Chapter 5 Principles of Chemical Reactions-MWH''''s Water Treatment - Principles and Design, 3d Edition

Sách tính toán công trình xử lý nước thải MWH''''s Water Treatment - Principles and Design, 3d Edition

Activity and Activity Coefficients

5-3 Thermodynamics of Chemical Reactions

Reference Conditions

Free Energy of Formation

Free Energy of Reaction

Free Energy at Equilibrium

Calculation of Free Energy of Formation Using Henry’s Constant

Temperature Dependence of Free-Energy Change

5-4 Reaction Kinetics

Reaction Rate

Rate Law and Reaction Order

Relationship between Reaction Rates

Rate Constants

Factors Affecting Reaction Rate Constants

Determination of Reaction Rate Constants

5-5 Determination of Reaction Rate Laws

Reaction Rate Laws for Individual Reaction Steps

Reaction Rate Expressions for Overall Reaction

Empirical Reaction Rate Expressions

5-6 Reactions Used in Water Treatment

Acid–Base Reactions

Precipitation–Dissolution Reactions

225

MWH’s Water Treatment: Principles and Design, Third Edition

John C Crittenden, R Rhodes Trussell, David W Hand, Kerry J Howe and George Tchobanoglous

Copyright © 2012 John Wiley & Sons, Inc.

Complexation ReactionsOxidation–Reduction Reactions

Problems and Discussion Topics References

Terminology for Chemical Reactions

Acid–basereactions

Reactions that involve the loss or gain of a proton Thesolution becomes more acidic if the reactionproduces a proton or basic if it consumes a proton.Acid/base reactions are reversible

Activationenergy

Energy barrier that reactants must exceed in order forthe reaction to proceed as written

Activity Ability of an ion or molecule to participate in a

reaction In dilute solution, the activity is equal tothe molar concentration For ions in solution, theactivity decreases as ionic strength increases.Activity coefficient Parameter that relates the concentration of a species

to its activity

Conjugate base A molecule that can accept a proton and is formed

when an acid releases a proton

Conversion Amount of a reactant that can be lost or converted to

products, normally given as a moles fraction

neither consumed nor produced by the reaction.Complex Species that is comprised of a metal ion and a ligand.Elementary

reaction

A chemical reaction in which products are formeddirectly from reactants without the formation ofintermediate species

Free energy Thermodynamic energy in a system available to do

chemical work Associated with the potential energy

of chemical reactions Also known as the Gibbsenergy

Heterogeneousreaction

A chemical reaction in which the reactants are present

in two or more phases (i.e., a liquid and a solid).Homogeneous

reaction

A chemical reaction in which all reactants are present

in a single phase

5 Principles of Chemical Reactions 227

Ionic strength A measure of the total concentration of ions in solution

An increase in the ionic strength increases nonidealbehavior of ions and causes activity to deviate fromconcentration

Irreversible

reaction

A chemical reaction that proceeds in the forwarddirection only, and proceeds until one of thereactants has been totally consumed

Ligand Anions that bind with a central metal ion to form soluble

complexes Common ligands include CN−, OH−,

Cl−, F−, CO32−, NO3−, SO42−, and PO43−,Oxidant A reactant that gains electrons in a oxidation/

Precipitation

reaction

A chemical reaction in which dissolved speciescombine to form a solid Precipitation reactions arereversible The reverse is a dissolution reaction, inwhich a solid dissolved to form soluble species

Parallel reactions Reactions that involve the concurrent utilization of a

reactant by multiple pathways

Reaction order The power to which concentration is raised in a

reaction rate law

Reaction rate law Mathematical description of rate of reaction It takes

the form of a rate constant multiplied by theconcentration of reactants raised to a power

Reductant A reactant that loses electrons in a oxidation/reduction

reaction

Reversible

reaction

A chemical reaction that proceeds in either the forward

or reverse direction, and reaches an equilibriumcondition in which products and reactants are bothpresent

Selectivity The preference of one reaction over another

Selectivity is equal to the moles of desired productdivided by the moles of reactant that has reacted

Series reactions Individual reactions that proceed sequentially to

generate products from reactants

Stoichiometry A quantitative relationship that defines the relative

amount of each reactant consumed and eachproduct generated during a chemical reaction

Chemical reactions are used in water treatment to change the physical,chemical, and biological nature of water to accomplish water quality objec-tives An understanding of chemical reaction pathways and stoichiometry isneeded to develop mathematical expressions that can be used to describethe rate at which reactions proceed Kinetic rate laws and reaction stoi-chiometry are valid regardless of the type of reactor under considerationand are used in the development of mass balances (see Chap 6) to describethe spatial and temporal variation of reactants and products in chemi-cal reactors Understanding the equilibrium, kinetic, and mass transferbehavior of each unit process is necessary in developing effective treatmentstrategies Equilibrium and kinetics are both introduced in this chapter,and mass transfer is discussed in Chap 7.

Topics presented in this chapter include (1) chemical reactions andstoichiometry, (2) equilibrium reactions, (3) thermodynamics of chemicalreactions, (4) reaction kinetics, (5) determination of reaction rate laws, and(6) chemical reactions used in water treatment Water chemistry textbooks(Benefield et al., 1982; Benjamin, 2002; Pankow, 1991; Sawyer et al., 2003;Snoeyink and Jenkins, 1980; Stumm and Morgan, 1996) may be reviewedfor more complete treatment of these concepts and other principles ofwater chemistry

5-1 Chemical Reactions and Stoichiometry

Chemical operations used for water treatment are often described usingchemical equations These chemical equations may be used to develop thestoichiometry that expresses quantitative relationships between reactantsand products participating in a given reaction An introduction to the types

of chemical reactions and reaction stoichiometry used in water treatmentprocesses is presented below

Types

of Reactions

Chemical reactions commonly used in water treatment processes can bedescribed in various ways For example, the reactions of acids and bases,precipitation of solids, complexation of metals, and oxidation–reduction

of water constituents are all important reactions used in water treatment

In general, reactions can be thought of as reversible and irreversible.Irreversible reactions tend to proceed to a given endpoint as reactantsare consumed and products are formed until one of the reactants is totallyconsumed Irreversible reactions are signified with an arrow in the chemicalequation, pointing from the reactants to the products Symbols commonlyused in chemical equations are described in Table 5-1 In the followingreaction, reactants A and B react to form products C and D:

Reversible reactions tend to proceed, depending on the specific conditions,until equilibrium is attained at which point the formation of products from

5-1 Chemical Reactions and Stoichiometry 229

Table 5-1

Symbols used in chemical equations

→ Irreversible reaction Single arrow points from the reactants to the

products, e.g., A+ B → C

 Reversible reaction Double arrows used to show that the reaction

proceeds in the forward or reverse direction,depending on the solution characteristics[ ] Brackets Concentration of a chemical constituent or

compound in mol/L{ } Braces Activity of a chemical constituent or compound

(s) Solid phase Used to designate chemical component present

in solid phase, e.g., precipitated calciumcarbonate, CaCO3(s)

(l) Liquid phase Used to designate chemical component present

in liquid phase, e.g., liquid water, H2O(l)(aq) Aqueous (dissolved) Used to designate chemical component

dissolved in water, e.g., ammonia in water,

NH3(aq)(g) Gas Used to designate chemical component present

in gas phase, e.g., chlorine gas, Cl2(g)x

→ Catalysis Chemical species, represented by x, catalyzes

reaction, e.g., cobalt (Co) is the catalyst in thereaction SO32−+1

2O2−→ SOCo 2 −

4

↑ Volatilization Arrow directed up following a component is

used to show volatilization of given component,e.g., CO32−+ 2H+ CO2(g)↑ +H2O

↓ Precipitation Arrow directed down following a component is

used to show precipitation of given component,e.g., Ca2++ CO 2 −

3  CaCO3(s)↓

Source : Adapted from Benefield et al., 1982.

the forward reaction is equal to the loss of products for the reverse reaction

For example, in Eq 5-1 the reactants A and B react to form products C and

D, whereas in Eq 5-2 the reactants C and D react to form products A and B:

The reactions presented in Eqs 5-1 and 5-2 can be combined as follows:

Theoretically, all reactions are reversible given the appropriate conditions;however, under the limited range of conditions typically experienced inwater treatment processes, some reactions may be classified as irreversiblefor practical purposes.

HOMOGENEOUS REACTIONS

When all the reactants and products are present in the same phase, the

reactions are termed homogeneous For homogeneous reactions occurring in

water, the reactants and products are dissolved For example, the reactions

of chlorine (liquid phase) with ammonia (liquid phase) and dissolvedorganic matter (liquid phase) are common homogeneous reactions

HETEROGENEOUS REACTIONS

When reacting materials composed of two or more phases are involved, the

reactions are termed heterogeneous The use of ion exchange media (solid

phase) for the removal of dissolved constituents (liquid phase) from water is

an example of a heterogeneous reaction used in water treatment Reactionsthat require the use of a solid-phase catalyst may also be consideredheterogeneous

Reaction

Sequence

An understanding of the sequence of reaction steps is needed for neering and control of reactions in water treatment reactors Chemicalreactions in water treatment can occur via a single reaction step or multiplesteps in a sequential manner In addition, reactions may occur in series orparallel or in a combination of series and parallel reactions Due to thediverse chemistry of water originating from surface and subsurface sources,many reactions occur during water treatment processes

engi-SERIES REACTIONS

The conversion of a reactant to a product through a stepwise process ofindividual reactions is known as a series reaction For example, reactant Aforms product B, which in turn reacts to form product C:

deter-5-1 Chemical Reactions and Stoichiometry 231

PARALLEL REACTIONS

Reactions that involve the concurrent utilization of a reactant by multiple

pathways are known as parallel reactions Parallel reactions may be thought

of as competing reactions In the reactions shown in Eqs 5-7 and 5-8,

reactant A is simultaneously converted to products B and C:

When there are competing parallel reactions such as those shown in Eqs 5-7

and 5-8, there is often a preferred reaction The preference of one reaction

over another is known as reaction selectivity For example, if Eq 5-7 were the

preferred reaction over Eq 5-8 due to the undesirable nature of product C,

product B would be the desired product, and the selectivity would be

defined as

S = moles of desired product formed, [B]

where S = selectivity, dimensionless

MULTIPLE REACTIONS

Many reactions in water treatment involve complex combinations of series

and parallel reactions, as shown in the following reactions:

For example, the reaction of ozone (O3) with bromide ions (Br−) in

groundwater occurs by the following three-step process:

In this series of reactions, ozone converts bromide to bromate (BrO3−),

which can be a health concern Reactions involving ozone are discussed in

more detail in Chaps 8, 13, and 18

Reaction Mechanisms

Many reactions proceed as a series of simple reactions between atoms,

molecules, and radical species A radical species is an atom or molecule

containing an unpaired electron, giving it unusually fast reactivity A radical

species is always expressed with a dot in the formula (e.g., HO•)

Inter-mediate products are formed during each step of a reaction leading up

to the final products An understanding of the mechanisms of a reaction

may be used to improve the design and operation of water treatment

processes

ELEMENTARY REACTIONS

Reaction mechanisms involving an individual reaction step are known aselementary reactions Elementary reactions are used to describe what ishappening on a molecular scale, such as the collision of two reactants.For example, the decomposition of ozone (in organic-free, distilled water)has been described by the following four-step process (McCarthy andSmith, 1974):

OVERALL REACTIONS

A series of elementary reactions may be combined to yield an overallreaction The overall reaction is determined by summing the elementaryreactions and canceling out the compounds that occur on both sides ofthe reaction For the elementary reactions shown in Eqs 5-15 to 5-18, theoverall reaction may be written as

The specific reaction mechanism and intermediate products that areformed cannot be determined from the overall reaction sequence In manycases the elementary reaction mechanisms are not known and empiricalexpressions must be developed to describe the reaction kinetics

Reaction

Catalysis

A catalyst speeds up a chemical reaction, but it is neither consumed norproduced by the reaction For a reaction between two molecules to occur,the molecules must collide with the proper orientation However, moleculeshave a tendency to move in ways that make the proper orientation less likely.For example, molecules move about their axis in two directions (called arotation and a translation) and they vibrate Adsorption and reaction on acatalyst surface reduce this motion and increase the local concentration ofreactant

Catalysts may be homogeneous or heterogeneous in nature geneous catalysts are dissolved in solution and speed up homogeneousreactions For example, cobalt, a homogeneous catalyst, is known to speed

Homo-up the following reaction, which is used to deoxygenate water for oxygentransfer studies (Pye, 1947):

SO32−+1O2

Co

5-1 Chemical Reactions and Stoichiometry 233

Example 5-1 Reactions for dissolution of carbon dioxide in water

The dissolution of carbon dioxide in water leads to the formation of several

different components Combine the following elementary reactions to

deter-mine the overall reaction with the initial product CO2and the final product of

CO32−:

CO2(g) CO2(aq)

CO2(aq)+ H2O H2CO3

H2CO3 HCO3−+ H+HCO3− CO 2 −

Heterogeneous catalysts speed up reactions at the interface of a liquid or

gas with a solid phase, even if all reactants and products are in a single

phase If the products and reactant are not adsorbed too strongly, reactions

at a surface can increase the rate of reaction, which demonstrates the

utility of heterogeneous catalysis Another purpose of catalysis is to improve

reaction selectivity and minimize the formation of harmful by-products

Reaction Stoichiometry

The amount of a substance entering into a reaction and the amount of a

substance produced are defined by the stoichiometry of a reaction In the

general equation for a chemical reaction, as shown in Eq 5-21, reactants A

and B combine to yield products C and D:

where a, b, c, d= stoichiometric coefficients, unitless

Using the stoichiometry of a reaction and the molecular weight of thechemical species, it is possible to predict the theoretical mass of reactantsand products participating in a reaction For example, calcium hydroxide[Ca(OH)2] may be added to water to remove calcium bicarbonate:

Ca(HCO3)2+ Ca(OH)2 2CaCO3(s)↓ +2H2O (5-22)

As shown in Eq 5-22, 1 mole of Ca(HCO3)2and 1 mole of Ca(OH)2react

to form 2 moles of CaCO3(s) and 2 moles of H2O The molecular weightscan be used to determine the theoretical mass of calcium hydroxide needed

to react with a specified mass of calcium bicarbonate and the amount ofcalcium carbonate formed, as shown in Example 5-2

Example 5-2 Determination of product mass using stoichiometry

For the reaction shown in Eq 5-22, estimate the amount of CaCO3(s) thatwill be produced from the addition of calcium hydroxide to water containing

50 mg/L Ca(HCO3)2 Use a flow rate of 1000 m3/d and determine thequantity of CaCO3(s) in kilograms per day Assume that the reaction proceeds

in the forward direction to completion

Solution

1 Write the chemical equation and note the molecular weight of thereactants and products involved in the reaction The molecular weightsare written below each species in the reaction

Ca(HCO3)2162

+ Ca(OH)2 74

3 Compute the mass of CaCO3(s) that will be produced each day

a Determine the mass of Ca(HCO3)2 removed each day:

Ca(HCO3)2removed= (0.050 g/L)(1000 m3/d)(1000 L/m3)

= 50,000 g/d

5-1 Chemical Reactions and Stoichiometry 235

b Estimate the amount of CaCO3(s) produced each day:

CaCO3(s) produced= [50,000 g Ca(HCO3)2/d]

× [1.23 g CaCO3(s)]/[gCa(HCO3)2](1 kg/103

g)

= 61.5 kg CaCO3(s)/d

Comment

In addition to estimating the amount of CaCO3(s) produced, it is also

possible to estimate the amount of calcium hydroxide that must be added

to water to bring about this reaction However, due to the nonideal nature

of water treatment processing, the amount of calcium hydroxide that is

required will exceed the stoichiometric amount, which is the minimum amount

needed

Reactant Conversion

As a reaction proceeds, reactants are converted into products At any

intermediate point during the reaction or when the reaction has reached

equilibrium, it is possible to determine the amount (in moles) of reactants

and products remaining if the stoichiometry and the amount of one of the

reactants present is known For example, consider the reaction shown in

Eq 5-21, in which a, b, c, and d are stoichiometric coefficients For this

reaction, the conversion may be determined for a reference reactant A and

written per mole of A by dividing by the stoichiometric coefficient a:

A+ b

aB c

aC+d

For the general reaction shown in Eq 5-23, all the reactants and products

can be related to the conversion of reactant A, XA, and the initial

concen-tration of A, assuming there is no volume change upon reaction (which is

valid for most water treatment problems):

moles of A present initially = NA0− NA

NA0

(5-24)where XA= conversion of reactant A

NA0= initial amount of reactant A, mol

NA= final amount of reactant A, mol

Equation 5-24 can be written in molar concentration units by dividing each

term by the volume in which the reaction is occurring Thus, Eq 5-24

written in concentration units is

XA= CA0− CA

where CA0= initial concentration of reactant A, mol/L

CA= final concentration of reactant A, mol/L

If the final amount and concentration of A, NA, and CAare written in terms

of the conversion, the following expressions are obtained:

For the reaction given in Eq 5-23, the final concentrations of B, C, and

D can be computed in terms of A The final amount of B, C, and D aredetermined by subtracting the product of moles of A reacted and thestoichiometric ratio of B, C, and D to A from the initial moles of B, C, and

D, as shown by the following expressions The final amount of B written interms of moles is shown below

NB= NB0− b

where NB= final amount of reactant B, mol

NB0= initial amount of reactant B, molThe final amount of B in terms of concentration is

where CC, CD= final concentration of reactants C and D, mol/L

CC0, CD0= initial concentration of reactants C and D, mol/LThe final concentration of the various species are related to one anotherand to the conversion, as summarized in Table 5-2 As illustrated onFig 5-1, the addition of a catalyst (or other change in the reaction condi-tions) may improve the selectivity and the reaction conversion for a giventime The conversion from reactant to product can eventually reach thethermodynamic limit of the reaction, as discussed in the following section

Initial Amount Initial Amount, Final Amount Concentration,

Present, mol mol Present, mol mol/L

Conversion with catalysis Thermodynamic limit to conversion

Figure 5-1

Improved reactant conversion with addition of catalyst.

5-2 Equilibrium Reactions

As discussed previously in this chapter, many of the reactions of significance

in water treatment processes are reversible reactions In other words,

reactions such as that shown in Eq 5-3 will not usually achieve complete

conversion of reactants to products but instead will reach a state of dynamic

equilibrium Dynamic equilibrium is characterized by a balance between

the continuous formation of products from reactants and reactants from

products If there is a change or stress to the system that affects the balance,

the amount of reactants and products present will change to accommodate

the stress This concept is known as Le Chatelier’s principle, which states

that a reaction at equilibrium shifts in the direction that reduces a stress to

the reaction For example, in Eq 5-21 if constituent A is removed from the

system, the equilibrium will shift to form more A In a chemical system, thedifference between the actual state and the equilibrium state is the drivingforce used to accomplish treatment objectives.

Equilibrium

Constants

When chemical reactions come to a state of equilibrium, the numerical value

of the ratio of the concentration of the products over the concentration ofthe reactants all raised to the power of the corresponding stoichiometric

coefficients is known as the equilibrium constant (K c) and, for the reactionshown in Eq 5-21, is written as

[C]c[D]d

where K c = equilibrium constant (subscript c used to signify

equilibrium constant based on species concentration)[ ]= concentration of species, mol/L

a, b, c, d= stoichiometric coefficients of species A, B, C, D,

respectivelyFor example, the ionization of carbonic acid, given previously as Eq 5-5, isshown as

H2CO3 HCO3−+ H+The equilibrium constant at 25◦C (neglecting nonidealities) for the reactionshown above may be written as

[H+][HCO3−]

The value of equilibrium constants and reactant and product tions are typically small and, therefore, are often reported in the literatureusing the operand ‘‘p,’’ which is defined as

where [i] = concentration of species i, mol/L

The reporting of the hydrogen ion activity as pH is a familiar example of

the p notation Similarly, an equilibrium constant K may be reported as

pK , which is defined as

Therefore, the K creported in Eq 5-33 may be written as

pK c = −log10K c= −log10(5.0 × 10−7)= 6.3 (5-36)

Ionic Strength In dilute solutions, the ions present behave independently of each other

However, as the concentration of ions in solution increases, the activity ofthe ions decreases because of ionic interaction The ionic strength may be

5-2 Equilibrium Reactions 239

Example 5-3 Dependence of chemical species on pH

A drinking water contains hypochlorous acid (HOCl) Using the following

relationship, determine the ratio of the hypochlorite ion (OCl−) to HOCl at

(a) pH 7.0 and (b) pH 8.0 (neglecting nonidealities):

HOCl OCl−+ H+The equilibrium constantKc for the dissociation of HOCl into OCl− and H+

(also known as an acid dissociation constant and typically reported asKa) is

2 Determine the ratio of [OCl−] to [HOCl] at the given pH values

a At pH 7.0, the hydrogen concentration [H+] is equal to 10−7 and

the equilibrium relationship is written as

(10−7)[OCl−]

[HOCl] = 10−7.5 [OCl−]

[HOCl] = 10−0.5 = 0.32

b At pH 8.0, the hydrogen concentration [H+] is equal to 10−8 and

the equilibrium relationship is written as

(10−8)[OCl−]

[HOCl] = 10−7.5 [OCl−]

[HOCl] = 100.5 = 3.2

Comment

As shown in the calculations above, the solution pH can have a significant

impact on the chemical species present As shown in Chapter 13, HOCl is a

more effective disinfectant than OCl− and is formed when chlorine is added

to water Consequently, it will be important to keep the pH 7 or less to

achieve the greatest level of disinfection for a given dose of chorine

For a given reaction, the value of the equilibrium constant, expressed in

terms of concentration, will depend on the temperature and ionic strength

of the solution It should be noted that the equilibrium condition shown in

Eq 5-32 is based on the concentration of the chemical species involved in

the reaction and may need to be adjusted for ionic activity, as discussed

below

determined using the equation (Lewis and Randall, 1921)

I =1 2



i

where I = ionic strength of solution, mol/L(M)

C i = concentration of species i, mol/L(M)

Z i = number of replaceable hydrogen atoms or their equivalent

(for oxidation–reduction reactions, Z is equal to the change

in valence)

If the concentration of individual species is not known, the ionic strengthmay be estimated from the total dissolved solids concentration using thecorrelation (Stumm and Morgan, 1996)

where TDS= total dissolved solids, mg/L

To account for nonideal conditions encountered due to ion–ion tions (e.g., at high ionic strength), an effective concentration term called

of pressure a temperature of 298.15 K (25◦C), elements in their lowestenergy level (e.g., O2 as a gas, carbon as graphite), and 1 molal hydrogen

ion (1 mole of hydrogen ion per 1000 g of water) Some recent chemicalreferences use 1 bar rather than 1 atm as the standard state, but thedifference is small (1 atm= 1.01325 bar) Nonetheless, when looking upvalues for free energy in reference tables, note whether 1 atm or 1 bar isused for the standard state The activity coefficient of a chemical in watermay be determined as discussed below

For ions and molecules in solution,

where {i} = activity or effective concentration of ionic species, mol/L(M)

γi= activity coefficient for ionic species

[i]= concentration of ionic species in solution, mol/L(M)

In general,γi is greater than 1.0 for nonelectrolytes and less than 1.0 forelectrolytes As the solution becomes dilute (applicable to most applica-tions in water treatment),γi approaches 1 and{i} approaches [i] In the

dilute aqueous solutions normally encountered in water treatment, activitycoefficients are assumed to be equal to 1

5-2 Equilibrium Reactions 241

For pure solids or liquids in equilibrium with a solution{i} = 1, and for

gases in equilibrium with a solution, the activity of species i is

where {i} = activity or effective gas pressure, atm

P i = partial pressure of i, atm

When reactions take place at atmospheric pressure (actually, much less

then its critical pressure), the activity of a gas is equal to its partial pressure

in atmospheres and the activity coefficient is 1.0

For solvents or miscible liquids in a solution,

where x i = mole fraction of species i

As the solution becomes more dilute, γi approaches 1 As stated above,

the activity coefficient generally is assumed to be 1 for the dilute solutions,

which are typical in water treatment

When a species in water is an electrolyte, the activity should be considered

but is usually ignored in routine calculations The activity coefficient

for electrolytes in solution with ionic strength less than 0.005 M may be

estimated from the Debye–H ¨uckel limiting law (Debye and H ¨uckel, 1923):

log10γi = −AZ2

where A= constant equal to 0.51 at 25◦C (Stumm and Morgan, 1996)

For more concentrated solutions up to I ≤ 0.1 M, the following

modifica-tion of the Debye–H ¨uckel equation, known as the Davies equation, can be

applied with acceptable error (Davies, 1967):

The Davies equation is typically in error by 1.5 percent and 5 to 10 percent

at ionic strengths between 0.1 and 0.5 M, respectively (Levine, 1988)

The constant A in Eq 5-43 depends on temperature and can be estimated

from the equation (Stumm and Morgan, 1996; Trussell, 1998)

A = 1.29 × 106

√2

where T = absolute temperature, K (273 +◦C)

Dε= dielectric constant (see Eq 5-45)

The dielectric constant may be determined using the equation (Harnedand Owen, 1958)

Dε∼= 78.54{1 − [0.004579(T − 298)] + [11.9 × 10−6(T− 298)2]

where T= absolute temperature, K(273 +◦C)

Therefore, the constant A for water at 0, 15, and 25◦C is 0.49, 0.5, and 0.51,respectively

The equilibrium relationship shown in Eq 5-32 may now be expressed

in terms of activities:

(γc[C])c(γd[D])d(γa[A])a(γb[B])b = {C}c{D}d

where K = equilibrium constant based on ionic activity (note absence of

subscript to signify activity basis)The corresponding equilibrium for Eq 5-23 is

{C}c/a{D}d/a

For most water supplies, the ionic strength is less than 5 millimole/L (mM)and the activity coefficients for monovalent ions are close to one Thecalculation of activity coefficients for solutions of different ionic strengths

is presented in the following example

Example 5-4 Determination of activity coefficients at different

ionic strengths

Calculate the activity coefficients of Na+, Ca2+, and Al3+ at ionic strengths

of 0.001, 0.005, and 0.01 M at 25◦C

Solution

1 Determine the activity coefficients for an ionic strength of 0.001 M at

25◦C using the Debye–H¨uckel limiting law (Eq 5-42):

5-3 Thermodynamics of Chemical Reactions 243

2 Determine the activity coefficients for an ionic strength of 0.005 M at

25◦C using the Debye–H¨uckel limiting law:

3 Determine the activity coefficients for an ionic strength of 0.01 M at

25◦C using the Davies equation (Eq 5-43):

The activity for all the ions decreases as the ionic strength increases As the

ionic strength of the solution increases, the impact of charge on the species

has a large influence on the value of the activity coefficient For example, as

ionic strength increased from 0.001 to 0.01, the activity coefficient for Na+

decreased by only about 6 percent as compared to Al3+, which decreased

by 46 percent

5-3 Thermodynamics of Chemical Reactions

Principles from equilibrium thermodynamics provide a means for

deter-mining whether reactions are favorable and are also used in process

design calculations to determine the final equilibrium state The difference

between the actual state and the equilibrium state is the driving force for

many processes and reactions Equilibrium thermodynamics can be used

to determine whether the treatment process is feasible, and the reactionkinetics, described in the following sections, will provide a basis for thetreatment device size

To determine whether a reaction will proceed (i.e., is thermodynamicallyfavorable), two fundamental thermodynamic criteria must be considered.The first thermodynamic criterion that must be satisfied is that the change

in entropy of the system and its surroundings must be greater than zerofor a reaction to proceed When evaluating chemical reactions, the entropyrequirement is typically satisfied, especially when heat is produced bythe reaction and, therefore, is not considered further in this text Thesecond thermodynamic criterion necessary for a reaction to proceed is therequirement that the change in free energy (final energy state minus initialenergy state) of the reaction must be less than zero

Reference

Conditions

To understand how the free energy of reaction changes as a reactionproceeds, it is useful to examine the total free energy of reaction as afunction of the reaction extent, as shown on Fig 5-2 Because the absolutefree energy of reaction cannot be determined easily, it is most common todetermine the change in free energy of a reaction The free energy of thereaction curve shown on Fig 5-2 is compared to a convenient set of standardconditions For example, a common definition of standard conditions is

as follows: (1) solids, liquids, and gases in their lowest energy state at 1 atm(or 1 bar); (2) solutes in solution referenced to a 1 molal hydrogen ionconcentration; and (3) a specified temperature, usually 25◦C For mostwater treatment applications, the molar concentration is essentially equal

to the molal concentration and a 1 M solution is 1 mole per 1000 g ofsolvent

Figure 5-2

Total free energy as function of the extent of the reaction.

5-3 Thermodynamics of Chemical Reactions 245

Free Energy

of Formation

The expression for free energy was developed by J W Gibbs and is often

referred to as the Gibbs free energy or Gibbs function G The free-energy

change of formation of a substance i is given by the expression

where G F ,i = free-energy change of formation of species i, kJ/mol

G F ,i◦ = free-energy change of formation per mole of i at standard

conditions, kJ/mol

R = universal gas law constant, 8.314 × 10−3kJ/mol· K

T = absolute temperature, K(273 +◦C)

{i} = activity of species i

Thermodynamic constants may be found in various reference books,

includ-ing Stumm and Morgan (1996) and Lange’s Handbook (Dean, 1992).

Free Energy

of Reaction

The free energy of a reaction can be calculated using the definition of

activity and the free-energy change of formation For this purpose, consider

the reaction shown in Eq 5-21, in which a, b, c, and d are stoichiometric

coefficients For this reaction, the free-energy criterion may be determined

for a reference reactant A and written per mole of A by dividing by the

stoichiometric coefficient a, as shown in Eq 5-23 and repeated here:

A+ b

aB c

aC+d

aD

The free-energy change is defined as the final state minus the initial state

(David, 2000; Dean, 1992; Poling et al., 2001) Therefore, the change in free

energy of a reaction is the sum of the free-energy change of each product

minus the sum of the free-energy change of the reactants, as shown in the

following expression written in terms of free-energy change per mole of A:

GRxn,A= −G F ,A−b

a G F ,B+c

a G F ,C+d

where GRxn,A= free-energy change of reaction per mole of A, kJ/mol

G F ,A= change in free energy of reactant A, kJ/mol

G F ,B = change in free energy of reactant B, kJ/mol

G F ,C= change in free energy of product C, kJ/mol

G F ,D= change in free energy of product D, kJ/mol

The free-energy change of the formation of each species, as defined in

Eq 5-49, may be substituted into Eq 5-49 for each reactant and product

to obtain the overall free-energy change for the reaction The resulting

expression for the free-energy change of the reaction is shown in the

where GRxn,A = free-energy change of reaction per mole of A, kJ/mol

{A} = activity of reactant A, mol/L{B} = activity of reactant B, mol/L{C} = activity of product C, mol/L{D} = activity of product D, mol/LThe free-energy change of the reaction per mole of A at standard conditions(25◦C and 1 atm pressure),GRxn,A◦ , can be written as

While a reaction is thermodynamically feasible whenGRxn,A< 0, the rate

at which a reaction will proceed is not known because reactants often have

to proceed through reactive intermediates that have a higher free energythan the reactants Alternately, ifGRxn,A> 0, the reverse reaction would

equal to the equilibrium constant K as shown below:

GRxn,A = GRxn,A◦ + RT ln{C}c/a{D}d /a

{A}{B}b/a



5-3 Thermodynamics of Chemical Reactions 247

Rearranging Eq 5-57 and solving for the equilibrium constant result in the

expression

The free energy, calculated using Eq 5-52, is actually the slope of a

tangent to the total free-energy curve shown on Fig 5-2, and equilibrium

is represented by the special case where the slope is zero This means that

GRxn,A is really the change in free energy that results from an infintesmal

conversion of A to products

Calculation

of Free Energy

of Formation Using Henry’s Constant

A difficulty often encountered when calculating the free energy of reaction

is that the free-energy change of formation per mole of A in the aqueous

phase, G F ,A,aq◦ , is needed to calculate GRxn,A, and the free energy of

formation may be reported for the gas phase, G F ,A,gas◦ However, the

relationship shown in Eq 5-59 can be used to develop an expression for

the free energy of formation of slightly soluble gases in the aqueous phase,

G F ,A,aq◦ , based on the free energy of formation in the gas phase,G F ,A,gas◦ :

GVol,A◦ = G F ,A,gas◦ − G F ,A,aq◦ = −RT ln HPC (5-59)

where GVol,A◦ = free-energy change of volatilization per mole of A at

standard conditions, kJ/mol

G F ,A,gas◦ = free-energy change of formation per mole of A in gas

phase at standard conditions, kJ/mol

G F ,A,aq◦ = free-energy change of formation per mole of A in

aqueous phase at standard conditions, kJ/mol

H PC= Henry’s law constant atm/(mol/L)

Equation 5-59 can then be rearranged to solve for the aqueous-phase

concentration of A as a function of the gas-phase concentration of A:

G F ,A,aq◦ = G F ,A,gas◦ + RT ln HPC (5-60)Consequently,G F ,A,aq◦ can be calculated fromG F ,A,gas◦ if HPCis known

Henry’s law is presented and discussed in detail in Chap 14

Temperature Dependence

of Free-Energy Change

Most reactions in water treatment do not occur at 25◦C because the water

temperature is usually lower The free-energy change at other temperatures

can be determined from the expression

GRxn◦T

The temperature-dependent standard enthalpy of reaction is defined as

HRxn◦ (T )=

T

T=298 KC p,Rxn dT + HRxn,298 K◦ (5-62)where C p,Rxn= change in heat capacity for the reaction, kJ/mol

HRxn,298K◦ = standard enthalpy at 298 KThe heat capacity term may be calculated using the equation (Poling et al.,2001)

To calculate C p,Rxn , the difference of each constant (A, B, C, and D)

between products and reactants needs to be calculated:

C p,Rxn = A + BT + CT2+ DT3 (5-64)For the reaction shown in Eq 5-23, the terms in Eq 5-64 are given by theexpressions

GRxn◦T

298 K = HRxn,298 K◦

1

298 K

(5-67)

At equilibrium, Eq 5-57 can be substituted into Eq 5-61 to yield thevan’t Hoff relationship, which may be used to determine the equilibrium

5-3 Thermodynamics of Chemical Reactions 249

constant (Keq) at different temperatures:

R

1

Using Eq 5-68, the linear relationship between ln K and 1/T can be

determined by plotting the function

◦ Rxn,298 K

For most reactions occurring in water treatment processes,HRxn◦ can be

assumed to be constant becauseHRxndoes not change significantly over

the temperature range encountered in water treatment (0 to 30◦C)

Example 5-5 Dependence of pH and free-energy change

on temperature

For the dissociation reaction of water, the free-energy change and enthalpy

change for each species in the reaction

H2O H++ OH−are as follows:

Calculate the pH of neutrality and free-energy change of the reaction at

10◦C Assume thatH◦Rxndoes not change with temperature

b Calculate the equilibrium constant at 25◦C (298 K) using Eq 5-58:

K= exp



−GRxn,H◦

2 ORT

1

283 K



= 3.0015 × 10−15

3 Calculate the pH at neutrality at 10◦C

At neutral conditions, [H+] is equal to [OH−[H+] [OH −]=Keq,283= 3.0015 × 10−15

[H+]= [OH −]=3.0015 × 10−15= 5.48 × 10−8

pH= pOH = −log5.48 × 10−8= 7.26

According to Le Chatelier’s principle, as the temperature decreases, the

reaction for the dissociation of water would be less favorable because it

takes energy to dissociate water; consequently, the equilibrium constant is

lower at 10◦C than at 25◦C

5-4 Reaction Kinetics

While thermodynamic calculations provide a means for estimating the

likelihood and maximum possible extent of a given reaction, they cannot

be used to determine the rate of the reaction This section discusses reaction

rate laws which describe how fast a reaction proceeds in the absence of

mass transfer limitations

Reaction Rate

The rate of a chemical reaction depends on the activity of the reacting

species and the temperature of the system As noted earlier for organic

compounds (nonelectrolytes) or monovalent ions in water, the activity is

nearly equivalent to concentration when the ionic strength is less than

0.005 M For divalent and trivalent ions, the activity coefficients should be

calculated to determine whether activity coefficients are needed

The rate of a reaction is expressed as the change in the concentration of

a constituent with time:

Rate of reaction= change in concentration

The reaction rate is used to describe the rate of formation of a product orthe rate of decomposition of a reactant As a reaction proceeds, the reactionrate changes; for example, as the concentration of reactants is decreased,the rate of a reaction may decrease The reaction rate usually changes as theconcentrations of the reactants and products change The rate of change isdescribed by an expression known as a rate law, as discussed in the followingsection.

Rate Law

and Reaction

Order

In the following discussion, the term rA is used to represent the reaction

rate It should be noted that dCA/dt is not the reaction rate law but is

obtained from a mass balance on a completely mixed batch reactor, whichhas a constant volume The subscript A is used to designate the speciesdescribed by the reaction rate The units of the reaction rate are given bythe expression

rA= moles A lost (−) or generated (+) due to reaction

L· s (5-71)

A negative or positive sign for the reaction rate indicates that species A iseither disappearing or appearing, respectively The following irreversiblereaction is used to develop the reaction rate

For an irrevisible reaction, the rate law depends on the concentrations

of reactants For the reaction shown in Eq 5-72, the rate law may takethe form

where rA= reaction rate, mol/L · s

k= reaction rate constant, units vary depending on reactionorder as discussed later

m, n= constants, unitlessThe concentration dependence of the reaction rate is accounted for

in the reactant exponents m and n and is known as the reaction order For Eq 5-73, the reaction order is m for species A and n for species B, and the overall reaction order is m + n The reaction order is typically

a small positive integer; however, it may also be negative, zero, orfractional

5-4 Reaction Kinetics 253

To simplify the expression, Eq 5-21 can be divided by the stoichiometric

coefficient a, which yields the expression

A+ b

aB c

aC+d

Assuming the reaction shown in Eq 5-23 is proceeding to the right, the

following relationship between the reaction rates can be written using the

stoichiometry from Eq 5-23:

The reaction rates for reactants A and B,−rAand−rB, are negative because

they are disappearing to form products, and the reaction rates for products

C and D, rC and rD, are positive because they are being produced If the

reaction rate for A is known, the stoichiometric coefficients, as given by

Eq 5-74, can be used to determine the reaction rate for reactant B and

products C and D The use of stoichiometric coefficients is illustrated in

the following example

Example 5-6 Determination of reaction rates using stoichiometry

Given the reaction 4Fe2++ O2+ 10H2O→ 4Fe(OH)3+ 8H+, estimate the

loss rates of oxygen and water and production rates of iron hydroxide and

acid when the rate of loss of Fe2+is 2× 10−7mol/L· min

Solution

1 Write the relevant reaction using the form shown in Eq 5-23 To

cancel out the coefficient for Fe2+, the reaction must be divided by 4,

resulting in the expression

= −rH2 O 5 2

= +rFe(OH)3

2

3 Estimate the loss rates for oxygen (−rO2) and water (−rH2O) given that

the rate of loss of Fe2+(−rFe2+) is 2× 10−7mol/L· min:

4 Estimate the production rates for iron hydroxide

+rFe(OH)3

and acid(+rH +) :

+rFe(OH)3 = −rFe2+ = 2 × 10−7mol/L · min

+rH += 2(−rFe2 +)= 4 × 10−7mol/L · min

Comment

For reactions that are given by Eq 5-23, the reaction rates are related to oneanother If one knows the rate of reaction of any at the components, thenreaction rates of the other components can be calculated using Eq 5-74

Rate Constants The units of the rate constant depend on the reaction order; but, it should

be noted that the units of the reaction rate r are always of the form mol/

L· s For a zero-order reaction, the rate constant would have the followingunits:

For Eq 5-77, the rate is first order in A and B and second order overall For

a third-order reaction, the rate constant would have the following units:

on temperature, pH, ionic composition, and other factors are essential forproper control of the reactions of interest

EFFECT OF TEMPERATURE AND CATALYSIS ON REACTION RATE CONSTANT

Reaction rate constants are known to be dependent on temperature Arelationship known as the Arrhenius equation is used to describe thetemperature dependence: