeding up of the reactions in critical solutions has also been reported,6−8 more reliable experimental observations and the theoretical analysis support the critical slowing down effect of the chemical reactions.1,9−17 This slowing down has been attributed to the thermodynamic or dynamic anomalisms;9,11,18 however, more likely these two factors are coexistent but difficult to be separated. As the thermodynamic effect was considered, a reaction rate was commonly expanded in a Taylor series at its chemical equilibrium, and only the linear term usually was kept to discuss the critical anomalous behavior of the reaction including the critical exponent, reflecting the strength of the critical slowing down of the reactivity. Obviously, the neglect of the higher order terms of the Taylor series is only acceptable near the equilibrium of the chemical reaction; unfortunately, in © 2012 American Chemical Society

this case the reaction rate is hard to determine with a required precision because it approaches zero at the chemical equilibrium. The usual method of determining the rate of reaction under conditions where the concentrations of reactants and products are far from equilibrium appears to contradict the neglect of higher order terms in the Taylor series in the analysis of the critical slowing down of the reaction. In recent years, the basic hydrolysis of crystal violet (CV) in aqueous media,19,20 micelles,21,22 alcoholic solutions,23 and microemulsions24,25 has been studied by UV spectrophotometry, and the kinetics of this reaction are now well-known. Focusing on the first-order reaction in this work, we measure reaction rates of CV at the initial reaction stage in the medium of a critical solution 2-butoxyethanol (2BE) + water, which has a lower critical solution temperature, by using a UV spectrophotometer and a microcalorimeter at a series of temperatures both far from and near the critical point. These measurements allow us to investigate the critical slowing down of the chemical reaction. This article is organized as follows. In the Theoretical Background, we show that for a first-order or pseudofirst-order reaction in both directions the reaction rates at the initial reaction stage may be used for analysis of the critical slowing down of the chemical reaction using a formalism that is limited to a first-order Taylor series described in this section. The methods of the measurements, the experimental procedures, and the results are described in the Experimental Section. We Received: November 11, 2012 Revised: December 27, 2012 Published: December 31, 2012 283

dx.doi.org/10.1021/jp3111502 | J. Phys. Chem. A 2013, 117, 283−290

The Journal of Physical Chemistry A

Article

dξ dc =− dt dt k+c ⎛ ∂ΔG ⎞ ⎟ (c − c ) + higher order terms = − e⎜ e RT ⎝ ∂c ⎠ce

discuss critical slowing down from both the thermodynamic and the dynamic points of view and evaluate the corresponding critical exponents in the Discussion. Finally, section 5 contains some concluding remarks.

2. THEORETICAL BACKGROUND A reversible chemical reaction converting reactants A1 and A2 into products A3 and A4 is written in the general form:

and ΔG = ΔG 0 + RT ln

k+

ν1 A1 + ν2 A 2 ⇄ ν3 A3 + ν4 A4

a−c c

(7)

where ce is the value of c at the chemical equilibrium. For a reaction system far from the critical point, the derivative of eq 7 gives ((∂ΔG)/(∂c))ce, which is substituted into eq 6 to give

(1)

k−

(6)

and the rate of the reaction, dξ/dt, is given by

dc k+a (c − ce) + higher order terms =− dt (a − ce)

dξ = k+a1ν1a 2ν2 − k−a3ν3a4ν4 dt ν ν ⎛ k− a3 3a 4 ⎞ = k+a1ν1a 2ν2⎜1 − + ν1 4ν2 ⎟ k a1 a 2 ⎠ ⎝ ⎡ ⎛ ΔG ⎞⎤ ⎟ = k+a1ν1a 2ν2⎢1 − exp⎜ ⎝ RT ⎠⎥⎦ ⎣

(8)

On the other hand, the rate of the reaction for a first-order reversible reaction can be written as dc = k−(a − c) − k+c dt

(2)

Using k− = k+((ce)/(a − ce)) at equilibrium, eq 9 becomes

where ξ is the extent of the reaction, ν1, ν2, ν3, and ν4 are the respective stoichiometric coefficients, ai (i = 1, 2, 3, 4) is the thermodynamic activity of the ith reacting species, ΔG is the instantaneous Gibbs free energy change for the reaction, T is the absolute temperature, R is the gas constant, and k+ and k− are the rate constants in the forward and reverse directions, respectively. The rate of the reaction can be expanded in a Taylor series about ξe, which is the value of ξ at the chemical equilibrium, and expressed near the equilibrium by8

ce dc k+a (a − c) − k+c = − (c − ce) = k+ dt (a − ce) (a − ce) (10)

Unlike eq 8, eq 10 was derived without approximation. Comparison of eq 10 with eq 8 indicates that the high order terms in eqs 4, 6, and 8 vanish without the assumption that the reaction occurs near the equilibrium. Thus, eq 6 becomes

k+(a1e)ν1 (a 2e)ν2 ⎛ ∂ΔG ⎞ dξ =− ⎜ ⎟ (ξ − ξe) ⎝ ∂ξ ⎠ξ dt RT

k+ce ⎛ ∂ΔG ⎞ dc ⎜ ⎟ (c − c ) = e dt RT ⎝ ∂c ⎠ce

e

+ higher order terms

(9)

(3)

(11)

Because eq 11 is valid when the reaction is near or far from equilibrium, kinetic data collected at the initial reaction stage can be used in studying the critical slowing down of the firstorder chemical reaction. As it is well-known that ((∂ΔG)/(∂c))ce approaches zero in a simple power law of the reduced temperature with a critical exponent f near the critical point,7 therefore ((∂ΔG)/(∂c))ce should be expressed by

aie

where is the equilibrium value of ai. In principle, as long as the extent of the reaction is not sufficiently close to the equilibrium, the higher order terms must be kept in eq 3. For a first-order or a pseudofirst-order reaction in both directions: k+

ν1 A1 ⇄ ν2 A 2 k−

f ⎧ ⎪ ⎜⎛ ∂ΔG ⎟⎞ ∝ − T − Tc ⎪ ⎝ ∂c ⎠c Tc e ⎪ ⎪ = −τ f (near the critical point) ⎨ ⎪ ⎛ ∂ΔG ⎞ a 1 ⎟ = − RT (far from the ⎪⎜ ⎝ ⎠ ce a − ce ⎪ ∂c ce ⎪ critical point) ⎩

Equation 3 becomes k+(a1e)ν1 ⎛ ∂ΔG ⎞ dξ =− ⎜ ⎟ (ξ − ξe) + higher order terms dt RT ⎝ ∂ξ ⎠ξ e

(4)

with a ν2 a ν2 k+ ΔG = ΔG + RT ln 2ν1 = −RT ln − + RT ln 2ν1 a1 k a1 0

(12)

where f characterizes the slowing down of a chemical reaction and was thought to be dependent on the number of inert components in the reaction system.8 There exists a crossover between the critical and the noncritical regions. Some crossover theories have been developed to include not only the analytic behavior far from the critical point, but also the nonclassical divergence of the thermodynamic derivatives in the critical region;26−28 however, these formalisms are rather too complex to be used in analysis of the critical slowing down of the

(5)

where ΔG is the standard Gibbs free energy change for a reaction. If a reaction system is an ideal dilute solution for all species, ν1 = ν2 = 1, we can introduce the molarity, c = a − ξ, measured in mol L−1 as the concentration of the reactant, and at the initial time, set c = a and the product concentration equal to zero; hence: 0

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that of wc = 0.295 and 0.294 reported by Schmitz et al.30 and Kim et al.,31 respectively. A small amount of NaOH and CV was added into the binary critical solution respectively to examine the effect of the concentration of NaOH and CV on the critical composition. No shift of the critical composition was observed for solutions containing up to 1.235 × 10−3 mol L−1 of NaOH and up to 1.439 × 10−5 mol L−1 of CV. In our kinetic measurements, the concentrations of NaOH and CV were less than 7.974 × 10−4 and 8.068 × 10−6 mol L−1; thus the effect of the NaOH and CV on the critical composition may be neglected. It is well-known that unlike the critical composition, the critical temperature of a binary solution is sensitive to the trace amounts of impurities. The critical temperature of a binary 2BE aqueous solution was carefully measured, which was 321.802 ± 0.011 K. However, the critical temperature was shifted to 321.647 ± 0.011 K after the reactant NaOH was added into the binary solution. Because the amount of the reactant CV was very small, the effect of the composition change during reaction on the critical temperature shift may be neglected. It was checked by measurement of the critical temperature of the reaction system after the reaction completed, which was 321.638 ± 0.011 K. This was consistent with that of the original 2BE/NaOH aqueous solution within the experimental uncertainty. Hence, this value was taken as Tc and used in calculations. 3.3. Sample Preparation. A NaOH aqueous solution with the concentration of NaOH being about 0.1 mol L−1 was prepared as a stock solution and titrated with potassium hydrogen phthalate to determine its accurate concentration. The NaOH stock solution then was diluted to 1.168 × 10−3 mol L−1 and mixed with 2BE to form a ternary aqueous solution denoted as (1) where the mass fraction of 2BE was 0.2953 and the concentration of NaOH was 7.974 × 10−4 mol L−1, respectively. A CV aqueous solution with the concentration of CV being 4 × 10−3 mol L−1 was prepared and mixed with 2BE to form a ternary aqueous solution denoted as (2) where the mass fraction of 2BE was also 0.2953 and the concentration of CV was 2.731 × 10−3 mol L−1. Both solutions (1) and (2) had the critical concentration, and as described above the mixture of 2.7 mL of solution (1) plus 8 μL of solution (2) as the reaction system was also a critical solution. 3.4. Kinetics Measurement. The reaction of CV+ + OH− is written as

chemical reaction. We propose a simpler formalism to describe this crossover phenomenon: ⎛ ∂ΔG ⎞ a 1 ⎜ ⎟ = − RT ·Φ(f , τ ) ⎝ ∂c ⎠c c e a − ce e

(13)

with f ⎧ (near the critical point) ⎪τ Φ=⎨ ⎪ ⎩1 (far from the critical point)

(14)

Thus, in the critical region, eq 11 can be rewritten as k+ce ⎛ ∂ΔG ⎞ dc ⎜ ⎟ (c − c ) = e dt RT ⎝ ∂c ⎠ce =−

k+a (c − ce) ·τ f (a − ce)

= −kobs(c − ce)

with the observed rate constant a kobs = k+ ·τ f (a − ce)

(15)

(16)

If a ≫ ce, then eq 16 becomes

kobs = k+τ f

(17)

Thus, the observed rate constant kobs can be affected by the thermodynamic factor through τf and by the dynamic factor through k+ in a critical region. In a noncritical region, Φ = 1 and kobs = k+ = kb, where kb is the noncritical rate constant. The temperature dependence of kb can be described by the Arrhenius equation. In the initial reaction stage, c ≫ ce, eq 15 becomes

dc = −kobsc dt

(18)

and the natural logarithm of eq 17 is ln

kobs = f ln τ k+

(19)

For a first-order chemical reaction, which goes essentially to completion, a ≫ ce, the kinetic data at a series of temperatures may be collected at the initial reaction stage to obtain the observed rate constant by using the linear relation of eq 18. The critical exponent of the reaction criticality may be deduced from the analysis of the temperature dependence of kobs both in the critical and in the noncritical regions.

k+

CV + + OH− ⇄ CVOH k−

(20)

The reaction is a pseudofirst-order reaction with respect to CV+ because the concentration of OH− was much higher than that of CV+ in our experimental design. 3.4.1. UV Spectrophotometry. An Agilent 8453 UV spectrophotometer with a photodiode array detector, capable of measuring absorbance in the whole wavelength range simultaneously, was used to monitor the progress of the reaction. The detector permitted three-wavelength spectrophotometry to be used to eliminate the influence of critical scattering in the study of the chemical reaction. A optical cell filled with 2.7 mL of solution (1) was placed in a thermostatted cell holder with a temperature stability of ± 0.1 K over the temperature range of (Tc − T) ≥ 2.48 K. The variations in the spectrum and the absorbance of CV+ at wavelength λ = 595 nm with time in the system were monitored in the noncritical range. As examples, Figure 1a and

3. EXPERIMENTAL SECTION 3.1. Materials. CV (analytic grade, Beijing Chemical Co.), NaOH (analytic grade, Tianjing Chemical Co.), and 2BE (Alfa Aesar, ≥99%) were used as received. Doubly distilled water was used for preparation of aqueous solutions throughout the study. 3.2. Determination of the Critical Composition and the Critical Temperature of the Reaction System. The critical mass fraction wc of 2BE in its binary aqueous solution was approached by fixing the mass of 2BE and adjusting the amounts of water to achieve identical volumes of two separating liquid phases at phase separation temperature.29 It was found that the value of wc was 0.2953 and in good agreement with 285



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Figure 1. (a) Variation of the absorption spectrum with time for alkaline fading of CV in 2BE + water at T = 319.16 K. (b) Plot of absorbance versus t for alkaline fading of CV in 2BE + water at λ = 595 nm and T = 317.68 K.

b shows the changes of the spectrum at T = 319.16 K and the absorbance at wavelength λ = 595 nm and at T = 317.68 K with the reaction time. As we can see from Figure 1, the maximum of the absorbance is not changed with the reaction time, and the absorbance of CV+ becomes constant after the reaction had proceeded for about 7 h, indicating that chemical equilibrium had been reached. For a first-order or a pseudofirst-order reaction in both directions, the absorbance (A) at any time is expressed as32 A = Ae + (A 0 − Ae) exp[−(k+ + k−)t ]

the critical scattering at each of the three wavelengths is linear with the wavelength. Proper choices of the setting absorption wavelengths are crucial, which are required to ensure that the absorbance due to the critical scattering in the wavelength range between λ1 and λ3 is linear with the wavelength and the values of ΔA for CV+ are sufficiently large. The values of λ1 and λ3 were carefully selected to be 590 and 620 nm, respectively, in our experiment. Figure 2 shows the linear relation between the

(21)

+

where A0 is the absorbance of CV at the initial time of the reaction, and Ae is the absorbance at the chemical equilibrium. The values of A0 and Ae in eq 21 were determined to be 0.438 and 0.010, and then (A0)/(A0 − Ae) = (a)/(a − ce) = 1.02. Thus, in the initial reaction stage, we neglect the contribution of ce in eq 15 and rewrite eq 18 by dc cv + = −kobsc CV + dt

(22)

with c = cCV+, where kobs is described by eq 17 in a critical region. For measurements of the absorbance in a temperature range of (Tc − T) ≤ 2.48 K, an optical cell filled with 2.7 mL of solution (1) was placed in the sample chamber of a metal bath where the temperature was controlled with a precision of ±0.005 K, and then 8 μL of solution (2) was added into the optical cell by using a microsyringe. In this temperature range close to the critical point, large fluctuation resulted in the strong scattering, which may be eliminated by the threewavelength spectrophotometry. The relative absorbance (ΔA) is expressed by33

Figure 2. Plot of the absorbance due to the scattering versus wavelength for 2BE + water at T = 321.56 K.

absorbance of the critical 2BE aqueous solution and the wavelength at 321.56 K. Figure 3 shows the variation of ΔA of the reaction system with time at 321.56 K. 3.4.2. Microcalorimetry. The measurement of reaction kinetics was also carried by using an isothermal titration microcalorimeter (ITC, TAM 2277-201, Thermometric AB, Sweden). This apparatus has sample and reference cells with volume of 4 mL. The temperature in the cells was controlled to within ±1 × 10− 3 K. To begin an experiment, 2.7 mL of solution (1) and 1 mL of solution (2) were transferred into the sample cell and a Hamilton microliter syringe, respectively. The solution in the sample cell was stirred at a stirring speed of 60 rpm. About 8 μL of solution (2) was added by the Hamilton microliter syringe after the sample reached thermal equilibrium. The calorimetric signals for CV+ + OH− reaction in 2BE aqueous solution with the critical concentration were automatically recorded at various reaction times for each of the reaction temperatures. As an example, Figure 4 shows a plot of typical calorimetric signal versus time at 317.715 K. The heats of mixing solutions

mA1 + nA3 m+n mελ1 + nελ3 ⎞ ⎛ ⎟c +l = ⎜ελ2 − cv ⎝ m+n ⎠

ΔA = A 2 −

= Δεc cv +l

(23)

where λ2 is the maximum absorption wavelength of CV , λ1 and λ3 are setting absorption wavelengths on each side of λ2; Ai is the absorbance at λi and ελi is the molar absorptivity at λi for i = 1, 2, and 3; m = λ2 − λ1, n = λ3 − λ2; c = cCV+ is the concentration of the reactant CV+; and l is the path length of the optical cell. The value of ΔA is independent of the influence of the critical scattering intensity when the absorbance due to +

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Figure 5. Typical plots of ln A and ln ΔA versus t for alkaline fading of CV in 2BE + water at T = 314.62 and 321.56 K, respectively. The points represent experimental results: (■) (Tc − T) > 2.5 K, (○) (Tc − T) < 2.5 K. The lines represent the results of the fits with eqs 24 and 26, respectively.

Figure 3. Plot of the relative absorbance versus t for alkaline fading of CV in 2BE + water at T = 321.56 K.

3.5.2. Analysis of the Data from Microcalorimetry Measurement. For a first-order reaction, the reduced extent ψ of the reaction is expressed by34 ln(1 − ψ ) = −kobst

(27)

where ψ = ξ/ξe. ξ and ξe are proportional to the heat Q produced in the reaction before time t, and the total heat Q∞ produced in the whole reaction, respectively, and thus ψ=

Q ξ = Q∞ ξe

(28)

A plot of ln(1 − ψ) versus t gave a straight line with a slope being kobs. As an example, Figure 6 shows such a plot and a good straight line evidencing the validity of the pseudofirstorder model and the precision of the experimental results.

Figure 4. Plot of typical calorimetric signal versus t for alkaline fading of CV in 2BE + water at T = 317.715 K.

(1) and (2) were estimated by mixing 2.7 mL of solution (1) and 8 μL of solution (2) without CV, or mixing 2.7 mL of solution (1) without NaOH and 8 μL of solution (2), which were found to be less than 0.4% of the reaction heat, and were neglected in the data treatment. 3.5. Data Analysis. All of the experimental data were collected in the one-phase region below the critical temperature and were analyzed as follows. 3.5.1. Analysis of the Data from UV Spectrophotometry Measurement. In a temperature range of (Tc − T) > 2.5 K, using the Lambert−Beer law and integrating eq 22 yields

ln A = ln A 0 − kobst

(24)

In a temperature range of (Tc − T) < 2.5 K, combination of eqs 22 and 23 yields

dΔA = −kobsΔA dt

Figure 6. The typical plot of ln(1 − ψ) versus t for alkaline fading of CV in 2BE + water at T = 320.649 K. (■) The experimental result; the line represents the result of the fit with eq 27.

(25)

Integration of eq 25 gives ln ΔA = ln ΔAo − kobst

(26)

The values of kobs at various temperatures from the two experimental methods are listed in Tables 1 and 2. The uncertainties in kobs for both UV spectrophotometry and microcalorimetry were estimated to be less than 3% in the temperature range far from the critical point and 7% near the critical point, respectively.

where ΔA0 is the relative absorbance at initial time. Therefore, the value of kobs can be obtained from the slope of the plot of ln A for (Tc − T) > 2.5 K or ln ΔA for (Tc − T) < 2.5 K versus time in the two temperature ranges described above. As an example, Figure 5 shows such plots at T = 314.62 and 321.56 K, respectively. 287



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Table 1. Values of Observed Rate Constant kobs at Various Temperatures for Alkaline Fading of CV in 2BE + Water As Determined by UV Spectrophotometry T/K 309.37 310.52 311.69 312.53 314.54 316.13 316.82 317.68 319.16 320.41 320.79 320.92 321.02 321.12 321.25 321.29 321.32 321.33 321.34 321.35

1000 kobs/s−1

T/K

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

321.36 321.37 321.40 321.41 321.42 321.43 321.44 321.45 321.46 321.49 321.50 321.51 321.52 321.53 321.56 321.57 321.58 321.59 321.60

1.21 1.31 1.39 1.49 1.71 1.78 1.89 2.01 2.21 2.41 2.33 2.26 2.21 2.04 2.15 2.01 2.07 2.10 2.11 2.09

0.04 0.04 0.04 0.05 0.05 0.05 0.06 0.06 0.07 0.07 0.07 0.07 0.18 0.18 0.18 0.17 0.18 0.17 0.17 0.17

1000 kobs/s−1 1.92 1.95 1.91 1.82 1.89 1.85 1.81 1.72 1.89 1.78 1.67 1.78 1.67 1.63 1.67 1.62 1.56 1.52 1.47

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.16 0.16 0.15 0.16 0.15 0.15 0.14 0.14 0.15 0.14 0.14 0.14 0.14 0.13 0.14 0.13 0.13 0.12 0.12

Figure 7. Plot of ln kobs versus 1/T for alkaline fading of CV in 2BE + water; kobs was measured by UV spectrophotometry and microcalorimetry: (■) from UV spectrophotometry, (●) from microcalorimetry. The line represents the result of the fit with eq 29.

experimental data were fitted to the Arrhenius equation in the temperature range of (Tc − T) > 1.24 K simultaneously to give Ea = 40.4 ± 1.6 kJ mol−1 and B = 12.5 ± 0.6. This straight line serves as the “background” for determining the critical effect. It is also clearly shown in Figure 7 that the slowing down effect of the reaction becomes significant in a temperature range of (Tc − T) < 0.6 K. According to eqs 15 and 17, the dynamic and thermodynamic factors of slowing down are represented by the vanishes of k+ and ((∂ΔG)/(∂c))ce as the temperature approaches the critical point, which results in the significant reduction of the observed rate constant kobs. The variation of the ((∂ΔG)/ (∂c))ce value near the critical point was explained by the Griffiths−Wheeler rule,2,35,36 which describes the universal behavior of the derivatives of thermodynamic variables in the critical region. According to the Griffiths−Wheeler rule, when the experimental conditions are such that the fixed thermodynamic variables consist of fields and two or more densities, then the derivative of a field with respect to a density, for example, ((∂ΔG)/(∂c))ce, will not go to zero as T approaches Tc; otherwise, if less than two densities are held fixed, the derivative should go to zero as a power of τf. If one density is held fixed, ((∂ΔG)/(∂c))ce should go to zero weakly, and the value of f is expected to be about 0.10. No fixed density would result in ((∂ΔG)/(∂c))ce vanishing strongly with the value of f being about 1. In our experiment, the pressure P and the temperature T are the appropriate fixed field variables. The appropriate density variables are the mass fractions wNaOH, wCV, wH2O, w2BE, and wCVOH of the components NaOH, CV, H2O, 2BE, and CVOH, respectively. The components NaOH, CV, and CVOH were involved in the chemical reaction of CV+ + OH−, and the ionization equilibrium of water also shifted due to the reduction of the sodium hydroxide concentration in the reaction. Only 2BE can be considered to be an inert, keeping constant in the reaction process,and thus, according to Griffiths and Wheeler, the critical slowing down effect for this reaction system should be a weak one. The value of k+ in eq 17 may be influenced by the critical reduction of the diffusion coefficients of the reactants and the products in the critical region, which is resulted from the anomaly of the viscosity of the critical reaction system and

Table 2. Values of Observed Rate Constant kobs at Various Temperatures for Alkaline Fading of CV in 2BE + Water As Determined by Microcalorimetry T/K 308.077 309.638 311.048 311.639 312.668 313.140 314.683 315.665 316.156 317.165 317.715 318.652 319.155 320.649 320.753 320.960 321.072 321.150

1000 kobs/s−1

T/K

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

321.185 321.232 321.264 321.289 321.312 321.336 321.360 321.385 321.408 321.432 321.457 321.480 321.503 321.527 321.552 321.575 321.600

1.21 1.29 1.41 1.47 1.19 1.61 1.78 1.89 2.01 2.11 2.12 2.19 2.49 2.45 2.38 2.35 2.21 2.22

0.04 0.04 0.04 0.05 0.05 0.05 0.05 0.06 0.06 0.06 0.06 0.07 0.07 0.16 0.16 0.16 0.15 0.15

1000 kobs/s−1 2.06 2.04 2.14 2.11 2.06 2.11 1.95 2.06 2.01 1.89 2.01 1.89 1.78 1.73 1.83 1.74 1.56

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.14 0.13 0.15 0.14 0.14 0.14 0.12 0.13 0.13 0.13 0.13 0.11 0.10 0.10 0.11 0.10 0.09

4. DISCUSSION The plots of ln kobs versus 1/T for two sets of the results obtained from UV spectrophotometry and microcalorimetry, respectively, are shown in Figure 7. It can be seen in Figure 7 that the values of kobs determined by the two methods are in accordance with each other within the experimental uncertainties. The Arrhenius equation is expressed by ln k b = Ea /RT + B (29) where Ea is the activation energy. Figure 7 shows a good linear relation between ln kobs and 1/T in a temperature range of (Tc − T) > 1.24 K, corresponding to the noncritical region, which confirms the validity of the Arrhenius equation. The two sets of 288



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exhibits a weak divergence at the critical point. In the reaction system we studied, the mixed solvent 2BE aqueous solution had the critical composition, and the viscosity η in this solution for a temperature range of (Tc − T) < 1.2 K was reported to be expressed by37 η /ηb = (Q 0ξ0τ −ν)z = (Q 0ξ0)z τ −y

(30)

where Q0ξ0 is constant, ηb is the background value of the viscosity determined in the noncritical region, z = 0.065 and v = 0.63 for the 3D-Ising model, and y = vz = 0.041. The diffusion coefficients of the reactants and the products in the critical mixed solvent are related to the viscosity of the solvent by the Stokes−Einstein equation:38

kT D= B 6πrη

Figure 9. Plot of ln k/kb versus ln τ for alkaline fading of CV in 2BE + water near the critical point determined by microcalorimetry. (■) The experimental result; the line represents the result of the fit with eq 33.

(31)

Combination of eqs 30 and 31 gives η D k+ ∝ b ∝ ∝ τy Db kb η

experimental design to gain some insight about the critical dynamic slowing down effect. (32)

5. CONCLUSIONS We determined the pseudofirst-order reaction rate constants kobs for alkaline fading of CV by both UV spectrophotometry and microcalorimetry at the initial reaction stage and at the various temperatures. It was found that when the temperature is far from the critical point, the values of kobs determined by the above two methods were in good agreement and well described by the Arrhenius equation. We showed that for such a reaction the reaction rates at the initial reaction stage may be used for analysis of the reaction rate slowing down in the critical region arising from both the dynamic and the thermodynamic sources, although the latter reflects the critical behavior at the chemical equilibrium. A simple crossover model was proposed and used to analyze the experimental kinetic data to obtain the critical exponents p, which control the slowing down effects from both the thermodynamic and the dynamic sources. It was found that the value of p was 0.158 ± 0.013 from UV spectrophotometry and in good agreement with the theoretical prediction of 0.151, while the value of p was 0.133 ± 0.012 from microcalorimetry. The slight lower value from microcalorimetry may be attributed to the stirring in the microcalorimeter that weakens the critical reduction of the diffusion coefficient.

Substituting eq 32 into eq 19 gives ln

kobs ∝ (f + y) ln τ = p ln τ kb

(33)

where kb is the background value of the rate constant in the noncritical region, and the temperature dependence of kb may be calculated by Arrhenius equation 29. According to eq 33, a plot of ln((kobs)/(kb)) versus ln τ yielded a straight line with a slope being the critical exponent p. Such plots for the experimental data obtained from UV spectrophotometry and microcalorimetry are shown in Figures 8 and 9. The least-

■

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Corresponding Author

*Phone: +86-21-64250804. Fax: +86-21-64252510. E-mail: [email protected]. Notes

Figure 8. Plot of ln k/kb versus ln τ for alkaline fading of CV in 2BE + water near the critical point determined by UV spectrophotometry. (■) The experimental result; the line represents the result of the fit with eq 33.

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (projects 20973061 and 21073061).

squares fits gave the values of 0.158 ± 0.013 and 0.133 ± 0.012 for UV spectrophotometry and microcalorimetry, respectively. The value of p obtained from UV spectrophotometry is in good agreement with the theoretical prediction of p (p = f + y = 0.11 + 0.041 = 0.151), but the value of p obtained from microcalorimetry is slightly lower. It may be attributed to the stirring in the microcalorimeter that weakened the critical reduction of the diffusion coefficient, while the sample in the sample cell of the UV spectrophotometer was not stirred in the

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