A novel proton transfer model of the closed equilibrium system H2O–CO2–CaCO3–NHX

Variation in the concentration of dissolved carbon dioxide [CO2]W causes changes in the solubility of limestone and in the pH of an equilibrium system. An elevation of the pH will shift the equilibrium of the reversible reaction NH 4 ⇌ NH3 + H towards the formation of free ammonia (NH3). This results in the inhibition of the activity of microorganisms that perform the biological wasteand reject-water treatment. The model of the system H2O–(CO2)W−CaCO3 was upgraded on the basis of proton transfer principles and taken as the basis for modelling the closed system H2O–(CO2)W–CaCO3–NH4Cl. The distribution of ions and molecules in the closed system H2O–(CO2)W–CaCO3–NHX is described in terms of a structural scheme. A novel proton transfer model was developed to calculate the pH, concentrations of the formed ions and molecules, and proton transfer parameters of the closed equilibrium system using an iteration method. In the formation of the equilibrium system H2O–(CO2)W−CaCO3, as a result of the dissolution of CaCO3, the CO  2 3 ions are released and these will accept a certain quantity of protons (   2 3 CO ] Δ[H ), which originate from two sources: the reversible dissociation of water ( O H2 ] Δ[H  ) or H2CO3 ( 3 2CO H ] Δ[H ), which is the product of the reaction between H2O and (CO2)W0. In case the final closed system H2O–(CO2)W–CaCO3–NH4Cl includes small initial concentrations of [CO2]W0, the main amount of protons (   4 NH ] Δ[H ) comes from the dissociation of NH 4  , or if there are higher concentrations of [CO2]W0, the source of protons is H2CO3 ( 3 2CO H ] Δ[H ). The developed models were experimentally validated.


INTRODUCTION
* Limestone bedrock, which consists mainly of CaCO 3 , affects the treatment of wastewater including reject water generated in biogas plants. The variation in the concentration of dissolved carbon dioxide [CO 2 ] W changes CaCO 3 solubility in and the pH of equilibrium systems. The equilibrium system H 2 O−(CO 2 ) W −CaCO 3 (where (CO 2 ) W stands for dissolved carbon dioxide) has been extensively studied [1−6], but no systemic interpretation of equilibrium reactions has been proposed. We have previously published the models of the equilibrium system H 2 O−(CO 2 ) W −CaCO 3 based on proton-centred acid-base equilibria for both open [7] and closed systems [8].
The aqueous systems with prevailing carbonaceous and ammoniacal species are present in many wastewaters of various types and origin. However, there are relatively few studies modelling interconnected equilibrium processes that affect wastewater treatment [9]. No proton-centred models have been proposed for interconnected equilibria in wastewaters. Ammonia nitrogen NH X -N (NH  4 -N together with NH 3 -N) is a pollutant present in high concentrations in the reject water from the anaerobic digestion of organic substrates and landfill leachate. The concentration of [NH X -N] is up to 2200 mg L -1 in the case of landfill leachate [10−12] and up to 1500 mg·L -1 in the reject water [13,14] that comes from the dewatering of anaerobically digested wastewater sludge [15,16].
In an anaerobic digester, the pH of the liquid phase is affected by carbonaceous equilibria. In case of limestone bedrocks, the water has been in contact with CaCO 3 . On the other hand, the liquid phase is exposed to the gas phase of which about a third is CO 2 [17]. The liquid phase in the anaerobic digester of the Tallinn Municipal Wastewater Treatment Plant has a pH value of 7.2 (±0.2) [18]. During dewatering the digested sludge, the reject water comes into contact with ambient air and the degassing of CO 2 from the liquid phase takes place, which causes the pH value of the reject water to increase to 8.1 (±0.2) [18,19], shifting the equilibrium of the reaction NH  4 ⇌ NH 3 + H + towards the dissociation of NH  4 and the formation of free ammonia (NH 3 ) [20−22].
In the case of anaerobic ammonium-oxidizing (anammox) bacteria, an elevated concentration of NH 3 -N (≥8 mg·L -1 ) acts as an inhibitor on biological wastewater treatment processes, including the activity of bacteria performing autotrophic nitrogen removal [20−24]. The lowest NH 3 toxicity threshold concentration, provided by Jung et al. (2007) [25], is 1.7 mgN·L -1 during the startup period, which is significantly smaller than the values typically observed in pilot and full-scale anammox reactors [23,26]. Zekker et al. (2012) [27] investigated short-and long-term effects of various bicarbonate and free ammonia concentrations on wastewater treatment in the biofilm process showing a decrease in the bacterial activity at very low and high bicarbonate concentrations. A similar inhibition effect was determined in batch tests, with a decrease of 50% in the specific anammox microorganisms' activity at NH 3 -N concentrations of 38 mg·L -1 , and long-term tests in sequencing batch reactors showed performance instability when the NH 3 -N concentration exceeded 20-25 mg·L -1 [23]. Thus, the adjustment of the pH is required in waste-and reject-water treatment systems in streams with a high content of [NH X -N] in cases when there is contact with limestone [15,18]. The concentration of NH 3 has so far been calculated from the empirical formula published by Anthonisen et al. in 1976, which is a function of the concentration of NH  4 , pH, and temperature [20]: where b W 6344 273 In equations (1) and (2), K b is the basicity constant of ammonia, K W is the ion-product constant of water, e is the base of the natural logarithm, and t the temperature in C.
Equation (1) does not take into account the effect of interactions between the carbonaceous and ammoniacal equilibrium processes, which alter the concentration of [NH 3 ]. Therefore, the actual concentration of [NH 3 ] may be significantly different from that calculated using equation (1). To calculate [NH 3 ] in a system having a high concentration of dissolved [NH X ] and carbonaceous species, a mathematical model needs to be developed. In the present study the model of the system H 2 O-(CO 2 ) W −CaCO 3 was upgraded on the basis of proton transfer principles and taken as the basis to model the closed system H 2 O-(CO 2 ) W -CaCO 3 -NH 4 Cl. Proton transfer parameters, pH, and equilibrium concentrations of ions and molecules in the liquid phase were calculated using the developed model, which was also experimentally validated.

DERIVATION OF THE THEORETICAL MODEL OF THE SYSTEM H 2 O-(CO 2 ) W -CACO 3 -NH 4 CL 2.1. The development of the proton transfer based model of the initial system H 2 O-(CO 2 ) W -CaCO 3
When solid CaCO 3 is added to the equilibrium system H 2 O-(CO 2 ) W with a summarized concentration of all inorganic carbon containing particles ∑[CO X ] W0 , including (CO 2 ) W , H 2 CO 3 , HCO  3 and CO    3 ] ions are released by the reversible dissolution of CaCO 3 [8].
The reversible dissociation reaction is as follows: It is always present in the water phase, which acts as a host solvent for other chemical reactions. If reactions involve H + or OHions, a strong coupling will arise between the solvent and solute reactions. The reversible dissociation reactions in aqueous media involve intermolecular proton transfer. Water dissociates into H + and OHions, and if the dissociation products of the solute can bind or release either H + and OHions, these reactions will always take place, changing the chemical equilibrium of the system. In the formation of the equilibrium system H 2 O-(CO 2 ) W −CaCO 3 , as a result of the dissolution of CaCO 3 , the CO  ions are released and they will accept a certain amount of protons ( ) formed in the reaction between (CO 2 ) W0 and H 2 O. Thus, the H + ions have a central role in the evolution of the equilibrium system and form a link between acid-base equilibrium processes. The results of the calculations before adding NH 4 Cl are presented in Table 1.
The data in Table 1 show that at the range of concentrations from [CO 2 ] W0  3.39×10 -7 to [CO 2 ] W0  33.9 mmol·L -1 in the closed equilibrium system H 2 O−(CO 2 ) W −CaCO 3 , the pH will increase up to a maximum value of 9.88 at the decreasing concentration of [CO 2 ] W0 [8]. The smaller the initial concentration of [CO 2 ] W0 , the fewer protons are produced due to the dissociation of H 2 CO 3 . Thus, the dissociation of water at lower concentrations of [CO 2 ] W0 produces the main amount of protons to form an equilibrium of the system. As a result of equilibrium reactions, the protons bind to CO   ions is nearly equal to the concentration of the [H + ] ions released by the dissociation of water. In this case, the dissociation of water and the solubility of CaCO 3 determine the equilibrium of the system. At a concentration of [CO 2 ] W0 = 2.45 mmol·L -1 , the pH is equal to 7.0. When the concentration of [CO 2 ] W0 exceeds 2.45 mmol·L -1 , the pH will fall below 7. The reaction between (CO 2 ) W0 and H 2 O produces H + ions, which partly participate in the reaction between CO  ion is an acceptor of protons. At higher [CO 2 ] W0 values, the concentration of the formed protons is elevated due to the dissociation of H 2 CO 3 . As a result, the reversible dissociation of water is suppressed, because the ion product of water (K W ) remains constant. The protons will be bound to OHions and will acquire a negative value. In this region the source of protons, which will react with CO  2 3 , is formed by the dissociation of H 2 CO 3 . In the region of [CO 2 ] W0 ≤1.00×10 -6 , has a negative value as a reversible dehydration of H 2 CO 3 is taking place.

Model of the final equilibrium system after adding NH 4 Cl to H 2 O-(CO 2 ) W -CaCO 3
To the equilibrium system H 2 O-(CO 2 ) W -CaCO 3 , with the initial concentrations of [CO 2 ] w0 (shown in Table 1), NH 4 Cl was added. As a result, a final equilibrium system H 2 O-(CO 2 ) W -CaCO 3 -NH 4 Cl was formed. The ions and molecules in the system were quantitatively distributed in accordance with the equilibrium constants as shown in Fig. 1. The values of the initial concentrations of dissolved [CO 2 ] W0 in the water phase were taken as constants to model the final closed system H 2 O-(CO 2 ) W -CaCO 3 -NH 4 Cl. The values of the equilibrium constants of the system under investigation are given as equations (4-9): and the molar balance equation of carbon containing particles is   3 ] in the final equilibrium system, it is necessary to solve nine equations: (4)- (12).
To decrease the number of unknowns, the composite constant K a1 was used. The first pseudo-acid ionization constant K a1 [7,34] is expressed as a product of the constants K h and K ' a1 : 7 1 al h a1 4.44 10 mol L .
´ On the basis of equations (9) and (12) The concentrations of variables in charge and molar balance equations were eliminated by replacing them by equilibrium constant equations. As a result of replacing the variables, the system is characterized by two equations with two unknown concentrations ( Equations (15) and (16) Table 2. The concentration of NH 4 Cl = 71 mmol·L -1 equalled the average measured concentration in the reject water of the pilot study by Rikmann et al. [18].
At the given range of concentrations ([CO 2 ] W0  33.9 mmol·L -1 to [CO 2 ] W0  3.39×10 -7 mmol·L -1 ), adding NH 4 Cl to the initial closed system H 2 O−[CO 2 ] W0 −CaCO 3 resulted in a reduction in the pH value of the formed equilibrium system H 2 O−[CO 2 ] W0 −CaCO 3 −NH 4 Cl presented in Tables 1 and 2. The smaller the initial concentration of (CO 2 ) W0 , the fewer protons were released from the dissociation of H 2 CO 3 , while the bulk of the protons that would react with CO  2 3 came from the dissociation of NH  4 . In the range of concentrations of [CO 2 ] W0 from 3.39×10 -7 to 3.39×10 -2 mmol·L -1 , the concentration of [HCO  3 ] ions is less than the concentration of H + ions released from the dissociation of NH  4 ( ) is equal to the concentration of NH 3 . In the closed system under investigation, at higher concentrations of [CO 2 ] W0 in the range from 2.45 to 33.9 mmol·L -1 , the dissociation of H 2 CO 3 will predominate and produce more protons have negative values as the dissociation of water is suppressed.

MATERIALS AND METHODS
In the initial phase of the experiment, MilliQ water was purged with argon (99.999% pure) to remove any residual (CO 2 ) W . To guarantee an equilibrium distribution of CO 2 between the gas and liquid phases, the aqueous phase was saturated by purging it with a gas mixture with a selected constant concentration of CO 2 until a stable pH value was achieved. The measured pH values were registered using a Jenway type 3520 pH meter and Dataway software. A one-litre closed reaction cell was kept at 25.0 ± 0.1 °C using a thermostat equipped with a pH and temperature measuring system (Mettler Toledo InLab ® Routine and Jenway model 027500 sensors, respectively). The gas mixtures were ordered from AGA Eesti Ltd. (stored as compressed gas in cylinders containing 1010 ± 2 ppm and 395 ± 2 ppm CO 2 ) or prepared using a Bronkhorst EL-FLOW ® controller-equipped digital thermal mass flow meter (stored in 25 L gas collection bags). The prepared mixture might have a higher degree of variation of CO 2 concentration; also, the gas flow from gas collection bags might be less uniform. The experiments were conducted in a closed system H 2 O-(CO 2 ) W -CaCO 3 -NH 4 Cl with 1000 mg of solid CaCO 3 (CAS # 471-34-1, 99.95% Sigma-Aldrich) and 3821 mg (~71 mmol·L -1 ) of NH 4 Cl (CAS # 12125-02-9, 99.5% Acros Organics, US), which corresponds to 1000 mg·L -1 NH X -N. The salts were added to the liquid in the reaction cell (volume 1 L, after purged with certain concentrations of [CO 2 ] using gas bags from AGA). At the end of the experiment, samples were taken and filtered through a 0.45 μm membrane filter. From the filtered samples, [Ca 2+ ] ion concentrations were determined by titration using a 0.001 M EDTA solution (other ions that might react with EDTA were not present). Dissolved CO 2 was measured using a Hach-Lange DR2800 spectrophotometer and LCK388 cuvettes.
Nessler's reagent, which forms a complex with ammonia, was used to determine NH X -N. The experiments were performed in triplicate for each initial concentration of [CO 2 ] W0 in a closed reaction cell, where CaCO 3 was added to the equilibrium system H 2 O−[CO 2 ] W0 , and a supplementary amount of CO  2 3 and Ca 2+ ions was released by the dissolution of CaCO 3 .

RESULTS AND DISCUSSION
As shown in Fig. 2, at the range of concentrations from [CO 2 ] W0  3.39  10 -5 mmol·L -1 to [CO 2 ] W0  33.9 mmol·L -1 , adding CaCO 3 to the equilibrium system caused the pH value to rise about 2-4 units depending on the initial concentration of [CO 2 ] W0 . The CO  After the pH of the equilibrium system H 2 O−[CO 2 ] W0 had stabilized following CaCO 3 addition, 71 mmol of NH 4 Cl was added to 1 L of the liquid phase, which approximately corresponds to the concentration of NH x -N in the reject water [15,18,19]. Consistently, as shown in Fig. 2, at the initial concentration of [CO 2 ] W0  3.39·10 -5 mmol·L -1 , the pH value decreased after NH4Cl was added to the initial system H 2 O-(CO 2 ) W0 -CaCO 3 from 9.89 ± 0.03 to 7.65 ± 0.02. The change in the pH value was 2.24 ± 0.02 units.   Table 3, experimentally measured final stabilized pH values of the closed equilibrium system H 2 O-[CO 2 ] W -CaCO 3 -NH X -N were generally close to the theoretical ones and the small differences in the obtained values might be due to the gas prepared-flow control system's accuracy.
The experimentally determined values of the concentration of Ca 2+ ions in the range of  0.11-5.82 mmol·L -1 were in accordance with the developed theoretical model ( Table 3). As seen in Fig. 2 and Tables 1 and 2, the experimental results corresponded to the theoretical model, which supported the validity of the developed novel proton transfer model linking different acid-base equilibria. The same approach can be applied to complex systems involving more acid-base equilibria (e.g. phosphoric acid equilibrium). Thus, the present experimental study confirmed the validity of the theoretical model we developed.
The developed model of equilibrium in the closed system H 2 O-(CO 2 ) W -CaCO 3 -NH 4 Cl can be used to determine concentrations of all components in the water phase over a wide range of natural and industrial conditions, based on the known concentrations of [CO 2 ] W0 in the water phase. The model allows one to assess the impact of anthropogenic processes in the environment and could be applied in environmental technology.

CONCLUSIONS
In this study of modelling the closed system H 2 O-(CO 2 ) W -CaCO 3 -NH 4 Cl, the system H 2 O-(CO 2 ) W −CaCO 3 was taken as the basis and upgraded using proton transfer principles. In the developed model, proton transfer parameters, pH, and the equilibrium concentrations of ions and molecules in the liquid phase were calculated.
In the formation of the equilibrium system H 2 O-(CO 2 ) W −CaCO 3 by the dissolution of CaCO 3 , CO  2 3 ions will release and accept protons ( ). The smaller the initial concentration of [CO 2 ] W0 , the fewer protons are produced on the basis of the dissociation of H 2 CO 3 . Thus, the dissociation of water at lower concentrations of [CO 2 ] W0 gives the main amount of protons to form the equilibrium of the system. As a result, protons will be bound to CO  2 3 ions and the pH of the formed equilibrium system will increase. In the system H 2 O−[CO 2 ] W −CaCO 3 -NH X -N, at lower initial concentrations of [CO 2 ] W0 the main source of protons is NH  4 ions. Experimentally measured data coincided well with the theoretically calculated values.
At higher initial concentrations of [CO 2 ] W0 the main source of protons is H 2 CO 3 . The concentration of toxic [NH 3 ] in the system is affected by the interaction between carbonaceous and ammoniacal equilibrium processes. The developed model of the equilibrium closed system H 2 O-(CO 2 ) W0 -CaCO 3  can be used to determine concentrations of all components in the water phase over a wide range of natural and industrial conditions, based on the known concentration of [CO 2 ] W0 in the water phase. The model allows one to assess the impact of anthropogenic processes on the environment and could be applied by environmental technologists in water and wastewater treatment. The model is useful in the development of innovative methods of measuring [CO 2 ] W and [NH 3 ] concentrations in water.