Transition-State Expansion: A Quantitative Model for Counterion Effects in Ionic Reactions

Summary

Ionic reactions are the most common reactions used in chemical synthesis. In relatively low dielectric constant solvents (e.g., dichloromethane, toluene), ions usually exist as ion pairs. Despite the importance of counterions, a quantitative description of how the paired 'counterion' affects the reaction kinetic is still elusive. We introduce a general and quantitative model, namely transition-state expansion (TSE), that describes how the size of a counterion affects the transition-state structure and the kinetics of an ionic reaction. This model could rationalize the counterion effects in nucleophilic substitutions and gold-catalyzed enyne cycloisomerizations.

Graphical Abstract

Highlights

• •A quantitative model for counterion effects was introduced
• •'Transition State Expansion' was used to describes how the size of a counterion affects the kinetics

Organic Synthesis; Organic Reaction; Chemical Reaction Kinetics

Introduction

Chemical synthesis plays a vital role in pharmaceuticals, materials, agrochemicals, and many other related fields. Ionic reactions are the most common reactions in chemical synthesis (Dugger et al., 2005). According to a comprehensive survey of Good Manufacturing Practice (GMP) bulk reactions run in research facilities between 1985 and 2002 (Dugger et al., 2005), more than 68% of all C-C bond formations are ionic reactions. Examples of ionic reactions include nucleophilic substitutions (S_N), β-eliminations (E₂), enolate-based reactions (e.g., Aldol, Mannich reactions), Friedel-Crafts reactions, and many transition metal-catalyzed reactions (e.g., cationic gold catalysis). The role of counterion and ion pairing in organic reactions and transition metal-catalyzed transformations has been extensively investigated (Macchioni, 2005; Riddlestone et al., 2018; Chen and Lancaster, 2013). However, a quantitative description of how the paired 'counterion' affects the reaction kinetic is still elusive. The selection of counterions in synthesis is still empirical and relies heavily on a trial-and-error approach.

In an ionic reaction, the solvent plays a crucial role in the existing state of ions. If solvents of high dielectric constant (e.g., dimethyl sulphoxide [DMSO], water) are used, the ions are often fully dissociated and solvated. In this case, the counterion will be far away from the reaction center, and the counterion effect should be small. In a solvent of relatively low dielectric constant such as dichloromethane and toluene, ionic pairs may be the dominant species (Macchioni, 2005). In this case, the counterion will be close to the reaction center, and the counterion may have a significant effect on the reactivity. Although there was much empirical information on the counterion effect on individual ion reactions, to the best of our knowledge, there is no general quantitative description of how much ion pairing will affect an ionic reaction. In most theoretical studies of ionic reactions, counterions are simply ignored (Macchioni, 2005). Herein, we introduce a general and quantitative model on how sizes of counterions affect the transition-state structure and the kinetics of ionic reactions.

Results

In a simplified representation of an ion reaction (Scheme 1A), a 'free' ion (A⁻) reacts with reactants (RCT) to give the product via transition state (TS-0). Because TS-0 includes both A⁻ and the reactants, TS-0 could be bigger than A⁻ itself. We name this phenomenon 'transition-state expansion' (TSE). As we discussed earlier, in a solvent of relatively low dielectric constant such as dichloromethane, ionic pairs may be the dominant species (Macchioni, 2005). The reaction of corresponding ion pair – [A⁻ C⁺] will go through transition states TS-1 (when the counterion C₁⁺ is small) or TS-2 (when the counterion C₂⁺ is big), both of which are also ion pairs (Schemes 1B and 1C). Because TS-1 and TS-2 include both A⁻ and the reactants (RCT), TS-1 and TS-2 will also be bigger than the original ion pair (A⁻ C⁺) (TSE). In other words, the distance between the center of positive charge and the center of negative charge will be longer (r₁+δ vs. r₁) if we assume ions are hard spheres. According to Coulomb's law, additional energy will be needed to overcome the attraction between cation and anion to reach transition states TS-1 and TS-2. Also, according to Coulomb's law, the relationship between potential energy (E) and distance (r) is not linear (Scheme 1D). The energy needed to expand an ionic pair the same distance (ε) decreases significantly over distance r (ΔE₂ < ΔE₁). In other words, for TS-2, the energy required to overcome the Coulombic attraction will be smaller than that of TS-1. It should be noted that the ion pair (A⁻ C⁺) can be a catalyst, reagent, or an intermediate.

Scheme 1.

Concept of Transition-State Expansion

We assume that all ion pairs and corresponding transition-state structures (TS-1 or TS-2) can be treated as structureless hard balls. The rate constants (k₁ and k₂) for the reaction in Scheme 1 can be expressed as Equation 2 (Eyring equation, where k_B is the Boltzmann constant and h is the Planck constant). Based on the above assumption, the activation energy difference between TS-1 and TS-2 will only be affected by the electrostatic component of the activation energies $\left({\Delta }^{‡}{H}_2^o-{\Delta }^{‡}{H}_1^o\right)$ (see Equation 5), which can be calculated from Coulomb's law (Equation 1). Other components of the activation energy, such as bond formation and bond rupture in the transition state, or change of solvation, are not influenced by the counterion. Based on those assumptions, the effect of the counterion ions [ln (k₂/k₁)] on the kinetics can be described by Equation 6. In Equation 6, there is a linear relationship between ln(k₂/k₁) and $\left[\left(\frac{1}{{\text{r}}_2}-\frac{1}{r_2+\delta }\right)-\left(\frac{1}{{\text{r}}_1}-\frac{1}{r_1+\delta }\right)\right].$ However, the TSE factor (δ) cannot be determined easily, so we cannot use this equation to verify our model. Because δ will be small compared to r₁ and r₂, then $\frac{1}{{\text{r}}_1}-\frac{1}{r_1+\delta }$ can be simplified as$-\frac{d\left(\frac{1}{r_1}\right)}{d{r}_1}\delta$, and Equation 6 can be simplified as Equation 7. Both r₁ and r₂ can be estimated using computational chemistry methods (see section 4 of Supplemental Information for more details), and k₁ and k₂ can be determined experimentally. Therefore, we could use Equation 7 to verify our kinetic model. We expect this model is especially suitable for reactions whose activation energy is dominated by the enthalpic contribution.

$$\Delta E=\frac{{\text{q}}_1{q}_2}{4\text{πε}}\frac{1}{r}$$ (Equation 1)

$$k=\frac{k_{B}T}{h}\exp \left(\frac{{\Delta }^{‡}{S}^o}{R}\right)\exp \left(-\frac{{\Delta }^{‡}{H}^o}{RT}\right)$$ (Equation 2)

$$\frac{k_2}{k_1}=\exp \left(\frac{{\Delta }^{‡}{S}_2^o-{\Delta }^{‡}{S}_1^o}{R}\right)\exp \left(-\frac{{\Delta }^{‡}{H}_2^o-{\Delta }^{‡}{H}_1^o}{RT}\right)$$ (Equation 3)

$${\Delta }^{‡}{S}_2^o-{\Delta }^{‡}{S}_1^o=0$$ (Equation 4)

$${\Delta }^{‡}{H}_2^o-{\Delta }^{‡}{H}_1^o=\frac{q_1{q}_2}{4\pi \text{ε}}\left[\left(\frac{1}{r_2+\delta }-\frac{1}{{\text{r}}_2}\right)-\left(\frac{1}{r_1+\delta }-\frac{1}{{\text{r}}_1}\right)\right]$$ (Equation 5)

$$\ln \left(\frac{k_2}{k_1}\right)=-\frac{q_1{q}_2}{4\pi \text{εRT}}\left[\left(\frac{1}{{\text{r}}_2}-\frac{1}{r_2+\delta }\right)-\left(\frac{1}{{\text{r}}_1}-\frac{1}{r_1+\delta }\right)\right]$$ (Equation 6)

$$\ln \left(\frac{k_2}{k_1}\right)=-\frac{q_1{q}_2}{4\pi \text{εRT}}\left(-\frac{d\left(\frac{1}{r_2}\right)}{d{r}_2}\delta +\frac{d\left(\frac{1}{r_1}\right)}{d{r}_1}\delta \right)=-\frac{q_1{q}_2\delta }{4\pi \text{εRT}}\left(\frac{1}{{\text{r}}_1^2}-\frac{1}{{\text{r}}_2^2}\right)=$$

$$\mathit{A}\left(\frac{1}{{\text{r}}_1^2}-\frac{1}{{\left(r_1+\Delta r\right)}^2}\right),\Delta r={\text{r}}_2-{\text{r}}_1,\mathbf{A}=-\frac{q_1{q}_2\delta }{4\pi \text{εRT}}$$ (Equation 7)

The above analysis is based on the assumption that a counterion does not affect the core structure of the transition state. In reality, the charged counterion should be able to distort the core structure of TS. We chose to begin our analysis by selecting one of the most synthetically important and also better-studied reactions—the bimolecular nucleophilic substitution reaction (S_N2) (Mikosch et al., 2008) (Scheme 2). The transition state in an S_N2 reaction involves the well-known Walden inversion (TS-A), where the anionic species TS-A shows a linear arrangement (Cl-C-Cl). We assume TS-A should be bigger than the starting anion (Cl⁻) (TSE). Mechanistic studies of S_N2 reactions that take into account the effects of ion pairing are comparatively few (Harder et al., 1995; Streitwieser et al. 1997, 2008; Chen et al., 2009; Streitwieser and Jayasree, 2007; Zheng et al., 2010; Westaway, 2011; Li et al., 2015; Laloo et al., 2016).

Scheme 2.

Analysis of Nucleophilic Reactions Using the Concept of Transition-State Expansion

(A) The transition state in an S_N2 reaction when the counter is ignored.

(B) The hypothetical transition state for an S_N2 reaction with a contact ion pair.

(C) Calculated transition state for an S_N2 reaction with a contact ion pair.

(D) A ball/spring model in the gravitational field as an analogy.

TS-B is our hypothetical transition state for an S_N2 reaction with a contact ion pair. TS-B should have similar geometry to TS-A, but with the counterion arranged equidistantly to minimize Coulombic potential energy (Scheme 2B). Due to the TSE,' the distance between the center of positive charge and the center of negative charge will be longer than the starting ion pair. According to Coulomb's law (Equation 1), compared to the reaction using a free ion (TS-A), additional energy will be needed to overcome the attraction between cation and anion to reach the transition state TS-B. The transition state obtained by quantum chemistry calculations (TS-C) (Scheme 2C) is similar to our hypothetical transition state TS-B. However, we can see that the geometry of TS-C is not the same as TS-A; the most noticeable change is that Cl-C-Cl is no longer linear due to the attraction to the counterion (NMe₄⁺). This bent or distorted transition state TS-C will be higher in energy than its linear counterpart TS-A. The energy difference can be obtained by DFT calculations: energy of TS-C (without a counterion) is higher than that of TS-A. A smaller counterion (e.g., Na⁺) will induce even more distortion.

Therefore, the total activation energy difference caused by a paired counterion (energy difference between TS-A and TS-C) will be the sum of two terms: expansion energy and distortion energy. Dealing with two energy terms could be a complicated task. We hypothesize that TS-C and TS-B have very similar potential energies because—from a molecular mechanics point of view (bonds are considered springs)—the bending of two chlorine atoms brings the center of the negative change nearer to the counterion (center of positive charge). This effect reduces the Coulombic potential energy, which in turn is converted into potential energy caused by the bent of C-Cl bond (distortion energy). We can use the mechanics of a ball/spring model in a gravitational field as an analogy (Scheme 2D). In this model, first, we attach two balls to each end of the spring, then we take away the supports; the balls will go down because of gravity, but the energy of the whole system (balls plus spring) will remain the same because the gravitational potential energy will be converted into the elastic potential energy of the springs. In our ball/spring model, the gravity is analogous to the Coulombic attraction between a counterion and TS-C, and the spring is analogous to the C-Cl bond. This assumption significantly simplifies our calculations on the influence of different counterions on the activation energy because we only need to take into consideration the expansion energy. This assumption will make it possible to give a general model for counterion effects.

Determination of Charge Separation in an Ion Pair

To verify the kinetic model using Equation 7, we need quantitative information about the distance between cation and anion and the relative ion radii to establish the correlation between the reactivity of ionic reaction and the size of ions. Although both atoms and ions do not have sharp boundaries, they are sometimes treated as if they were hard spheres with radii such that the sum of ionic radii of the cation and anion gives the distance between the ions in a crystal lattice. The distance between two ions in an ionic crystal can be determined by X-ray crystallography (Shannon, 1976). However, although X-ray crystallography gives the distance between ions, it does not indicate where the boundary is between those ions, so it does not directly give ionic radii. However, the distance obtained from X-ray crystallography (solid state) is not necessarily the distance in the solution phase (ion pair). Spectroscopic methods and dynamic simulations have been proposed and used to quantify average interionic distances in solution (Correa and Cavallo, 2006; Zuccaccia et al., 2001; Macchioni et al., 2003). Furthermore, these data from X-ray crystallography and spectroscopic methods are limited; usually, only data for specific ions are available. To get a simple, general, and reliable method to estimate the distance between cation and anion in an ion pair, we used a computational chemistry method. All calculations were conducted using DFT B3LYP/6-31G∗ (gas phase). For ionic pairs with symmetric ions (e.g., chloride, BF₄^-), it is simple to determine the center of the cation and anion, so it is easy to determine the distance between cation and anion based on the ion pair's equilibrium geometry (Figure 1A). For example, the distance between Na⁺ and Cl⁻ is 2.369 Å (Figure 1A). The direct method described above is simple to use. However, it is challenging to apply to ion pairs that contain unsymmetrical ions (e.g., AcO⁻Li⁺) because it is difficult to identify the center of the ions. In this case, we used the energy method (Figure 1B). We can calculate energies of AcO⁻Li⁺ ion pair and individual ions (AcO⁻ and Li⁺) at their equilibrium geometries. ΔE will be the Coulombic potential energy caused by ion pairing; according to Coulomb's law, we can calculate r (distance) (Figure 1A). For most ion pairs, the two methods (direct and energy) give similar results; for example, for Na⁺Cl⁻ ion pair, the direct method gives 2.37 Å, and the energy method gives 2.34 Å.

Figure 1.

Equilibrium Geometry and Energies of AcO⁻Li⁺ Ion Pair and Individual Ions

Kinetic Studies of Model Ionic Reactions

The first model of ionic reactions that we studied was the nucleophilic substitution reaction in both of its variances, S_N2, and S_N1. We began our experimental investigation with a typical S_N2 reaction, namely, the substitution of bromide by acetate in n-octyl bromide (Figure 3). The solvent we used was acetic acid because all the metal acetate salts studied had good solubility in this solvent and also because acetic acid has a low dielectric constant (ε = 6.15) and low solvation power. These metal acetates exist as non-dissociated ion pairs in this solvent (Jones and Griswold, 1954; El-Sherbini, 1996). We measured k₁ (M⁺ = Li⁺) and k_n (M⁺ = Na⁺, K⁺, Cs⁺, Me₄N⁺, Bu₄N⁺) and found that the reaction rate increased significantly with an increase in the size of the counterion (relative rates ranged from 1.0 to 28.5).

Figure 3.

Description of S_N2 Reaction Using Equation 7 (r₁ = 1.75 Angstrom).

In theory, we could use Equation 7 to describe the counterion effects. Although r₁ (the charge separation in Li⁺ OAc⁻) in Equation 7 could be calculated from Figure 1 via Coulombic energy in a vacuum, in the reaction mixture, r₁ may be influenced by factors such as solvation and temperature, so it cannot be easily determined with great certainty. To simplify the calculation, we did linear least-squares regression (LSR) fitting using eq-7 for various possible r₁ (from 0.5 Å to 12 Å). In other words, we tried to find the 'best' r₁, which leads to the smallest fitting error using Equation 7. The XY scattering graph between the coefficient of determination (R²) of LSR fitting vs. r₁ was shown in Figure 2. The optimal r₁ is between 1.70 Å and 2.1 Å. This value of r₁ is consistent with our calculation based on Coulombic energy in a vacuum (r₁ = 1.75 Å, Figure 1). When r₁ is determined, we found an excellent linear relationship using Equation 7.

Figure 2.

The Relationship between the Coefficient of Determination (R²) of LSR Fitting and r₁

From the report by Bordwell et al (Olmstead and Bordwell, 1980), metal acetates will exist as dissociated ions in a solvent of high dielectric constant like DMSO (ε = 46.7). As predicted by our model, the reactions of dissociated ions will have a small counterion effect. When we switched to DMSO as the solvent, the counterion effect became minimal indeed (range of k_n/k₁ = 0.7 to 1.2, Figure 4A).

Figure 4.

Nucleophilic Substitution Reactions in Various Solvents

In S_N1 reactions, counterions are not directly involved in the rate-determining step (carbon cation formation); we expect the counterion effect will be small. Our experimental results confirmed this expectation. We chose a typical S_N1 reaction—the reaction of adamantyl bromide with metal acetates. This S_N1 reaction was still fast when bigger counterion ions (Na⁺ < K⁺ < Cs⁺ < Me₄N⁺ < Bu₄N⁺) were used, but the counterion effect was small (range of k_n/k₁ = 1.0 to 1.9, Figure 4B).

In the paragraphs above, we investigated two very typical S_N1 and S_N2 reactions, but not all S_N reactions are either S_N1 or S_N2. A case in point is the reaction of benzyl bromide with nucleophiles. We found that our quantitative study of counterion effects can also serve as a diagnostic tool: if the counterion effect is significant (k_n/k₁ close to a typical S_N2 reaction), then the reaction will have a strong S_N2 character. If the counterion effect is weak, the reaction will have a strong S_N1 character. Indeed, we proved that this was the case experimentally. When we investigated the counterion effect in the reaction of benzyl bromide with metal acetates, we found that their kinetic effects were closer to the reaction in Figure 4C. Therefore, the mechanism of this reaction still possesses a stronger S_N2 character, but an S_N1 mechanism may operate at the same time.

It also should be pointed out that in many ionic reactions, especially in transition metal catalysis, multiple equilibria and several reactive intermediates might be involved in the catalytic cycle. Therefore, detailed mechanisms are challenging to establish, and in many cases, the rate-determining step may not begin with a simple M⁺A⁻ ion itself, as shown in Scheme 1. For example, in cationic metal catalysis, a cationic metal may form a p or s complex with one reactant (RCT) first, and the rate-determining step may be the rearrangement of this complex without the involvement of a second reactant (Scheme 3). In the transition state, bond formation and bond rupture (in both cases, there will be bond elongation) will occur, causing a possible increase in the size (or charge separation between the centers of negative charge and positive charge) of the transition-state structure. It also should be noted that a counterion can be either a pure spectator or directly involved in the corresponding transition state.

Scheme 3.

Transition-State Expansion in a Rearrangement Step

Although most people would consider nucleophilic substitutions (e.g., S_N2) and gold-catalyzed enyne cycloisomerizations to be very different reactions, we have been able to explain their counterion effects using the similar treatment of the nucleophilic substitution (Figure 6). Unlike nucleophilic substitutions, gold-catalyzed reactions have a more complex mechanism, and the rate-determining step is often not clear. As a result, we could not calculate the r1 based on the structure of the transition state. Therefore, the determination of r₁ is not straightforward. However, we still could estimate r₁ based on the kinetic data similar to we did in S_N2 reactions. First, we did linear LSR fitting of Equation 7 for various possible r₁ (from 1.0 Å to 12 Å). The XY scattering graph between the coefficient of determination (R²) of LSR fitting vs. r₁ was shown in Figure 5. The optimal r₁ is around 3.0 Å, which is reasonable for typical ionic gold complexes. To estimate the distance between cation and anion in a gold complex, we did the geometry optimization of one of the simplest cationic gold complexes (Me₃P-Au⁺BF₄^-) using Gaussian 09 (see Supplemental Information section 4.3); the distance between Au and B atom is around 3.1 Å. It is generally accepted that the bond between Au and OTf in L-Au-OTf is not ionic, and our model is for ionic specie. As a result, although we measured the kinetic data for OTf-based reactions, we did not use it in our model (Equation 7).

Figure 6.

Cationic Gold-Catalyzed Cycloisomerization of 1,6-enyne (Calculations Are Based on r₁ = 3.0 Å, Data for OTf⁻ Was Not Used).

Figure 5.

The Relationship between the Coefficient of Determination (R²) of LSR Fitting and r₁ in the Gold-Catalyzed Enyne Cycloisomerization

Our model can also explain many isolated literature observations of transition metal-catalyzed reactions. Although quantitative kinetic data are not available in those literature reports, their findings consistently showed that larger counterions produce faster reaction rates. Selected examples include the copper-catalyzed Diels-Alder reaction (Scheme 4A) (Evans et al., 1999), the palladium-catalyzed CO/styrene copolymerization (Scheme 4B) (Macchioni et al., 1999), the rhodium-catalyzed additions of boronic acid to an aldehyde (Scheme 4C) (Moreau et al., 2001), and the iridium catalyzed hydrogenation of alkenes (Scheme 4D) (Smidt et al., 2004).

Scheme 4.

Selected Literature Examples Showing Primary Counterion Effects

We hypothesize that the counterion has a limited influence on the structures of the transition state in most cases. However, the above statement does have an exception. The interaction between a pairing counterion (X⁻) and the corresponding transition structure is a long-range electrostatic attraction, which is usually relatively weak. While the long-range electrostatic attraction of a counterion may exhibit limited interaction with most atoms or groups, if the transition-state intermediate contains an active proton (e.g., O-H^d+), which has the smallest mass and is highly charged, then a counterion may have a significant impact (Scheme 5) via proton shuttling (Bandini et al., 2012). In these cases, the ability of a counterion to mediate the proton transfer (hydrogen bonding basicity) may play an important role. Indeed, in our previous investigation on the counterion effects on the gold-catalyzed reaction, we found that this was indeed the case (Lu et al., 2017).

Scheme 5.

A Transition State Involves an Active Proton Transfer

Conclusion

In summary, we have postulated a general model, namely 'transition-state expansion,' to describe the influence of the counterion on the kinetics of ionic reactions. This model describes how the size of a counterion affects the transition-state structure and the kinetics of an ionic reaction in a quantitative manner. Further implications and applications of this model are under investigation in our laboratory.

Limitations of the Study

Among many ionic reactions, only nucleophilic substitutions and cationic gold-catalyzed reactions were studied.

Data and Code Availability

Procedure for experiments and characterization data for products are available in Supplemental Information.

Methods

All methods can be found in the accompanying Transparent Methods supplemental file.

Published: October 23, 2020