Calculation of Hydrogen Bonding Enthalpy Using the Two-Parameter Abraham Equation

Abstract

In this work, an approach to the calculation of hydrogen bonding enthalpies is proposed. It employs the correlation proposed by M.H. Abraham, establishing the connection between the equilibrium constant (K_HB) and acidity (${\alpha }_2^{\mathrm{H}}$) and basicity (${\beta }_2^{\mathrm{H}}$) parameters: log K_HB = 7.354 · ${\alpha }_2^H$ · ${\beta }_2^H$ − 1.099. Hydrogen bonding enthalpy (Δ_HBH) is found using the compensation relationship with Gibbs energy (Δ_HBG): Δ_HBG = 0.66 · Δ_HBH + 2.5 kJ·mol⁻¹. This relationship enables the calculation of the enthalpy, Gibbs energy and entropy of hydrogen bonding. The validity of this approach was tested against 122 experimental hydrogen bonding enthalpies values available from the literature. The root mean square deviation and average deviation equaled 1.6 kJ·mol⁻¹ and 0.5 kJ·mol⁻¹, respectively.

1. Introduction

There is a continuing interest in the hydrogen bonding phenomenon [1,2,3,4]. This is evidenced by the growing number of publications using hydrogen bonding (HB) as a keyword. However, it is worth highlighting that the number of direct experimental studies on the thermodynamic parameters of hydrogen bond formation has been declining over the last 3 decades. Among the reasons for this is the technically complex and time-consuming measurement procedure. Several approaches for the simple prediction of the thermodynamic functions of HB have been proposed [5,6,7,8]. These works complement the existing data on the Gibbs energies (Δ_HBG) and enthalpies (Δ_HBH) of acid-base complexation. However, the use of these methods is limited by the available constants characterizing the proton acceptor and proton donor abilities.

Recently [9,10], we developed an approach for the calculation of complexation enthalpies (protic acid-base and charge-transfer complexes) using the experimental values of the equilibrium constants at 298.15 K. The determination of complexation enthalpies is a far more difficult procedure, compared with that of the Gibbs energies. This is evidenced by the better reproducibility of the Gibbs energies [1,9,10], and it naturally follows from the fact that most binding enthalpies values are found from the temperature dependences of the equilibrium constants (K_HB) [1,4].

The purpose of this work is to test the predictive capability of a successful method for K_HB prediction developed by Abraham et al. combined with the approach for Δ_HBH calculation using K_HB values.

2. Methodology

Below, the complex formation equilibrium (1) between a proton donor, A-H, and proton acceptor, B, is considered:

$$\mathrm{A}-\mathrm{H}+\mathrm{B}\stackrel{K_{\mathrm{HB}}}{\rightleftarrows }\mathrm{A}-\mathrm{H}\cdots \mathrm{B}$$

(1)

The Gibbs energy of hydrogen bonding is related to the equilibrium constant of reaction (1) as follows:

$${\mathrm{\Delta }}_{\mathrm{HB}}G=-RT\ln K_{\mathrm{HB}}$$

(2)

The hydrogen bonding enthalpy can be determined from direct calorimetric experiments [11,12,13] or from the temperature dependence of K_HB, along with the entropy (Δ_HBS) [1,4]:

$$\ln K_{\mathrm{HB}}=-{\displaystyle \frac{{\mathrm{\Delta }}_{\mathrm{HB}}H}{R}}\cdot {\displaystyle \frac{1}{T}}+{\displaystyle \frac{{\mathrm{\Delta }}_{\mathrm{HB}}S}{R}}$$

(3)

K_HB depends on the proton acceptor ability of B and proton donor ability of A-H. Since the 1980s, several empirical two-parameter approaches have been proposed for predicting K_HB by multiplying the descriptors characterizing A-H and B, particularly by Raevsky et al. [6,7] and Abraham et al. [8]. M.H. Abraham and co-workers developed the acidity and basicity scales based on the analysis of 1:1 complexation constants in tetrachloromethane of (a) a set of acids with reference bases [14] and (b) a set of bases with reference acids [15]. In order to construct the scales, (1) two arbitrary acids (set a) or bases (set b) were chosen with the fixed acidity (log $K_{\mathrm{A}}^{\mathrm{H}}$) and basicity constants (log $K_{\mathrm{B}}^{\mathrm{H}}$), (2) the log K_HB values were correlated with the latter fixed log $K_{\mathrm{A}}^{\mathrm{H}}$ and log $K_{\mathrm{B}}^{\mathrm{H}}$ parameters to characterize the reference bases (set a) and acids (set b), and (3) the log $K_{\mathrm{A}}^{\mathrm{H}}$ and log $K_{\mathrm{B}}^{\mathrm{H}}$ values were obtained for the rest of the proton acceptors and proton donors from the correlations with log K_HB. As a result, the logarithm of the concentration constant of hydrogen bonding (K_HB,c) in tetrachloromethane can be found using Equation (4):

$$\log K_{\mathrm{HB},\mathrm{c}}=(7.354\pm 0.019)\cdot {\alpha }_2^{\mathrm{H}}\cdot {\beta }_2^{\mathrm{H}}-(1.094\pm 0.007)$$

(4)

where ${\alpha }_2^{\mathrm{H}}$ = (log $K_{\mathrm{A}}^{\mathrm{H}}$ + 1.1)/4.636 and ${\beta }_2^{\mathrm{H}}$ = (log $K_{\mathrm{B}}^{\mathrm{H}}$+1.1)/4.636 are re-scaled acidity and basicity parameters. For 1312 acid-base systems, the root mean square deviation of Equation (4) was 0.09 log units [8]. The acidity (${\alpha }_2^{\mathrm{H}}$) and basicity (${\beta }_2^{\mathrm{H}}$) parameters have been derived for a wide range of organic compounds [14,15,16], which makes Equation (4) quite effective in the estimation of K_HB values. This methodology was later developed and fruitfully applied in the studies of hydrogen bonding and fluid phase equilibria by M.H. Abraham and his colleagues, particularly by W.E. Acree, Jr. [17,18,19], to whom the present issue is devoted.

In recent works [20,21], we studied the compensation relationship between the Gibbs energies and enthalpies of solvation and vaporization of organic non-electrolytes at 298.15 K and distinguished four types of solute–solvent systems, exhibiting different types of the Gibbs energy–enthalpy relationship, depending on their hydrogen bonding capabilities. Aromatic and short-chained aliphatic compounds incapable of hydrogen bonding dissolved in non-hydrogen-bonded solvents followed a general linear trend. Whenever hydrogen bonds were formed between a solute and non-hydrogen-bonded solvent, the Gibbs energies of solvation appeared to be systematically more positive; the slope of the relationship stayed the same. Based on these findings, we derived Equation (5), connecting the Gibbs energies and enthalpies of hydrogen bonding [9]:

$${\mathrm{\Delta }}_{\mathrm{HB}}G/\left(\mathrm{kJ}\cdot {\mathrm{mol}}^{-1}\right)=0.660 \cdot {\mathrm{\Delta }}_{\mathrm{HB}}H/\left(\mathrm{kJ}\cdot {\mathrm{mol}}^{-1}\right)+2.5 \cdot n$$

(5)

where n is a number of hydrogen bonds in acid-base complex. Equation (5) was also applied to charge-transfer complexes with iodine and interhalogens [10]. The validity of Equation (5) was confirmed by comparing the complex formation enthalpies calculated using Equation (6) with the literature data on 293 acid-base systems [9] and 152 charge-transfer complexes:

$${\mathrm{\Delta }}_{\mathrm{HB}}H/(\mathrm{kJ}\cdot {\mathrm{mol}}^{-1})={\displaystyle \frac{{\mathrm{\Delta }}_{\mathrm{HB}}G/(\mathrm{kJ}\cdot {\mathrm{mol}}^{-1})-2.5\cdot n}{0.660}}$$

(6)

The root mean square deviation was 1.2 kJ·mol⁻¹ for the hydrogen-bonded complexes and 1.4 kJ·mol⁻¹ for the complexes with iodine and interhalogens.

It is worth highlighting that, in our previous studies of solvation thermodynamics [20,21], the mole-fraction scale was adopted, so Δ_HBG in Equations (5) and (6) is also a mole-fraction-scale-based quantity. The relationships between mole-fraction and concentration constants and the respective Gibbs energies of hydrogen bonding under infinite dilution conditions are given by Equations (6) and (7):

$$K_{\mathrm{HB}}=K_{\mathrm{HB},\mathrm{c}}/V_{\mathrm{m}}^{\mathrm{S}}$$

(7)

$${\mathrm{\Delta }}_{\mathrm{HB}}G={\mathrm{\Delta }}_{\mathrm{HB}}{G}_c+RT\ln \left({\displaystyle \frac{V_{\mathrm{m}}^{\mathrm{S}}}{\mathrm{L}\cdot {\mathrm{mol}}^{-1}}}\right)$$

(8)

where $V_{\mathrm{m}}^{\mathrm{S}}$ is the molar volume of solvent at 298.15 K.

In this work, Equation (6) was combined with Equation (4) to obtain the hydrogen bonding enthalpies in tetrachloromethane directly from the acidity and basicity parameters of the proton donor and proton acceptor. Taking into account the molar volume of tetrachloromethane of 0.0965 L·mol⁻¹ and combining Equations (4) and (6), one comes to a two-parameter Equation (9) for the calculation of Δ_HBH:

$${\mathrm{\Delta }}_{\mathrm{HB}}H/(\mathrm{kJ}\cdot {\mathrm{mol}}^{-1})=-63.61\cdot {\alpha }_2^{\mathrm{H}}\cdot {\beta }_2^{\mathrm{H}}+3.03$$

(9)

It was tested against 122 literature values of the hydrogen bonded enthalpies determined in tetrachloromethane medium. The quality of the prediction was judged based on the root mean square deviation (RMS) and average deviation (AD) between the calculated and literature values given by Equations (10) and (11):

$$RMS=\sqrt{{\displaystyle \frac{\Sigma {[{\mathrm{\Delta }}_{\mathrm{HB}}H(\mathrm{calc})-{\mathrm{\Delta }}_{\mathrm{HB}}H(\mathrm{lit})]}^2}{N-1}}}$$

(10)

$$AD={\displaystyle \frac{\Sigma [{\mathrm{\Delta }}_{\mathrm{HB}}H(\mathrm{calc})-{\mathrm{\Delta }}_{\mathrm{HB}}H(\mathrm{lit})]}{N}}$$

(11)

3. Results

The experimental data on the Δ_HBH values were collected from comprehensive books by Joesten and Schaad [1] and Laurence and Gal [3], as well as original research papers. For some compounds, the specific interaction enthalpies derived using the data on solvation enthalpies in the inert solvents and the pure base according to Ref. [22] are provided. The comparison between the calculated and literature data, along with the Δ_HBG found using Equations (4) and (8), is shown in

Table 1. The Gibbs energies of hydrogen bonding at 298.15 K calculated according to Equation (4), the enthalpies of hydrogen bonding calculated according to Equation (9), and the literature values.

Proton Donor	Proton Acceptor	$\frac{-{\Delta }_{\mathbf{HB}}\mathit{G}\left(\mathbf{Eq}. 4\right)}{\mathbf{kJ}\cdot {\mathbf{mol}}^{-1}}$a	$\frac{–{\Delta }_{\mathbf{HB}}\mathit{H}\left(\mathbf{Eq}.9\right)}{\mathbf{kJ}\cdot {\mathbf{mol}}^{-1}}$ a	$\frac{-{\Delta }_{\mathbf{HB}}\mathit{H}\left(\mathbf{lit}.\right)}{\mathbf{kJ}\cdot {\mathbf{mol}}^{-1}}$	$\frac{\Delta }{\mathbf{kJ}\cdot {\mathbf{mol}}^{-1}}$b	Ref.
Phenol	Benzene	3.1	8.5	6.5	2.0	[1]
	Toluene	3.1	8.5	6.9	1.6	[1]
	m-Xylene	3.6	9.2	8.7	0.5	[1]
	p-Xylene	3.6	9.2	9.0	0.2	[1]
	Mesitylene	4.3	10.4	9.2	1.2	[1]
	Acetophenone	11.6	21.4	19.7	1.7	[1]
		11.6	21.4	21.1	0.3	[1]
	Cyclohexanone	13.7	24.5	22.7	1.8	[22]
	Butanone	12.4	22.6	21.8	0.8	[1]
	Acetone	11.9	21.8	19.7	2.1	[23]
		11.9	21.8	22.3	−0.5	[22]
		11.9	21.8	21.3	0.5	[24]
	Methyl acetate	10.9	20.3	18.8	1.5	[1]
	Ethyl acetate	10.9	20.3	21.8	−1.5	[1]
	Butyrolactone	11.9	21.8	20.5	1.3	[1]
	Dimethylformamide	18.2	31.4	28.7	2.7	[1]
		18.2	31.4	29.1	2.3	[22]
	Propanal	10.9	20.3	18.0	2.3	[1]
	Dimethylacetamide	19.2	32.9	30.8	2.1	[1]
		19.2	32.9	30.3	2.6	[22]
	Pyridine	15.2	26.8	27.2	−0.4	[1]
		15.2	26.8	26.8	0.0	[22]
	Diethyl ether	10.9	20.3	20.1	0.2	[1]
	Benzonitrile	7.9	15.7	13.8	1.9	[1]
		7.9	15.7	19.3	−3.6	[1]
	Tetrahydrofuran	11.6	21.4	22.1	−0.7	[1]
		11.6	21.4	23.0	−1.6	[1]
	Chlorocyclohexane	2.1	6.9	8.2	−1.3	[1]
	1-Bromohexane	2.6	7.7	6.7	1.0	[1]
	1-Chlorobutane	2.1	6.9	7.2	−0.2	[1]
		2.1	6.9	9.3	−2.4	[1]
	1-Bromobutane	2.6	7.7	7.2	0.5	[1]
	1-Iodobutane	3.3	8.8	5.4	3.4	[1]
	1-Iodohexane	3.3	8.8	7.3	1.5	[1]
	Diethyl sulfide	7.6	15.3	15.1	0.2	[1]
	Di-n-butyl sulfide	7.6	15.3	14.2	1.1	[1]
	Benzaldehyde	9.4	18.0	18.0	0.0	[1]
4-Fluorophenol	Benzene	3.3	8.7	7.3	1.4	[3]
		3.3	8.7	7.7	1.0	[3]
	Toluene	3.3	8.7	7.9	0.8	[3]
	p-Xylene	3.8	9.5	8.1	1.4	[3]
	Mesitylene	4.6	10.7	8.8	1.9	[3]
	Pentamethylbenzene	4.8	11.1	10.3	0.8	[3]
	Hexamethylbenzene	5.1	11.5	10.8	0.7	[3]
	1-Heptene	1.4	5.9	6.8	−0.9	[3]
	Diethyl ether	11.5	21.1	24.1	−3.0	[3]
		11.5	21.1	22.7	−1.6	[3]
	Acetophenone	12.2	22.3	20.8	1.5	[3]
	Acetone	12.5	22.7	22.4	0.3	[3]
	Butanone	13.0	23.6	21.1	2.5	[3]
		13.0	23.6	22.9	0.7	[3]
	Cyclohexanone	14.4	25.6	24.3	1.3	[3]
	Benzaldehyde	9.9	18.7	18.6	0.1	[3]
	Methyl formate	9.6	18.3	18.0	0.3	[3]
	Ethyl formate	9.6	18.3	18.0	0.3	[3]
	Ethyl acetate	11.5	21.1	20.8	0.3	[3]
	Methyl acetate	11.5	21.1	20.8	0.3	[3]
	Nitrobenzene	7.0	14.3	11.5	2.8	[3]
	Chlorocyclohexane	2.2	7.1	8.7	−1.6	[1]
	Bromocyclohexane	3.8	9.5	8.2	1.3	[1]
	1-Chlorobutane	2.2	7.1	8.1	−1.0	[1]
	1-Bromobutane	2.7	7.9	7.6	0.3	[1]
	1-Iodobutane	3.5	9.1	6.5	2.6	[1]
	Benzonitrile	8.3	16.3	17.5	−1.2	[3]
	Dimethyl sulfide	7.2	14.7	13.0	1.7	[3]
	Ethanethiol	5.9	12.7	10.4	2.3	[3]
	Diethyl sulfide	8.0	15.9	14.7	1.2	[3]
	Dibutyl sulfide	8.0	15.9	15.5	0.4	[3]
	Pyridine	16.0	28.0	29.6	−1.6	[3]
Chloroform	Pyridine	3.5	9.0	10.0	−1.0	[1]
	Tetrahydrofuran	2.6	7.7	9.2	−1.5	[22]
	Triethylamine	4.5	10.7	13.5	−2.8	[22]
	Benzene	0.4	4.4	7.1	−2.7	[1]
	Diethyl ether	2.4	7.4	7.2	0.2	[22]
	Acetone	2.6	7.8	7.3	0.5	[22]
	Dimethylformamide	4.2	10.2	11.8	−1.6	[22]
	Dimethylacetamide	4.5	10.6	13.0	−2.4	[1]
	Dimethyl sulfoxide	4.3	10.4	10.5	−0.1	[22]
	Ethyl acetate	2.4	7.4	7.7	−0.3	[1]
		2.4	7.4	7.4	0.0	[22]
	Diethyl sulfide	1.6	6.2	7.1	−0.9	[1]
	Acetonitrile	1.5	6.1	4.9	1.2	[22]
	Nitromethane	1.5	6.1	3.7	2.4	[22]
Pyrrole	Acetonitrile	4.0	9.8	10.3	−0.5	[25]
	Anisole	3.7	9.4	7.4	2.0	[25]
	Benzene	1.6	6.1	5.7	0.4	[25]
	Benzonitrile	4.3	10.3	10.3	0.0	[25]
	Chlorobenzene	0.6	4.6	4.1	0.5	[25]
	Cyclohexanone	7.5	15.2	13.7	1.5	[25]
	Dimethylformamide	10.1	19.1	16.6	2.5	[25]
	Dimethyl sulfoxide	10.4	19.5	18.0	1.5	[25]
	Ethyl acetate	6.0	12.8	13.1	−0.3	[25]
	HMPA c	13.8	24.7	24.0	0.7	[25]
	Nitrobenzene	3.6	9.2	8.8	0.4	[25]
	Nitromethane	4.0	9.8	8.8	1.0	[25]
	Pyridine	8.4	16.5	16.2	0.3	[25]
	Tetrahydrofuran	6.4	13.5	15.2	−1.7	[25]
	Toluene	1.6	6.1	6.1	0.0	[25]
N-methylaniline	Dimethylformamide	4.8	11.1	9.9	1.2	[22]
	Dimethyl sulfoxide	5.0	11.3	9.9	1.4	[22]
	Ethyl acetate	2.8	8.0	5.4	2.6	[22]
	Pyridine	4.0	9.8	8.5	1.3	[22]
Methanol	Benzylamine	12.6	22.8	22.3	0.5	[22]
	Triethylamine	13.8	24.7	24.1	0.6	[22]
Butan-1-ol	Dimethylformamide	11.0	20.5	19.2	1.3	[1]
	Dimethylacetamide	11.7	21.5	18.7	2.8	[1]
	Triethylamine	11.8	21.7	23.1	−1.4	[1]
	Pyridine	9.2	17.7	18.4	−0.7	[1]
	Diethyl ether	6.5	13.7	16.3	−2.6	[1]
		6.5	13.7	13.6	0.1	[22]
	Anisole	4.1	9.9	7.1	2.8	[1]
	Tetrahydrofuran	7.0	14.4	12.8	1.6	[1]
	Acetophenone	7.0	14.4	15.9	−1.5	[1]
	Butanone	7.5	15.1	12.8	2.3	[1]
	Cyclohexanone	8.3	16.3	17.2	−0.9	[1]
2-methylbutan-2-ol	Dimethylformamide	9.19	17.7	15.1	2.6	[22]
	Dimethyl sulfoxide	9.45	18.1	15.0	3.1	[22]
	Ethyl acetate	5.41	12.0	9.3	2.7	[22]
	Triethylamine	9.84	18.7	19.7	−1.0	[22]
Hexanol	Benzylamine	10.7	20.1	21.3	−1.2	[22]
	Pyridine	9.2	17.7	15.6	2.1	[22]
	Tetrahydrofuran	7.0	14.4	12.3	2.1	[22]

^a The ${\alpha }_2^{\mathrm{H}}$ and ${\beta }_2^{\mathrm{H}}$ values [16,26] used for the calculation are listed in Table S1 of Supplementary Materials. ^b The difference between columns 4 and 5. ^c Hexamethylphosphoramide.

.

The data on 122 acid-base pairs were collected in total. Among 50 proton acceptors, aromatic and aliphatic amines, amides, ethers, esters, ketones, nitriles, aromatic hydrocarbons, nitro-compounds, halohydrocarbons, sulfides, and hexamethylphosphoramide were considered. The proton donors included phenol and 4-fluorophenol, pyrrole, methanol, butan-1-ol, hexan-1-ol, 2-methylbutan-2-ol, N-methylaniline, and chloroform. When collecting the enthalpies values, we attempted to cover more systems (pyrrole, N-methylaniline, 2-methylbutan-2-ol as acids; amines—alcohols pairs) that were not included in our recent work [9] devoted to the calculation of hydrogen bonding enthalpies from the experimental binding constants.

4. Discussion

For the 122 acid-base systems considered, RMS equaled 1.6 kJ·mol⁻¹ and AD was 0.5 kJ·mol⁻¹. The deviations greater than 3 kJ·mol⁻¹ are rather inconsistent with other available data. Although the maximum positive deviation of 3.4 kJ·mol⁻¹ can be seen for the 1-iodobutane–phenol pair, excellent agreement was observed for 1-bromobutane and 1-iodohexane. One of the values from the Ref. available for the benzonitrile–phenol pair [1], for which the deviation is minimal and equals −3.6 kJ·mol⁻¹, deviates from another value by 5.2 kJ·mol⁻¹.

When analyzing the deviations between the calculated and experimental values, one should consider the possible sources of uncertainties for each magnitude. First, the expected uncertainty of Equation (4) of 0.09 log units contributes ±0.8 kJ·mol⁻¹ to Δ_HBH found using Equation (6) or (9). The uncertainty of Equation (6) itself was earlier estimated as 2 kJ·mol⁻¹ [9]. On the other hand, the uncertainty of the Δ_HBH measurement (usually due to K_HB variation with temperature) depends on the error in K_HB. It was earlier evaluated at the level of ca. 0.05–0.1 log units [15]. Then, one can estimate the uncertainty in Δ_HBH determination from its temperature dependence, similarly to Refs. [27,28]. Assuming that the temperature uncertainty is 0.01 K and three measurements at 288–308 K are performed, the respective standard deviation of Δ_HBH would equal 1–2 kJ·mol⁻¹. The issues with maintaining infinite dilution conditions and temperature effects on extinction coefficients during spectroscopic measurement may lead to far greater errors in Δ_HBH, which is reflected in poor agreement between data on some acid-base systems studied multiple times, e.g., phenol-pyridine. In this case, standard deviation of the literature Δ_HBH values exceeds 4 kJ·mol⁻¹. Based on these estimates, the agreement within 3 kJ·mol⁻¹ for most compounds listed in Table 1 and a RMS of 1.6 kJ·mol⁻¹ can be considered excellent.

Thus, the comparison between the hydrogen bonding enthalpies calculated according to Equation (9) and the experimental data confirms the consistency between the compensation relationship given by Equation (5) and Abraham’s two-parameter equation (4). The acidity and basicity parameters of Equation (4) are readily available for more than 600 compounds [26], or 3.6·10⁵ acid-base pairs. It is reasonable to expect that Equation (9) can be a useful tool for estimating Δ_HBH in numerous acid-base systems not studied before without measuring K_HB. Hydrogen bonding entropies can also be estimated from the calculated Gibbs energies and enthalpies.

One can recall the two- or multi-parameter models [6,7,29,30] for estimating complex formation enthalpies. They rely on the empirical acidity and basicity parameters correlated to the experimental enthalpic data. An important difference of the present study is that Equations (4), (5) and (9) do not follow from the experimental hydrogen bonding enthalpies at all, and their consistency with Δ_HBH in the literature is demonstrated independently.