#### 2.1. 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.

#### 2.2. 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) =

${K}_{H}^{0}$ = 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 [CS] = [CS]

_{0} − [CO

_{2}-CS], 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.

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 Δ

G° values that are not exergonic enough. Hence, we also studied the hypothetical reaction R14h with Δ

G° = −9 kcal/mol. For this model system, we set Δ

S° (R14h) = Δ

S° (R14a) and obtain Δ

H° (R14h) = −19.67 kcal/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) 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),

Figure 4 and

Figure 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}), Δ

G is the Gibbs Free Enthalpy of the capture reaction at

T, and Δ

G = Δ

G° for the special case of

T =

T°. This ratio shows a linear dependence on the CO

_{2} pressure, and temperature affects the slope (but not the curvature).

The Gibbs free enthalpy of the capture reaction obviously will vary with temperature because of the entropy term, Δ

G = Δ

H −

T·Δ

S. The reaction enthalpy Δ

H(

T) and reaction entropy Δ

S(

T) depend on temperature only modestly and, for the temperature variations considered here, they can be approximated very well by Δ

H(

T) = Δ

H° and Δ

S(

T) = Δ

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 –Δ

_{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}.

For –Δ

_{sol}H/

R > 0, it is clear that CO

_{2} will be better solvated at low temperatures and less solvated at temperatures higher than 298.15 K. 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.

#### 2.3. 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.

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 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.

#### 2.4. 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 Δ

G° = −6.59 kcal/mol for HR1 (as for R14a), Δ

G° = −9 kcal/mol for HR2, and Δ

G° = −11 kcal/mol for HR3. Within each set, we vary Δ

H° in steps of 5 kcal/mol. For HR1 and HR2, we use Δ

H° values of −15, −20, and −25 kcal/mol, respectively, and the Δ

S° values follow. The systems HR3 were constructed to share the Δ

S° values with HR2 and the Δ

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, 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.

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 Δ

G° = −6.59 kcal/mol (R14a, HR1) is highly ineffective at

T° and above, but as

Figure 3b shows, at

T° − 20 (278.15 K) 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 Δ

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 ΔG° = −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.