Is a Solution the System Chemistry

Annual Reports in Computational Chemistry

David M. Rogers , ... Susan B. Rempe , in Annual Reports in Computational Chemistry, 2012

Abstract

Solution equilibria are at the core of solvent-catalyzed reactions, solute separations, drug delivery, vapor partitioning and interfacial phenomena. Molecular simulation using thermodynamic integration or perturbation theory allows the calculation of these equilibria from parameterized force field models; however, the statistical many-body nature of solution environments inevitably complicates molecular interpretations of these phenomena. If our goal is molecular understanding in addition to prediction, then the statistical thermodynamic theories designed for mechanistic insight from structural analyses are especially important. In this report, we survey recent advances in the thermodynamic analysis of rigorous local structural models based on chemical structure.

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Surface Complexation Modelling

J.A. Mielczarski , O.S. Pokrovsky , in Interface Science and Technology, 2006

3. Calculations

Homogeneous solution equilibria as well as surface speciation were calculated for each solution composition in the system SiO₂(s) – H₂O – HNO₃ – NaNO₃ using the MINTEQA2 code and MINTEQA2 thermodynamic database [54]. This program combines surface equilibria, homogeneous solution equilibria, and mass balance calculation. The set of equations obtained is solved iteratively by the Newton-Raphson method.

The activity coefficients of surface species were set equal to 1. The activity coefficients of free ions and charged complexes were calculated using the Davies equation, which is standard in MINTEQA2. Although the Davies equation is usually applied for low ionic strength media, its validity for predicting the activity coefficients in solutions of high ionic strength (i.e., seawater at 0.7 m) has been demonstrated for divalent ions [55, 56].

The standard state for the aqueous species is the hypothetical 1 molar solution. The standard state chosen for surface species is a concentration of 1 molar for the adsorbed species and zero surface potential. At our experimental conditions (0 < pH < 3 and [Si < 30 mg/L), all solutions were strongly undersaturated with respect to silica solid phases (i.e., log Ω < 0.2 where Ω is the ratio of ion activity product in solution to ion activity product in the equilibrium with solid phase, K^o _sp) and all silicon was present in the form of non-polymerized Si(OH)₄ ^o species.

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FLOTATION | Reagent Adsorption On Phosphates

P. Somasundaran , L. Zhang , in Encyclopedia of Separation Science, 2000

Water Chemistry of Flotation Reagents

Long chain fatty acids such as oleic acid are among the commonly used reagents for the flotation of oxides, silicates and salt-type minerals. Flotation of these minerals using fatty acids is affected greatly by solution properties such as pH, since weakly acidic fatty acids undergo association interactions that can influence their adsorption and flotation properties. For example, oleic acid species will undergo dissociation to form ions (Ol⁻) at high pH values and exist as neutral molecules (HOI) at low pH value. In the intermediate region, the ionic and the neutral molecular species can associate to form ion–molecule complexes ((Ol)₂H⁻). As the surfactant concentration is increased, micellization or precipitation of the surfactant can occur in the solution. In addition, surfactant species can associate to form other aggregates such as the dimer ( ${\text{Ol}}_2^{2-}$ ) in premicellar solutions. Also, long chain fatty acids such as oleic acid have very limited solubility, which is a sensitive function of pH. The pH of precipitation of oleic acid calculated as a function of total oleate is shown in Figure 1 .

Figure 1. pH of oleic acid precipitation. (From Morgan LJ, Ananthapadmanabhan KP and Somasundaran P (1986) Oleate adsorption on hematite: problem and methods. International Journal of Mineral Processing 18: 39. Copyright: Elsevier Scilence.)

The solution equilibria of oleic acid (HOl) are expressed as below:

HOl(liquid) = HOl(aq)

pK sol = 7.6   (K sol: solubility product)

HOl(aq) = H+ + Ol−

pK a = 4.95   (K a: acid dissociation constant)

${\text{2Ol}}^{-}={\left(\text{Ol}\right)}_2^{2-}$

pK d = −3.7   (K d: dimerization constant)

HOl + Ol− = (Ol)2H−

pK ad = − 5.25   (Kad: acid–soap formation constant)

The species distribution of oleic acid as a function of pH based on the above equilibria at a given concentration is shown in Figure 2 . It can be seen from this figure that:

Figure 2. Oleate species distribution as a function of pH. Total oleate concentration = 3 × 10⁻⁵ mol L⁻¹. (From Ananthapadmanabhan KP and Somasundaran P (1980) Oleate chemistry and hematite flotation. In: Yarar B and Spottiswood DJ (eds) Interfacial Phenomena in Mineral Processing, p. 207. New York: Engineering Foundation.)

1.

The pH of the precipitation of oleic acid at the given concentration is 7.45.

2.

The activities of oleic monomer and dimer remain almost constant above the precipitation pH and decrease sharply below it.

3.

The activity of the acid–soap (Ol)₂H⁻ exhibits a maximum in the neutral pH range.

The surface activities of the various surfactant species can be markedly different from each other. It has been estimated that the surface activity of the acid–soap (Ol)₂H⁻ is five orders of magnitude higher than that of the neutral molecule (HOl) and about seven orders of magnitude higher than that of the neutral molecule (HOl) and about seven orders of magnitude higher than that of the oleate monomer Ol⁻.

The existence of salt will also affect the surfactant–solution equilibria by changing the surface activities of the various surfactant species, the critical micelle concentration and the solubility of the surfactant, and the solvent properties of the solution.

It is clear that, to understand the adsorption of reagents on solids, the effects of concentration, pH, ionic strength and activities of the various possible reagent species on the adsorption process need to be taken into account.

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ELEMENTAL SPECIATION | Waters, Sediments, and Soils

G.E. Batley , S.C. Apte , in Encyclopedia of Analytical Science (Second Edition), 2005

Modeling Speciation in Waters

There are a number of computer codes available for calculating solution equilibria and the speciation of trace metals in natural waters. Programs such as MINEQL and MINTEQA2 are now widely used by aquatic scientists. Similar models (e.g., GEOCHEM and SOILCHEM) are used by soil scientists. The effective use of such computational models requires critical knowledge of key reactions and judicious selection of the appropriate stability constants. Recent research endeavors in this area have been directed toward describing metal interactions with natural organic matter (NOM). This is an immense challenge owing to the polydisperse, heterogeneous nature of NOM. Biotic ligand models (BLM) have recently been developed which predict acute metal toxicity to some aquatic organisms. BLM models assume that metal toxicity is related to free metal ion binding to a specific ligand at the cell surface of unicellular organisms or at the gill surface in fish. The models take into account proton, calcium, and magnesium ion competition for the binding sites and assume a simple relationship between the bound metal concentration and toxicity.

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Elemental Speciation | Waters, Sediments, and Soils☆

Graeme E. Batley , Simon C. Apte , in Encyclopedia of Analytical Science (Third Edition), 2019

Modeling Speciation in Waters

There are a number of computer codes available for calculating solution equilibria and the speciation of trace metals in natural waters. ^{3 , 13} Programs such as Visual MINTEQ, PHREEQC and MINEQL   + are now widely used. The effective use of such computational models requires critical knowledge of key reactions and judicious selection of the appropriate stability constants. Recent research endeavors in this area have been directed toward describing metal interactions with natural organic matter (NOM), with models such as NICA-Donnan and Windermere Humic Acid (WHAM VI). This is a significant challenge owing to the polydisperse, heterogeneous nature of NOM. The biotic ligand model (BLM) has been developed to predict acute and chronic metal toxicity to some aquatic organisms. The model assumes that metal toxicity is related to free metal ion binding to a specific ligand at the cell surface of unicellular organisms or at the gill surface in fish. It takes into account proton, calcium, and magnesium ion competition for the binding sites and assumes a simple relationship between the bound metal concentration and toxicity.

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Reactions of ions and radical ions

E.T. Denisov , ... G.I. Likhtenshtein , in Chemical Kinetics: Fundamentals and New Developments, 2003

8.8.1 Method of temperature jump

The method is based on the following principle. The temperature of the equilibrium solution is increased very quickly, which removes the system from equilibrium. The kinetics of establishment of the new equilibrium state is monitored by high-speed spectrophotometry. The corresponding kinetic characteristics are calculated from experimental data. For the one-stage reversible reaction, the temperature change ΔT results in the corresponding change in the equilibrium concentration of the ith component Δc_i

(8.22) $\Delta c_i={\left(\frac{\partial c_i}{\partial \text{In}K}\right)}_{T,r}\frac{\Delta H}{R{T}^2}\Delta T$

The first technique for studying equilibria by this method was created by M. Eigen in 1959. The temperature increase is achieved most often by the discharge of a high-voltage capacitor through an aqueous solution. The temperature increases due to the friction of ions moving in the electric field. The rate of the temperature increase depends on the RC product (where R is the resistance of the cell, and C is the capacitance of the capacitor) and is described by the formula

(8.23) $\Delta T=\Delta T_{\infty }[1-\exp (-2t/RC\left)\right]$

The temperature jump technique is characterized by the following parameters (Fig. 8.7): the voltage on the capacitor plates is 10—100 kV, RC = 1÷10 ms, the cell volume is 0.2÷20 cm³, and ΔT = 6÷8 K. High-speed spectrophotometry is used to monitor a change in the reactant concentration. The method enables one to measure relaxation times from 1 to 10⁻⁶ s. High requirements are imposed on the reaction cell in the temperature jump technique. The cell must withstand the high-voltage discharge, provide the uniform heating of the liquid, and allow one to use an optical equipment for spectral monitoring.

Fig. 8.7. Scheme of the temperature jump technique: 1, high-voltage generator; 2, kilovoltmeter and trigger; 3, monochromator; 7, photoamplifier; and 8, differential amplifier and oscillograph.

The reaction cell is usually prepared from Plexiglas, electrodes are brass, and the light is transmitted through quartz rods. Uniform heating is provided by parallel electrodes embracing the reaction cell. Fast heating is achieved by a pulse generator of microwaves. Due to electric relaxation, the liquid absorbs microwaves with a certain frequency and is rapidly heated. Water absorbs the microwave radiation at a frequency of 10¹⁰ s⁻¹, and this allows one to rise the temperature by 1 K for 1 ms. Absorption spectrophotometry is the most popular method of detection.

For a single relaxation process, the kinetics of the ith component is described by an exponential law

$c_{io}-c_i=(c_{io}-c_{i\infty })[1-\exp (-t/\tau \left)\right]$

where τ is the relaxation time.

For small changes in the concentration

$\Delta I={\epsilon }_{i}l\Delta c_i$

where I is the change in intensity of the light absorption, l is the reactor length, and i is the molar absorption coefficient.

Data are processed on a computer. The method is widely used for studying the kinetics of the acid-base equilibrium, intermolecular transfer, formation of metal complexes, electron transfer reactions, and enzymatic catalysis. The method measures rate constants of up to 10¹¹ 1/(mol s).

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Studying molecular-scale protein–surface interactions in biomaterials

In Characterization of Biomaterials, 2013

6.4 Models of protein adsorption and adsorption isotherms

In studying protein adsorption, it is common to express adsorbed protein mass on a solid surface as a function of equilibrium solution concentration of protein. Parameters of protein adsorption can then be determined by fitting the data into the classic Langmuir model, which was originally used to describe gas adsorption. ^{6 , 20} Two characteristic features of protein adsorption data are commonly observed: high mass of surface-bound protein at small bulk protein concentration, and irreversible adsorption. This implies that proteins show preferred binding to interfaces and the protein/adsorbent interaction is a fairly complex mechanism. Other models have also been proposed to account for the protein adsorption data, which largely depends on the protein and surface under study.

Orientation of adsorbed protein has a significant effect on the amount of adsorbed protein, which varies from protein to protein. Protein adsorption does not normally reach monolayer coverage, in part because of the protein–protein interaction of adsorbed proteins. Incomplete coverage also means that protein molecules have space to unfold and spread once they reach the surface.

Over the years, principles developed for polymer adsorption have been used to describe protein adsorption. ²⁸ For instance, if proteins are considered as heterogeneous polymers consisting of segments, the protein–adsorbent interaction can be thought of as multiple segmental attachment. Protein adsorption can only occur when the energy gain of protein/adsorbent is outweighed by that of adsorbent/solvent. In comparison, the driving force for protein desorption requires the simultaneous detachment of all the bound segments. This is highly improbable, which explains why dilution per se is not strong enough to induce protein desorption. In fact, adsorbed proteins make a large number of surface contacts. For example, infrared spectroscopy reveals that up to 20% of fibrinogen backbone polypeptide chain is bound to the adsorbent. Although polymer theories can explain protein adsorption to some extent, the overall results are less than satisfactory, due partly to the fact that the chemical and physical properties of proteins are vastly different and systematic prediction is quite difficult, compared to that of polymers. Besides, protein structures vary a lot more from molecule to molecule than those of polymers.

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Simple models for nonlinear states of double-helix DNA

Raimo A. Lohikoski , Aatto Laaksonen , in Modern Methods for Theoretical Physical Chemistry of Biopolymers, 2006

17.2 ENGLANDER MODEL - BASE TWIST AROUND THE BACKBONE

The history of simple nonlinear DNA models goes back to the year 1980 and the hydrogen–deuterium exchange measurements of dsDNA molecules in an equilibrium solution performed by Englander and co-workers [3]. These experiments showed a dynamical existence of open states in the helical structure of dsDNA, where the open states were moving from site to site. The open state was a base pair or a sequence of adjacent base pairs where the hydrogen bonding in the base pair is broken or at least stretched so much that the canonical dsDNA structure could not exist anymore.

Owing to lack of direct information about the conformation, it was assumed that several adjacent base pairs could be simultaneously open. Further it was assumed that the open segment existed as a mobile unit free to move along the double helix. Based on the experimental data a guesstimate was made that the average length of the opened regime was roughly ten base pairs, leading to a proposition of thermal soliton excitations in the dsDNA. A soliton is a 'packet' which is usually related to a vibrational motion. It is formed as a combination of both linear and nonlinear properties of a system, and has the characteristic property of being radiationless or compact coherent entities over long periods of time or large distances. By definition, a slightly milder form of a soliton is a solitary wave which is allowed to deform slowly.

In a biological context, where large amplitude motions are typical, and consequently nonlinear interactions effective, solitons or solitary excitations were considered as a realistic possibility. One of the first proposed models for the open conformation of dsDNA is shown in Fig. 17.1. It consists of two parallel rods having pendula (bases) pointing towards each other in a minimum energy configuration. The assumed excitation was thought to be a twist or rotation of the bases around one of the two backbone strands. Physically this model is a pure mechanical device. Each base is a mass point attached via a rod to the related backbone. The backbone itself can be described by a spring resisting the twist. Base–base stacking interactions in the helix direction and a limited torsional movement around the P–O bonds are described by a (stiff) torsional elasticity constant K. Short-range attractive interactions come from base-pair hydrogen bonds and solvation. Owing to a mathematical convenience this potential was represented in a gravitational form V (ϕ_n ) = mgr(1 – cos ϕ_n ), where ϕ_n is a deviation of the nth rod from its equilibrium angle position. The Hamiltonian for this first DNA model was given as:

Fig. 17.1.. The Englander model. (A) The equilibrium state of a dsDNA. The sugar–phosphate backbones are shown by two horizontal springs having pendula to describe the bases. (B) A single strand, capable or torsional oscillations is shown with an idealistic soliton excitation.

(1) $H=\underset{n}{\Sigma }\left\{\frac{1}{2}m{r}^2{\varphi }_n^2+\frac{1}{2}K{\left({\varphi }_n-{\varphi }_{n-1}\right)}^2+\mathrm{mgr}\left(1-\text{cos}{\varphi }_n\right)\right\}$

This Hamiltonian can be treated following the normal procedure of mechanics [4] by deriving the equations of motion and solving them. However, in attempts to solve the equations these early and simple modeling works relied on a fairly drastic approximation that the motion from one base pair to the next was assumed to be slow enough to allow the use of the continuum limit approximation. This then made it possible to obtain analytical mathematical solutions, often even with other simplifying pre-suppositions.

The Englander model is another case of the sine-Gordon model, which is one of the well-known nonlinear equations producing soliton solutions. Because this is a very crude model in describing the actual dsDNA molecule in a given state we will not go into the details of the model but refer to the original paper by Englander and co-workers [3].

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Sediments, Diagenesis and Sedimentary Rocks

R.S. Arvidson , J.W. Morse , in Treatise on Geochemistry (Second Edition), 2014

9.3.2.2 Carbonic Acid System and Basic Solution Equilibria

It is useful at this point to make some brief remarks regarding the carbonic acid system as it applies to calculations involving carbonate mineral–solution equilibria. Calculation of the extent of over- or undersaturation of a fluid with respect to, for example, calcite involves first specifying the reaction:

[1] ${\mathrm{CaCO}}_3\left(\mathrm{s}\right)\stackrel{K_{\mathrm{cc}}}{\rightleftharpoons }{\mathrm{Ca}}^{2+}{+\mathrm{CO}}_3^{2-}$

At equilibrium, the ratio of activities of products over reactants is equivalent to the equilibrium constant for calcite, K _cc:

[2] $K_{\mathrm{cc}}=\frac{a_{{\mathrm{Ca}}^{2+}}{a}_{{\mathrm{CO}}_{{}_3}^{2-}}}{a_{{\mathrm{CaCO}}_3}}$

Because the pure solid has unit activity, the extent of disequilibrium can be evaluated by comparing the expression in eqn [2] with that prevailing under arbitrary conditions (*), using Ω to represent this ratio:

$\begin{array}{l}\mathit{\Omega }\equiv \frac{a_{{\mathrm{Ca}}^{2+}}^{*}{a}_{{\mathrm{CO}}_3^{2-}}^{*}}{K_{\mathrm{cal}}}\\ ={\mathrm{e}}^{\mathrm{\Delta }{G}_{\mathrm{r}}/\mathit{RT}}\end{array}$

where ΔG _r denotes the Gibbs free energy computed for the reaction as written in eqn [1]. Thus Ω  =   1 and ΔG _r  =   0 denotes saturation, Ω  >   1 and ΔG _r  >   0 in oversaturated solutions (i.e., the dissolution as written in eqn [1] is not favored thermodynamically), and Ω  <   1 and ΔG _r  <   0 in undersaturated solutions. Because we are dealing with systems other than those at infinite dilution, we need to compute activity coefficients on some scale. Activity coefficients are complex (and model-dependent) functions of ionic strength and solution composition, and are described in detail in Morse and Mackenzie (1990). Two basic approaches are employed: ion association and ion interaction models. An ion association model explicitly recognizes the importance of ionic complexes and pairs in determining the availability of free species (e.g., Garrels and Thompson, 1962). The assumption is that the stoichiometric molality of a component i in an aqueous solution can be represented as the sum of free ion and N_j complexed species molalities:

[3] $m_{\mathrm{T}}=m_i+{\displaystyle \sum _j^{N_j}{m}_{\mathit{ij}}^{*}}$

Computationally, this approach is comparable to the search for a 'basis' set that is the minimal number of components required to fully describe the composition of the system (e.g., see Reed, 1982). Free species and complexed species (i.e., those dependent on the availability of the free ion) can thus appear as linear combinations of these basis components through their reaction stoichiometry. As a simple example, the aqueous species OH⁻ can be represented as an aqueous complex formed by a linear combination of basis components H⁺ and H₂O, with stoichiometric coefficients of −   1 and 1, respectively:

${\mathrm{OH}}^{-}\stackrel{K_{{\mathrm{OH}}^{-}}}{\rightleftharpoons }-{\mathrm{H}}^{+}+{\mathrm{H}}_2\mathrm{O}$

By mass action,

$\begin{array}{l}{m}_{{\mathrm{OH}}^{-}}=\frac{a_{{\mathrm{H}}_2\mathrm{O}}{a}_{{}^{{\mathrm{H}}^{+}}}^{-1}}{{\gamma }_{{\mathrm{OH}}^{-}}{k}_{{\mathrm{OH}}^{-}}}\\ =\frac{a_{{\mathrm{H}}_2\mathrm{O}}}{{\gamma }_{{\mathrm{OH}}^{-}}{a}_{{\mathrm{H}}^{+}}}{K}_{{\mathrm{H}}_2\mathrm{O}}\end{array}$

Simple rearrangement thus shows K _OH ⁻ to be the reciprocal of the dissociation constant for water $(K_{{\mathrm{H}}_2\mathrm{O}})$ . Similar algebraic manipulations allow a set of dissociation constants to be recomputed using a compact system of basis components. The molality m_j of a dependent, nonbasis (ion pair or complex) species j may be represented by

$m_j=\frac{1}{{\left(\mathit{\gamma K}\right)}_j}{\displaystyle \prod _i^{N_i}{a}_i^{{\mathbf{A}}_{\mathit{ij}}}}$

where A _ij is the matrix of stoichiometric coefficients for each 'dissociation' reaction, a_i is the activity of basis component i, and γ_j and K_j denote the relevant activity coefficient and equilibrium constant, respectively.

The ion association approach has limitations: its use of single ion activity coefficients, typically computed using an extended Debye–Huckel formalism, makes its application to concentrated, high ionic strength solutions problematic. Uncertainties in the temperature and pressure dependencies of dissociation constants for aqueous complexes also limit its application at elevated temperatures and pressures. The model's principal strength derives from its explicit, straightforward representation of solution components, as it allows insight into the concentration and activity of the individual species that can potentially participate in mineral reactions. However, as electrolyte concentrations increase, the specific binary or ternary 'interactions' of ions become more important than their simple contribution to ionic strength. In addition, as electrolytes reach sufficient concentration, the issue of whether two ions are free or complexed (or simply close enough to have a mutual effect on one another) may become somewhat arbitrary. These conditions are more adequately described by semi-empirical ion interaction models, of which the Pitzer (1973) model is the most extensive and well-supported one. In this approach, the total Gibbs free energy of mixing solutes and solvent water is represented as a virial expansion in ionic strength, parameterized by binary and ternary ion interaction terms. To be clear, the Pitzer model does not necessarily deny that ion association or ion pair formation takes place, but simply provides for an alternative means of dealing with this property (see, e.g., the recent review in Marcus and Hefter, 2006). The elegance of Pitzer's approach is that interaction parameters can be measured separately in otherwise simple electrolytes, and then used to compute equilibria in complex, multicomponent solutions. Activity and osmotic coefficients are then expressed in terms of these (temperature- and ionic strength-dependent) parameters as summations over all possible interactions. The Pitzer model is now the basis for many calculations involving concentrated brines and has been extended to higher temperatures and pressures as well. Its application to the carbonic acid system in seawater was importantly extended for the carbon system in seawater by He and Morse (1993).

The union or intersection of these two very different approaches to activity coefficient (γ) representation is resolved by their relationship to thermodynamic activity (a), combining the expression for total molality m _T from eqn [3]:

$\begin{array}{l}a=\{\begin{array}{l}m\cdot \gamma \\ m_{\text{T}}\cdot {\gamma }_{\text{T}}\end{array}\\ {\gamma }_{\mathrm{T}}=\gamma \frac{m}{m+{\displaystyle {\sum }_j^{N_j}{m}_j^{*}}}\end{array}$

where γ _T denotes the total activity coefficient.

Carbonic acid (H₂CO₃) is a weak acid, and the carbonic acid system includes CO₂ as a gas phase. The exchange of CO₂ gas with the solution will cause the hydrogen ion concentration (as well as total dissolved carbon) to vary by the following stepwise reactions (all species are aqueous unless otherwise indicated, and equilibrium constants are given at 25   ᵒC):

$\begin{array}{l}{\mathrm{CO}}_2\left(\mathrm{g}\right)\stackrel{\mathrm{p}{K}_{\mathrm{h}}=1.468}{\rightleftharpoons }{\mathrm{CO}}_2\\ {\mathrm{CO}}_2{+\mathrm{H}}_2\mathrm{O}\stackrel{K_0}{\rightleftharpoons }{\mathrm{H}}_2{\mathrm{CO}}_3\\ {\mathrm{H}}_2{\mathrm{CO}}_3\stackrel{K_1}{\rightleftharpoons }{\mathrm{HCO}}_3^{-}+{\mathrm{H}}^{+}\\ {\mathrm{H}}_2{\mathrm{CO}}_3^{-}\stackrel{\mathrm{p}{K}_2=10.329}{\rightleftharpoons }{\mathrm{CO}}_3^{2-}+{\mathrm{H}}^{+}\end{array}$

Because true H₂CO₃ is quite small compared to aqueous CO₂, in some texts they are lumped together as H₂CO₃*. In addition, the true ionization constant for H₂CO₃ is not well known (Adamczyk et al., 2009; Berg and Patterson, 1953; Wissbrun et al., 1954), and thus the two reactions involving H₂CO₃ are commonly combined and collapsed to give

${\mathrm{CO}}_2{+\mathrm{H}}_2\mathrm{O}\stackrel{\mathrm{p}{K}_{1^{\prime }}=6.325}{\rightleftharpoons }{\mathrm{HCO}}_3^{-}+{\mathrm{H}}^{+}$

If the carbonic acid system lacked a gas phase, then specifying the total molal concentration of dissolved carbon would sufficiently describe the system. However, the potential for CO₂ exchange requires that we furnish an additional constraint in addition to dissolved carbon concentration. In practice, any pair of four principal measurements can be made to constrain this system: pH, total alkalinity (the sum of all titratable bases in solution in equivalents), total dissolved carbon (DIC   ≡   CO₂  +   H₂CO₃  +   HCO₃ ⁻  +   CO₃ ^{2   −}), or partial pressure of carbon dioxide (pCO₂). Algebraic relations involving these terms are given in Morse and Mackenzie (1990). In saturation state calculations, activities computed as shown above are relevant for the bulk solution; at the mineral surface itself, the distribution of surface and near-surface species reflects both hydration and hydrolysis reactions as well as the kinetics of the dissolving or precipitating mineral itself, and thus these may differ significantly from the bulk solution.

The above description is relevant to arbitrary solutions. In seawater, concentration ratios of major ions vary in a more or less constant fashion with respect to salinity, leading to the concept of a 'constant ionic medium.' Exceptions to this rule occur, but these are associated with either some sort of closure of the system with respect to the open marine environment (e.g., precipitation losses accompanying evaporation in restricted lagoons and the evolution of pore waters during diagenesis) or, alternatively, proximity to riverine sources, hydrothermal vents, etc. Thus, in marine systems, a far more specialized scheme has been adopted to describe the carbonic acid system, one that is parameterized entirely on the basis of salinity (as well as temperature and pressure or depth). This 'seawater' system discards the system of explicit activity coefficients and thermodynamic constants described above and substitutes either 'stoichiometric' or 'apparent 'constants. Because the evolution of sophisticated thermodynamic descriptions of seawater is relatively recent (facilitated by the widespread proliferation of computer programs for equilibrium calculations), this seawater system has been in use for over 40 years. The thermodynamic constants (K) described above are replaced with either apparent or stoichiometric equivalents, typically denoted as K′ or K*. These constants are 'apparent' because all of the complexation and departure from ideality described by the activity coefficients are completely assimilated within the terms themselves. In place of the formal thermodynamic concentration scales (e.g., molality, mol per kg-solvent), the molinity scale (mol per kg-seawater) is used. Although this system affords ease of use, it sheds no light on the details of the solution chemistry, and thus on interactions of seawater with carbonate mineral surfaces.

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Lattice Boltzmann method: application for MHD nanofluid hydrothermal behavior

Mohsen Sheikholeslami , Davood Domairry Ganji , in External Magnetic Field Effects on Hydrothermal Treatment of Nanofluid, 2016

4.3.1 BGK model equation

The collision operator Ω _k mentioned earlier is in general a complex nonlinear integral. The idea is to linearize the collision term around its local equilibrium solution. Ω _k is often replaced by the so-called BGK (Bhatnagar, Gross, Krook, 1954 [21]) collision operator

(4.17) ${\Omega }_k=-\frac{1}{\tau }\left(f_k-f_k^{\text{eq}}\right)$

where τ is the rate of relaxation toward local equilibrium and $f_k^{\text{eq}}$ is the equilibrium distribution function. The collision operator is equivalent to the viscous term in the Navier–Stokes equation. The rate of relaxation τ is constant for all ks. This scheme is called the single-time relaxation scheme. All nodes relax with the same time scale τ [20]. The Navier–Stokes equation can be recovered from a Chapman–Enskog expansion [18]. This gives the kinematic viscosity υ in terms of the single relaxation time τ.

(4.18) $\upsilon =\left(\tau -0.5\Delta t\right)c_{\text{s}}^2$

where $c_{\text{s}}=\frac{1}{\sqrt{3}}\frac{\Delta x}{\Delta t}$ is the sound speed in the lattice, Δx is the lattice spacing, and Δt is the time increment. Usually both Δx and Δt are set to unity. After introducing the BGK approximation, the Boltzmann equation (without external forces) can be written as

(4.19) $\frac{\partial f}{\partial t}+\overrightarrow{e}.\overrightarrow{\nabla }f=-\frac{1}{\tau }\left(f-f^{eq}\right)$

The form of the Boltzmann equation in a specific direction would be

(4.20) $\frac{\partial f_k}{\partial t}+\overrightarrow{e_k}.\overrightarrow{\nabla }{f}_k=-\frac{1}{\tau }\left(f_k-f_k^{\text{eq}}\right)$

We could say that this equation is the heart of LBM. It is the most popular kinetic model and replaces the Navier–Stokes equation in CFD simulations. The completely discretized equation with the time step Δt and space step $\Delta \overrightarrow{x_k}=\overrightarrow{e_k}\Delta t$ is

(4.21) $\begin{array}{ll}\text{Collision}\text{step}:\hfill & \overline{f_k}\left(\overrightarrow{x_k},t+\Delta t\right)=f_k\left(\overrightarrow{x_k},t\right)-\frac{\Delta t}{\tau }\left(f_k-f_k^{\text{eq}}\right)\hfill \\ \text{Streaming}\text{step}:\hfill & f_k\left(\overrightarrow{x_k}+\overrightarrow{e_k}\Delta t,t+\Delta t\right)=\overline{f_k}\left(\overrightarrow{x_k},t+\Delta t\right)\hfill \end{array}$

Collision is not defined explicitly any more as in LGA. In the LBM, collision of the fluid particles is considered as relaxation toward a local equilibrium [22]. During the computation, there is no need to store both $f_k\left(\overrightarrow{x_k},t+\Delta t\right)$ and $f_k\left(\overrightarrow{x_k},t\right)$ [23] (Table 4.1).

Table 4.1. Comparison Between the Navier–Stokes Equation and Lattice Boltzmann Equation

Navier–Stokes equation $\rho \left(\frac{\partial \overrightarrow{u}}{\partial t}+\left(\overrightarrow{u}.\overrightarrow{\nabla }\right)\overrightarrow{u}\right)=-\overrightarrow{\nabla }p+\mu {\nabla }^2\overrightarrow{u}$	Lattice Boltzmann equation $\frac{\partial f}{\partial t}+\overrightarrow{e}.\overrightarrow{\nabla }f=-\frac{1}{\tau }\left(f_k-f_k^{\text{eq}}\right)$
Second-order PDE	First-order PDE
Need to treat the nonlinear convective term $\overrightarrow{u}.\overrightarrow{\nabla }\overrightarrow{u}$	Avoids convective term; convection becomes simple advection
Need to solve the Poisson equation for the pressure p	Pressure p is obtained from equation of state

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Is a Solution the System Chemistry

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