Thermochemistry of a Biomimetic and Rubisco-Inspired CO 2 Capture System from Air

In theoretical studies of chemical reactions the reaction thermochemistry is usually reported for the stoichiometric reaction at standard conditions (∆G ̋, ∆H ̋, ∆S ̋). We describe the computation of the equilibrium concentrations of the CO2-adducts for the general capture reaction CO2 + Capture System Õ CO2-adduct (GCR) and the rubisco-type capture reaction CO2 + Capture System Õ CO2-adduct + H2O (RCR) with consideration of the reaction CO2(g) Õ CO2(aq) via Henry’s law. The resulting equations are evaluated and graphically illustrated as a function of atmospheric CO2 concentration and as a function of temperature. The equations were applied to the thermochemistry of small molecule rubisco-model reactions and series of additional model reactions to illustrate the range of the Gibbs free enthalpy for the effective reversible capture and of the reaction entropy for economic CO2 release at elevated temperature. A favorable capture of free enthalpy is of course a design necessity, but not all exergonic reactions are suitable CO2 capture systems. Successful CO2 capture systems must allow for effective release as well, and this feature is controlled by the reaction entropy. The principle of using a two-pronged capture system to ensure a large negative capture entropy is explained and highlighted in the graphical abstract. It is hoped that the presentation of the numerical examples provides useful guidelines for the design of more efficient capture systems.


Introduction
Continuous atmospheric CO 2 tracking started more than 50 years ago with CO 2 measurement at Mauna Loa and resulted in the "Keeling Curve," the iconic demonstration of the seasonal variations of CO 2 concentrations and the steady rise of the annual average CO 2 concentration [1,2].NOAA's (National Oceanic and Atmospheric Administration) CarbonTracker [3] employs a global network of CO 2 mole fraction observations to study the CO 2 exchange between the biosphere and the atmosphere as a function of time [4,5].Comprehensive carbon inventories have been under development by the US Carbon Cycle Science Program [6] and include space-based CO 2 observation (i.e., Orbiting Carbon Observatory, ).While considerable uncertainties remain with regard to accounting for all CO 2 , all reports consistently show that the global atmospheric CO 2 concentrations have been growing steadily, and at Mauna Loa the average annual mean increased from 316.0 ppm in 1959 to its value of 404.83 ppm at the time of this writing [2].The Intergovernmental Panel on Climate Change (IPCC [8]) reported in its fifth assessment report in 2014 (AR5 [9]) an average rate of increase in CO 2 of 1.4 ppm/year.Data by the Energy Information Agency (EIA, [10]) show worldwide CO 2 emissions from the consumption of coal were 13.8 billion metric tons in 2012, up from 8.0 billion metric tons two decades earlier, and there is no reason to expect this trend to stop any time soon.
We have been interested in the development of rubisco-based biomimetic systems for reversible CO 2 capture from air.Our design of the chemical CO 2 capture and release systems is informed by the understanding of the binding of the activator CO 2 ( A CO 2 ) in rubisco (Figure 1) and the general strategy has been described [37,38].Rubisco (ribulose 1,5-bisphosphate carboxylase/oxygenase, RuBisCO) catalyzes the addition of CO 2 and water to RuBP (d-ribulose 1,5-bisphosphate) in the photosynthetic carbon assimilation via the Calvin-Bassham-Benson cycle, and results in two molecules of 3-PGA (3-phospho-d-glycerate) and 0.5 O 2 [39,40].Nearly all carbohydrate production in the biosphere depends on rubisco catalysis and rubisco is the most abundant protein on Earth.Form I rubisco contains eight large (L) and eight small (S) subunits, and the crystal structure of Spinacia oleracea [41,42] provided an early example of a hexa-decameric rubisco [43].Form II rubisco lacks small subunits, generally occurs as L 2 , and is exemplified by the bacterial rubisco of Rhodospirillum rubrum [44].The mechanism of rubisco catalysis has been studied in detail [45,46] and these studies have established that rubisco must be activated by carbamylation of active-site lysine (Lys) with an activator CO 2 ( A CO 2 ).The carbamate formed is stabilized both by complexation to Mg 2+ and by NH¨¨¨OC hydrogen-bonding (Figure 1) and, importantly, the rubisco A CO 2 capture (RCR) occurs with the release of a water molecule from the capture system.
In theoretical studies of reaction thermochemistry one usually computes the reaction enthalpy ∆H ˝and the reaction entropy ∆S ˝for the stoichiometric reaction at standard conditions (1 atm, 25 ˝C) and determines the reaction Gibbs free enthalpy via ∆G ˝= ∆H ˝´T ˝¨∆S ˝.This knowledge then allows for the computation of equilibrium concentrations, although this step is not usually applied in theoretical studies.For reactions with a severely limited reagent, one needs to consider the consequences of the non-stoichiometry on product formation.With an atmospheric CO 2 concentration of about 400 ppm, questions about the equilibrium concentration of CO 2 capture reactions have been asked frequently and they are entirely warranted.Hence, in this paper, we describe the determination of CO 2 capture equilibria as a function of CO 2 concentration.In addition, we describe temperature dependence of the CO 2 capture equilibria with a focus on the importance of the reaction entropy for effective CO 2 release at elevated temperature.

Capture Reaction Systems Studied
We analyzed the thermochemistry of a rubisco-based small molecule model of the formation of N-methylcarbamic acid (NMCA) by addition of methylamine (CH3NH2) to CO2, considering substrate and product stabilization by an active-side carbonyl model and complexation by a model metal complex ML2 [37].The model system is described by 14 reactions and the thermochemistry was determined at the B3LYP/6-31G* level for the natural case of magnesium M 2+ = Mg 2+ .Reaction R1 is the formation of NMCA (H3CNHCOOH) by addition of methylamine to CO2 in the gas phase.Three effects of the environment need to be considered in the active site of rubisco (Figure 1) and these are (A) the stabilization of the product by hydrogen-bonding between NMCA and a model carbonyl compound; (B) the complexation of NMCA by a model metal complex ML2; and (C) the forced replacement of ligand water by CO2 at the metal center.
Importantly, the complexation of CO2 to the metal cation M 2+ requires the replacement of a water molecule, and our analysis showed that this H2O/CO2 replacement is an essential feature of rubisco thermochemistry.The exchange of a ligand water molecule by a CO2 molecule (reaction R10) is highly endergonic and this H2O/CO2 replacement penalty is required to lower the overall exergonicity of the CO2 capture reaction and to allow overall reversibility.Reaction R14 models the capture of CO2 and the addition of the Mg 2+ -complexed CO2 to a pre-positioned, carbonyl-aggregated methylamine CH3NH2•(A/K) to form the Mg 2+ -complexed and carbonyl-aggregated NMCA, (A/K)•NMCA•ML2.
We evaluated the RCR-reaction R14 with the model carbonyl formaldehyde (FA, R14a) and acetone (Ac, R14b), respectively.Molecular models of the CO2-adducts 17a (R14a) and 21a (R14b) are shown in Figure 2, and the associated thermochemical parameters are listed in Table 1.

Capture Reaction Systems Studied
We analyzed the thermochemistry of a rubisco-based small molecule model of the formation of N-methylcarbamic acid (NMCA) by addition of methylamine (CH 3 NH 2 ) to CO 2, considering substrate and product stabilization by an active-side carbonyl model and complexation by a model metal complex ML 2 [37].The model system is described by 14 reactions and the thermochemistry was determined at the B3LYP/6-31G* level for the natural case of magnesium M 2+ = Mg 2+ .Reaction R1 is the formation of NMCA (H 3 C´NH´CO´OH) by addition of methylamine to CO 2 in the gas phase.Three effects of the environment need to be considered in the active site of rubisco (Figure 1) and these are (A) the stabilization of the product by hydrogen-bonding between NMCA and a model carbonyl compound; (B) the complexation of NMCA by a model metal complex ML 2 ; and (C) the forced replacement of ligand water by CO 2 at the metal center.
Importantly, the complexation of CO 2 to the metal cation M 2+ requires the replacement of a water molecule, and our analysis showed that this H 2 O/CO 2 replacement is an essential feature of rubisco thermochemistry.The exchange of a ligand water molecule by a CO 2 molecule (reaction R10) is highly endergonic and this H 2 O/CO 2 replacement penalty is required to lower the overall exergonicity of the CO 2 capture reaction and to allow overall reversibility.Reaction R14 models the capture of CO 2 and the addition of the Mg 2+ -complexed CO 2 to a pre-positioned, carbonyl-aggregated methylamine CH 3 NH 2 ¨(A/K) to form the Mg 2+ -complexed and carbonyl-aggregated NMCA, (A/K)¨NMCA¨ML 2 .
We evaluated the RCR-reaction R14 with the model carbonyl formaldehyde (FA, R14a) and acetone (Ac, R14b), respectively.Molecular models of the CO 2 -adducts 17a (R14a) and 21a (R14b) are shown in Figure 2, and the associated thermochemical parameters are listed in Table 1.

Equilibrium Concentrations with Non-Stoichiometric Conditions
The calculation of thermodynamic data for a given reaction with ab initio methods uses stoichiometric concentrations of the reagents.In the case of passive carbon capture, such an assumption does not accurately depict the relative concentrations of the reagents.Passive capture systems must operate at low pressures of CO2, including the current atmospheric concentration (~400 ppm) [47].
Also, Henry's Law must be considered for aqueous carbon capture systems, because only dissolved CO2 is available for binding.Henry's Law describes the proportion of gaseous CO2 that enters the aqueous phase at equilibrium, illustrated in reaction (R15) and described by Equation (1).KH(T) is constant for a given gas and a given temperature; for CO2 at room temperature this value has been experimentally determined to be KH(298.15)= = 0.034 [48].Using Henry's Law and the atmospheric pressure of CO2, it is then simple to determine an approximate concentration of CO2 in the aqueous phase of a passive capture system.

Equilibrium Concentrations with Non-Stoichiometric Conditions
The calculation of thermodynamic data for a given reaction with ab initio methods uses stoichiometric concentrations of the reagents.In the case of passive carbon capture, such an assumption does not accurately depict the relative concentrations of the reagents.Passive capture systems must operate at low pressures of CO 2 , including the current atmospheric concentration (~400 ppm) [47].
Also, Henry's Law must be considered for aqueous carbon capture systems, because only dissolved CO 2 is available for binding.Henry's Law describes the proportion of gaseous CO 2 that enters the aqueous phase at equilibrium, illustrated in reaction (R15) and described by Equation (1).K H (T) is constant for a given gas and a given temperature; for CO 2 at room temperature this value has been experimentally determined to be K H (298.15) = K 0 H = 0.034 [48].Using Henry's Law and the atmospheric pressure of CO 2 , it is then simple to determine an approximate concentration of CO 2 in the aqueous phase of a passive capture system.In this work, we primarily focus on the simple capture reactions (R1) and the rubisco capture reactions (R14).The product formation can be calculated using the equations for the equilibrium constants K eq for reactions (GCR) and (RCR), respectively.Defining the concentration of the capture system at equilibrium where [CO 2 -CS] is the concentration of the CO 2 -adduct formed and [CS] 0 is the initial concentration of the capture system, we obtain equations (2-GCR) and (2-RCR).
The isotherms in Figure 3 show the concentration of the CO 2 -adduct formed [CO 2 -CS] as a function of the CO 2 pressure.This relationship produces isothermal curves with their asymptotic boundaries determined by the initial concentration of the capture system.
C 2016, 2, 18 5 of 14 In this work, we primarily focus on the simple capture reactions (R1) and the rubisco capture reactions (R14).The product formation can be calculated using the equations for the equilibrium constants Keq for reactions (GCR) and (RCR), respectively.Defining the concentration of the capture system at equilibrium , where [CO2-CS] is the concentration of the CO2-adduct formed and [CS]0 is the initial concentration of the capture system, we obtain equations (2-GCR) and (2-RCR).
The isotherms in Figure 3 show the concentration of the CO2-adduct formed [CO2-CS] as a function of the CO2 pressure.This relationship produces isothermal curves with their asymptotic boundaries determined by the initial concentration of the capture system.(Figure 3d, green curve) and a very reasonable capture fraction of 0.492 (Table 1) at [CO2] = 400 ppm.
Temperature affects the curvature of the fraction (CO2-adduct) = f([CO2]) isotherms in Figure 3 and we will discuss the T-dependence in more detail below (see Section 2.3).Instead of the fraction (CO2adduct), Figures 4 and 5 show the ratios [CO2-CS]/[CS] as a function of [CO2].The ratios were computed with equations 3-GCR and 3-RCR, where R is the gas constant (1.9858775 cal•mol −1 •K −1 ), ΔG is the Gibbs Free Enthalpy of the capture reaction at T, and ΔG = ΔG° for the special case of T = T°.This ratio shows a linear dependence on the CO2 pressure, and temperature affects the slope (but not the curvature).The fraction of CO 2 -adduct differs greatly in magnitude depending on the application of the GCR or RCR equations.The same thermochemical parameters afford much higher fractions computed with the GCR equation than with the correct RCR equation (Figure 3).The fractions computed with the RCR equations for reactions R14a (red) and R14b (blue) at [CO 2 ] = 400 ppm are listed in Table 1, and the respective values of 0.016 and 0.003 are very small.Obviously, our simplified theoretical model produces ∆G ˝values that are not exergonic enough.Hence, we also studied the hypothetical reaction R14h with ∆G ˝= ´9 kcal/mol.For this model system, we set ∆S ˝(R14h) = ∆S ˝(R14a) and obtain ∆H (R14h) = ´19.67kcal/mol.With R14h, we achieve a much more effective capture at room temperature (Figure 3d, green curve) and a very reasonable capture fraction of 0.492 (Table 1)  This ratio shows a linear dependence on the CO 2 pressure, and temperature affects the slope (but not the curvature).
C 2016, 2, 18 6 of 14 The fraction of CO2-adduct differs greatly in magnitude depending on the application of the GCR or RCR equations.The same thermochemical parameters afford much higher fractions computed with the GCR equation than with the correct RCR equation (Figure 3).The fractions computed with the RCR equations for reactions R14a (red) and R14b (blue) at [CO2] = 400 ppm are listed in Table 1, and the respective values of 0.016 and 0.003 are very small.Obviously, our simplified theoretical model produces ΔG° values that are not exergonic enough.Hence, we also studied the hypothetical reaction R14h with ΔG° = −9 kcal/mol.For this model system, we set ΔS° (R14h) = ΔS° (R14a) and obtain ΔH° (R14h) = −19.67kcal/mol.With R14h, we achieve a much more effective capture at room temperature (Figure 3d, green curve) and a very reasonable capture fraction of 0.492 (Table 1)     The Gibbs free enthalpy of the capture reaction obviously will vary with temperature because of the entropy term, ∆G = ∆H ´T¨∆S.The reaction enthalpy ∆H(T) and reaction entropy ∆S(T) depend on temperature only modestly and, for the temperature variations considered here, they can be approximated very well by ∆H(T) = ∆H ˝and ∆S(T) = ∆S ˝.However, in our aqueous system temperature also significantly affects Henry's constant.This variation is described by the van't Hoff equation (Equation ( 4)), where the change in enthalpy of dissolution is a constant for a given gas.For CO 2 the value is -∆ sol H/R = 2400 K.With this value, Equation ( 5) can be used to determine the value of the Henry's constant at temperature T for CO 2 .

5)
For -∆ sol H/R > 0, it is clear that CO 2 will be better solvated at low temperatures and less solvated at temperatures higher than 298.15K.This drives capture to be more effective at lower temperatures and release to be more effective at higher temperatures.The consequences of the T-dependence of Henry's constant are illustrated for specific cases in Figure 3, where solid lines were computed with T-dependence of Henry's Law and dashed lines without.In every panel, the dashed curves always are closer to the T ˝-curve than the solid curves, that is, the load difference will always be increased by the T-dependence of Henry's constant.

Temperature Dependence of Capture Systems: Loading and Unloading
This illustration in Figure 6 is a rough schematic to consider the scale of the CO 2 capture in a gedankenexperiment (German: thought experiment).This apparatus is comprised of several square trays containing thin layers of aqueous capture solution.In order to capture large amounts of CO 2 , the most important aspect is the volume of the capture system solution (CSS).To illustrate that a large volume is desired as well as a large surface area, a rack of shallow baths of solution was chosen to balance these two goals in a simple way.These thin layers are an attempt to acknowledge that the diffusion is a relevant and an important factor, but the schematic system is not optimized for diffusion because the main purpose of the gedankenexperiment is the thermodynamic analysis of slow processes.Because capture from ambient air occurs at very high dilutions of CO 2 , it will not be a fast capture regardless.Of course, for fast capture at concentrated sources, the amount of CO 2 captured is controlled by kinetics and sophisticated chemical engineering solutions have been described that include more complicated gas/liquid contactor systems [49][50][51][52] and counter-current flow systems [53].These systems vastly increase surface area-to-volume ratios and provide for surface renewal opportunities and facile gas diffusion.However, for passive capture from air the simplified apparatus of Figure 6 suffices to obtain a rough estimate of the amount of CO 2 that would be collected per load cycle.
Load " FWpCO 2 q ˆrCO 2 ´CSs 400 ˆVCSS The load is determined by the volume of the capture system solution, the initial concentration of the amine capture system, and the temperature of the aqueous capture system.The Load can be calculated simply using Equation ( 6), where Load is the mass of captured CO 2 in g, FW(CO 2 ) is the formula weight for CO 2 , 44.01 g/mol, [CO 2 -CS] 400 is the equilibrium concentration of product (mol/L) when the partial pressure of CO 2 is 400 ppm, and V CSS is the volume of the capture system solution (L).For an ideal capture system, the load should be highly dependent on the temperature so that T-variation may afford release under economic conditions.The fraction captured per cycle can be determined by comparison of the load data at capture and release temperatures (Table 1).
For a numerical illustration, we consider the trays of the apparatus to have a surface areas of 1 m 2 and to hold a solution with a depth of 2.5 cm.For a standard system of 10 layers of these trays, this C 2016, 2, 18 8 of 14 example results in a total solution volume of 250 L. Assuming an initial concentration of the capture system solution [CS] 0 = 0.1 M, the load is calculated using Equation (6).To find the concentration of the product, the initial concentration of the capture system solution is multiplied by the fraction of CO 2 -adduct reported in Table 1.The amount of CO 2 collected in one capture cycle, the cycle load, equals the difference of the loads at the capture and release temperatures.With apparatus parameters being equal, the cycle load parallels the difference in the CO 2 -adduct fractions at capture and release temperatures (cycle load fraction).In the case of capture system R14h, the free enthalpy is low enough to ensure binding at T ˝and the data in Table 1 show the fractions of CO 2 -adduct to be 0.492 (T ˝), 0.068 (T ˝+ 20), and 0.007 (T ˝+ 40).The fractions show that most CO 2 will be released when the temperature is raised by 20 ˝, and that nearly all of the CO 2 will be released when the temperature is raised by 40 ˝.The amount of CO 2 collected per cycle for a capture temperature of T ˝and release temperatures of T ˝+ 20 and T ˝+ 40 are 0.47 kg and 0.53 kg, respectively.For a numerical illustration, we consider the trays of the apparatus to have a surface areas of 1 m 2 and to hold a solution with a depth of 2.5 cm.For a standard system of 10 layers of these trays, this example results in a total solution volume of 250 L. Assuming an initial concentration of the capture system solution [CS]0 = 0.1 M, the load is calculated using Equation (6).To find the concentration of the product, the initial concentration of the capture system solution is multiplied by the fraction of CO2-adduct reported in Table 1.The amount of CO2 collected in one capture cycle, the cycle load, equals the difference of the loads at the capture and release temperatures.With apparatus parameters being equal, the cycle load parallels the difference in the CO2-adduct fractions at capture and release temperatures (cycle load fraction).In the case of capture system R14h, the free enthalpy is low enough to ensure binding at T° and the data in Table 1 show the fractions of CO2-adduct to be 0.492 (T°), 0.068 (T° + 20), and 0.007 (T° + 40).The fractions show that most CO2 will be released when the temperature is raised by 20°, and that nearly all of the CO2 will be released when the temperature is raised by 40°.The amount of CO2 collected per cycle for a capture temperature of T° and release temperatures of T° + 20 and T° + 40 are 0.47 kg and 0.53 kg, respectively.

Variations of Thermodynamic Parameters-Guidelines for Capture System Design
We modeled three sets of hypothetical reactions HR1−HR3 to explore the effectiveness of capture and release in dependence of the Gibbs free enthalpy and the capture reaction entropy (Table 1).The sets differ in the Gibbs free enthalpy with ΔG° = −6.59kcal/mol for HR1 (as for R14a), ΔG° = −9 kcal/mol for HR2, and ΔG° = −11 kcal/mol for HR3.Within each set, we vary ΔH° in steps of 5 kcal/mol.For HR1 and HR2, we use ΔH° values of −15, −20, and −25 kcal/mol, respectively, and the ΔS° values follow.The systems HR3 were constructed to share the ΔS° values with HR2 and the ΔH° values follow.With these parameters, we used equation 2-RCR to compute the fractions of CO2adduct at [CO2] = 400 ppm and T°, T° + 20, and T° + 40, and the results are listed in Table 1.
Reaction HR1 is not sufficiently exergonic to produce adequate capture and Table 1 shows that all fractions are relatively low.Reactions HR2 and HR3, with their higher binding free enthalpies, demonstrate adequate capture, and these systems are illustrated in Figure 7.
Figure 7 shows the fractions of CO2-adduct remaining bound at T° + 40 for the series of reactions HR2 and HR3, and this figure reveals the important role of entropy in the release process.As the

Variations of Thermodynamic Parameters-Guidelines for Capture System Design
We modeled three sets of hypothetical reactions HR1´HR3 to explore the effectiveness of capture and release in dependence of the Gibbs free enthalpy and the capture reaction entropy (Table 1).The sets differ in the Gibbs free enthalpy with ∆G ˝= ´6.59 kcal/mol for HR1 (as for R14a), ∆G ˝= ´9 kcal/mol for HR2, and ∆G ˝= ´11 kcal/mol for HR3.Within each set, we vary ∆H ˝in steps of 5 kcal/mol.For HR1 and HR2, we use ∆H ˝values of ´15, ´20, and ´25 kcal/mol, respectively, and the ∆S ˝values follow.The systems HR3 were constructed to share the ∆S ˝values with HR2 and the ∆H ˝values follow.With these parameters, we used equation 2-RCR to compute the fractions of CO 2 -adduct at [CO 2 ] = 400 ppm and T ˝, T ˝+ 20, and T ˝+ 40, and the results are listed in Table 1.
Reaction HR1 is not sufficiently exergonic to produce adequate capture and Table 1 shows that all fractions are relatively low.Reactions HR2 and HR3, with their higher binding free enthalpies, demonstrate adequate capture, and these systems are illustrated in Figure 7.
Figure 7 shows the fractions of CO 2 -adduct remaining bound at T ˝+ 40 for the series of reactions HR2 and HR3, and this figure reveals the important role of entropy in the release process.As the reaction becomes more exentropic, the cycle yield improves in all cases, but the capture effectiveness is more complicated.For the most exentropic reaction of the HR2 series (HR2c), the fraction of CO 2 -adduct remaining at T ˝+ 40 is 0.003 and the unloading is highly effective.Even for the system HR2a, with its exentropicity mirroring that of the GCR R1, the unloading remains efficient: the fraction remaining is only 0.018 and the cycle load difference relative to HR2c is small (3.3%).Comparison of the HR2 series (Figure 7a) shows that at atmospheric pressure, the difference in capture due to entropy is functionally low.
In the more exergonic series of hypothetical reactions (HR3), the cycle load efficiency increases as with HR2 as the system becomes more exentropic.However, with the more exergonic reaction HR3, C 2016, 2, 18 9 of 14 the effect of entropy is drastically enhanced.Reaction HR3a is shown to have a fraction of CO 2 -adduct of 0.269 at T ˝+ 40 compared to HR3b with a fraction of 0.119 remaining.This corresponds to a 21.5% increase in cycle load for the more exentropic reaction.Further, the fraction of CO 2 -adduct remaining for HR3c at the same temperature is 0.050.This is an 8.1% increase over HR3b and a 31.4% increase over HR3a.As illustrated by the increase in cycle load percentage across each series, the entropy effects for the more exergonic HR3 system are enhanced by a factor of nearly 10 over HR2.In the more exergonic series of hypothetical reactions (HR3), the cycle load efficiency increases as with HR2 as the system becomes more exentropic.However, with the more exergonic reaction HR3, the effect of entropy is drastically enhanced.Reaction HR3a is shown to have a fraction of CO2adduct of 0.269 at T° + 40 compared to HR3b with a fraction of 0.119 remaining.This corresponds to a 21.5% increase in cycle load for the more exentropic reaction.Further, the fraction of CO2-adduct remaining for HR3c at the same temperature is 0.050.This is an 8.1% increase over HR3b and a 31.4% increase over HR3a.As illustrated by the increase in cycle load percentage across each series, the entropy effects for the more exergonic HR3 system are enhanced by a factor of nearly 10 over HR2.
After studying these hypothetical systems, it becomes pertinent to determine which would provide the best capture and the capture temperature plays an important role.We have already shown that ΔG° = −6.59kcal/mol (R14a, HR1) is highly ineffective at T° and above, but as Figure 3b shows, at T° − 20 (278.15K) the system will have a CO2-adduct fraction near 0.2.If we were to capture at T° − 20 and release at T°, the system will yield a relatively small but efficient cycle load.Even more useful at low temperatures would be a reaction with ΔG° = −9 kcal/mol (R14h, HR2).The most important feature of this hypothetical reaction is its variability around T°.The fraction of CO2-adduct captured of 0.492 at T° and [CO2] = 400 ppm by this reaction is almost exactly between the two asymptotes.Because of this, it will vary greatly in capture with even small differences in temperature, as can be seen in Figure 3d.This great variability causes these reactions to be useful for low temperatures while they are less useful for higher temperatures.Considering R14h at T° − 20 the fraction of CO2-adduct is greater than 0.9 at [CO2] = 400 ppm (Figure 3d).Raising the temperature to T° + 20 yields the fraction 0.068 (Table 1) and an excellent cycle load fraction of over 0.8.Even with a temperature variation from T° (CO2-adduct fraction = 0.492) to T° + 20, a cycle load fraction of 0.4 is obtained, a good collection that requires minimal heating.
However when considering a capture at T° + 20 (45 °C), and release at T° + 40, the cycle load fraction will fall to 0.06, a very poor collection.Thus for higher temperature climates, a system with a more robust exergonicity is desired, for example ΔG° = −11 kcal/mol (HR3).HR3b at T° + 20 has a CO2 capture fraction of 0.623, and increasing the temperature to T° + 40 (capture fraction of 0.119) yields a cycle load fraction of 0.504, a decent collection.Even better, for a capture at T° (capture fraction of 0.966) with release at T° + 40, the cycle load fraction is an excellent 0.847.At T° reaction HR3 already has a capture fraction of 0.966.Since the system cannot load significantly more CO2 during capture, there is no benefit to using this system in a climate with a temperature cooler than 25 After studying these hypothetical systems, it becomes pertinent to determine which would provide the best capture and the capture temperature plays an important role.We have already shown that ∆G ˝= ´6.59 kcal/mol (R14a, HR1) is highly ineffective at T ˝and above, but as Figure 3b shows, at T ˝´20 (278.15K) the system will have a CO 2 -adduct fraction near 0.2.If we were to capture at T ˝´20 and release at T ˝, the system will yield a relatively small but efficient cycle load.Even more useful at low temperatures would be a reaction with ∆G ˝= ´9 kcal/mol (R14h, HR2).The most important feature of this hypothetical reaction is its variability around T ˝.The fraction of CO 2 -adduct captured of 0.492 at T ˝and [CO 2 ] = 400 ppm by this reaction is almost exactly between the two asymptotes.Because of this, it will vary greatly in capture with even small differences in temperature, as can be seen in Figure 3d.This great variability causes these reactions to be useful for low temperatures while they are less useful for higher temperatures.Considering R14h at T ˝´20 the fraction of CO 2 -adduct is greater than 0.9 at [CO 2 ] = 400 ppm (Figure 3d).Raising the temperature to T ˝+ 20 yields the fraction 0.068 (Table 1) and an excellent cycle load fraction of over 0.8.Even with a temperature variation from T ˝(CO 2 -adduct fraction = 0.492) to T ˝+ 20, a cycle load fraction of 0.4 is obtained, a good collection that requires minimal heating.
However when considering a capture at T ˝+ 20 (45 ˝C), and release at T ˝+ 40, the cycle load fraction will fall to 0.06, a very poor collection.Thus for higher temperature climates, a system with a more robust exergonicity is desired, for example ∆G ˝= ´11 kcal/mol (HR3).HR3b at T ˝+ 20 has a CO 2 capture fraction of 0.623, and increasing the temperature to T ˝+ 40 (capture fraction of 0.119) yields a cycle load fraction of 0.504, a decent collection.Even better, for a capture at T ˝(capture fraction of 0.966) with release at T ˝+ 40, the cycle load fraction is an excellent 0.847.At T ˝reaction HR3 already has a capture fraction of 0.966.Since the system cannot load significantly more CO 2 during capture, there is no benefit to using this system in a climate with a temperature cooler than 25 ˝C; in fact, the efficiency would suffer because more heat would be required to liberate the CO 2 .
Our home state, Missouri, has great fluctuations in temperature through the seasons, so using a mixture of different capture systems may be most effective.In Columbia, Missouri, the average temperature in December is 0.78 ˝C which would indicate that a system with thermodynamic parameters similar to R14h would be best, while the average temperature in July is 25.4 ˝C, indicating that a reaction like HR3 would be considerably more effective [54].For coastal or equatorial regions with less variation in temperature, the use of one capture system would be more feasible.
The considerations emphasize the significance of the magnitude of the entropy of the capture reaction.The large magnitude of reaction entropy of the capture reaction is a direct consequence of the presence of two capture moieties in the active site of rubisco and we offer an analogy with the illustration shown in Figure 8.
C 2016, 2, 18 10 of 14 temperature in December is 0.78 °C which would indicate that a system with thermodynamic parameters similar to R14h would be best, while the average temperature in July is 25.4 °C, indicating that a reaction like HR3 would be considerably more effective [54].For coastal or equatorial regions with less variation in temperature, the use of one capture system would be more feasible.
The considerations emphasize the significance of the magnitude of the entropy of the capture reaction.The large magnitude of reaction entropy of the capture reaction is a direct consequence of the presence of two capture moieties in the active site of rubisco and we offer an analogy with the illustration shown in Figure 8.

Conclusions
In theoretical studies of chemical reactions with modern methods of electronic structure theory the reaction thermochemistry is usually reported for the stoichiometric reaction at standard conditions (1 atm, 25 °C) and the equilibrium constant Keq is related to the reaction Gibbs free enthalpy via ΔG° = ΔH° − T°•ΔS°.We described the computation of the equilibrium concentrations of the CO2adducts for the general capture reaction GCR and the rubisco-type capture reaction (RCR) considering the atmospheric CO2 concentration via Henry's law and as a function of temperature.The equations were applied to the thermochemistry of rubisco-based small molecule model reactions R1, R14a, and R14b, and three series of model reaction HR1-HR3 to illustrate the importance of the

Conclusions
In theoretical studies of chemical reactions with modern methods of electronic structure theory the reaction thermochemistry is usually reported for the stoichiometric reaction at standard conditions (1 atm, 25 ˝C) and the equilibrium constant K eq is related to the reaction Gibbs free enthalpy via ∆G ˝= ∆H ˝´T ˝¨∆S ˝.We described the computation of the equilibrium concentrations of the

Figure 1 .
Figure 1.CO2 capture in the active site of rubisco.

Figure 1 .
Figure 1.CO 2 capture in the active site of rubisco.

Figure 2 .
Figure 2. Molecular model of aggregates 17a and 21a formed between the magnesium formate MgL2 complexed trans-trans isomer of N-methylcarbamic acid and model carbonyl formaldehyde (FA) or acetone (Ac), respectively.(Ball and Bond Type models are shown and color indicates elements: oxygen: red; nitrogen: blue; carbon: grey; hydrogen: yellow; magnesium: lime).

Figure 2 .
Figure 2. Molecular model of aggregates 17a and 21a formed between the magnesium formate MgL 2 complexed trans-trans isomer of N-methylcarbamic acid and model carbonyl formaldehyde (FA) or acetone (Ac), respectively.(Ball and Bond Type models are shown and color indicates elements: oxygen: red; nitrogen: blue; carbon: grey; hydrogen: yellow; magnesium: lime).

Figure 3 .
Figure 3. Equilibrium concentrations of complexed product of a general capture reaction (GCR) (left) and a rubisco-type capture reaction (RCR) (right) at five temperatures from 278.15 to 318.15 K.The color of the central curve (298.15K) corresponds to the same system in Figures 3 and 4. Panels (a) and (b) describe the thermochemistry of R14a with ∆G ˝= ´6.59 kcal/mol, ∆H ˝= ´17.26 kcal/mol, and ∆S ˝= ´35.81 cal/(mol¨K).Panels (c) and (d) describe the thermochemistry of R14h with ∆G ˝= ´9.00 kcal/mol, ∆H ˝= ´19.67 kcal/mol, and ∆S ˝= ´35.79 cal/(mol¨K).Solid lines were computed with T-dependence of Henry's Law and dashed lines were computed without.
at [CO 2 ] = 400 ppm.Temperature affects the curvature of the fraction (CO 2 -adduct) = f([CO 2 ]) isotherms in Figure 3 and we will discuss the T-dependence in more detail below (see Section 2.3).Instead of the fraction (CO 2 -adduct), Figures 4 and 5 show the ratios [CO 2 -CS]/[CS] as a function of [CO 2 ].The ratios were computed with equations 3-GCR and 3-RCR, where R is the gas constant (1.9858775 cal¨mol ´1¨K ´1), ∆G is the Gibbs Free Enthalpy of the capture reaction at T, and ∆G = ∆G ˝for the special case of T = T ˝.
at [CO2] = 400 ppm.Temperature affects the curvature of the fraction (CO2-adduct) = f([CO2]) isotherms in Figure 3 and we will discuss the T-dependence in more detail below (see Section 2.3).Instead of the fraction (CO2adduct), Figures 4 and 5 show the ratios [CO2-CS]/[CS] as a function of [CO2].The ratios were computed with equations 3-GCR and 3-RCR, where R is the gas constant (1.9858775 cal•mol −1 •K −1 ), ΔG is the Gibbs Free Enthalpy of the capture reaction at T, and ΔG = ΔG° for the special case of T = T°.This ratio shows a linear dependence on the CO2 pressure, and temperature affects the slope (but not the curvature).

Figure 6 .
Figure 6.Schematic diagram of the carbon capture apparatus.

Figure 6 .
Figure 6.Schematic diagram of the carbon capture apparatus.

Figure 8 .
Figure 8.The genius of the rubisco-capture system stems from the large reduction of system entropy associated with bringing the two tethered capture moieties (the Mg 2+ site and the lysine amino group) together on CO2 capture.The principle is well exemplified by the image of two leashed dogs biting on the same bone.In the absence of the bone, each dog enjoys great freedom in their accessible range (illustration on top).Biting on the same bone curtails the accessible range and the freedom of the dogs' mobilities (bottom).

Figure 8 .
Figure 8.The genius of the rubisco-capture system stems from the large reduction of system entropy associated with bringing the two tethered capture moieties (the Mg 2+ site and the lysine amino group) together on CO 2 capture.The principle is well exemplified by the image of two leashed dogs biting on the same bone.In the absence of the bone, each dog enjoys great freedom in their accessible range (illustration on top).Biting on the same bone curtails the accessible range and the freedom of the dogs' mobilities (bottom).

Table 1 .
Thermochemical data for capture systems.

Table 1 .
Thermochemical data for capture systems.1 1Gibbs free enthalpy ∆G ˝and enthalpy ∆H ˝in kcal/mol.Entropy ∆S ˝in cal/(mol¨K).