1 Introduction

The original Cooks’ kinetic method was a landmark for comparison of thermochemical data by tandem mass spectrometry [1–4]. It constitutes a simple approach for the determination of thermochemical data such as proton affinities (PA) and gas-phase acidities (ΔH° acid ). Such data for functionalized compounds (A0) are obtained relative to reference compounds (Ai) with known PA or ΔH° acid values. In the following discussion, only equations corresponding to acidity measurements will be considered, but are easily generalized. The method is based on the competitive dissociations of heterodimers (Scheme 1).

Scheme 1
scheme 1

ᅟ

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Originally, the method was applied to compounds presenting the same chemical function and thus entropic effects were limited. This standard Cooks’ method is generally usable when (1) A0 and Ai are mono-functional compounds, (2) the isomeric dimers [(A0-H), Ai]– and [A0, (Ai-H)]– are interconvertible (i.e., there is a low energy barrier between them), (3) T eff is the same (or very close) for each dissociative heterodimer used for the ΔH° acid (A0) determination, and (4) the entropy variation associated with proton transfer is negligible [ΔΔS° acid (A0, Ai) ≈ 0]. According to the latter requirement, structural and physicochemical similarities for A0 and Ai appear to be necessary. This last condition limits the choice of reference compounds. Fenselau and coworkers proposed an extended approach using references that are similar between themselves but different from the studied compounds [5]. In this case, entropic effects are considered as constant, ΔΔS° acid (A0, Ai) ≈ constant. In theory, this constant could hold any value, however, it has been found that when the entropy variation difference between analyte and reference is too large (ΔΔS° acid is greater than 35–40 J mol–1 K–1), the kinetic method does systematically under- or overestimate ΔH° acid and ΔΔS° acid [6, 7].

In the extended approach, experiments must be carried out under different activation conditions yielding changes in effective temperature in order to obtain the ΔH° acid (A0) value. Then, the kinetic method (for gas-phase acidity determination) can be rationalized by Equations (1) and (2) with the introduction of the GA app term (apparent gas-phase acidity) by analogy with the GB app (apparent gas-phase basicity) introduced by Wesdemiotis [8]. It should be stressed that the effective temperature (T eff ) is a fictional Maxwell-Boltzmann distribution representation of the internal energy of the activated dissociating dimeric ion (within the fragmentation time-window of the analyzer) and is not really equivalent to a thermodynamic temperature [9–11].

$$ ln\frac{{I{{{\left[ {{A_i}-H} \right]}}^{-}}}}{{I{{{\left[ {{A_0}-H} \right]}}^{-}}}}\approx \ln \frac{{{k_i}}}{{{k_0}}}\approx -\frac{{\Delta H{{^{\circ}}_{acid }}\left( {{A_i}} \right)}}{{R{T_{eff }}}}+\frac{{G{A^{app }}\left( {{A_0}} \right)}}{{R{T_{eff }}}} $$
(1)
$$ G{A^{app }}\left( {{A_0}} \right)=\Delta H{{^{\circ}}_{acid }}\left( {{A_0}} \right)-{T_{eff }}\Delta \Delta S{{^{\circ}}_{acid }}\left( {{A_0},{A_i}} \right) $$
(2)

Armentrout later established improved mathematical treatments of the extended method, the alternative and statistical methods, which avoid correlation in the quantities derived from Equation (1) [12]. With the alternative method, GA app(A0) and T eff are determined from the x-intercept and slope of the first kinetic method plot: ln(k i /k 0 ) versus ΔH° acid (Ai). Then, a second plot, GA app(A0) versus T eff , allows the determination of ΔH° acid (A0) and ΔΔS° acid (A0, Ai), which correspond to the y-intercept and the slope of the line, respectively, according to Equation (2). It should be noted that the ln(k i /k 0 ) versus ΔH° acid curves obtained at various activation conditions should ideally present a common crossing point called the “isothermal point” [13] or more accurately “isoequilibrium point” [7] and its abscissa corresponds to ΔH acid (A0). It should be noted that ΔΔS°(A0, Ai) is negligible when the isoequilibrium point is in proximity to the x-axis. Importantly, the ΔΔS°(A0, Ai) values are not exactly constant for dissociations of each noncovalent dimer, which means that a true isoequilibrium point cannot actually exist. Generally, if the distribution in ΔΔS°(A0, Ai) values is not too large, uncertainties in the ΔH acid (A0) value obtained are minimized because these effects are averaged by the linearization in the GA app(A0) versus T eff curves.

The statistical method [12] more rigorously removes cross-correlation in the second kinetic method plots by introducing ΔH° acid (Ai)avg, the average gas-phase acidity of all references used. Now the plots ln(k i /k 0 ) versus [ΔH° acid (Ai)-ΔH° acid (Ai)avg] obtained for each collision energy yield straight lines according to Equation (3). From the slope, -1/RT eff , and y-intercept, [GA app(A0)-ΔH° acid (Ai)avg]/RT eff , the [GA app(A0)-ΔH° acid (Ai)avg]/RT eff versus (1/RT eff ) relation ideally yields a linear dependence. Both the ΔH° acid (A0) and ΔΔS°(A0, Ai) values are consequently easily deduced because the slope and the y-intercept represent [ΔH° acid (A0)-ΔH° acid (Ai)avg] and -ΔΔS°(A0, Ai)/R, respectively, as shown by Equation (4).

$$ \ln \frac{{{k_i}}}{{{k_0}}}=\frac{{G{A^{app }}\left( {{A_0}} \right)-\Delta {H^\circ} _{acid}{{{\left( {{A_i}} \right)}}_{avg }}}}{{R{T_{eff }}}}-\frac{{\Delta {H^\circ}_{acid}\left( {{A_i}} \right)-\Delta {H^\circ}_{acid}{{{\left( {{A_i}} \right)}}_{avg }}}}{{R{T_{eff }}}} $$
(3)
$$ \frac{{G{A^{app }}\left( {{A_0}} \right)-\Delta {H^\circ}_{acid}{{{\left( {{A_i}} \right)}}_{avg }}}}{{R{T_{eff }}}}=\frac{{\Delta {H^\circ}_{acid}\left( {{A_0}} \right)-\Delta {H^\circ}_{acid}{{{\left( {{A_i}} \right)}}_{avg }}}}{{R{T_{eff }}}}-\frac{{\Delta \Delta S^{\circ}\left( {{{\mathrm{A}}_0},\ {{\mathrm{A}}_{\mathrm{i}}}} \right)}}{R} $$
(4)

Again, this approach assumes that ΔΔS°(A0, Ai) values are constant, when in fact, they cannot be.

Although Cooks’ kinetic method is widely used because of its apparent simplicity and ease of use, it has also been subject to criticism by its abusive use. The main reason for its large success is that the kinetic method can be applied using almost all commercial tandem mass spectrometers. In the literature, several nonlinear effects have been observed upon the use of the kinetic method [6, 11, 14, 15]. Indeed, Drahos and Vékey have shown that the first kinetic method plot [ln(k i /k 0 ) versus ΔH° acid (Ai)] is not strictly linear [14]. Such nonlinearity necessarily induces a reduction in the accuracy of the basicity/acidity measurements as they are based on assumed linear regression analyses. This deviation appears to be particularly significant for small molecules presenting a large entropic effect and is reinforced at higher activation energies. Nonlinearities have been also observed in the second kinetic method plot of the alternative approach, GA app(A 0 ) versus T eff [11]. These non-linear effects have been interpreted as resulting from artifacts occurring at high or low energy conditions [11] or resulting from the existence of various isomeric forms within the dimer [15]. Significantly, nonlinear effects probably exist in many other studies but are masked by the cross-correlation inherent in the second kinetic plot of the conventional extended method, as demonstrated below.

In this paper, we are interested in more thoroughly exploring nonlinear effects occurring in the second plot [GA app(A 0 ) versus T eff] used for the alternative approach to the extended kinetic method, and even more apparent in the second plot of the statistical approach. Two examples are discussed. The first example concerns an experimental determination of ΔH° acid of substituted phenols, compounds that have been already widely investigated to determine their thermochemical properties (basicity [16], acidity [17, 18], cation and electron affinities). The results of this study compelled us to examine the second example, which concerns proton affinity determination of theoretical model molecules for which the thermochemical properties are known exactly. It is shown that even in this ideal case, the approximations required to apply the extended kinetic method still result in nonlinear plots of the data.

2 Materials and Methods

2.1 Chemicals

All compounds were purchased from Sigma-Aldrich (Saint-Quentin Fallavier, France). Substituted phenols and references (Table 1) were diluted in methanol at a concentration of 70 pmol μL–1 for experiments carried out on the triple quadrupole and 30 pmol μL–1 with the other mass spectrometers. In order to improve ionization efficiency, 0.1 % (vol/vol) of triethylamine was added to the solution.

Table 1 Molecular Weights and ΔH°acid of the Phenols (R-C6H4-OH) and Carboxylic Acid (R′-COOH) References
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2.2 Mass Spectrometry

All experiments were performed using electrospray ionization operated under the negative ion mode. Desolvation conditions were relatively soft to preserve the deprotonated noncovalent heterodimers. The solutions were infused into the ESI source using a syringe pump (flow rate of 160–400 μL h–1). Three different mass spectrometers have been used: (1) a triple quadrupole (QQQ) instrument (Quattro I; Micromass, Manchester, United Kingdom), (2) a quadrupole ion trap (Esquire 3000; Bruker Daltonics, Bremen, Germany), and (3) a hybrid Qq-TOF (QSTAR Pulsar Hybrid QqTOF; Applied Biosystems, Courtaboeuf, France). The competitive dissociations of the studied noncovalent heterodimers were carried out under CID conditions with a collision voltage range of 0.2– 0.8 V p-p for the ion trap mass spectrometer, 0–26 V for the QQQ instrument, and 0–13 V for the Qq-TOF hybrid mass spectrometer. Helium was used as collision gas for the ion trap mass spectrometer (Pcollision cell = 10–3 mbar), whereas argon was used with the QQQ instrument (Pcollision cell = 5 × 10–5 mbar), and nitrogen for the Qq-TOF hybrid mass spectrometer (arbitrary value of 1 for pressure in collision cell). (For the QQQ instrument, the pressure was kept as low as possible in order to be close to single collision conditions while still obtaining sufficiently abundant product ions at lower collision energies. The estimated average number of collisions is close to unity.) Weighted orthogonal distance regression (ODR) using the ODRPACK suite of programs [19] was used for PA and ΔΔS° calculations as implemented in the ODRFIT program described elsewhere [7]. Uncertainties in these values are quoted as 95 % confidence intervals.

2.3 Theoretical Calculations

Density functional theory (DFT) calculations were performed using Gaussian 03 software [20]. Optimized geometries of the neutrals and anions were conducted at the B3LYP/aug-cc-pVDZ level of theory followed by single point calculations using the aug-cc-pVTZ basis set and B3LYP, B3P86, and MP2(full) levels of theory. Final values are corrected for zero point energy and thermal effects using a rigid rotor/harmonic oscillator approximation to yield ΔH° 298 values.

3 Results

3.1 Experimental Gas-Phase Acidity Results of Substituted Phenols

Each deprotonated heterodimer, comprising an analyte (halogenated phenol) and a reference (carboxylic acid), was prepared with an electrospray ionization source. Table 1 lists the various species considered and their ΔH°acid values taken from the literature. Our experimental source conditions were tuned in order to avoid the dissociation of noncovalent heterodimers in the limit of their lifetimes. Figure S1a in the Supplemental Material presents the mass spectrum obtained for an ortho-fluoro phenol/valeric acid mixture with the QQQ instrument. The proton-bounded heterodimer [A0+Ai-H]– (m/z 213) is produced in good yield reflecting its gas-phase stability under the soft desolvation conditions used. The distribution of homo- and heterodimeric m/z 203, 213, 223 ions is not statistical, a feature that is relevant to differences in their relative stabilities [21]. Low-energy collision-induced dissociations (CID) of the [A0+Ai-H]– heterodimers yielded the two corresponding competitive deprotonated molecules [A0-H]– (m/z 111) and [Ai-H]– (m/z 101) (Supplementary Figure S1b). It should be stressed that consecutive decompositions were observed in certain cases. Strikingly, several carboxylic acid references yielded significant secondary generation of product ions produced by loss of carbon dioxide from 4-pentenoic and trans-2-pentenoic acid anions and induced by losing ethene molecule from ethoxyacetic acid anion. These ions were included in the data analysis by simply adding them to the primary product ion signal.

These experiments have been repeated under various activation conditions for all substituted phenols. In this paper, we present data obtained primarily by using laboratory frame collision energies from 2 to 22 eV in the collision cell of the QQQ mass spectrometer, a range for which nonlinear effects could be observed in some systems. The use of laboratory frame energies is primarily one of convenience because the same energy is required for all systems shown in a kinetic plot, which is of course why most presentations in the literature also follow this protocol. It is probably more appropriate to examine the data on the same center-of-mass frame scale, as these are the energies truly available for the reaction being considered and differences among references can occur especially if they differ appreciably in mass. However, in order to have the same energy for all systems, this requires interpolation of the data, which also introduces some smoothing. For the systems examined here, we have performed this operation and found that it does not significantly change the character of the results, although a slight improvement in the range of linearity was obtained (data not reported). This latter result suggests that the additional work associated with the energy scale conversion may be worthwhile, but we proceed using the more convenient and conventional laboratory scale energies.

Figure 1a and b present the plot of ln(k i /k 0 ) versus ΔH° acid (Ai) for ortho-fluorophenol and para-fluorophenol, respectively. In the following discussions, only these two isomers are described in detail but all data are reported in Table 2. For the ortho-fluoro derivative (Figure 1a), the lines for all activation energies cross at one point: the isoequilibrium point; however, there is no single isoequilibrium point for the para-isomer (Figure 1b). It should be noted that only collision energies at or below 22 eV are reported in these curves and used for the acidity determinations. Indeed, above this energy, the effective temperature (T eff ) value does not rise smoothly with the collision energy, as seen in Figure 1c and d.

Figure 1
figure 1

Plots of ln(ki/k0) versus ΔH° acid (Ai) obtained from experiments performed with the triple quadrupole for (a) ortho-fluorophenol and (b) para-fluorophenol. Plots of T eff versus Ecollision for (c) ortho-fluorophenol and (d) para-fluorophenol

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Table 2 Experimental Values of ΔH° acid and ΔΔS°(A0, Ai) obtained using the Alternative and Statistical Methods and A Comparison with Both Theoretical and Literature Values of ΔH° acid
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The alternative and statistical treatments [12] were used to determine the ΔH° acid (A0) values (Figure 2). According to the alternative approach, the GA app(A0) versus T eff plot for ortho-fluorophenol (Figure 2a) is characterized by a straight line that evolves in the T eff range from 450 to 1100 K. The intercept and slope of this line provides ΔH° acid (A0) = 1446 kJ mol–1 and ΔΔS° = −9 J mol K–1, respectively. In contrast, two slopes are observed for the para-fluorophenol over the same T eff range. The first line was in the 500–850 K range and the second between 850 and 1100 K (Figure 2c), corresponding to acidity values of 1447 and 1454 kJ mol–1, respectively. The gas-phase acidity of this compound was also investigated by using the statistical approach, known to remove errors in the second plot of [GA app(A0)-ΔH° acid (Ai)avg]/RT eff versus (1/RT eff ) [12]. In the case of ortho-fluorophenol (Figure 2b), the statistical approach again leads to a [GA app(A0)-ΔH° acid (Ai)avg]/RT eff versus (1/RT eff ) plot characterized by a single straight line. For para-fluoro phenol, the statistical approach also yields two slopes (Figure 2d), but the slope difference between the two lines is reinforced. (In this case, note that the two slopes have different signs, indicating that the relative acidity is lower than ΔH° acid (Ai)avg = 1451.6 kJ/mol at low energies and higher at high energies.) Both the alternative and statistical approaches give the same ΔH° acid values (as expected) within 1 kJ mol–1, well within the estimated experimental errors (Table 2). It can also be noted that an ODR analysis provides nearly identical results when applied over the same energy ranges for both ortho- and para-fluorophenol. These results are in agreement with the unique isoequilibrium point observed in Figure 1a for ortho-fluorophenol, whereas careful scrutiny of Figure 1b shows two (or more) isoequilibrium points for para-fluoro phenol.

Figure 2
figure 2

Alternative approach: Plots of GA app versus T eff for (a) ortho-fluorophenol and (c) para-fluoro phenol. Statistical approach: Plots of [GA app – ΔH° acid (Ai)avg]/RT eff versus [1/RT eff ] for (b) ortho-fluorophenol and (d) para-fluorophenol. Data obtained using the triple quadrupole with low (2–14 eV) and high (14–22 eV) energies

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If the original extended method [5] is applied to these data, the GA app(A0)/RT eff versus 1/RT eff plot appears to be a straight line with a regression coefficient close to R2 = 1.0000 for both fluoro phenols. From this approach, one experimental acidity value is obtained, 1446 ± 5 and 1449 ± 5 kJ mol–1, for ortho- and para-fluorophenols, respectively. The very good regression coefficient obtained by this method is a result of the self-correlation of the data [12]. Hence, the original extended method does not allow the observation of these two slopes for para-fluorophenol, a feature that probably explains why such behavior has not often been described previously. However, this phenomenon has been observed previously, e.g., two isoequilibrium points can be distinguished on the ln(k i /k 0 ) versus ΔH° acid (Ai) plots for the acidity measurements of urea in the literature [22], corresponding to two acidities values: 1514 and 1518 kJ mol–1. In this case, the use of the original extended method prevented the authors from observing two lines in the second kinetic method plot. Whereas this difference of 4 kJ mol–1 for urea could easily be a result of experimental uncertainties, in the present case, the change in the gas-phase acidity values is more significant for para-fluoro phenol at 7 kJ mol–1. This behavior is not unique as shown in Table 2, where two experimental acidities are characterized for para-fluoro as well as ortho, meta, and para-chloro phenols, with differences of 4–9 kJ mol–1. Only ortho-fluoro phenol presents a single acidity value. (Interestingly, even though the ortho-fluoro phenol results appear linear in all data formats, if the low and high data are independently analyzed, they also yield different ΔH° acid values: 1444.6 ± 2.4 kJ mol–1 and ΔΔS° = −11.7 ± 3.9 J mol–1 K–1 for 2 – 14 eV, and ΔH° acid  = 1451.1 ± 8.7 kJ mol–1 and ΔΔS° = −3.7 ± 9.4 J mol–1 K–1 for 14–22 eV, with a difference in acidities of 6.5 kJ mol–1 that is comparable to the other cases.) Results obtained from the alternative, statistical, and ODR methods are essentially the same when analyzed over the same energy ranges, with differences less than 1 kJ mol–1 (as expected).

In the case of para-fluorophenol, the ΔH° acid value obtained at lower effective temperatures is smaller than that recorded at higher effective temperatures. As the origin of the two experimental acidity values is unclear, we wondered if this phenomenon is inherent to assumptions required by the extended kinetic method. First, good linear plots were obtained for the ln(k i /k 0 ) versus ΔH° acid (Ai) curves, Figure 1b, suggesting that T eff is similar for each dimer system examined. Moreover, the ΔΔS° acid (A0, Ai) differences are not too large (~20 J mol–1 K–1 for para-fluoro phenol and smaller for all other complexes) such that systematic errors in the determination of thermochemical parameters are not expected [6, 7]. The Ai compounds (i.e., aliphatic and alicyclic carboxylic acids) were chosen as references because they are monofuctionnal and similar between themselves. Therefore it is expected that the entropy change differences (ΔΔS° acid ) between the substituted phenols and the references are approximately constant (isoentropic conditions). However, as discussed by Ervin and Armentrout [7], even with such references, the constant entropy exchange assumption is never exactly true.

Similar experiments have also been carried out for the para-fluorophenol system using a quadrupole ion trap and a hybrid quadrupole-time-of-flight instrument (Qq-TOF) (Figure 3 and Supplementary Figure S2). Note that these data do not include the cyclopentylacetic acid reference because it cannot be used with the ion trap or Qq-TOF instrument as their dynamic range is not high enough, i.e., only deprotonated cyclopentylacetic acid is observed in the decomposition of the [C6H11COO– H+ –OC6H4 pF] dimer. This helps explain why the double slope in the QQQ data is not as pronounced in Figure 3 (although still clearly present) as in the comparable plots of Figure 2c and d. Figure 3 shows that the ion trap yields a low T eff , around 300 K, that does not rise significantly with the resonant collision energy, as generally observed with this instrument [23–25]. The Qq-TOF is equipped with the LINACTM collision cell, that allows trapping of the ions in order to increase the duty cycle [26]. Interestingly, in this condition, T eff is between that of the quadrupole ion trap and the triple quadrupole. The lowest T eff value is very close to that obtained with the ion trap, but T eff rises significantly with the increase of the collision energy (up to 410 K). Alternative and statistical analyses of the Qq-TOF data yield ΔH° acid (A0) = 1451 ± 5 kJ mol–1, midway between the values obtained for the low and high T eff ranges in the QQQ instrument. In the present case, the use of either statistical or alternative methods to analyze the ITMS, Qq-TOF, and low-energy QQQ data together yield a value for ∆H° acid (pFC6H4OH) of 1447.9 ± 0.3 kJ mol–1 (1447.3 ± 0.3 kJ mol–1 for a weighted fit), in agreement with the value obtained for the lower collision energy range of the QQQ data, Table 2. Because of the more limited range of temperatures accessed by these alternative experimental results, they do not exhibit the nonlinearities evident in the QQQ data. In particular, the ion trap data alone have a very small T eff range (ΔT eff  ≈ 7 K here), such that accurate thermochemical data cannot be obtained using these data alone. Hence, similar studies were not pursued for systems other than para-fluorophenol.

Figure 3
figure 3

Kinetic method plots for para-fluorophenol obtained with ion trap (ITMS), Qq-TOF, and triple quadrupole (TQ). (a) Alternative approach and (b) statistical approach. The linear regression line in black includes all data points with the exception of the higher energy TQ points. Linear regression lines in blue, red, and green correspond to the Qq-TOF, low-energy TQ, and high-energy TQ data, respectively. Fitting parameters are provided in all cases. These curves have been plotted without the cyclopentylacetic acid

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3.2 Results from the Literature and from DFT Calculations

The ΔH° acid values experimentally obtained for substituted phenols were compared with those obtained by DFT calculations and experimental values reported in the literature (Table 2). ΔH° acid values from the literature presented here were measured by means of the equilibrium method. From the two major acidity reference scales, we choose those determined by McMahon and Kebarle, as this work includes most of the species studied in the present work [27]. In the case of ortho-fluorophenol, the experimental value obtained here is in very good agreement with that reported in the literature. For the p-F and o-Cl phenols, the literature values are in better agreement with the value corresponding to the higher T eff range. In contrast, for m-Cl and p-Cl phenols, the literature values are in better agreement with the experimental line corresponding to the lower T eff range. In all cases, values obtained at both low and high T eff ranges agree with the literature values within experimental uncertainties.

To obtain additional clues to explain our experimental results, quantum chemical calculations were carried out highlighting the relative acidity associated with the M → [M-H]– + H+ equation. Overall, experimental and theoretical 298 K acidity values present a fairly good correlation (Table 2), with mean absolute deviations of 7 ± 4 kJ mol-1 for the B3LYP and MP2(full) approaches and 4 ± 1 kJ mol–1 for the B3P86 calculations. The signed deviations show that the former approaches are systematically low by about 4 %, whereas the B3P86 predictions are evenly distributed about the experimental results. Notably, the quantum chemical calculations provide no obvious explanation for the existence of two slopes (or two acidities) in the experimental results. In particular, this phenomenon cannot be explained by the existence of several isomers as observed for other systems [28]. An implicit assumption of the kinetic method is that the dissociating dimer has only one structure. Lorenz and Rizzo have shown that for dimers of multifunctional amino acids, this is not always the case and different isomers can be experimentally evidenced [28]. In the present work, all monomers are monofunctional, such that the existence of various isomers is unlikely. Although different conformers can be considered, they will not differ in their deprotonation sites. Complications associated with different conformers of deprotonated carboxylic acid dimers have been explored by Ervin and coworkers and were used to explain some inconsistencies in the acidity measurements [29]. This work also demonstrated that only the ground state Z conformers of the carboxylic acids (associated with having the hydroxyl group be oriented cis relative to the carbonyl) contribute appreciably to the dissociation, whereas the E conformers (trans hydroxyl orientation) are sufficiently high in energy (>20 kJ mol–1) that their contributions to the dissociation are negligible. Additional calculations performed here confirm that the E conformers of the other acids also lie 19–21 kJ/mol above the Z conformers. These calculations also show that the lower energy Z conformers of the acids are the conformer involved in the ground state complexes with the phenols.

3.3 Double Slopes: Reality or Artifact?

In principle, the existence of the two slopes could be a result of different experimental issues occurring either at low or high activation conditions. At low collision energy, product ion abundances may be too low to measure accurately, in particular if the difference in branching ratio is large. It can be postulated that ln(k i /k 0 ) should not be larger than 4, corresponding to the measurement of a relative intensity of 2 % for the less abundant fragment ion produced from the decomposition of the noncovalent dimer. This value could be adjusted depending on the dynamic range of the instrument used. At high collision energy, several effects may contribute to the anomalies. First, consecutive decompositions may take place; although this should be overcome by taking into account the intensity of the consecutive product ions by adding their abundance to that of the corresponding primary deprotonated monomer. This presumes that there are no collection efficiency differences in the primary and secondary product ions. Second, fragmentation efficiency may level off or be reduced at high activation energy. In Figure 1c and d, it can be observed that T eff rises almost linearly with the collision voltage up to about 20 eV. After 20 eV, the effective temperature levels out or declines with increasing collision energy, whereas in theory, the T eff values should continue to increase [10, 30]. A continued quasi-linear increase is not expected theoretically, because of angular momentum constraints associated with bimolecular collisions. This high energy behavior alone cannot explain the double slopes observed in the second kinetic method plots of Figure 2, which avoid the high-energy ranges associated with this behavior. Third, m/z discrimination may occur at high collision energy that can lead to errors in the determination of the k i /k 0 ratio. This consideration is particularly important in the case of quadrupole ion trap instruments because of the low-mass cut-off corresponding to about 20 %–33 % of the precursor ion m/z [31]. Other discrimination effects may be related to ion scattering losses in QQQ and Qq-TOF instruments [32], where product ions with relatively high kinetic energy are not collected and detected. Such behavior may be involved in the evolution of T eff with the collision energy in Figure 1c and d. However, because this effect is controlled by the relative masses of the dimer fragments, it cannot explain the difference in behavior between the ortho-fluoro and para-fluoro phenol systems.

The number of references and the available acidity range has a great influence on the determination of the slope of the ln(k i /k 0 ) versus ΔH° acid plot. Our ortho-fluorophenol kinetic plot was done using 8 references with about 30 kJ mol–1 of acidity range, Figure 1a. By contrast, the para-fluorophenol data, Figure 1b, uses only five references because some references such as ethoxyacetic acid have gas-phase acidity and entropy differences that are too large. In order to compare these results in a more rigorous way, three of the references can be removed from ortho-fluorophenol plot, but the curves and isoequilibrium crossing point obtained remain largely unchanged.

3.4 Modeling of the Two Competitive Channels for the Different Systems Studied

Instead of analyzing the QQQ data using the kinetic method approach, the kinetic energy dependence of the dissociation yields can be modeled using the statistical methods developed for analysis of energy-resolved collision-induced dissociation [33–36], which includes Rice-Ramsperger-Kassel-Marcus (RRKM) statistical rate theory [37, 38] to evaluate kinetic and competitive shifts. The CRUNCH Fortran program [36] was used to model the competitive threshold collision-induced dissociation (TCID) of the para-fluorophenol/carboxylic acid deprotonated heterodimers. Using CRUNCH, fits for the normalized A1 and A0 experimental ion peak intensities (logarithmic scale) plotted as a function of the center-of-mass collision energy were analyzed and reproduced (see Supplementary Figure S3).

For these simulations, the ion beam kinetic energy distribution was estimated by analysis of experimental retarding potentials. A relatively large kinetic energy distribution (FWHM) equal to 1.7 eV and a shift in the zero of 0.7 eV towards lower energy values compared to the nominal applied energy were observed. This broadening probably results from the first quadrupole because ions are not extracted only from the center. The offset in energy is also typical and can be the result of many factors. The dominant factor is probably a field in the source region where the zero voltage for the ions is established, but additional factors could include contact potential differences between the source and the region where the ions are measured and aberrations in the focusing.

For this modeling, the total internal energy of the energized complex E* is given by its initial thermal energy distribution [39, 40] plus the energy ε acquired from the collision with argon gas. This process is modeled with the empirical energy-transfer distribution function Equation (5). E is the relative collision energy in the center-of-mass reference frame, σ0 is an energy-independent scaling factor related to the total collision cross section, and N is an adjustable parameter that describes the efficiency of translational to internal energy transfer [29, 41].

$$ {P_e}\left( {\varepsilon, E} \right)={\sigma_0}N\frac{{{{{\left( {E-\varepsilon } \right)}}^{N-1 }}}}{E} $$
(5)

The probability for dissociation, P D (ε), is given by 1 – exp[−k total (E*)τ], where k total (E*) is the total unimolecular rate constant of dissociation as given by RRKM theory, the reaction occurs only when E* ≥ E 0, and τ is the time available for dissociation (here 10-4 s). Branching into the two pathways is controlled by k j (E*) / k total (E*) where k total (E*) = k 1 (E*) + k 2 (E*) for channels j = 1 and 2. The transition states for product formation are treated as orbiting TSs (loose, i.e., located at the centrifugal barrier). Overall the model for the experimental cross sections of a particular channel j is given by Equation (6),

$$ {\sigma_j}(E)=\left. {{{\mathrm{S}}_0}(j)\int {\left[ {{{{{k_j}\left( {E^*} \right)}} \left/ {{{k_{total }}\left( {E^*} \right)}} \right.}} \right]}\ {P_D}\left( \varepsilon \right)} \right){P_e}\left( {\varepsilon, \mathrm{E}} \right)\mathrm{d}\varepsilon $$
(6)

where S0(j) is a scaling factor for channel j. This equation can be compared to the data after convoluting over the kinetic and internal energy distributions of the reactants.

The optimized critical energy values (E 0 ) and the differences of critical energy values (ΔE 0 ) for the two competitive channels are reported in Table 3. Thermal enthalpy corrections from 0 to 298 K (calculated with the rigid-rotor/harmonic oscillator approximation) are included to obtain corresponding ΔΔH° 298 values. For the [CH3COO– H+ –OC6H4 pF] dimer, two different variations of the loose transition states were used (Table 3 and Supplementary Figure S3). First, independent adjustable scaling factors (model a) are employed to improve the fit to the experimental data and allow fitting over a wider collision energy range. Optimal values of S 0(1) = 1 and S 0(2) = 0.33 were obtained from simulations of the two competitive decomposition channels. Such scaling factors suggest that at least one methyl rotor gets locked down by interaction with another part of the molecule in the heterodimer reactant. Second, the methyl rotor of CH3COOH and phenyl rotor of HOC6H4 pF are explicitly treated as hindered rotors rather than as vibrations and the methyl rotor of CH3COO─ is treated as a free rotor (see Supplementary Figure S3 and Supplementary Table S1), which leads to an optimized scaling factor of S 0(2) = 0.07 (model b).

Table 3 Analysis of the Dissociation of [R′COO─ H+ ─OC6H4pF] Dimers with TCID Modelinga
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From Table 3, one can notice that only small differences in the ΔE 0 values are measured depending on the model used (−0.034 and −0.043 eV). These differences are reduced slightly once the thermal enthalpy corrections to convert the thresholds to ΔΔH 0 298 values are applied (−3.7 and −4.5 kJ mol–1) because the hindered rotors have different heat capacities than when treated as vibrations. Because of these small differences, the alternate approaches involving hindered and free rotations were not considered for any but the acetic acid dimer. The utility of using the same approach can be partially assessed by noting that the ΔΔH 0 298 range between the two extreme references (acetic acid and cyclopentylacetic acid, -13 kJ mol–1, Table 1, with a relative uncertainty probably near 4–6 kJ mol–1) is close to that obtained from model a on both compounds, -9.7 ± 1.1 kJ mol–1, Table 3.

Although not the major point of the present work, it is worth noting that the results of Table 3 can be used to derive a value for the acidity of para-fluoro phenol. If the 298 K acidities of the five references from Table 1 are combined with the ΔΔH°298 values, we obtain five values for ΔH°298,acid (para-fluoro phenol) ranging from 1450 to 1455 kJ mol–1. The simple average of these five values is 1452.0 ± 4.2 kJ mol–1, which does not include the uncertainties in the reference values (either relative or absolute), and their weighted average is 1450.6 ± 0.9 kJ mol–1 (uncertainties are two standard deviations). These determinations agree nicely with the literature value of 1455 ± 10 kJ mol–1, as well as with the kinetic method approaches explored here, although it is interesting that the weighted average lies midway between the values obtained from the low and high T eff ranges, Table 2. Notably, the weighted average agrees very well with the ODR result, within the 1 kJ mol–1 uncertainty.

3.5 Modeling of the A1 over A0 Ratios for the Different Studied Dimers

Figure 4 shows the experimental ratios of Ai over A0 ion peak intensities (logarithmic scale) as a function of the center-of-mass energy corresponding to collision voltages taken every 2 volts from 2 to 30 V in the laboratory frame for the five studied deprotonated dimers involving para-fluoro phenol. Calculated ratios obtained from the statistical modeling with Equation 6 and the parameters of Table 3 are also shown using full lines. For most of the dimers, the models reproduce the data relatively well through most of this energy range, whereas there are larger deviations, especially at higher energies. The range of data that seems most reproducible is highlighted by the dashed rectangle in Figure 4. The deviations between experiment and TCID modeling are observed for collision energy values above about 3 E0 or E lab  = 20 eV) and also at the lowest internal energies. Similar deviations have also been observed using more sophisticated instrumentation. For instance, guided ion beam studies of the [CH3COO– H+ –OC6H5]– dimer, where the threshold is 0.98 ± 0.30 eV, deviate from the TCID model for energy values lower than 0.7 eV and higher than 2.5 eV in the center-of-mass frame [42].

Figure 4
figure 4

ln(A i /A 0 ) versus E CM for the different studied dimers [R-COO– H+ –OC6H4 pF]. Points are experimental results and lines are obtained by TCID modeling. The rectangle indicated by the dashed line corresponds to the energy range and product ratios presenting good agreement between experimental and theoretical data. The 3E 0 dashed line is approximately equal to three times the calculated critical energy

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If the TCID model data shown in Figure 4 are plotted according to the kinetic method approaches over the range used in the analyses above, (i.e., 2–22 eV) the results in Figure 5a and b are obtained, where they are compared with the original data (reproduced from Figure 2c and d). As noted above, nonlinear deviations are observed in the second kinetic method plots when using the experimental data (e.g., GA app(A0) versus T eff (alternative method) and [GA app(A0)-ΔH avg (Ai)]/RT eff versus 1/RT eff (statistical method). Such nonlinear deviations are not observed from the corresponding plots that use the TCID model data (fitted curves of Figure 4). In part, this is because the TCID modeling explicitly evaluates variations in the entropy terms from system to system. Notably, the ΔH° acid value obtained from the TCID model, 1451 ± 5 kJ mol–1, matches that obtained from the ODR analysis of the original data over the same range of energies, Table 2. Given this conclusion, it is not surprising that when the kinetic plots are reconstructed considering only the area presenting good agreement between experiment and TCID simulations (removal of the cyclopentylacetic acid reference data and restriction to a collision voltage range from 4 – 18 V in the laboratory), the nonlinear effects largely disappear for both alternative and statistical approaches (Figure 5c and d). Under these new conditions, one can notice that experimental data and calculated curves are nearly superimposable. It should be noted that removing only the cyclopentylacetic acid (keeping the full energy range) still yields a double slope as shown in Figure 3, although it is significantly attenuated. On the other hand, keeping the cyclopentylacetic acid reference but restricting the energy range between 2 and 18 V maintains a strong double slope (in practice, corresponding to the removal of the two highest energy points in Figure 5a and b). This shows that both the energy range and the references used can influence the nonlinear effects. This can be seen in Figure 4, where the “good” data points correspond to those within the dashed rectangle, as determined by agreement between the data and the ideal trends. Importantly, if the ODR model is used, the same ΔH° acid values are obtained (deviations ≤0.6 kJ mol–1) for both 2–22 and 4–18 eV energy ranges with or without the cyclopentylacetic acid reference. Furthermore, the ODR values lie within their 1.2 kJ mol–1 uncertainty (Table 2) of the values obtained from analyses of the TCID model data in Figure 5. The same conclusions hold for the ΔΔ S° values derived in all cases.

Figure 5
figure 5

Experimental and TCID modeling of the second kinetic method plot for [R-COO– H+ –OC6H4 pF] heterodimers; (a) and (c) alternative approach and (b) and (d) statistical approach. Parts (a) and (b) show curves obtained for collision voltages of 2–22 V. Parts (c) and (d) show curves obtained using a restricted voltage range of 4–18 V and removing the cyclopentylacetic acid reference

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For the dimers studied here, the possible coexistence of different isomers resulting from multiple hydrogen bonding interactions seems unrealistic and hence cannot explain the nonlinear effects observed. As noted above, dissociation to the higher energy E conformer of the carboxylic acid neutral has been demonstrated to contribute little to the observed dissociation [29]. Thus, these nonlinear effects are interpreted as resulting from artifacts occurring both at high and low energy conditions. Even in well-controlled guided ion beam mass spectrometry measurements, low-energy deviations can occur and are ordinarily the result of the uncertainties in the data, which are relatively large near threshold because the ion signals approach zero. At higher energies, collisions with rare gases can become more impulsive such that the statistical energy deposition assumed in the TCID modeling may no longer be appropriate. In the experiments performed here, divergences for low-energy values can also occur as a result of DC offset voltages between the Q1 and Q3 analyzers and the h2 collision cell, which lead to higher energy collisions in these regions [41]. The lower-order fields in a hexapole (and especially a quadrupole collision region) introduce perturbations in the kinetic energy distributions of the reactant ions that are much larger than those of the octopole guide used in guided ion beam instruments. These phenomena are probably reinforced by the higher pressure conditions used here compared with those in well-controlled guided ion beam mass spectrometry measurements [43], in order to obtain adequate product ion signal intensity. Note that as the systems get larger and the E 0 threshold values get higher (both of which are true for the [C6H11COO– H+ –OC6H4 pF] dimer), the sensitivity to multiple collision processes is enhanced especially for lower energy values. At these low collision energies, multiple collisions enhance the dissociation probability of larger complexes and those with higher critical energies more than other complexes, whereas at high energies, pressure effects are less important because dissociation of all complexes can be efficiently induced by a single collision, which is much more likely than subsequent collisions.

3.6 Double Slopes Can Be Real: The Proton Affinity of a Model Molecule

In order to better understand the origin of double slopes in the second kinetic method plots, a theoretical system was investigated. The model system was originally proposed by Drahos and Vekey [14] and also used by Ervin and Armentrout [7]. The model proton bound dimer presents 141 degrees of freedom with a mass of 243 u. The proton affinity of the unknown compound is exactly 985 kJ mol–1 and the seven model references are molecules with identical frequencies and variable proton affinity (every 5 kJ mol–1 from 945 to 975 kJ mol–1). The entropy variation between each dissociation channel is ΔΔS° 298  = 65 J mol–1 K–1. In this way, perfect data with no uncertainties are obtained and the energy behavior is well defined. Figure 6 presents the first kinetic method plot obtained from ten center-of-mass kinetic energies ranging from 0.5 to 5 eV every 0.5 eV (equivalent to 3.5 to 35 eV in the laboratory frame with argon as the target gas). Note that there is extensive curvature in the lines, especially for the higher energy data. The first plot (Figure 6a) shows linear regression analyses of the data for each individual collision energy. Note that the isoequilibrium point shifts smoothly with kinetic energy. The ODR analyses of all the data (Figure 6b), of the five lowest collision energies (Figure 6c), and of the five highest collision energies (Figure 6d) are also shown [19]. It should be pointed out that the ODR plot requires the existence of an isoequilibrium point where the lines for all data cross. The ODR plot with all data included (Figure 6b) gives good agreement with the correct input values, whereas the other two analyses deviate somewhat from the correct values.

Figure 6
figure 6

The logarithm of ratios of cross sections, ln(σ1/σ2), is plotted as a function of the reference proton affinity PA298(Bi) for the model system for CID at 10 relative collision energies from 0.5 (diamonds at top) to 5.0 (diamonds at bottom) eV: (a) linear fits to each collision energy, (b) orthogonal distance regression (ODR) fits of all data, (c) ODR fits of low energy data, and (d) ODR fits of high energy data. In each panel, the + symbol shows the input values of PA = 985 kJ mol–1 and ΔΔS = 65 J mol–1 K–1. The solid circle shows the isoequilibrium point in panels (b)–(d) for which the PA and ΔΔS values are provided

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From the linear regression analyses (slopes and intercepts) of Figure 6a, data can be acquired to perform kinetic method analyses using the original extended (Figure 7a), the statistical (Figure 7c), and alternative (Figure 7d) methods. (In all cases, the data analysis is of properly weighted data. Notably, the error bars in Figure 7d are not included because they are sufficiently large to span much of the area of the plot (e.g., the lowest temperature point is T eff  = 319 ± 4 K and GA app = 964 ± 19 kJ mol–1 and the highest point is T eff  = 654 ± 37 K and GA app = 945 ± 77 kJ mol–1). Figure 7b also shows the value of T eff as a function of the center-of-mass collision energy. Note that in the original extended plot, the data look very linear although analysis of only the low and high-energy data (five points each) yield different PA and ΔΔS° values. The apparent linearity is again a reflection of the cross-correlation between the GA app/RT eff and 1/RT eff coordinate. In contrast, when this correlation is removed according to the statistical or alternative approaches, the plots present significant curvature. Linear fits of the low and high-energy data (five points each) yield different PA and ΔΔS° values, but the same values are obtained for all three approaches (as expected). In addition, these values agree approximately with the ODR results for the low and high energies (Figure 6c and d). If all the data are analyzed (resulting linear fits not shown in Figure 7), the results are again similar to those obtained with the ODR method, Figure 6b (see comparison in Table 4) although ODR yields the most accurate results compared with the input parameters. This result shows the potential of the ODR approach, which forces the lines to cross at a single point, although this approach would no longer reveal the deviations from linearity observed in the second kinetic method plots, such as those presented in Figure 2c and d.

Figure 7
figure 7

Extended kinetic method plots obtained from the data of Figure 6a. (a) Original extended method, (b) Teff versus Ecollision, (c) statistical method, and (d) alternative method. All data were generated using input values of PA = 985 kJ mol–1 and ΔΔS = 65 J mol–1 K–1. In parts a, c, and d, lines show weighted linear regression fits to the data at low and high energies (five points each) and the resultant PA (kJ mol–1) and ΔΔS (J mol–1 K–1) values obtained are shown

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Table 4 Data Obtained from the Theoretical Modela
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Another interesting aspect of the ODR results in Table 4 is the very large uncertainties found for the high energy range (3–5 eV), which occurs because the branching ratios do not vary appreciably over this range. Thus, to obtain precise thermodynamic information, it is critical to include low energies (or temperatures) where the differentiation between the two product channels is maximized. This is equivalent to obtaining data in the threshold region for energy-resolved CID data.

3.7 How to Improve the Experimental Results

Various considerations may be proposed to improve the accuracy of measurements obtained by kinetic method measurements [2, 6, 7]. The quality of the references is probably the crucial point [44]. Ideally, the references should include values both above and below the molecule of interest. A series of references that are really homologous (allowing more constant ΔΔS° terms to be maintained) are necessary [5]. The proton affinity (or acidity) of each reference should be well defined, which should allow values of very good relative accuracy to be obtained. A series of references from which thermochemical data was obtained using the same method and from the same laboratory is preferable. This is an important point because absolute errors as reported in the NIST webbook are often in the 10–15 kJ mol–1 range [45]. If these errors are really random, the use of the kinetic method to acquire accurate thermochemistry is impossible because this range also corresponds to the typical maximum ΔH° difference between the reference and the compound of unknown ΔH°, as illustrated in Figure 1. Larger ΔH° differences will lead to branching ratios between the two possible decomposition pathways that are too large to measure accurately. In practice, if the references are of the same origin, the relative error may be much lower, on the order of 2–5 kJ mol–1. Ideally, these uncertainties should be included in the data treatment, where the relative errors are more relevant for the determination of error bars in the x axis of the first kinetic method plot. In addition, the number of references used must be taken into consideration [46]. Three references is probably a minimum, but the use of five or more references is preferable. One advantage of using a relatively large number of references is that bad references that are uncorrelated may be more easily identified. In Figure 4, it is shown that deviation from ideal behavior is strong at low energy for cyclopentylacetic acid and at high energy for acetic acid and propionic acid, because they have the lowest E 0 values.

The use of multiple references is not always possible but increasing the effective temperature range is also in principle a good way to increase the measurement accuracy [23]. In essence, this is one fundamental reason why energy-resolved CID data is generally a superior approach to the acquisition of thermodynamic data on such systems, i.e., the data are fit over a wide range that covers the threshold (or room temperature region) to very high energies. In addition, because these data can be modeled using a statistical approach, entropic effects are included explicitly for all individual systems, rather than forcing an average entropic factor on related but distinct systems. In such cases, establishing what data are “reliable” is more apparent, although still not foolproof. In the absence of such energy-resolved data, an efficient approach appropriate for commercial instruments is to use different instruments characterized by different time windows and yielding different kinetic shifts, as illustrated in Figure 3. For instance, a quadrupole ion trap is characterized by a relatively long time window (1–10 ms), and therefore tends to yield decompositions under low activation conditions [47, 48]. In this case, measured T eff values are close to 300 K. QQQ instruments are characterized by a time window of about 100 μs, and generally lead to higher T eff values in the range of 400 to 1000 K. Although a quadrupole ion trap alone does not allow a large change in T eff , together with a triple quadrupole, they offer a significant increase in the T eff range [23]. Finally the data obtained on the Qq-TOF instrument are complementary as T eff is in the 300–400 K range. Therefore, this instrument appears interesting in the context of the kinetic method and, in particular, together with a QQQ instrument. It should be noted that the increase of the T eff range may be at the cost of the ΔH° range of references because it is important to use the same set of references with all instruments to avoid systematic errors. Thus, if the T eff is too low or too high, significant changes in the branching ratio can occur, such that some references cannot be used with all instruments, as noted above for the cyclopentylacetic acid reference, which cannot be used with the ion trap or Qq-TOF instruments. Finally, it should be noted that the double slope was only clearly observed with the QQQ data because this instrument is the only one that yields a large T eff range.

The use of the ODR analysis for analyzing kinetic method plots avoids correlation among derived parameters and allows all the data to be included and accurate uncertainties to be properly evaluated. For the model system, comparison to the theoretical data demonstrates that the ODR fit (with all data) gives the most accurate determination of PA (Table 4). However, because ODR demands an isoequilibrium point, this method may hide deviations related to experimental artifacts (e.g., ion scattering and ion discrimination). The ln(k i /k 0 ) versus E CM plots (Figure 4) appear to be very good markers of the validity of the results (although this is most obvious when compared to the statistical analysis). Such plots were originally included in the first paper of Fenselau and coworkers on the extended kinetic method [5]. As shown by the model data in Figure 4, all the curves should tend toward a common value at high kinetic energies corresponding to the isoequilibrium point. Deviations from this ideal behavior indicate errors in the experimental data and may provide information about what energy ranges can be trusted.