4. EQUILIBRIUM CHEMICAL MODELING

Everything that can be invented has been invented

-Charles Duell, U.S. Patent Officer, 1899

강의 내용 4.1 서론 4.2 평형 원칙 4.2.1 물의 이온화 4.2.2 산과 염기 4.2.3 열역학적 자료에 따른 평형상수 4.2.4 평형상수에 대한 온도와 압력 보정 4.2.5 활성도 보정 4.2.6 화학평형 문제의 풀이 4.3 수치해석 방법 4.4 표면착화물화와 흡착

4.4.1 Kd 흡착모델 4.4.2 Langmuir 흡착모델 4.4.3 Freundlich 흡착모델 4.4.4 이온교환모델 4.4.5 확산이중층모델 4.4.6 일정 정전용량 모델

4.5 평형모델에서의 침전과 용해 4.6 평형모델에서의 산화 환원 반응 4.7 컴퓨터 모델 과제물

4.1 INTRODUCTION

Reaction kinetics are sometimes fast relative to transport processes. In these cases, it is valid to assume that the system is at chemical equilibrium and to calculate the concentrations of all species in solution.

Even when reactions are slow, sometimes we may assume that chemical equilibrium applies as a limiting case . Acid-base and complexation reactions are usually fast relative to tran

sport reactions. Dissolution and precipitation reactions are variable (sometimes they a

re quite slow). Redox reactions are generally slow.

In mass balance equation, the "Reactions" term refers to chemical, physical, or biological kinetics that take place within the control volume, representing a reaction rate times the volume.

Figure 4.1 Schematic flowchart of a chemical equilibrium submodel within a fate and transport mass balance model.

4.2 EQUILIBRIUM PRINCIPLES4.2.1 Ionization of Water

Water undergoes self-ionization to form hydrated protons and hydroxide ions

Table 4.1 Ion Product of Water

According to the mass law, we can express the ion product of water at chemical equilibrium with an equilibrium constant, Kw

Its value at different temperatures, which can be determined from the following equation, is given in Table 4.1

A combination pH electrode measures the pH relative to the junction potential of {H3O+}.

Water can act as either an acid or a base depending on the reactants.

4.2.2 Acids and Bases An acid can react with water or a base to form an acid and its conjugate base, and

a base can react with water or an acid to form its conjugate acid and a base. Acid is defined as any substance that can donate a proton to another substance, a

nd a base is defined as any substance that can accept a proton from another substance. (Bronsted).

HAc represents acetic acid (CH3COOH). HCl is a strong acid so its equilibrium reaction goes to virtual completion.

Acidity constants Ka and basicity constants Kb can be defined using equilibrium equations:

HA} is the acid; {B} is the base. { } - activities in solution, [ ] - concentrations.

Table 4.2 gives a compilation of some common acidity and basicity constants at 25 .℃

Activity of water in dilute solution is constant (mole fraction is unity 55.4 M H2O at 25 ) ℃ hydration of the proton in the above equation can be neglected.

Activity of water = 1.0 as a solvent. Thermodynamic convention sets Kb=1 for the below equation, which corresponds to a free-energy change G = 0: △

the equation can rewritten as follows (aluminum and protons coordinate H2O molecules)

Max coordination number of Al is 6 (2 × valence), it coordinates six solvent molecules (H2O) around it in solution. It is an acid because of its tendency to donate a proton into solution (pK = 4.9 in Table 4.2).

Table 4.2 Acidity and Basicity Constants of Acids and Bases in Aqueous Solutions at 25 ℃

a In order of decreasing acid strength. b In order of Increasing bale strength.

4.2.3 Equilibrium Constants from Thermodynamic Data

Gibbs' free energy: for calculation of equilibrium constants. Definition: energy content of a system is equal to the heat content minus its state of randomness . G is Gibb's free energy (kJ/mol), H is its enthalpy (heat content) (kJ/K), T is absolute temperature (K), S is the entropy (kcal/K).

For a chemical reaction: ΔG is related to ΔH and the change in entropy (TΔS) at constant temperature and pressure as follows below.

ΔG>0 the reaction will not occur as written, ΔG<0 the reaction proceeds spontaneously to the right , ΔG=0 the reaction is at chemical equilibrium.

The relation of free energy change to thermodynamics for the equation below is described further.

νj are the stoichlometric coefficients of the reactants. . At chemical equilibrium: G=0 the following equation .

K is the equilibrium constant, Q is the reaction quotient , Q=K at chemical equilibrium.

For reactions not at chemical equilibrium:

There are a few thermodynamic conventions for use of the above equations. • Aqueous species are expressed as activities in solution (usually in mol/L or mol/k

g)• Pure solids and the so1vent (H2O) have activities equal to 1.

• Gas components are expressed in units of partial pressure.

Change in free energy: from its free energy of formation of the products and reactants and the mass law equation. Q is the reaction quotient at any time in the reaction , G0 is the Gibbs standard free energy of the reaction products minus reactants.

Find log10K from ΔGof at 25 for the reaction℃

Example4.1 Calculation of K from standard Free Energies of formation

Solution: Chemistry texts have a thermodynamic database that gives values of ΔGo

f for all products and reactants for the dissociation of carbonic acid. Here, we interpret the problem to be for H2CO3

*, including “true” molecular H2CO3(aq) and CO2H2O. RT=(0.008314 kJ mol-1 K-1)(298.15 K) at 25℃.

혹은

4.2.4 Temperature and Pressure Correction for Equilibrium Constants Equilibrium constant can be adjusted for temperature change using van’t Hoff appr

oximation. Gibbs' equation that relates equilibrium constants to thermodynamics:

For small changes in temperature ΔS≈0 standard enthalpy ΔGo= ΔHo. Integrating the above equation and rearranging, we have van’t Hoff’s equation.

Pressure influence on equilibrium constants can be neglected in most cases of natural waters, except where p>= 10-1000 atm in seawater standard molar volumes of the products and reactants can be used to calculate K as a function of pressure according to the following equation at p = const:

ΔVo is the change in partial molar volume of the reaction at standard state (cm3 mol-1) (P = 1 atm)

Find the change in K acidity constant of H2CO*3(aq) from Ex. 4.1 at 10

˚C.

Example 4.2 Calculation of K at Different Temperatures

Solution: Find the standard enthalpies for the products and the reactants from a table of thermodynamic data.

At the lower temperature, the acidity constant is smaller 0.85 times then K value at 25 ˚C: because the reaction is endothermic and ΔHo

f is positive; higher temperatures greater acidity constant (stronger acid) and the reaction proceeds farther to the right.

Use van Hoff’s eq. to solve for the new equilibrium constant at 10˚C.

4.2.5 Activity Corrections The theory of ideal solutions: there is no interaction between individual

species Real solutions (particularly solutions of ionic species in water): these

conditions are not met. There are electrostatic interactions between charged ions.

Activity coefficient: ratio of the activity of a species to its concentration

In general: fA of uncharged species ~ 1 in dilute solutions and > 1 in concentrated solutions, because much of the water in concentrated solutions is involved in the hydration shells of ions and less water is available to solvate uncharged species.

From a purely thermodynamic point of view there is no way in which the activity coefficient of a single ion can be measured. But for dilute solutions it is convenient to use single-ion activity coefficients

Debye-Huckel theory: a model that allows activity coefficients to be calculated on the basis of the effect ionic interactions should have on the free energy. Table 4.3: different equations proposed for the estimation of individual activity coefficients.

Table 4.3 Activity Coefficients of Individual Ions

{H+} is the activity used in chemical equilibrium expressions for acids and bases; can be replaced by an activity coefficient times a concentration f

H[H+]. When measuring with a combination pH electrode, the measurement is closest to an activity measurement, not concentration.

Table 4.4 Parameter for Ion Diameter, a, and Activity Coefficient

In exact calculations with the carbonate system in a groundwater (I=5x10-3M) we need activity corrections. pH is measured operationally with standard buffers. We apply our corrections to the equilibrium constants, K’, and then make our calculations numerically or graphically as before

Example 4.3 Activity Corrections for the Carbonate System

Using the Guntelberg approximation :

For K2, the divalent carbonate ion has a valence Z = +2.

Find the activity coefficient of dissolved oxygen Q2(aq), in seawater (I=0.7M) Morel and Herlng give the general empirical equation for neutral molecules:

Example 4.4 Activity Coefficient of Nonelectrolytes

f is the activity coefficient of the neutral molecule and I is the ionic strength, M. At I < 0.1M, the activity correction is less than 2% . For molecular oxygen in seawater, f=1.17, according to the equation above.

Experimental studies of dissolved oxygen in seawater the following (more accurate) equation:

f=1.24. Neutral molecules have f > 1.0 in electrolyte solutions and this is known as the "salting out" effect.

4.2.6 Solving Chemical Equilibrium Problems

Weak acid in solution-acetic acid, CH3COOH, 0.01 M at 25 . (HAc: to d℃esignate acetic acid). The relevant equilibrium equation is defined by the acidity constant Ka, given in Table 4.2 as pKa = 4.7. The system is defined at the zero level as HAc and H2O.

Other relevant equations are the mass law equation for the ionization of water, the mass balance on total acetate in solution, the charge balance, and the proton condition.

: four unknown concentrations or activities H+, OH-, Ac-, HAc

Equilibrium expressions use activities { } rather than concentrations [ ] we must replace the activities with their activity coefficients times the concentration of each species.

Ex. 4.4: activity coefficient of a neutral molecule at such low ionic strength is approximately 1.0, {HAc}≈[HAc]

Activity coefficients are dependent on the ionic strength which cannot be calculated until we know all the concentrations in solution in numerical chemical equilibrium models, we must specify the ionic strength or correct the equilibrium constants for ionic strength effects iteratively

Approximate solution is possible for the above eq’s neglecting activity coefficients for the moment . HAc is a weak acid [H+]>>[OH-].

Let x=[H+]=[Ac-] [HAc]=0.01-x, substitute into the equilibrium expression .

The activity coefficients are very near to 1.0 (ideal dilute solution) in this case (a weak acid dissolved in water with no other electrolytes present).

The approximate solution is pH 3.36 and, [H+]=[Ac-]=4.37x10-4M which seems to be quite accurate because we can check the error using the above equation and the ionization of water.

Using the Guntelberg activity coefficient formula we find

The equilibrium constant corrected for activity coefficients:

The corrected ionization constant Kw’ is 1.001x10-14 .

Substitute the above equation into the mass balance equation.

Combining the above equations and solving.

The above equation is a third-degree polynomial. The easiest way to solve it algebraically is by trial and error. The Newton-Raphson method can solve this iteratively using a computer.

4.3 NUMERICAL SOLUTION TECHNIQUE

Consider our example of 0.01M acetic acid from the previous section. First, in setting up a numerical solution we need to determine the number of s

pecies present in solution. In this case, we have four: Species : HAc, Ac-, OH-, and H+

Second, we need to determine a minimum number of species necessary to solve the system of equilibrium equations -independent variables, or components. We have two equilibrium equations, the acid dissociation of HAc with equilibrium constant Ka and the ionization of water with equilibrium constant Kw.

We have two equations and two unknowns only need two components (independent variables) to specify the system : let us choose HAc and H+.

Next, we need to write chemical equations for the four species in terms of the components:

Assume that all activity coefficients are 1.0 all equilibrium constants must be corrected for activity coefficients (conditional stability constants) as in Ex. 4.3. Here, with 0.01 M HAc in H2O , the activity corrections are negligible.

Formation of HAc: Formation of Ac-: Formation of OH-: Formation of H+:

The final step before solving the matrix equation: an initial estimate of the [H+]to get the program started, and the mass balance equation.

In MacQL, MINEQL, MINTEQ we provide this information by specifying the mode of each component: whether the concentration specified is the total or free concentration.

The total concentration of acetate is 10-2 (at the bottom of the column), and the initial guess for [H+] is 10-7 M.

The programs will recognize the concentrations specified to be the mass balance HAc+Ac -=0.01M and pH=7 (initial guess), and the distribution of species is calculated as a function of pH.

In matrix notation: { } - one-dimensional arrays (column vectors); [ ] - two-dimensional matrices. .

In matrix notation, the above equations in a general form become

{C*} = the column vector of log species concentration, n dimension[A]  = the matrix of stoichiometric coefficients, nxm{X*} =  the column vector of log component concentrations, m dime

nsion{K*} = the column vector of log equilibrium constants, n dimensionn     = the number of species m    = the number of components

In our case the number of species is 4, the number of components is 2 . The material balance equations are determined in the computer progr

am from the modes that were specified and the concentrations given at the bottom. There is exactly one material balance equation for each component.

The material balance equations can also be written, in general, in matrix notation. The equation is cast in terms of differential error term that will be calculated by the Newton-Raphson method until it is within an acceptable closure tolerance.

T[A] = the transposed stoichiometric coefficient matrix , m x n{C}  = the column vector of species concentration , n dimension

{CTOT} = the column vector of component total concentrations, m dim{Y}  = the column vector of errors or remainders, in the material balance equatio

n for each component, m dimension

Log concentration of each species is computed from the matrix equation based on an initial guess for the concentration of the components.

The error, or remainder, is estimated from the above equation. Newton-Rhapson method is applied to solve iteratively.

Z = square matrix (mxm) that is the Jacobian of Y with respect to X = ∂Y/∂X

ΔX = column vector for the improvement in component concentrations, m dimension, Xoriginal – Ximproved

Y = the column vector, which is the remaining error in the material  balance equations , m dimension

The Jacobian operator can be written in terms of the stoichiometric coefficients a ij.

ΔX can be found by inversion of Z matrix.

A criterion for convergence is

When the error term is within closure limits, the program stops.

Figure 4.2 Flowchat of a typical chemical equilibrium program with Newton-Raphson iterative solution technique.

Table 4.5 Mathematical Description of Chemical Equilibrium Problem

4.7 COMPUTER MODELS 4.7.1  MacµQL

Developed by Müller for Macintosh computers.

User-friendly interface; interactive base. Smallest of the chemical equil

ibrium models discussed here, but runs on the same principles as the o

thers (mass law equations + mass balance equations, solved iteratively

using a Newton-Raphson numerical technique).

The code is small because it does not include a thermodynamic databa

se.

Does not include activity or temperature corrections; the user must sp

ecify these by specifying K' conditional equilibrium constants if the ion

ic strength and temperature of the solution are known. No large editor or output data manager, but it is possible to create the

database in EXCEL for import to MacµQL, and to export the output data tables to Cricket Graph, EXCEL, of Kaleida Graph, etc.

4.7.2 MINEQL+ 3.0

Was developed by Schecher and McAvoy beginning with the MINEQL+ model of Westall and co-workers .

Runs on DOS operating systems with Windows and includes a database manager for graphical output.

Interactive input and output relatively simple to plot field data and/or laboratory results with model output for comparison .   

MINEQL+ includes the two- layer surface complexation model together with important thermodynamic data on hydrous ferric oxides (HFO) .

The thermodynamic database is the most extensive in MINEQL+. It includes all of the data from MINTEQA1 that came from the WAT

EQ model of the USGS, and it includes the MINEQL  database plus some other .

4.7.3 MINTEQA2

Has been supported by the U.S. Environmental Protection Agency Environmental Research Laboratory, Athens, Georgia. Originally developed by Battelle Pacific Northwest Laboratory

Took the best database that was available from WATEQ, the USGS model, and wed it to the numerical engine of MINEQL .

The most recent development: the addition of a user-friendly interactive program, PRODEF2 - helps assemble a coherent input database without making obvious mistakes.

There are no graphics capabilities within PRODEFA2, but files can be exported to other software programs for plotting .

Wide array of choices available for adsorption in MINTEQA2, including the simple equilibrium distribution coefficient .

All three models are well-documented and suitable for chemical equilibrium problems in natural waters (acid-base, precipitation-dissolution, complexation, surface complexation, and/or redox).

Table 4.6 is a summary of the three selected models in this section. .    

표 4.6

화학평형모형의 비교

과제물

대표적인 화학평형모델과 모델에 대해 설명하라 . 평형상수와 열역학적 이론의 관계를 설명하라 . Van’t Hoff 식과 활성도 계수를 설명하라 . 아세트산의 화학평형문제를 풀어라 . 위의 문제를 수치해석적으로 풀어라 . 위의 문제를 프로그램을 작성하여 풀어라 . 위의 문제를 대표적인 화학평형모델을 사용하여 풀어라 . MINTEQA2 을 설명하고 운영한 예를 제출하여라 . 모든 과제물은 e-Stream 을 이용하여 동영상파일 , 슬라이드파일 ,

제목의 3 개창으로 구성하여 수강생 개인의 웹 서버에 구축되어야 함 .