Study of tert-butyl bromide hydrolysis in the 1-propanol/water system using the fundamental thermodynamic equation of chemical reactivity

Abstract

In previous work we have demonstrated the viability of the fundamental thermodynamic equation of chemical reactivity and have outlined the procedure for analyzing intrinsic activation parameters using this equation. We have recently developed a three-step interpolation technique for evaluating the functional dependencies of these parameters in terms of the system (state) variables. In this work we apply this technique to the hydrolysis reaction of tert-butyl bromide in the 1-propanol/water solvent system, and we use the Kirkwood–Onsager equation to model the electrostatic term in the fundamental equation. Analyses under isobaric conditions show that the intrinsic activation entropy and Kirkwood–Onsager parameter depend strongly on the system variables, particularly the relative permittivity. We also analyze the standard (non-intrinsic) activation entropies and enthalpies for iso-mole fraction and isodielectric conditions. These analyses provide a fundamental thermodynamic platform for distinguishing solvation-shell and bulk electrostatic effects on reaction rates.

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Introduction

The approach we use in our kinetic analyses is primarily based on Eyring's theory,^1–3 but is extended using an expression we call the fundamental thermodynamic equation of chemical reactivity (the fundamental equation for short).^4–6 The form of the equation used in this work includes intrinsic terms for the pressure (P), temperature (T), solvent mole fraction (X, for either component), and solvent relative permittivity (ε_r). The intrinsic activation terms are defined and described in more detail in the Fundamental aspects section.

To illustrate the efficacy of the fundamental equation, we recently presented an analysis for the hydrolysis of tert-butyl chloride in acetonitrile/water (t-BuCl-MeCN/W) between 20 and 50 °C and a water mole fraction (X_water) between 0.660 and 0.790.⁷ Although the rate data was successfully analyzed using the fundamental equation, the results showed irregular kinetic behavior. One possible reason for this is the tert-butyl chloride reaction may occur by a mixed S_N1/S_N2 mechanism. But a more likely reason is the complex physical characteristics of the acetonitrile/water (MeCN/W) system affect the reaction rates in odd ways.

For this work we selected tert-butyl bromide (t-BuBr) as the bromide ion is an excellent leaving group, ensuring the mechanism is S_N1. In addition, we replaced acetonitrile with 1-propanol, as the 1-propanol/water (1-PrOH/W) system is less complex and good literature data is available. We conducted the rate studies for this system (t-BuBr-1-PrOH/W) between 20 and 50 °C and X_water between 0.450 and 0.600. We chose a lower range for X_water because tert-butyl bromide reacts faster than tert-butyl chloride, and there is an upper limit to the reaction rates that we can accurately measure with our instrumental system. As a useful complement to these studies, we use the standard activation thermodynamic equation to analyze the iso-mole fraction and isodielectric rate data. As we will show, these analyses prove to be most valuable in assessing “regional” solvent effects.

As part of our analysis using the fundamental equation, we employ a three-step protocol which we introduced and discussed in our most recent article.⁸ This protocol includes a point-by-point interpolation technique for determining functionalities for the activation parameters, and for generating activation parameter grid equations using what we call “layered polynomials”. Because of the issues previously identified, we were unable to obtain parameter grid equations for the t-BuCl-MeCN/W system. But as expected the reaction rates proved to be more systematic for the t-BuBr-1-PrOH/W system, and for this system we were able to generate parameter grid equations. This article represents an important milestone, as this is the first reaction system that we have been able to implement the complete protocol.

Foundational aspects

The fundamental equation, its interpretation, and restrictions

The fundamental equation in terms of P, T, X, and ε_r is:^4–7in which and are the intrinsic activation volume and entropy, respectively, and the last two terms are the intrinsic activation solvent terms. Intrinsic terms and parameters are those for which one variable (the primary variable) is represented as variable, and all other variables (the complementary variables) are represented as constant. For example, the primary variable for is T, and P, X, and ε_r are the complementary variables.

The rate-determining activation process for solvent-phase unimolecular reactions is illustrated using the following color-coded schematic:

in which Rs represents the reactant-state molecule, Ts represents the transition-state structure, the purple zone represents the solvation shell encasing the Rs, the orange zone represents the bulk solvent surrounding the Rs solvation shell, the blue zone represents the solvation shell encasing the Ts, and the red zone represents the bulk solvent surrounding the Ts solvation shell. As defined in this article, the reactant state (RsS) is a comprehensive term that includes the Rs, its solvation shell, and the surrounding bulk solvent. The TsS is similarly defined.

The terms in eqn (1) are described in more detail as follows using schematic (2). is the difference in molar volume between the Ts and Rs under isothermal/iso-mole fraction/isodielectric conditions , and is the difference in molar entropy between the Ts and Rs under isobaric/iso-mole fraction/isodielectric conditions . and include the structural changes as the Rs transforms into the Ts, but not the structural changes associated with the solvent. The Rs and Ts are of course affected by the solvent. The last two terms in eqn (1) encompass the effects of the structural changes in the solvent. In particular, is the free energy associated with the activation solvent term, , and encompasses the free energy difference between the Ts and Rs solvation shells (). has a meaningful interpretation only if it is a continuous, well-defined function of X. ΔG^‡_P,T,X is the free energy associated with the activation solvent term, , and encompasses the free energy difference associated with the different alignments of the bulk solvent dipoles surrounding the Ts and Rs solvation shells ().

There are a couple of experimental restrictions associated with the application of eqn (1). The first relates to the concentration of the reactant. The electric field strength for small charge distributions decreases radially as the reciprocal of the distance squared . As such, the electrostatic influence on the bulk solvent is expected to become insignificant beyond a short distance from the Rs and Ts. This can be ensured if the Rs concentration is low enough so that the RsS and TsS do not interact with each other. For our experiments the concentration of tert-butyl bromide did not exceed 0.0025 mol dm⁻³, which is approximately equivalent to one reactant molecule encased in a 5-nanometer radius solvent sphere. Secondly, no single term in eqn (1) can be analyzed independently because no single variable can be experimentally varied while the others remain constant.^7,8 While it is possible that variables can be individually varied using simulation methods, this subject is not explored in this article.

Analysis of the linearized isobaric/iso-mole fraction equation using the Kirkwood–Onsager equation to model the electrostatic term

We have routinely model the electrostatic term in eqn (1) using the following basic form of the Kirkwood–Onsager equation:^9,10in which and Q is a convenient substitution parameter (we call the Kirkwood–Onsager parameter) given by:

N _A is Avogadro's number, ε₀ is the vacuum permittivity, Å is the Angstrom, and D is the Debye unit. In eqn (4), μ_Rs is the dipole moment for Rs, r_Rs is its effective radius, μ_Ts is the dipole moment for Ts, r_Ts is its effective radius, and the summation is over the number of Rs molecules. The summation contains only one term for unimolecular reactions as is the case for S_N1 and E1 mechanisms. We note that the Kirkwood–Onsager parameter is intrinsic, but for brevity we present it as Q rather than Q_P,T,X.

Using the Kirkwood–Onsager equation, and assuming and Q are constant, the isobaric/iso-mole fraction expression from eqn (1) is readily integrated to yield:

in which the subscript “0” denotes reference values. Oddly, non-linear regression analyses of eqn (5) can yield good correlation coefficients even if and Q are not constant. Whether or not these parameters are constant can be determined by regression analysis of the following differential form of eqn (1): and Q are constant if the plot of is linear. Otherwise, the functionalities for Q and are evaluated using the two-point interpolation technique described in our most recent article.⁸

The reaction rate law

Whether the mechanism is S_N1 or S_N2, the overall hydrolysis reaction of tert-butyl bromide is:

(CH₃)₃CBr + H₂O → (CH₃)₃COH + H⁺ + Br⁻ (7)

The simplest form of the first-order rate law is:

[(CH₃)₃CBr] = [(CH₃)₃CBr]₀e^−kt (8)

in which k is the rate constant and t is time. We note that eqn (8) applies for a wide range of mechanisms. This includes, among others, S_N1 and E1 for any type of solvent, and S_N2 and E2 if the reactive solvent component is in high concentration, thereby rendering a pseudo-first order rate.

We monitored the reactions using conductivity measurements, so eqn (8) must be recast in terms of conductivity. The following expression, derived in the supplemental section, is the rate law in terms of the solution conductivity, κ:

κ = C − Be^{−k(t−t′)} (9)

in which C and B are substitution constants that represent more elaborate expressions, and t′ is an initial induction time required to ensure the conductivity rate is not affected by the autoionization equilibria of the water and protic cosolvent. We discuss how this induction time is factored into the experimental protocol in the supplemental section. Many of the physical parameters in the expressions for C and B are not known or easily estimated.¹¹ Hence, regression values for these parameters cannot be readily compared with independent estimates.

For our solvent systems the protons that form in the reaction are instantaneously distributed as hydronium and 1-propyloxonium (CH₃CH₂CH₂OH₂⁺) ions. We have considered the possibility that the 1-propyloxonium and bromide ions may react in an S_N2 mechanism to form water and 1-propylbromide, in which case the charge neutralization will introduce systematic error in the analysis of eqn (9). However, the concentrations of these ions never exceed ∼0.0015 mol dm⁻³, so we have presumed this reaction is negligible. Non-linear regression analyses consistently yield excellent correlation coefficients, suggesting this assumption is valid.

Grid equation for the 1-PrOH/W binary solvent system

The application of eqn (1) requires a solvent grid equation that relates the state variables for the solvent system used in the studies. Since we have conducted all experiments at atmospheric pressure, the grid equation that we use only relates T, X, and ε_r.

The following “two-layer polynomial” was used to fit Akerlof's data¹² for the 1-PrOH/W system at one atmosphere between 20 and 60 °C and 0.27 ≤ X_water ≤ 0.93:

ε_r = (aX_water³ + bX_water² + cX_water + d)T + a′X_water³ + b′X_water² + c′X_water + d′ (10)

in which the primed and unprimed lower-case letters are the regression constants. Two-layer polynomials contain two independent variables in which the polynomial parameters of one variable (in this case T) are represented as polynomial functions of the second variable (in this case X_water). The values for the regression constants are shown in Table 1. The relative permittivity range associated with the ranges for T and X_water is 18.1 (at 60 °C and X_water = 0.27) to 66.5 (at 20 °C and X_water = 0.93). The absolute differences between the calculated and experimental values for ε_r vary from 0.00 to 0.26, for which the average is 0.10. Unfortunately, we cannot compare these differences with experimental errors because Akerlof's article does not include an error analysis.

Table 1 Regression constants for eqn (10) using Akerlof's data.¹² Lower-case unprimed and primed letters, starting with a and a′, respectively, are used for all the polynomial expressions in this article

Parameters	a (°C^−1)	b (°C^−1)	c (°C^−1)	d (°C^−1)	a′	b′	c′	d′
Values	−0.308483	0.137074	−0.055573	−0.120757	159.0647	−160.0074	73.27014	14.68355

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We calculated solvent “data” using eqn (10) for specific experimental conditions, and then fitted the data using first- and second-order polynomials. The purpose for this analysis was to reduce eqn (10) to more manageable forms. Table 2 shows the solvent terms analyzed for the various experimental conditions, the values for the regression constants, and the expressions for the solvent derivative terms.

Table 2 Polynomial fits to the “data” calculated from eqn (10) for the conditions shown herein. Also included are the solvent derivative terms with their best-fit polynomial parameters. The correlation coefficients vary from 0.99993 to exactly 1.00000

Experimental conditions and solvent variable ranges		Solvent polynomials; solvent derivative terms; and values for the regression constants
	X water
		a (K^−1)	b
Isobaric/iso-mole fraction; 20.0 to 50.0 °C, 22.4 ≤ εr ≤ 32.0	0.450	−0.146118	69.6605
	0.500	−0.152835	72.9468
	0.550	−0.161181	77.0709
	0.600	−0.171386	82.2152

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Experimental conditions and solvent variable ranges		Solvent polynomials; solvent derivative terms; and values for the regression constants
	T (°C)
		a	b	c
Isobaric/isothermal; 0.450 ≤ Xwater ≤ 0.600, 22.4 ≤ εr ≤ 31.1	25.0	81.7998	−52.6067	33.2182
	40.0	76.5680	−49.6344	30.7479
	50.0	73.0802	−47.6529	29.1010

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Experimental conditions and solvent variable ranges		Solvent polynomials; solvent derivative terms; and values for the regression constants
	ε r
		a (K^−2)	b (K^−1)	c
Isobaric/isodielectric; 22.0 to 50.0 °C, 0.459 ≤ Xwater ≤ 0.598	26.75	−3.71280 × 10^−5	0.0278818	−4.53570

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Experimental protocol and data analysis

Reaction solutions were made using appropriate amounts of distilled water and anhydrous 99.9% 1-propanol (Alfa Aesar), which was stored over 3A molecular sieves (Thermo Scientific). 98+% tert-butyl bromide stabilized with potassium carbonate (Alfa Aesar) was used without purification. Conductivity measurements were made using an Oakton Con 450 probe and conductivity meter, which displayed the conductivity to four digits. Temperatures for the reaction mixtures were maintained within ±0.01 °C using a Fisher Scientific Isotemp circulating thermostat immersed in a water bath. About 18 milliliters of 1-propanol/water solution were added to a small glass vial with a magnetic stir bar, and the vial was immersed in the water bath such that the solution was well below the bath water surface. The probe was snuggly inserted into a cork with a hole drilled into it, and then the cork with the inserted probe was placed over the opening of the vial. The magnetic stirrer was routinely adjusted to give a moderate circulating vortex at the surface of the solution. Two small drops of tert-butyl bromide (about 5 mg, equivalent to ∼0.002 mol dm⁻³) were dripped into the solution, and conductivity and time were monitored manually. To ensure a suitable induction time, monitoring began when the conductivity was at least 50 micro-siemens (µS). The reactions were monitored from 2.0 to 2.5 half-lives. The maximum conductivity measurements varied from 340 at lower X_water to 590 µS at higher X_water. Rate data was generally taken in triplicates, and standard deviations were generally less than 1% of the average experimental values. More than three trials were conducted if the standard deviation was greater than 1%. Inasmuch as this is a well-documented reaction that has been studied rather extensively,^13–16 the reaction products were not verified.

The rate data (as κ vs. t) was analyzed by non-linear regression analysis of eqn (9) using Curve Expert Basic 2.2.3 (Copyright 2020, Daniel G. Hyams), which does not calculate regression errors. Correlation coefficients for the plots varied from 0.99990 to 0.99996, and the half-lives, which were smaller at higher T and X_water, varied from 1.9 to 116 minutes. Once a rate constant was analyzed, the activation free energy was calculated using the following form of the Eyring equation:^1–3

in which R is the gas constant, k_B is Boltzmann's constant, h is Planck's constant, and is 1.2502 × 10¹² K⁻¹ min⁻¹. The transmission coefficient, not shown in the expression, was assumed to be one for all the reactions.

Analyses of the rate data and generation of the parameter grid equations for Q and

For ease of recognition and comparison, the plots and graphs in some of the figures presented in this article are color coded as follows: isobaric/iso-mole fraction – green, isobaric/isodielectric – blue, and isobaric/isothermal – red. Isobaric/iso-mole fraction data for X_water = 0.450, 0.500, 0.550, and 0.600, shown in Fig. 1, was fitted with second-order polynomials (as ΔG^‡_P,Xvs. T), and isobaric/isodielectric data for ε_r = 26.75, also shown in Fig. 1, was fitted with a second-order polynomial (as vs. T). Isobaric/isothermal data for 25.0, 40.0 and 50.0 °C, shown in Fig. 2, was fitted with second-order polynomials (as ΔG^‡_P,Tvs. X_water). The regression values for the polynomial fits are shown in the table insets.

Fig. 1 Isobaric/iso-mole fraction data plots (green) for the following water mole fractions (from top to bottom) and relative permittivity ranges (from left to right in parentheses): 0.450 (26.8 to 22.4), 0.500 (28.1 to 23.6), 0.550 (29.8 to 25.0), and 0.600 (32.0 to 26.8). Also included is the isobaric/isodielectric data plot (blue) for ε_r = 26.75, for which X_water varies from 0.459 and 0.598 (left to right). The rate constants were determined using non-linear regression analyses of eqn (9) with the rate data. Values for ΔG^‡ were calculated using eqn (11) and then fitted with second-order polynomials (ΔG^‡ = aT² + bT + c). The polynomial regression values are shown in the table inset. The correlation coefficients for the polynomial fits vary from 0.999999990 to 0.999999996 for the isobaric/iso-mole fraction data, and the coefficient is 0.99998 for the fit of the isobaric/isodielectric data. The error bars (black horizontal dashes) represent ± one standard deviation.

Fig. 2 Isobaric/isothermal data plots and the relative permittivity ranges (from left to right in parentheses) for 25.0 °C (bottom, 26.1 to 31.1), 40.0 °C (middle, 23.9 to 28.6), and 50.0 °C (top, 22.4 to 26.8). The rate constants were determined using non-linear regression analyses of eqn (9) with the rate data. Values for ΔG^‡_P,T were calculated using eqn (11) and then fitted using second-order polynomials (ΔG^‡_P,T = aX_water² + bX_water + c). The polynomial regression constants are shown in the table inset, and the correlation coefficients vary from 0.99993 to 0.999992.

Fig. 3 shows the plots using eqn (6) and the linear regression fits for the data for the four mole fractions, and the table inset shows the average fitted values for Q and . The plots are obviously not linear, indicating that Q and depend on T and ε_r. Furthermore, the slopes and intercepts are not constant, indicating Q and also depend on X. The degree to which Q and depend on these variables, and whether these dependencies are explicit or implicit, cannot be determined from these plots. However, the narrative presented in the Discussion section gives some insight into the functionalities.

Fig. 3 Plots of for the following water mole fractions and relative permittivity ranges (in parentheses): 0.450 () (26.8 to 22.4), 0.500 () (28.1 to 23.6), 0.550 () (29.8 to 25.0), and 0.600 () (32.0 to 26.8). The linear regression values for Q and are shown in the table inset. The correlation coefficients are all 0.994.

Sets of values for Q and from each plot were evaluated using the two-point interpolation technique described in our latest publication.⁸ However, as the isobaric/iso-mole fraction data was measured at 5.0° increments, the interpolated values are midway between points, and the applicable temperature range is reduced to 22.5–47.5 °C. Fig. 4 and 5 show the isobaric/iso-mole fraction plots of Q vs. ε_r and T, respectively, and Fig. 6 and 7 show the plots of vs. ε_r and T, respectively. The interpolated values were fitted with polynomials, and the regression values are shown in the table insets.

Fig. 4 Isobaric/iso-mole fraction plots of Q vs. ε_r for X_water = 0.450 (), 0.500 (), 0.550 (), and 0.600 (), and the second-order polynomials fits (Q = aε_r² + bε_r + c). The values for Q were calculated using the two-point interpolation technique, and the polynomial regression values are shown in the table inset. The temperature range is 47.5 to 22.5 °C (from left to right) for all the graphs, and the correlation coefficients are all 0.9999997.

Fig. 5 Isobaric/iso-mole fraction plots of Q vs. T for the following mole fractions and relative permittivity ranges (from left to right in parentheses): X_water = 0.450 () (26.5 to 22.8), 0.500 () (27.8 to 23.9), 0.550 () (29.4 to 25.3), and 0.600 () (31.5 to 27.3), and the second-order polynomial fits (Q = aT² + bT + c). The values for Q were calculated using the two-point interpolation technique, and the polynomial regression values are shown in the table inset. The correlation coefficients are all 0.9999997.

Fig. 6 Isobaric/iso-mole fraction plots of vs. ε_r for X_water = 0.450 (), 0.500 (), 0.550 (), and 0.600 (), and the linear fits . The values for were calculated using the two-point interpolation technique, and the polynomial regression values are shown in the table inset. The temperature range is 47.5 to 22.5 °C (from left to right) for all the graphs, and the correlation coefficients are all exactly 1.0000.

Fig. 7 Isobaric/iso-mole fraction plots of vs. T for the following mole fractions and relative permittivity ranges (from left to right in parentheses): X_water = 0.450 () (26.5 to 22.8), 0.500 () (27.8 to 23.9), 0.550 () (29.4 to 25.3), and 0.600 () (31.5 to 27.3), and the linear fits . The values for were calculated using the two-point interpolation technique, and the polynomial regression values are shown in the table inset. All correlation coefficients are exactly 1.0000.

There are two components (C) and one phase (P) for miscible binary solvents. The degrees of freedom (F) calculated from the phase rule (F = C − P + 2) are F = 2 − 1 + 2 = 3. If a system variable is constant, F reduces to 2. Hence, under isobaric conditions the parameter grid equations can be expressed in terms of any two of the three system variables (T, X, and ε_r). In this work, the activation parameters were more conveniently expressed in terms of T and X_water as these variables were systematically varied in the experiments. The sets of regression values shown in the insets in Fig. 5 and 7 were used to generate the following two-layer polynomial expressions for Q and , respectively:

Q = (aX_water³ + bX_water² + cX_water + d)T² + (a′X_water³ + b′X_water² + c′X_water + d′)T + a″X_water³ + b″X_water² + c″X_water + d″ (12)

Eqn (12) and (13) are the parameter grid equations in polynomial form, and they apply within the following variable ranges: 22.5 to 47.5 °C and 0.450 ≤ X_water ≤ 0.600. Table 3 shows the regression values for these polynomial expressions. Our choice of casting Q and in terms of T and X_water does not imply Q and are independent of ε_r. In fact, we will show that Q and are strongly dependent on ε_r.

Table 3 Regression values for the parameter grid equations for Q(X,T) and (X,T) (eqn (12) and (13), respectively)

Parameters	Q(X,T)
a	−0.063035 D^2 Å^−3 K^−2	0
b	0.086762 D^2 Å^−3 K^−2	0
c	−0.035738 D^2 Å^−3 K^−2	0
d	0.0048123 D^2 Å^−3 K^−2	0
a′	49.2959 D^2 Å^−3 K^−1	88.8375 J K^−2 mol^−1
b′	−67.8818 D^2 Å^−3 K^−1	−111.6677 J K^−2 mol^−1
c′	27.9455 D^2 Å^−3 K^−1	39.9966 J K^−2 mol^−1
d′	−3.7595 D^2 Å^−3 K^−1	−4.47914 J K^−2 mol^−1
a″	−9777.467 D^2 Å^−3	−33 834.827 J K^−1 mol^−1
b″	13 470.500 D^2 Å^−3	43 053.124 J K^−1 mol^−1
c″	−5541.811 D^2 Å^−3	−15 797.997 J K^−1 mol^−1
d″	744.711 D^2 Å^−3	1817.954 J K^−1 mol^−1

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Q and can be graphed for iso-mole fraction and isothermal conditions by straightforward application of eqn (12) and (13), respectively, and they can be graphed for isodielectric conditions using the solvent grid equation (eqn (10)) to correlate T and X_water. However, as ε_r was not systematically varied in our studies, acceptable variable ranges for T and X_water under isodielectric conditions must be determined using what we call range graphs, which are generated from the solvent grid equation. For isobaric/isodielectric conditions, range graphs consist of a pair of ε_rvs. X_water graphs for the highest and lowest temperatures over the experimental mole fraction range, or alternately a pair of ε_rvs. T graphs for the highest and lowest mole fractions over the experimental temperature range. Fig. 8 shows the ε_rvs. X_water range graphs for 22.5 °C (the lowest temperature) and 47.5 °C (the highest temperature). The range for X_water must be within these two graphs to ensure T is between 22.5 and 47.5 °C.

Fig. 8 ε _r vs. X _water range graphs generated from the solvent grid equation (eqn (10)) for 22.5 °C (top graph) and 47.5 °C (bottom graph). The double-headed arrow shows the permissible range for X_water at ε_r = 26.75. The significance of this arrow is revealed in the Discussion section.

Expressions for the standard activation entropies and enthalpies

The derivative of the standard activation thermodynamic expression for isobaric conditions is:

Numerical analyses (not given here) show that for both iso-mole fraction and isodielectric conditions. Therefore, it immediately follows that and . Using these thermodynamic identities, along with eqn (1) and the Kirkwood–Onsager equation, expressions for ΔS^‡_P,X and can be straightforwardly derived as follows:

We note that eqn (15) is eqn (6). We also note that the intrinsic term, , is common to both equations, indicating its fundamental importance for the reaction coordinate. Fig. 9 shows plots and graphs of ΔS^‡_P,X and vs. T, and graphs of vs. T under isobaric/iso-mole fraction and isobaric/isodielectric conditions. Fig. 10 shows the plots and graphs of ΔH^‡_P,X and vs. T.

Fig. 9 Data plots and graphs of ΔS^‡_P,X and vs. T, and graphs of vs. T for isobaric/iso-mole fraction and isobaric/isodielectric conditions for the hydrolysis of the t-BuBr-1-PrOH/W system. The plots and graphs are described as follows (date points and graphs (green), with the relative permittivity ranges from left to right in parentheses): X_water = 0.450 () (26.8 to 22.4), 0.500 () (28.1 to 23.6), 0.550 () (29.8 to 25.0), and 0.600 () (32.0 to 26.8); (graphs from bottom to top (green)): X_water = 0.450, 0.500, 0.550, and 0.600; . (data points and graph (blue)): ε_r = 26.75 and 0.471 ≤ X_water 0.590 (from left to right); (graph (blue)): ε_r = 26.75. The data points for ΔS^‡_P,X and are generated using the two-point interpolation method, and the graphs are generated from the derivatives of the polynomials in the ΔG^‡vs. T graphs in Fig. 1. The graphs for are the same as those in Fig. 7, and the graph for is generated using eqn (13).

Fig. 10 Data plots and graphs of ΔH^‡_P,X and vs. T for the hydrolysis of the t-BuBr-1-PrOH/W system. The water mole fractions for the isobaric/iso-mole fraction graphs (green) and the relative permittivity ranges from left to right in parentheses are: X_water = 0.450 () (26.8 to 22.4), 0.500 () (28.1 to 23.6), 0.550 () (29.8 to 25.0) and 0.600 () (32.0 to 26.8). For the isobaric/isodielectric graph (blue), ε_r = 26.75 and 0.471 ≤ X_water ≤ 0.590 (from left to right). The data points and graphs for ΔH^‡_P,X and are generated from the standard expression, ΔH^‡ = ΔG^‡ + TΔS^‡.

Recall that encompasses the internal structural changes as the Rs transforms into the Ts. As such, is expected to be similar in magnitude under iso-mole fraction and isodielectric conditions, which is the case. The activation solvent terms, and , encompass the effects of the solvent on the rate. In particular, the former term correlates the bulk solvent effect, and the latter term correlates the solvation shell effect. There are two ways the solvent affects the Rs and Ts. First, the solvent, which in this context includes the solvation shell and bulk solvent, can cause the Ts to shift in either direction along the reaction coordinate. Secondly, the solvent can stabilize or destabilize the Ts relative to the Rs. These effects are synergistic in the following sense. If the Ts is stabilized, the reaction coordinate tends to shift toward the intermediate or product state. If it is destabilized, it tends to shift toward the Rs. The solvation shell and bulk solvent molecules structurally respond to the developing Ts in two basic steps that occur simultaneously. The first step is the breaking of the intermolecular forces, rendering more free molecules and small clusters, and the second step is the real time alignment of the dipoles of these molecules and clusters with the transient Ts dipole. The first step is structure breaking, endothermic, and incurs an increase in entropy. The second step is a type of electrostriction that is exothermic and incurs a decrease in entropy. Hence, the signs of the activation parameters determine which step dominates the thermodynamics of the reaction.

Discussion

One of the primary goals in this and our last few publications has been to accurately model the Kirkwood–Onsager equation using eqn (1). As we have discussed, terms must be paired in the experimental analysis. The electrostatic term can in principle be coupled with any term, but the best choice in the absence of a good solvation shell model is the intrinsic activation entropy term. With this in mind, the expression for Δ(ΔG^‡_P,X) from eqn (1) is:

Fig. 11 shows graphs of Δ(ΔG^‡_P,X), , and vs. T for X_water = 0.550. The relationship between T and ε_r, necessary for evaluating eqn (17), is given in Table 2. It is noteworthy that the activation entropy and electrostatic terms largely compensate each other, rendering a minimal change for ΔG^‡_P,X over the temperature range. This result, which stems from the fact that , is a testament to the profound effect solvents can have on reaction rates.

Fig. 11 Color-coded graphs of Δ(ΔG^‡_P,X), the activation entropy term, , and the electrostatic term, , vs. T for X_water = 0.550. Ranges for ε_r, , and Q are, respectively from left to right, 29.4 to 25.4, 85.7 to 48.7 J mol⁻¹ K⁻¹, and 11.3 to 7.3 D² Å⁻³. and Q are graphed in Fig. 5 and 7, respectively, and the integrals are numerically evaluated.

The expression for , shown below, enables us to turn our attention to the mole fraction term.

Fig. 12 shows graphs of , , and for ε_r = 26.75. The relationship between T and X_water is given in Table 2. The expression for , needed for the entropy term, is evaluated using eqn (13), and the mole fraction term is evaluated by difference using eqn (18). In previous articles we have used an empirical equation to model the mole fraction term, but this is not necessary or desirable if parameter grid equations exist.

Fig. 12 Color-coded graphs of , the activation entropy term, , and the mole fraction term, , vs. T for ε_r = 26.75. The range for X_water is 0.462 to 0.587 (from left to right), in accordance with the range graph in Fig. 8. The expression for is evaluated using eqn (13), and the mole fraction term is evaluated by difference using eqn (18).

The contrast in the graphs in Fig. 11 and 12 is noteworthy. Whereas the slope for Δ(ΔG^‡_P,X) vs. T is slightly positive, it is substantially negative for vs. T. The intrinsic activation entropy term is similar in the two figures, so the contrast lies predominantly in the difference between the activation electrostatic and mole fraction terms. The positive slope for the electrostatic term in Fig. 11 indicates the Ts becomes less interactive with the bulk electrostatic environment as T increases. However, given the inverse correlation between ε_r and T, this effect likely depends more on ε_r than T. Stated differently, this effect is largely implicit in T, and explicit in ε_r. In contrast, the negative slope for the mole fraction term in Fig. 12 indicates the Ts becomes more interactive with the solvation shell as T increases. Since X_water and T directly correlate under isodielectric conditions, this effect is largely implicit in T, and explicit in X_water. Specifically, even though the Rs is only slightly polar, due to entropy effects its solvation shell probably becomes a bit more enriched with water as X_water increases. This enrichment destabilizes the Rs, but stabilizes the emerging Ts. The synergistic effect from these effects leads to a rather substantial negative slope for vs. T. The results described herein are consistent with the plots and graphs in Fig. 1.

The plots and graphs in Fig. 9 and 10 provide further support to our narrative in the previous paragraphs. Among other things, these figures unveil convincing evidence that the Ts for tert-butyl bromide hydrolysis in polar solvents is much looser than the Rs, which is consistent with an S_N1 or E1 mechanism. One anomalous feature in Fig. 9 is the graph for under isobaric/isodielectric conditions. The shape of this graph no doubt reflects subtle effects associated with the solvation shell, but any explanation for this behavior is conjectural in the absence of a good solvation shell model. However, it is noteworthy that the change in over the temperature range is noticeably less for isodielectric conditions, indicating that depends more strongly on ε_r than on T or X. As Q and correlate with each other (see caption for Fig. 11), we can draw the same conclusion for Q. Table 4 lists the signs for the various activation thermodynamic parameters and summarizes the structural implications associated with these parameters.

Table 4 Signs for the activation thermodynamic parameters and the mechanistic and structural implications for t-BuBr hydrolysis in the 1-PrOH/W solvent system over the variable ranges 22.5 to 47.5 °C and 0.450 ≤ X_water ≤ 0.600

Activation parameter	Sign	Implications and comments
	Positive	Indicates the formation of a loose Ts consistent with SN1 and E1 mechanisms. This result rules out SN2 and E2 mechanisms.
ΔS^‡P,X	Negative	Indicates a significant negative contribution from the alignment of the bulk solvent dipoles with the transient Ts dipole.
	Positive	As , there is a dominant positive contribution from structure breaking which likely occurs between the contact layer of the solvation shell and the Ts.
ΔH^‡P,X	Positive	This result is due largely to the energy required for stretching the C–Br bond in the Ts and aligning the bulk solvent dipoles.
	Large positive	As , there is a large energy requirement for solvation shell structure breaking, which is consistent with .

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A brief comparison of the results of these studies with our earlier studies for the t-BuCl-MeCN/W system is in order. The range for Q is larger for the t-BuCl-MeCN/W system (4.3 to 19.3 D² Å⁻³)⁸ than for the t-BuBr-1-PrOH/W system (4.4 to 12.2 D² Å⁻³). The range for is comparable for the two systems, but starts significantly lower for the t-BuCl-MeCN/W system (−18.7 to 37.4 vs. 42.3 to 92.3 J K⁻¹ mol⁻¹). The experimental ranges for X_water and ε_r are different for the two systems (0.660 ≤ X_water ≤ 0.790 and 48.3 ≤ ε_r ≤ 63.2 for t-BuCl-MeCN/W; 0.450 ≤ X_water ≤ 0.600 and 22.4 ≤ ε_r ≤ 34.7 for t-BuBr-1-PrOH/W). However, we believe the different ranges for Q and in the two systems are based more on electronic and structural differences, and less on the different variable ranges and solvent systems. There are several contributing factors for the different ranges for Q, including the different electronegativities for the halogens, the different carbon-halogen bond lengths, and perhaps even a different degree of solvation. The significantly lower range for in the t-BuCl-MeCN/W system might reflect a shorter and stronger Ts carbon-halogen bond for this system. However, because part of this range is negative, a better explanation is there are stronger intermolecular interactions and tighter Ts vibrational modes for the t-BuCl-MeCN/W system. These tighter modes may also contribute to an enhanced Ts dipole moment, which is likely a contributing factor for the larger range for Q in this system.

Conclusion

Unlike the t-BuCl-MeCN/W system,^7,8 we successfully generated the parameter grid equations for Q and for the t-BuBr-1-PrOH/W system. This was possible largely because reaction rates in the 1-PrOH/W solvent system are more systematic, which is likely due to less structural irregularities compared to the MeCN/W system.^17–19 The takeaway here is that the proper choice of a binary solvent is essential if the desired goal is generating parameter grid equations.

Analyses of the reaction systems we have studied to date show that Q and strongly correlate with each other. This correlation is expected as these parameters both reflect structural changes associated with the formation of the Ts along the reaction coordinate. Q and depend more strongly on ε_r than X_water and T for the variable ranges presented in this work. Further investigation using reactions having a variety of mechanisms is needed to determine how these parameters vary with different mechanisms.

Eqn (15) and (16) provide a valuable and hitherto untapped tool for analyzing the “regional” activation entropies, . The intrinsic activation entropy, , which is common to both equations, encompasses structural changes as the Rs transforms into the Ts under the influence of a solvent (schematic (2)). In a vacuum, are of course no longer relevant, and , which reduces simply to ΔS^‡_P, encompasses structural changes along the reaction coordinate without the influence of a solvent. As we have seen, can be positive or negative depending on the nature of the solvent–solute interactions, but ΔS^‡_P is expected to only be positive for hypothetical gas-phase S_N1 and E1 reactions. For solution-phase reactions, the implicit terms, and , encompass effects from the bulk electrostatic environment and solvation shell, respectively. These terms are correlated with and ΔS^‡_P,X through the non-intrinsic solvent terms, and , respectively. The solvent terms are completely independent of the reactants, and in general are expected to be larger if the structures and polarities of the solvent components are very different, and smaller if they are more similar.

Investigators over the past several decades have routinely obtained inconsistent and conflicting results when analyzing rate data using activation solvent models such as the Kirkwood–Onsager equation. We submit that the chief problem is not with the models, but with the conventional analysis methods. Rate data has traditionally been analyzed by linear regression analyses of the Kirkwood function, .²⁰ The plots used in such analyses are generated by varying T under iso-mole fraction conditions, or by varying X using binary solvents under isothermal conditions. Many types of reactions have been analyzed using the Kirkwood function, most notably the well-studied Menschutkin reactions,^21–25 which seem to serve as a baseline databank for testing theories and reactivity models. Good correlations have been obtained for some reactions,^21,26 but for many reactions either the Kirkwood plots are not linear or the data is so scattered as to preclude any useful interpretation.^24,25,27,28 The overarching problem, which until now has not been recognized, is that reactions in solvent systems cannot properly be analyzed using “stand-alone” solvent models. Solvent models are intrinsic, and as we have shown, intrinsic activation terms must be analyzed in pairs using fundamental equations such as eqn (1).^5–8

The accuracy of the results presented in this work is fundamentally premised on the quality of the Kirkwood–Onsager equation. While precision is certainly not required, this equation must correctly track the general effects from the bulk electrostatic environment for our discussions to be valid. Unfortunately, for reasons previously discussed, historical work with the Kirkwood function provides no solid evidence for the efficacy of the Kirkwood–Onsager equation. However, analyzing a variety of reaction systems using eqn (1) and the Kirkwood–Onsager equation to model the electrostatic term may in time provide a large enough database to undergird a sense of accuracy.

Conflicts of interest

There are no conflicts to declare.

Data availability

Supplementary information (SI): derivation of the rate law in terms of solution conductivity. See DOI: https://doi.org/10.1039/d5cp03924j.

The raw data used for the analyses in this article is available upon request.

Acknowledgements

We acknowledge the use of equipment and chemicals supplied by the Department of Mathematics and Science, Blue Mountain Christian University, Blue Mountain, Mississippi, USA.

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