- freely available
Int. J. Mol. Sci. 2009, 10(6), 2809-2837; doi:10.3390/ijms10062809
Abstract: The citric acid cycle (CAC) is the central pathway of energy transfer for many organisms, and understanding the origin of this pathway may provide insight into the origins of metabolism. In order to assess the thermodynamics of this key pathway for microorganisms that inhabit a wide variety of environments, especially those found in high temperature environments, we have calculated the properties and parameters for the revised Helgeson-Kirkham-Flowers equation of state for the major components of the CAC. While a significant amount of data is not available for many of the constituents of this fundamental pathway, methods exist that allow estimation of these missing data.
The citric acid cycle (CAC), also known as the Krebs or tricarboxylic acid cycle, is a fundamental pathway in intermediary metabolism among all the domains of life. Organisms that use the CAC (or especially the reverse or reductive CAC) are represented among the most deeply-rooted autotrophic hyperthermophilic Archaea and Bacteria and the most derived of organotrophic evolutionary lineages [1–4]. The key role of iron-sulfur proteins, thioester intermediates, and the reductive use of the CAC in hyperthermophilic autotrophs link the CAC to prebiotic theories of energy metabolism and abiotic carbon fixation at deep-sea hydrothermal vents [5–7]. Because of the fundamental metabolic roles and evolutionary importance of the CAC and the reverse CAC, it is of interest to understand the conditions constraining the many reactions of which it may be composed under the considerable range of physical and chemical environments where it functions. Most thermodynamic data available for the substrates used in the CAC are incomplete and have only been determined at the 25 °C and 0.1 MPa standard state. Therefore it is difficult to predict accurate reaction thermodynamics well beyond those conditions, including those likely to have hosted the emergence of life [7–10]. For examination of CAC reactions under the relatively extreme high-temperature and high-pressure conditions where life can thrive and may have originated, and to determine the geochemical environments where prebiotic conditions may have been favorable, the high pressure and temperature thermodynamic parameters of the substrates must be determined.
To evaluate the standard Gibbs free energy ( ) of reaction for a given set of products and reactants at temperatures and pressures beyond the 25 °C – 0.1 MPa reference conditions in an aqueous system, the standard Gibbs free energy of formation ( ) for each substance at non-standard temperature and pressure must be calculated. Most thermodynamic data available for aqueous organic molecules beyond the 25 °C and 0.1 MPa reference conditions are generally for compounds that are either potential growth substrates or metabolic by-products. Only in a few works are there data available for compounds that are intermediates in the ubiquitous biochemical pathways found in nature. The general lack of empirical data for aqueous organic compounds at elevated temperature and pressure has lead to the development of various equations of state to describe the behavior of various thermodynamic properties at non-standard temperature and pressure [11,12]. The revised Helgeson-Kirkham-Flowers equations of state [11,13], along with methods for the estimation of high pressure and temperature thermodynamic properties [14–20], can be used to predict accurately reaction thermochemical properties for aqueous organic compounds well within the thermal and pressure range of the possible environments where these reactions may be expected to occur in biological processes. In addition to aqueous organic compounds [13,18,21,22], the revised HKF equations of state have been used to calculate accurately the thermodynamic reaction properties, without the benefit of high pressure and temperature data, for aqueous inorganic  and organic ions [13,23], and inorganic electrolytes  at subcritical temperatures and pressures up to 500 MPa. Such versatility makes it one of the more useful tools for evaluating geochemical and biochemical reaction properties in a wide variety of environments, as the revised HKF allows calculation of species and reaction properties among minerals, gases and aqueous species. This provides a framework for understanding many interdependent metabolic and geochemical processes .
Data for reactions depicted in Figure 1 are used as a suggestion for a prebiotic reductive CAC. However, estimation of the thermochemical properties of the thioester intermediates may also function as proxies in lieu of high pressure and temperature reaction data for coenzyme A or other thioester intermediates found in Archaea.
The revised HKF equation of state uses the standard  , aqueous partial molal entropy (S0aq), partial molal volume (V0), and constant pressure molal heat capacity ( ), along with fitting parameters that integrate the change in the partial molal property, into the at the desired temperature and pressure conditions. In regard to the revised- HKF equation of state, the 25 °C – 0.1 MPa properties are referred to as the reference state, i.e. , and the calculated partial molal property at P and T, the standard state, i.e. . For substances for which incomplete thermodynamic data are available, the use of group contribution, or additivity, algorithms has become the most pragmatic method available in light of the vast number of naturally occurring and synthetic organic compounds. Group additivity relationships have been used to generate 25 °C, 0.1 MPa reference state thermodynamic properties and group values. These estimation methods have been used for pure phase gas, liquid and solid organic compounds [27–29] and aqueous neutral and ionic organic compounds. Namely among the aqueous species are n-alkanes, n-alkenes, n-alcohols, n–alkanones , aldehydes , amino acids [18,22], carboxylic, hydroxy and dicarboxylic acids, and their respective ions , as well as numerous biochemically-relevant organic compounds (e.g.[31,32]).
2.1. Strategy used for the estimation of missing reference state values
The approach taken to estimate reference state values herein is to utilize methods using state variables and reaction data, where available, to calculate missing values. In absence of state or reaction data, the group contribution method is used, with emphasis on attaining data for the closest structural analogues as a base structure. In doing so, the fewest group values are used to modify the base structure as a means of decreasing the probability of error accumulation. The selection of methods used to estimate reference state data are summarized in Figure 2 using the neutral species as examples.
2.2. Calculation of reference state and estimation of missing values
The values selected for neutral and ionic organic species are shown in Table 1. The for organic acids and ions for which there were no data available were calculated from the ionization constants (pKa), critically compiled in , and of the respective ion or acid for which there were reliable data. The only acid-anion pair for which values were unavailable was succinyl thioester. Therefore, estimation of its was necessary and attained through the addition of the for the (>C=O) and (–S-) groups to pentanoic acid-pentanoate for the succinyl thioester acid-anion pair. The group value was calculated from the difference in the y-intercepts for regression equations between n-alkanones and (n-1) n-alkanes (both taken from reference ). The was calculated similarly, from the difference in y-intercepts from dialkylsulfides  and n-alkanes , with the same number of carbon atoms. Data for other components of the citric acid cycle (ethyl thiol and acetic acid; see Figure 1) are available in the literature [16,23].
The estimation of high temperature and pressure thermochemical properties for cis-aconitate and isocitrate, for the reactions between citrate and α-ketoglutarate, were not calculated due to deficient reference state data. Since citric acid and its ions are the only feasible tricarboxylated base structures available to use for the estimation, this approach was rejected to avoid circularity. Furthermore, the primary goal of estimating the high pressure and temperature parameters for CAC reactions calculated herein is primarily driven to examine these reactions under the near equilibrium conditions that are present in anoxic systems. Extant microbes that use the CAC reductively to fix carbon generally only contain a partial cycle where the pathway is used to supply precursors for biosynthesis via succinyl-Coenzyme A or α-ketoglutarate [2–4,45].
2.3. Calculation of the reference state entropy (S0 aq) and estimation of missing values
The S0aq values selected are shown in Table 1. If S0aq values were not available, third-law entropies were calculated from the aqueous enthalpy and Gibbs free energy values along with the sum of entropies of the elemental constituents using Equation (1):
The aqueous formation entropies for H-α-ketoglutarte−1 and α-ketoglutarte−2 were estimated by assuming similar ΔionS from succinic acid and its mono- and divalent anions (from Shock ). The carbonyl group S0aq contribution value (54.3 J mol−1) used for oxaloacetic acid and its respective anions was calculated from the difference in S0aq between α-ketoglutaric acid and succinic acid from Shock :
To calculate the S0aq needed for acetyl thioester the value of S0aq (-S-) was calculated from the difference in aqueous entropies between ethyl sulfide (see Appendix A, Equation 21), and n-butane .
The S0aq (>C=O) value used for succinyl thioester and its anion was estimated from the difference (68.15 J mol−1 K−1) in y-intercepts between n-alkanones and (n-1) n-alkanes, both from Shock and Helgeson . The S0aq (>C=O) value used here was chosen since the value derived from an n-alkanone may be more likely to represent the value of a subterminal carbonyl adjacent to a sulfur atom, as opposed to the α-carbonyl adjacent to a carboxylic acid, as was used above for oxaloacetic acid.
2.4. Calculation of the reference state partial molal volume (V°aq) and estimation of missing values
The chosen partial molal volume values are shown in Table 1. The group value V0 (>C=O) used for the oxaloacetate series was estimated by the difference in V0 between α-ketoglutaric acid  and succinic acid  such that:
This V0 (>C=O) was used for pyruvic acid and pyruvate as well since it may rather well represent the volume of an α-carbonyl (V0 ~5 cm3 mol−1) adjacent to a carboxylic acid, rather than n-alkanone’s carbonyl with a much larger volume of (~14–15 cm3 mol−1 [30,31]).
The V0 for fumarate−2 was used to estimate the V0 of fumaric acid and the H-fumarate−1 anion by assuming the same ΔionV relationship for fumaric acid and its ions as for succinic acid from Criss and Wood , and H-succinate−1 or succinate−2 anion, from Shock , as demonstrated in Figure 2. The same relationship, using the ΔionV (13.13 cm3 mol−1) between lactic acid and lactate from Shock  was used to estimate the V0 for pyruvate from its acid.
The partial molal volume of the (-S-) group for a thioester was estimated by the addition of the value of V0 (-S-) from a value published by Lepori and Gianni  to the V0 of 2-butanone taken from Shock and Helgeson . The V0 of succinyl thioester and its anion were estimated by the addition of V0 (-S-)  and V0 (>C=O) to the V0 of pentanoic acid and pentanoate, respectively (both from Shock ). The V0 (>C=O) value was estimated to be the difference between the V0 of n-alkanones and (n-1) n-alkanes (both taken from Shock and Helgeson ).
2.5. Calculation of the reference state heat capacity (C°p aq) and estimation of missing values
The standard molal isobaric heat capacities at the reference temperature and pressure are shown in Table 1 and estimation procedures in Figure 2. The of pyruvate ion was assumed to have the same difference in from its acid, as did the lactic acid-lactate pair , of 167.4 J mol−1 K−1. The for oxaloacetic acid was estimated by adding the value (−52.0 J mol−1 K−1) from that reported by Cabani et al.  to that of malonic acid . For the estimation of values for both H-oxaloacetate−1 and oxaloacetate−2 ions, it was assumed that the difference from the acid was the same as between succinic acid and it respective anions from Shock . This assumption was also used to estimate values of the H-mono- and divalent anions of malate, fumarate, and α-ketoglutarate. The of fumaric acid was estimated through subtracting the values for the difference (70.7 J mol−1 K−1) in y-intercepts between n-alkanes from n-alkenes (both values being from Shock and Helgeson ) from that of succinic acid. For α-ketoglutaric acid, the value (−52.0 J mol−1 K−1) reported by Cabani et al.  was added to that of succninc acid. The for succinyl thioester was estimated from addition of the , and the (−81.2 J mol−1 K−1), both calculated in this work, to the of pentanoic acid from Shock . The for succinyl thioester was calculated from the difference in y-intercepts of C2,C4,C6 n-diaklysulfides  and C2,C4,C6 n-alkanes . The value (−135.2 J mol−1 K−1) was calculated to be the difference between 2-pentanone and n-butane, both from Shock and Helgeson . This value for was chosen for succinyl thioester because a carbonyl from an n-alkanone may be more likely to represent the in a thioester, as opposed to that from a α-carbonyl adjacent to a carboxylic acid. The resulting value of 216.1 J mol−1 K−1 is in close agreement with the sum of group values provided by Cabani et al. , using the (-S-) calculated here, of 216.4 J mol−1 K−1. The of the succinyl thioester anion was assumed to have the same difference from its acid as that between pentanoic acid and pentanoate . The of acetyl thioester was estimated by adddition , as described above, to that of 2-butanone from Shock and Helgeson .
2.6. Extrapolation of reference state data to high pressures and temperatures
Starting with 25°C, 0.1 MPa reference state data, calculating the at the elevated P and T involves the integration of Equation (5) as:
The revised-HKF model allows for the incorporation of the changes in the partial molal volume and heat capacity as described in Appendix B.
2.7. Estimation of the temperature and pressure effects on the partial molal volume of aqueous organic species: the non-solvation contribution
The partial molal volume of a substance in the revised HKF model is defined by Equation (24). At temperatures of ≤ ~150°C, at the water-saturation vapor pressure (Psat), the solute-dependent contribution (the non-solvation volume, ) to the partial molal volume term dominates the partial molal volume. The term is calculated from Equation (25) utilizing fitting parameters (a1, a2, a3, and a4) to integrate the term into V0 at a desired pressure and temperature. The a1 and a2 parameters have been generated from empirical data gathered at high pressure and temperature conditions for a variety of compounds [11,15–20,46,47]. The a1 variable is correlated to a high degree with the of a wide range of neutral and charged aqueous organic compounds with a variety of functional groups (Figure 3) and can therefore be used to estimate the values of a1 for compounds with structural homology to those with high pressure and temperature data for which there are no volumetric data beyond the reference state.
For the range of compounds regressed in Figure 3, the line is defined by:
The a2 parameter is somewhat more dependent on the functional group characteristics of a particular molecule. In Figure 4 the of aldehydes, hydroxy acids, carboxylic acids, and dicarboxylic acids, and their respective anions, are plotted against a2.
The data are fit by the line
The a2 parameter for all dicarboxylic acids, and dicarboxylate anions, citric acid and its anions, pyruvic acid and pyruvate, and succinyl thioester and its anion were estimated with Equation (7). Since the only apparently large departures from the line in Equation (7) are for shorter- chained carboxylate and hydroxylate anions, a fit for hydroxylates was considered separately to calculate the a2 parameter for pyruvate. The y-intercept for hydroxylates (Equation (9)) is slightly lower with a steeper slope than Equation (8):
However, if the y-intercept value for n-alcohols vs. n-alkanones can be considered comparable to that of the hydroxy and carbonyl groups in hydroxy- and α keto-acids, respectively, then pyruvate’s y-intercept would shift toward the line in Equation (7). Therefore the a2 value for pyruvic acid using Equation (7) was retained. The a2 parameter for acetyl thioester was estimated with the line defined by Equation (8) for n-alcohols and n-alkanones.
All a4 parameters were generated, as suggested by Shock and Helgeson and Shock [18,23], using the correlation with the a2 fitting parameter. The a3 parameter was then calculated by solving the rearranged non-solvation molal volume term ( ) of Equation (25) at 25 °C and 0.1 MPa.
2.8. Estimation of temperature and pressure effects on the isobaric heat capacity of aqueous organic species: The non-solvation contribution
The non-solvation and solvation heat capacity contributions to the partial molal heat capacity are defined in Equation (27). The non-solvation contribution term of Equation (27) is expanded in Equation (28) and combines the influence of pressure on the with incorporation of a substance’s a3 and a4 fitting parameters, from above, and the influence of temperature by the use of two heat capacity fitting parameters, c1 and c2. The c2 parameter correlates closely with the reference state for compounds with similar functional groups (Figure 5).
Therefore, from this correlation high temperature and pressure data from structurally similar compounds can be used to predict the c2 parameter of molecules for which no high temperature heat capacity data are available. Figure 5a shows the plot of c2 vs. values, taken from Shock , for organic acid anions. The regression of these values for the range of compounds shown is described by the line in Equation (10):
2.9. Estimation of temperature and pressure effects on the partial molal properties of aqueous organic species: The solvation contribution
At temperatures of ≥~150°C, at the water-saturation vapor pressure (Psat), the solvent-dependent contribution to the partial molal volume (Equation (26)) and heat capacity (Equation (29)) terms begin to dominate each partial molal function. The conventional (ω) and effective (ωe) Born coefficients, respectively, are used to describe the substance-specific solvation properties of an ionic species or electrolytes, and neutral species (see Appendix B).
2.10. Calculation of the conventional Born coefficient (ω) for ionic species
The ω of ionic species has been demonstrated to have a strong correlation with the  and can therefore be calculated from this quantity. In Figure 6 ω is plotted against the for a variety of mono-, di- and trivalent inorganic and organic anions (taken from Shock and Helgeson (1988) ) using relationships in Equations (30–32) to calculate ω, which was also used to calculate the ω for all anions in this work (data also shown in Figure 6).
2.11. Calculation of the effective Born coefficient (ωe) for neutral species
The effective Born coefficient used to describe the solvation contribution of neutral species in the revised-HKF model has been calculated from [13,22,23] and the Gibbs free energy of hydration (ΔhydG0) (Appendix C) in the absence of high temperature and V0 data. It has been demonstrated from empirical data that ωe generally has a negative value for low- molecular weight neutral organic compounds. The negative value is the result of the inflection of and V0 values towards positive-infinity near the critical point of water. As the properties of a set of molecules become increasingly polar, the solute-solvent interaction increases. In the case of some neutral inorganic polyhydroxyl compounds such as aqueous silica, of which the hydrated form is thought to be H4SiO4, and boric acid (H3BO3), both demonstrate electrolyte-like behavior and thus positive ωe values. The mechanism associated with this phenomenon is thought to arise from water-solute versus water-water competition near the critical point , where solutes that are associated with more solvent molecules than is the solvent itself, and will have and V0 values that approach −∞. The volatility of a substance in comparison to water is also thought to have influence over this process as well. This issue is discussed in detail by Amend and Plyasunov , where predictions were made concerning the near-critical point behavior of carbohydrates. It is clear, through their discussion and others, that the relationship between ωe and solvation at higher temperatures is not an obvious one. For instance, the non-electrolyte amino acid proline displays low-temperature solvation behavior that is best described with a negative value for ωe. As the temperature increases, however, proline’s solvation is best fit with a positive ωe. However, this type of solvation behavior may be particular to the proline zwitterion due to its particularly asymmetrical dipole and spatial arrangement of hydrophobic and hydrophilic moieties . If the high density of polar functional groups, as in polyhydroxy compounds, is to be viewed as an indicator of near-critical behavior of compounds with hydroxyl functionalities, then for dicarboxylic acids, oxalic acid with an ωe that displays neutral behavior may be the closest analogue.
In Table 1, different values for ωe are displayed as calculated from the correlations with the ΔhydG0 (Equation (44)) and entropy (Equation (45)), as has been done previously. Also included is a systematic estimation of ωe using values provided by Shock and Shock and Helgeson [18,23] with results shown in Figure 7 in comparison to other series of neutral organic compounds.
The values of these estimations were used for ωe in all the neutral compounds in this work. The ωe for pyruvic acid was estimated by taking the ωe from lactic acid and subtracting the difference between n-propanol and acetone:
The ωe for oxaloacetic acid was estimated from the value for malonic acid with addition of the carbonyl group value such that:
Malic acid’s ωe value was estimated by from succinic acid from:
The ωe for α-ketoglutaric acid also used succinic acid and carbonyl value for oxaloacetic acid:
Since there are currently no analogous compounds (tricarboxyl) for citric acid, its ωe was estimated to be the same as a C6 compound using the slope and y-intercept for dicarboxylic acids. The value for succinyl thioester was estimated by:
In consideration of consistency with previous works and the pragmatic aspects of the temperature ranges to be expected to be relevant for these compounds, negative values for ωe were chosen. In addition, for practical purposes, since the pKa of the strongest acid among these compounds is 2.49 (pyruvic acid from reference ), at pH above 3–4 the acids of these compounds will not usually be germane in writing reactions.
3.1. Analysis of the possible error associated with methods used in estimating reference and standard state parameters
Since the motivation for this work is to estimate the high temperature and pressure thermochemical parameters for compounds for which there was no data, the credibility of the resulting values need be examined. The effects of errors on the reference state data, as well as for values calculated beyond the reference state, can be analyzed by the comparison of expected calculations, using the estimated values, with calculations from values in which a sensible error is incorporated. Figure 8a demonstrates the influence of the over- and underestimation of ωe on the partial molal volume of propanoic acid along Psat calculated using Equations (25) and (26) with revised HKF parameters from Shock . Although the reference state is unaffected by ωe, the error in the calculated increases with temperature as the partial molal volume is influenced increasingly by the solvation term, although at higher pressures the error is diminished (Figure 8b). However, as we are most concerned with providing data for reaction thermodynamics at elevated pressure and temperature, the effects of inaccurate estimation of the are of primary concern. As can be seen in Figure 8c, the revised-HKF method is quite insensitive to even a 2-fold under- or over estimation of ωe, with a maximum of ~0.3% relative error at 350°C. The insensitivity to errors from the estimation methods can be further tested by swapping the HKF parameters of one compound for another, while using the original . To demonstrate this, for compounds in Figure 8d the original values were retained while the remaining values (S0aq, a1–a4, c1, c2, and Born coefficients) were mutually swapped from another compound: pyruvic acid and propanoic acid, H-oxaloacetate−1 and H-α-ketoglutarate−1, and oxaloacetate−2 and α-ketoglutarate−2 (all values are from Table 2 or Shock  for propanoic acid) and used to solve Equation (39). The largest error (~4% relative at 350 °C) is encountered when the equation of state parameters from propanoic acid are replaced with those of pyruvic acid. Considering that there is a relative difference (compared to pyruvic acid) of 15%, 24%, and 12% between the S0aq, V0, and of propanoic acid, respectively, this level of error is quite tolerable, as group additivity estimations generally give relative errors of ~5% for these classes of organic compounds . At 150°C the error is approximately half that at 350°C and is likely to fall below the analytical errors encountered when quantifying the concentrations of these substances.
3.2. Estimation of equilibrium constants at high temperatures and pressures
As an example of the usefulness of these data, values of the logarithm of the equilibrium constant (log K) for acid dissociation reactions were calculated at different temperatures from the values at Psat using Equations (38) and (39) and:
4. Concluding Remarks
Using the thermodynamic data that have either been measured experimentally or estimated through methods described and provided in this paper for the constituents of the citric acid cycle, we can begin to place the fundamental biological process of energy transfer into a geochemical context. With the data and parameters presented in this paper, we can for the first time calculate thermodynamic reaction properties for the citric acid cycle under hydrothermal conditions, whence life may have emerged. Furthermore, we can use calculations such as the ones described above to evaluate the energy cycles of microorganisms that live at elevated temperatures and pressures and thus gain insight into the conditions necessary for the initiation of these cycles. We are now also able to evaluate quantitatively the steps in the reverse or reductive citric acid cycle, which may have preceded the more modern oxidative citric acid cycle as the primary energy transfer mechanism for life. Such calculations are facilitated through use of the computer program SUPCRT92 .
In addition, we can gain some insight into the conditions (including factors such as pH, temperature, pressure and concentrations of the chemical components) on the early Earth that may have facilitated the initiation of the central metabolic pathways such as the reductive and oxidative citric acid cycles in biological energy systems by evaluating quantitatively the energy gained through various metabolic reactions.
For example, careful examination of Figure 9 reveals that the logarithms of the equilibrium constants of many of the deprotonation reactions involving components of the citric acid cycle vary by up to an order of magnitude over the known temperature range for life (currently up to 122°C). Because these reactions are functions of pH, changes in the equilibrium constants over the temperature range will change the range of pH at which they would be thermodynamically favorable. A full evaluation of the geochemical parameters attending any environment would be required to determine the effect each would have on reaction favorability and the viability of the citric acid cycle.
We would like to thank Drs. Yuan Gao and Marie-Paulez Bassez for the invitation to submit this article to the Special Issue on the Origin of Life. This manuscript has benefited from many discussions with colleagues over the years on thermodynamic properties of aqueous biomolecules, including Everett Shock, Karyn Rogers, the late Harold Helgeson, Andrey Plyasunov, Laurent Richard, Jan Amend, Doug LaRowe and Jeff Dick. We thank James Holden for discussions of metabolic pathways among life in extreme environments. Financial support has been provided by the School of Earth and Space Exploration at Arizona State University and an NSF RIDGE Postdoctoral Fellowship to PD-B, the Department of Geological Sciences at the University of Missouri, and NASA Astrobiology: Exobiology and Evolutionary Biology Grant NNX07AT84. This is Le BAR contribution #1.
The revised HKF eos combines substance specific structural or non-solvation ( ) and solvent dependent or solvation- partial molal properties to predict the conventional standard molal properties(μ0) of aqueous species:
The non-solvation contribution to the standard molal term is a summation of the intrinsic property of a substance and its affect the solvent structure in its local vicinity and is the dominant contribution to the partial molal quantity at temperatures of ~≤ 150 °C. The solvent contribution to the standard molal term is a factor of the intrinsic properties of water and the Born transfer properties of solvation . In effect, for a given structural moiety of a compound the greater the degree or hydrophilicity (i.e. compounds that form hydrogen bonds and/or ions), will reflect in the collapse of the local solvent structure, whereas more hydrophobic moieties will necessitate the formation of cavities within the solvent. The solvation contribution to a standard molal property has the greatest affect when the solvent (H2O) is undergoing the greatest degree of change in permitivity, volume, and heat capacity as these functions are influenced by high temperature and pressure. The non-solvation and solvation contribution to the partial molal volume for ions, electrolytes, and neutral aqueous organic species is defined by:
For neutral organic species, the non-solvation term partial derivative function (∂ω/∂P)T is taken to be zero, simplifying the term to −ωeQ since the effective Born coefficient is used.
The combination of non-solvation and solvation function for integration of the heat capacity function is:
The conventional Born coefficient of an ion was calculated by first utilizing the correlation of the standard partial molal entropy with the effective electrostatic radius (re) of the jth aqueous species (from ) at 1 bar and 25 °C, such that:
The factor is the absolute Born coefficent for the hydrogen ion equal to 0.5387 × 105 cal mol−1 at 1 bar and 25 °C . The partial and second derivative functions for the conventional Born coefficient for ionic species at T and P were calculated with Equations (33–35) from  using X1 and X2 values using Equations (36) and (37) (from ), respectively.
The variable η corresponds to the quantity equal to N0e2 / 2 (N0 is Avogadro’s number 6.02252 × 1023 mol −1, and e is the absolute electronic charge (esu) of 4.80898× 10−10), k and Z are the kth electrolyte and charge of the its j th species, respectively (from ). The term g is the solvent dependent function from .
Calculation of the effective Born coefficient (we) from ΔhydG0
The effective Born coefficient (ωe) for neutral organics was calculated form the correlation described by  with the Gibbs free energy of hydration (ΔhydG0) and the Henry’s constant of a neutral species. Since the Henry’s constant (KH) approximates the equilibrium constant  for a reaction:
Then using the correlation from :
Calculation of the effective Born coefficient (we) from
Aqueous entropy values have been used to estimate the effective Born coefficient for different classes of neutral organic compounds [18,22,23]. The correlation used for the neutral compounds in this study were determined by Shock :
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|ΔGf0 a||S0 b||V0 c||CP0 b||a1 d,ee||a2 d||a3 e||a4 f,hh||c1 g||c2 f||ωb|
|pyruvic acid||−489.1j||179.9 k||54.6 h||114.6 h||3.9096||5745.0 ff||2.478||−140079||24.645||384086 jj||−129222 mm|
|pyruvate−1||−474.9 I||171.5 k||41.5 t||−52.8 bb||3.2601||4220.3 ff||17.617||−133776||22.278||−178281 ii||42208 ll|
|oxalaoacetic acid||−838.3 j||287.5 h||72.4 h||108.7 h||4.8998||7178.9 ff||24.687||−146006||27.207||377592 jj||−52158 mm|
|H-oxalaoacetate−1||−823.7 j||233.1 p||60.3 u||−77.7 cc||4.3045||6672.1 ff||10.473||−143911||−10.600||−181300 ii||330604 ll|
|oxalaoacetate−2||−798.7 I||107.9 q||46.7 v||−328.1 cc||3.8109||5513.3 ff||13.850||−139121||−176.230||−211639 ii||1139579 ll|
|malic acid||−891.6 k||283.8 h||82.8 w||227.7 w||5.5023||8481.8 ff||21.621||−151393||117.243||509492 jj||−75348 mm|
|H-malate−1||−872.4 l||227.7 r||69.4 x||41.2 dd||4.8234||7890.2 ff||6.924||−148947||106.200||−166880 ii||344473 ll|
|malate−2||−843.1 l||126.8 r||55.7 x||−209.1 dd||4.3174||6702.4 ff||10.385||−144037||−62.817||−197220 ii||1170226 ll|
|fumaric acid||−645.8 k||261.1 k||78.8 h||154.7 h||5.2807||8963.5 ff||39.867||−153384||58.786||428516 jj||−95650 mm|
|H-fumarate−1||−628.1 k||203.3 k||65.4 y||−31.8 dd||4.6074||7383.1 ff||8.401||−146851||38.350||−175733 ii||373932 ll|
|fumarate−2||−601.9 k||105.4 k||51.7 y||−282.1 dd||4.0998||6191.7 ff||11.873||−141925||−131.082||−206072 ii||1185066 ll|
|α-ketoglutaric acid||−842.3 j||315.1 h||89.0 z||173.4 h||5.8421||9216.6 ff||24.128||−154430||78.884||449296 jj||−35121 mm|
|H-α-ketoglutarate−1||−829.4 j||243.5 s||75.6 x||−13.1 dd||5.1687||8700.7 ff||4.562||−152298||51.025||−173461 ii||292131 ll|
|α-ketoglutarate−2||−802.0 I||136.0 s||61.9 x||−263.4 dd||4.6661||7520.8 ff||8.000||−147420||−117.057||−203800 ii||1117934 ll|
|citric acid||−1243.4 m||329.4 l||113.6 w||322.5 w||7.2438||12247.7 ff||39.901||−166961||195.456||614557 jj||−23333 mm|
|H2-citrate−1||−1226.3 k||286.2 k||98.1 aa||187.9 k||6.4344||11671.9 ff||−4.096||−164580||241.056||−149109 ii||248464 ll|
|H-citrate−2||−1199.2 k||202.3 k||88.5 aa||0.84 k||6.1522||11009.3 ff||−2.165||−161841||131.407||−171775 ii||1038339 ll|
|citrate−3||−1162.7 k||92.1 k||72.0 aa||−254.8 k||5.4914||9458.3 ff||2.355||−155429||−40.909||−202760 ii||1874470 ll|
|succinyl thioester||−496.6 n||394.5 h||140.5 h||216.1 h||8.7769||15562.9 ff||115.513||−180666||113.931||496645 jj||−13856 mm|
|succinyl thioester−1||−468.9 o||292.0 h||133.7 h||78.0 h||8.4641||16436.3 ff||−17.979||−184277||133.088||−162424 ii||187278 ll|
|acetyl thioester||−140.1 n||400.1 h||107.3 h||255.5 h||6.9314||10305.7 gg||36.887||−158933||171.102||342400 kk||−160625 mm|
a-kJ mol−1.b- J mol−1 K−1.c-cm3 mol−1.d-J mol−1 bar−1.e- J mol−1.f-J K mol−1.g- J mol−1 K−1.h-value as described in Table 1.i- from .j-calculated from pKa values from  and of oxaloacetate−2 from above.k-from .l-from .m-from .n-estimated as described in Figure 2.o-estimated as was succinyl thioester in Figure 2, except using pentanoate, taken from , as the base structure.p- estimated as was oxaloacetic acid in Figure 2, except using H-malonate−1, taken from , as the base structure.q- estimated as was oxaloacetic acid in Figure 2, except using malonate−2, taken from , as the base structure.r-calculated from of malic acid in above table using the ΔionS from .s-estimated from of a-ketoglutaric acid from above table assuming the same ΔionS as between succinic acid and its respective ions in .t-estimated from V0 of pyruvic acid in above table assuming the same ΔionV as between lactic acid and lactate, in .u-estimated as was oxaloacetic acid in Figure 2, but using H-malonate−1, from  as the base structure.v- estimated as was oxaloacetic acid in Figure 2, but using malonate−2, from  as the base structure.w-from .x-estimated by assuming the same ΔionV as between succinic acid, taken from  and its respective ions, taken from .y-from .z-from .aa-from .bb-estimated from of pyruvic acid in above table assuming the same as between lactic acid and lactate in .cc-estimated as described in Figure 2 for oxaloacetic acid but using respective of H-malonate−1 or malonate−2.dd-estimated form the acid in the above table assuming the same as between succinic acid and the respective −1 or −2 ion from .ee – estimated using Equation (6)ff-estimated using Equation (7).gg- estimated using Equation (8).hh-calculated from the a2 parameter in above table as described by .ii- calculated using Equation (10).jj- calculated using Equation (11).kk- calculated using Equation (12).ll-calculated from in above table using Equations (30–32) from .mm-estimated as described in text.
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