Development of a Thermodynamic Model Using a Speciation Framework

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Development of a Thermodynamic Model using a speciation framework: Illustration on HNO3 – H2O system Ravi Kanth MVSR, Pushpavanam Subramaniam, Shankar Narasimhan, and Narasimha Murty B Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b05361 • Publication Date (Web): 22 Mar 2018 Downloaded from http://pubs.acs.org on March 22, 2018

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Industrial & Engineering Chemistry Research

Development of a Thermodynamic Model using a speciation framework: Illustration on HNO3 – H2O system Ravi Kanth MVSR,a Pushpavanam S,*, b Shankar Narasimhan,b and Narasimha Murty Ba a

Nuclear Fuel Complex, Department of Atomic Energy, Hyderabad, India,

500062 b

Department of Chemical Engineering, Indian Institute of Technology, Madras,

India, 600036 *

Corresponding author. Tel.: +91-44-22574161; Email address: [email protected] Abstract

A rigorous thermodynamic framework based on speciation analysis is developed for an aqueous solution of nitric acid in this work. The model uses experimentally measured ‘extent of dissociation’ for determining the thermodynamic dissociation constant of nitric acid in aqueous solutions at 25°C. The activity coefficients of H+ and NO3- ions are modeled using Specific Ion Interaction Theory (SIT). SIT parameters for H+ and NO3- ionic activity coefficients and the parameters for activity coefficient of undissociated HNO3 are obtained using the proposed methodology. An important contribution of our approach is that the model developed is applicable over the concentration range of 0 – 22 M nitric acid which is of practical relevance. The model predictions of concentrations are consistent with data reported in the literature. 1. Introduction Metals like Plutonium, Zirconium, Uranium and Hafnium have a significant importance in the nuclear field. Typically nitrate salts of these metals are extracted from aqueous nitric acid solutions using Tri-n-Butyl Phosphate (TBP) as solvent. Dissociation of nitric acid in water plays a key role in modeling these metal extraction processes. 1 - 8 The performance of these processes is determined by the dissociation of nitric acid in water. Knowledge of HNO3 dissociation is also important in the fields of nuclear fuel production, nuclear fuel re-processing, 5 marine aerosols 9 and ozone layer depletion. 10

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Nitric acid dissociation in water has been extensively studied. The dissociation has been reported as complete in dilute solutions. 11 - 16 The extent of dissociation decreases as the concentration of HNO3 increases. 2, 17 Recently, a detailed review on the reported data for HNO3 dissociation was carried out by Levanov et. al. 18 The dissociation constant values of nitric acid (K0HNO3) reported by most researchers are in the range 10 – 40 and Levanov et. al. 18 estimated the value as 35.5. There is hence uncertainty in the reported value of dissociation constant. In all the reports published till date, the activity coefficients of H+ and NO3- are modeled assuming complete dissociation. This assumption is valid only at the lower concentration range of HNO3. 16 - 18 The molar activity coefficient thus modeled under this assumption was termed as ‘Stoichiometric activity coefficient (yS)’. Molar activity coefficients of H+ ions (yH+) and NO3ions (yNO3-) are reported in the form of ‘Molar mean ionic activity coefficient (y±)’. This is modelled using ‘extent of dissociation (ߙ)’ as y± =

y  y H + y NO − =  S  3 α 

(1)

Extent of dissociation (ߙ) is defined as the ratio of nitric acid present in dissociated form to the total nitric acid present in dissociated and undissociated forms. The reported mean ionic activity coefficient (y±) and the molar activity coefficient of undissociated nitric acid (yHNO3) follow a non-symmetric notation. This is also known as the Henry’s law standard state. 19, 20 In this standard state, activity coefficient of a solute in a solution tends to unity for lower concentrations of solute. Accordingly for dilute solutions of nitric acid, the activity coefficient values of y± are close to unity. They show an asymptotically increasing trend with increasing concentration. 16 - 18 Mean ionic activity coefficient and activity coefficient of nitric acid were modelled using osmotic coefficient models13, 16 or using empirical models. 18 At higher concentrations of acid (22 M), the degree of dissociation of HNO3 is reported to be as low as 0.011. However the molar mean ionic activity coefficient under these conditions is reported as 1800. dissociation.

18

17

A similar increasing trend is reported in the recent study on HNO3

At higher concentrations of HNO3, the concentrations of H+ and NO3- ions are

low as the dissociation is low. Therefore, to be consistent with the choice of Henry’s law standard state, the mean ionic activity coefficients should tend to unity as the concentration of the nitric acid approaches the anhydrous limit of 23.87 M. The reported activity coefficients data 2 ACS Paragon Plus Environment

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do not conform to this. There is hence a need to develop activity coefficient models for H+, NO3and undissociated HNO3 based on the non-symmetric notation explicitly accounting for the extent of dissociation. In order to achieve this objective, the equilibrium concentrations of ionic species are required to be modeled. Recently we have discussed a rigorous speciation framework to predict equilibrium concentrations of a system when the thermodynamic parameters (equilibrium constant and activity coefficient models) are available. 21 In the present work, we extend that approach and discuss the estimation of the thermodynamic parameters from experimental data in the same framework. It is demonstrated that the thermodynamic parameters can be simultaneously obtained along with equilibrium concentrations under this framework. In this paper, we use the Specific Ion Interaction Theory (SIT) model as it explicitly models the activity coefficients of the ions as a function of the ionic strength and ensures consistency in that these activity coefficients tend to unity as the concentration of the ions goes to zero. Besides it can be used to model all ionic species such as the metal-ion complexes in extraction of nuclear materials and therefore a single theoretical framework can be used. Estimates of the interaction parameters in the SIT model will also be useful for modelling the thermodynamics of the extraction process. This paper focuses on applying a robust thermodynamic modeling framework for an aqueous solution of nitric acid at 25°C. Specifically, the dissociation constant of nitric acid and the model parameters for activity coefficients of H+, NO3- and undissociated HNO3 in the aqueous solution are estimated from experimental dissociation data. We first describe the procedure adopted for estimating the thermodynamic parameters for an aqueous nitric acid (HNO3-H2O) system using the speciation framework in section 2. The speciation model of HNO3-H2O system and the activity coefficient models used in this work are described in this section. We discuss the predictions of the model in section 3. 2. Methodology for Parameter estimation A methodology for computing equilibrium concentrations of various species present in single and multi-phase systems was developed recently based on a speciation framework.

21

This

requires the thermodynamic parameters as inputs. In this section we extend the methodology to 3 ACS Paragon Plus Environment


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estimate the thermodynamic parameters from experimental data. We demonstrate its application on the HNO3-H2O system. Consider a system with NS species present in aqueous phase. The concentrations of all the species can be determined using NS independent equations. If there are NR independent reactions (NR < NS) associated with the formation of each species, then we have NR equations from reaction equilibria. The number of component species (NC) forming the basis set of all the species in the system is given by NC = NS - NR. Mass conservation and charge (electro-neutrality) balance equations of these NC component species constitute the remaining set of independent equations. The reaction equilibria for NR independent species formation reactions constituting NR equations is given by NC

log10 K ∗j = log10 a j − ∑ν ji log10 ai ;

for all the reactions j = 1 to N R

(2)

i =1

Here K*j is the equilibrium formation constant of the jth species obtained by expressing the reaction as formation of species j from the selected ‘component species’, νji is the stoichiometric coefficient of the ith component in the formation reaction of the jth species, and ai is the activity of ith species which is expressed in terms of its molar concentration (Ci) as

ai = C i yi

(3)

where, yi is the molar activity coefficient of the ith species. Conservation of mass for the component species is invoked to generate the remaining set of NC equations. The total moles of ith component present in the system Ti, is conserved and this yields NS

∑ν

ji

C j Q = Ti ;

for all components i = 1 to N C

(4)

j =1

where, Q is the volume of the phase in which species j is present. The aqueous electrolytic solution is electrically neutral therefore charge balance given by equation (5) is applicable,


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(5)

NS

∑ z jC j = 0 j =1

Here zj represents the charge on the species j. It has been proven that the proton balance is analogous to electro-neutrality. 22, 23 Therefore, either charge balance or proton balance (but not both) constitute an independent equation. In this work, we used proton balance and the charge balance is not imposed. A unique feature of this robust and fast algorithm proposed by Ravikanth et. al. 21 is scaling the concentration variable using logarithm of the concentration. The method therefore has the advantage of ensuring convergence with initial guesses which are far away from equilibrium without imposing any non-negativity constraints on the species concentrations. The mass balances of component species in terms of logarithmic concentrations are expressed as NS

∑ν

ji

(10

log10 C j Q

) = Ti ;

for all the components i = 1 to N C

(6)

j =1

NR equations from reaction equilibria and NC equations from mass and charge balances of the component species form a complete set of NS equations. These are simultaneously solved for obtaining the concentrations of NS species using the routine ‘fsolve’ in MATLAB. This methodology is based on using the thermodynamic parameters such as equilibrium constants and activity coefficient models for computing the equilibrium concentrations of the species present. We now extend this methodology for simultaneously estimating the parameters of the thermodynamic model along with the equilibrium concentrations using experimentally measured data of concentrations. In the case of the HNO3-H2O system, the model parameters which are estimated include equilibrium constants at zero ionic strength (K0HNO3), activity coefficient model parameters of H+, NO3- and undissociated HNO3. The algorithm used to estimate thermodynamic parameters using experimental measured concentrations is described in Figure 1. The steps in the box within dashed lines discuss how to obtain the equilibrium concentrations when the thermodynamic parameters are known. We see that the thermodynamic parameters can



be estimated iteratively using the experimental data. This involves adding an extra loop to the algorithm developed earlier. 21 Guess the thermodynamic parameters, K0 and activity coefficient model parameters List all species in the system, NS List all the independent species formations, NR Determine the number of components, NC (NC = NS – NR) Collect all the equilibrium constants, K0

Input amount of components, T Evaluate equilibrium constants, K*and Stoichiometric coefficient matrix, A

Guess the concentrations of components Compute guess of logarithmic concentrations for all the species Initialize Newton – Raphson solver having logarithmic species concentrations as initial guess

1. Compute the Ionic strength and activity coefficients of all the species. 2. Simultaneously solve a. the reaction equilibria, and b. the mass and charge balances

Output logarithmic species concentrations are converted to species equilibrium concentrations

Minimizing the function residuals as compared to experimental measurements Figure 1: Proposed methodology for parameter estimation. This consists of an optimization loop within which the speciation calculation is embedded. 6 ACS Paragon Plus Environment

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Application to HNO3-H2O system The following species (NS = 5) are considered to be present at equilibrium in this system: H+, OH-, H2O, NO3- and HNO3. The reactions and reaction equilibria of the system are shown in Table 1. This is used for computing the equilibrium concentrations of species in HNO3-H2O system. Table 1: Reaction equilibria of Nitric Acid - water system at 25°C K0 (25°C)

Reaction 0

Kw H2O ← → H + + OH− K0

HNO 3 ←HNO 3 → H + + NO3−

Reaction

Reference

1×10-14

(R1)

Zemaitis Jr. et. al. 24

K0HNO3

(R2)

To be estimated

This system involves two independent reactions (NR), so the number of components (basis set) to be chosen is three (NC = NS − NR). We selected H+, NO3- and H2O as the basis set. The final solution for equilibrium concentrations is independent of the choice of the basis set. However, it is preferable to choose the basis set as the dominant species. This helps in obtaining good initial guesses for the concentrations.

23

The formation of other species was expressed as species

formation reactions from the selected basis components. The equilibrium constant values for the formation reactions of non-component species are obtained as K*.

21

The elements of

stoichiometric matrix (A) and the equilibrium constant values for the formation reactions of noncomponent species (log10 K*) are provided in Table 2. Table 2: Stoichiometric coefficient matrix and equilibrium constants for Nitric Acid - water system Species

H+

NO3-

H2 O

log10 K*

OH-

-1

0

1

-14

HNO3

1

1

0

To be estimated

H+

1

0

0

-

NO3-

0

1

0

-

H2 O

0

0

1

-



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Activity coefficient Models The activity coefficients of ions, H+, NO3- and OH- are modeled using the Specific Ion Interaction Theory (SIT).

25 - 27

The motivation for using SIT theory is that the activity

coefficients of ions naturally follow non-symmetric notation. SIT Model was introduced by Bronsted 25, implemented by Scatchard 26 and Guggenheim 27 for several systems. This model accounts for electrostatic (long range) interactions and nonelectrostatic (short range) interactions. Long-range interactions are accounted through the Debye-Hückel term, which is dominant in dilute solutions. For concentrated electrolyte solutions, short-range interactions are accounted using a linear dependence on concentration. Accordingly, the molar activity coefficient yj of an ion j of charge zj in a solution of molar ionic strength IC is described by

log y j

= − z 2j D + ∑ε ' ( j, k )Ck

(7)

k

The summation in equation (7) extends over all ions present in the system. The dependency of ε’ on ionic strength is considered to be very negligible and is ignored. Here Ck is the molar concentration of species k and ε’(j, k) is the molar ion-interaction parameter. This parameter describes the short-range interactions between the aqueous species j and k, and is non-zero only for ions of opposite charges. D is the Debye-Hückel term and is given by

D =

(8)

A IC 1 + Ba j IC

The value of A is taken as 0.509 L1/2 mol−1/2 and Baj as 1.5 L1/2 mol−1/2. IC, the molar ionic strength is defined as:

I C = 0.5∑Ci zi2

(9)

i


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Following this, the molar activity coefficients of the ions H+ (yH+), NO3- (yNO3-) and OH- (yOH-) are given as:

(

)

(

)

(11)

(

)

(12)

3

)

− D + ε ' H + , NO3− C H +

log y NO −

=

log yOH−

= − D + ε ' H + , OH − CH +

3

(

− D + ε ' H + , NO 3− C NO − + ε ' H + , OH − C OH −

=

log y H +

(10)

For this system, the activity coefficient of OH- (yOH-) is assumed to be unity and ε’(H+, OH-) is not considered as the concentration of OH- prevailing in this system is very low (of the order of 10-13 M). Therefore, only the ion interaction parameter ε’(H+, NO3-) is to be estimated. The molar activity coefficient of undissociated HNO3 (yHNO3) is modeled as

log yHNO 3

2 = a CHNO3 + b CHNO 3

(13)

Here a, b are model parameters which are to be estimated. CHNO3 is the stoichiometric molar concentration of nitric acid which is the total nitric acid present in both dissociated and undissociated forms. Activity of water (aH2O) is calculated using the Pitzer equation in a strong electrolyte/ionic medium (NX) as 24 log a H 2 O = −

2 φ [NX ] 55 .51

(14)

Here Φ is the osmotic coefficient in a strong ionic medium 24 and [NX] is the concentration of the ionic medium NX. Estimation of model parameters Nitric acid dissociation constant, K0HNO3 along with ε’(H+, NO3-), a and b are the parameters which have to be estimated. The experimental data for degree of dissociation of HNO3 in water (αExp) was reported by several authors. 11 -

15, 17, 18

α is defined as



α =

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(15)

CNO− 3

CNO− + CHNO3 3

The dissociation data reported by Davis and Bruin 17 at 25°C over the range 0 – 22 M nitric acid concentration is used in this work for estimating the four parameters. This dissociation data was obtained through partial pressure measurements of nitric acid solutions. A guess value of the four parameters is assumed. The speciation framework

21

is used for

predicting concentrations of all the species at equilibrium. Using the predicted concentrations of species, degree of dissociation of HNO3 in water (αcalc) is obtained using equation (15). The model parameters are estimated by minimizing the objective function (E) E

=

∑ ((α

Exp

− α Calc ) / α Exp )

(16)

2

3. Results and discussion The estimated best fit model parameters are K0HNO3 = 17.12 mol.L-1, a = 0.147 L.mol-1, b = 0.015 L2.mol-2 and ε’ (H+, NO3-) = 0.106 L.mol-1. The fit obtained using these parameters is reasonable (R2 = 0.99) as depicted in Figure 2. The algorithm was also found to be robust and to converge to the same values for different initial guesses of the parameters. The value of K0HNO3 obtained in this work is within the data reported by a majority of researchers in literature. 18


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1 0.8 0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

Exp

Figure 2: Comparison of experimental and calculated extent of dissociation (αExp and αCalc) using best fit parameters. Model Validation The concentration of free nitrate ion in the HNO3 – H2O system at various HNO3 concentrations has been measured by several authors using Raman Spectroscopy. 11 -

15

This data is considered

for validating our approach. The concentrations of free nitrate as a function of total or stoichiometric HNO3 concentration (over the range 0 – 22 M) predicted from this model are in good agreement with experimental data as indicated in Figure 3.



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6 5 4 3 2 1 0

0

5 10 15 20 Stoichiometric HNO 3 in aq, M

25

Figure 3: Variation in free nitrate concentration with stoichiometric HNO3 (M). (__ solid line) predictions from this work, (◊) Exp data Redlich et. al., 11 (∆) Exp data Irish et. al., 12 (○) Exp data Raus et. al., 13 (×) Exp data Hlushak et. al., 14 (□) Exp data Minogue et. al. 15 The mean ionic activity coefficient and activity coefficient of undissociated HNO3 reported in literature 17, 18 are depicted in Figure 4. We see that the molar activity coefficient exponentially increases and becomes unbounded in the concentration range of 0 – 22 M. This is inaccurate as the concentrations of ions (free H+ and NO3-) are low at high concentrations of nitric acid (Figure 3). The corresponding mean ionic activity coefficient must approach unity at high concentrations under the non-symmetric notation. This is not captured by model predictions reported in the literature. 17, 18


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104

102

100

0

5 10 15 20 Stoichiometric HNO3 in aq, M

25

Figure 4: Reported dependence of molar activity coefficients on stoichiometric HNO3 (M). (__ solid line) mean ionic activity coefficient data of Davis and Bruin, 17, (-.-. dashdot line) activity coefficient of HNO3 data of Davis and Bruin, 17, (--- dashed line) mean ionic activity coefficient data of Levanov et. al., 18 , (….. dotted line) activity coefficient of HNO3 data of Levanov et. al. 18 The mean ionic activity coefficient and activity coefficient of undissociated HNO3 predicted by the model developed in the present work are depicted in Figure 5 for the concentration range of 0 – 22 M. It can be seen that the ionic activity coefficients computed by this model follow nonsymmetric notation (for low H+ ion concentrations, yH+ tends to unity). More importantly they do not show an increasing trend as reported in earlier works. 17, 18 The molar activity coefficient of undissociated HNO3 (yHNO3) predicted using our approach follows the non-symmetric notation as depicted in Figure 5. Accordingly, yHNO3 approaches unity as the nitric acid concentration approaches zero. The molar concentration of free H+ and NO3ions approaches zero for pure nitric acid. The K0HNO3 is finite and non-zero in this limit. To ensure this at higher HNO3 concentrations, yHNO3 decreases and approaches zero for pure nitric acid. This behavior has not been reported in the literature to the best of our knowledge.



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2.5 2 1.5 1 0.5 0

0

5 10 15 20 Stoichiometric HNO 3 in aq, M

25

Figure 5: Modelled variation in Molar activity coefficients with stoichiometric HNO3 (M). (__ solid line) mean ionic activity coefficient, (….. dotted line) activity coefficient of HNO3. Conclusions In this work we have proposed a rigorous framework for estimating thermodynamic parameters and demonstrated its application on aqueous solutions of nitric acid. The model is shown to predict the experimental data in the concentration range of 0 – 22 M HNO3. A speciation calculation framework is used for estimating thermodynamic parameters of HNO3 – H2O system. The molar activity coefficients of H+ and NO3- are modeled using SIT Theory. Thermodynamic parameters for aqueous nitric acid solution at 25°C are determined using data from literature. K0HNO3 (25°C) is estimated to be 17.12 mol.L-1. The molar ion interaction parameter, ε’(H+, NO3) estimated to be 0.106 L.mol-1 ensures asymptotic consistency of model in the HNO3 concentration range of 0 – 22 M. The molar activity coefficient of undissociated HNO3 molecules is modeled as log y HNO 3

2 = 0.147 C HNO3 − 0.015 C HNO . 3

It is found that following the speciation framework aided in explicitly accounting for ‘extent of dissociation’ and obtaining consistent results for the entire range of 0 – 22 M nitric acid concentration. The activity coefficients thus modeled do not require any empirical corrections with respect to dissociation as is being practiced.


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While we have used SIT theory in this work to model the activity coefficient of ions present in the solution, it is also possible to use other theories to estimate the interaction parameters in the frame work developed. The results of this work can be useful for modeling nitric acid dissociation in multi-phase operations such as solvent extraction of nitric acid using TBP as solvent. It can also be used for modeling nitric acid dissociation in systems containing other metal nitrate species. Such systems arise in the context of nuclear fuel manufacture and re-processing. References: 1. Berg, J. M.; Veirs, D. K.; Vaughn, R. B.; Cisneros, M. A.; Smith, C. A. Plutonium(IV)

mononitrate and dinitrate complex formation in acid solutions as a function of ionic strength. J. Radioanal. Nucl. Chem. 1998, 235, 25–29. 2. Hesford, E.; McKay, H. A. C. THE EXTRACTION OF NITRATES BY TRI-nBUTYLPHOSPHATE (TBP) - Part 3. Trans. Faraday Soc., 1958, 54, 573-586. 3. Mishra, S.; Ganesh, S.; Velavendan, P.; Pandey, N. K.; Mallika, C.; Mudali, U. K.; Natarajan, R. Thermodynamics of Solubility of Tri-n-butyl Phosphate in Nitric Acid Solutions. Advanced Chemical Engineering Research 2013, 2 (3), 55 - 60. 4. Neck, V.; Altmaier. M; Fanghänel. Th. Ion interaction (SIT) coefficients for the Th4+ ion and trace activity coefficients in NaClO4, NaNO3 and NaCl solution determined by solvent extraction with TBP. Radiochim. Acta 2006, 94, 501 - 507. 5. Tkac, P.; Paulenova, A.; Vandegrift, G. F.; Krebs, J. F. Modeling of Pu(IV) Extraction from Acidic Nitrate Media by Tri-n-butyl Phosphate. J. Chem. Eng. Data 2009, 54, 1967–1974. 6. Davis, W. Jr. Thermodynamics of Extraction of Nitric Acid by Tri-n-Butyl Phosphate– Hydrocarbon Diluent Solutions:I. Distribution Studies with TBP in Amsco 125-82 at Intermediate and Low Acidities. Nucl. Sci. Engg. 1962, 14, 159-168. 7. Chaiko, D. J.; Vandegrift, G. F. A thermodynamic model of nitric acid extraction by tri-n butyl phosphate. Nucl Technol 1988, 82, 52 - 55. 8. Sarsfield, M. J.; Sims, H. E.; Taylor, R. J. Modelling Np(IV) Solvent Extraction between 30% Tri-Butyl Phosphate and Nitric Acid in the Presence of Simple Hydroxamic Acids. Solvent Extraction and Ion Exchange 2011, 29, 49 - 71.



9. Brimblecombe, P.; Clegg, S. L. The solubility and behaviour of acid gases in the marine aerosol. Journal of Atmospheric Chemistry 1988, 7 (1), 1 - 18. 10. Bianco, R.; Wang, S.; Hynes, J. T. Theoretical Study of the Dissociation of Nitric Acid at a Model Aqueous Surface. J. Phys. Chem. A, 2007, 111 (43), 11033 - 11042. 11. Redlich, O.; Duerst, R. W.; Merbach. A. Ionization of Strong Electrolytes. XI. The Molecular States of Nitric Acid and Perchloric Acid. The Journal of Chemical Physics 1968, 49, 2986. 12. Irish, D. E.; Puzic, O. Raman Spectral Study of the Constitution and Equilibria of Nitric Acid and d-Nitric Acid. Journal of Solution Chemistry 1981, 10 (6), 377 - 393. 13. Ruas, A.; Pochon, P.; Simoninb, J.; Moisya, P. Nitric acid: modeling osmotic coefficients and acid–base dissociation using the BIMSA theory. Dalton Trans. 2010, 39, 10148-10153. 14. Hlushak, S.; Simonin, J. P.; Sio, S. D.; Bernard, O.; Ruas, A.; Pochon, P.; Janb, S. Speciation in aqueous solutions of nitric acid. Dalton Trans. 2013, 42, 2853-2860. 15. Minogue, N.; Riordan, E.; Sodeau, J. R. Raman Spectroscopy as a Probe of LowTemperature Ionic Speciation in Nitric and Sulfuric Acid Stratospheric Mimic Systems. J. Phys. Chem. A 2003, 107, 4436-4444. 16. Hamer, W. J.; Wu, Y. C. Osmotic coefficients and mean activity coefficients of uni‐ univalent electrolytes in water at 25° C. J. Phys. Chem. Ref. Data 1, 1047 (1972) 1972, 1 (4), 1047 - 1100. 17. Davis, W. Jr.; Bruin, H. J. D. New activity coefficients of 0–100 per cent aqueous nitric acid. J. Inorg. Nucl. Chem 1964, 26 (6), 1069 - 1083. 18. Levanov, A. V.; Isaikina, O. Y.; Lunin, V. V. Dissociation Constant of Nitric Acid. Russian Journal of Physical Chemistry A 2017, 91 (7), 1221–1228. 19. Bart, H. Reactive Extraction; Springer, 2001. 20. Sandler, S. I. Chemical, Biochemical, and Engineering Thermodynamics, 4th ed.; John Wiley and Sons, Inc.: United States of America, 2006. 21. Ravi Kanth, M. V. S. R.; Pushpavanam, S.; Narasimhan, S.; Murty, B. N. A Robust and Efficient Algorithm for Computing Reactive Equilibria in Single and Multiphase Systems. Ind. Eng. Chem. Res. 2014, 53 (39), 15278 – 15286. 22. Brezonik, P. L.; Arnold, W. A. Water Chemistry: An Introduction to the Chemistry of Natural and Engineered Aquatic systems; Oxford University press, 2011. 16 ACS Paragon Plus Environment

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Development of a Thermodynamic Model Using a Speciation Framework

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