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

5
5-1

Principles
of Chemical
Reactions

Chemical Reactions and Stoichiometry
Types of Reactions
Reaction Sequence
Reaction Mechanisms
Reaction Catalysis
Reaction Stoichiometry
Reactant Conversion

5-2

Equilibrium Reactions
Equilibrium Constants
Ionic Strength
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

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.

225

226

5 Principles of Chemical Reactions
Complexation Reactions
Oxidation–Reduction Reactions

Problems and Discussion Topics
References

Terminology for Chemical Reactions
Term

Definition

Acid
Acid–base
reactions

A molecule that is capable of releasing a proton.
Reactions that involve the loss or gain of a proton. The
solution becomes more acidic if the reaction
produces a proton or basic if it consumes a proton.
Acid/base reactions are reversible.
Energy barrier that reactants must exceed in order for
the reaction to proceed as written.
Ability of an ion or molecule to participate in a
reaction. In dilute solution, the activity is equal to
the molar concentration. For ions in solution, the
activity decreases as ionic strength increases.

Parameter that relates the concentration of a species
to its activity.
A molecule that can accept a proton and is formed
when an acid releases a proton.
Amount of a reactant that can be lost or converted to
products, normally given as a moles fraction.
A species that Speeds up a chemical reaction, but is
neither consumed nor produced by the reaction.
Species that is comprised of a metal ion and a ligand.
A chemical reaction in which products are formed
directly from reactants without the formation of
intermediate species.
Thermodynamic energy in a system available to do
chemical work. Associated with the potential energy
of chemical reactions. Also known as the Gibbs
energy.
A chemical reaction in which the reactants are present
in two or more phases (i.e., a liquid and a solid).
A chemical reaction in which all reactants are present
in a single phase.

Activation
energy
Activity

Activity coefficient
Conjugate base
Conversion
Catalyst
Complex

Elementary
reaction
Free energy

Heterogeneous
reaction
Homogeneous
reaction


5 Principles of Chemical Reactions

Term

Definition

Ionic strength

A measure of the total concentration of ions in solution.
An increase in the ionic strength increases nonideal
behavior of ions and causes activity to deviate from
concentration.
A chemical reaction that proceeds in the forward
direction only, and proceeds until one of the
reactants has been totally consumed.
Anions that bind with a central metal ion to form soluble
−
complexes. Common ligands include CN , OH− ,
2−
−

2−
3−
Cl− , F− , CO3 , NO3 , SO4 , and PO4 ,
A reactant that gains electrons in a oxidation/
reduction reaction.
A chemical reaction in which electrons are transferred
from one molecule to another. Also known as a
redox reaction. Redox reactions are irreversible.
A chemical reaction in which dissolved species
combine to form a solid. Precipitation reactions are
reversible. The reverse is a dissolution reaction, in
which a solid dissolved to form soluble species.
Reactions that involve the concurrent utilization of a
reactant by multiple pathways.
The power to which concentration is raised in a
reaction rate law.
Mathematical description of rate of reaction. It takes
the form of a rate constant multiplied by the
concentration of reactants raised to a power.
A reactant that loses electrons in a oxidation/reduction
reaction.
A chemical reaction that proceeds in either the forward
or reverse direction, and reaches an equilibrium
condition in which products and reactants are both
present.
The preference of one reaction over another.
Selectivity is equal to the moles of desired product
divided by the moles of reactant that has reacted.
Individual reactions that proceed sequentially to
generate products from reactants.

A quantitative relationship that defines the relative
amount of each reactant consumed and each
product generated during a chemical reaction.

Irreversible
reaction
Ligand

Oxidant
Oxidation/
reduction
reaction
Precipitation
reaction

Parallel reactions
Reaction order
Reaction rate law

Reductant
Reversible
reaction

Selectivity

Series reactions
Stoichiometry

227

228

5 Principles of Chemical Reactions

Chemical reactions are used in water treatment to change the physical,
chemical, and biological nature of water to accomplish water quality objectives. An understanding of chemical reaction pathways and stoichiometry is
needed to develop mathematical expressions that can be used to describe
the rate at which reactions proceed. Kinetic rate laws and reaction stoichiometry are valid regardless of the type of reactor under consideration
and are used in the development of mass balances (see Chap. 6) to describe
the spatial and temporal variation of reactants and products in chemical reactors. Understanding the equilibrium, kinetic, and mass transfer
behavior of each unit process is necessary in developing effective treatment
strategies. 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 and
stoichiometry, (2) equilibrium reactions, (3) thermodynamics of chemical
reactions, (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 reviewed
for more complete treatment of these concepts and other principles of
water chemistry.

5-1 Chemical Reactions and Stoichiometry
Chemical operations used for water treatment are often described using
chemical equations. These chemical equations may be used to develop the
stoichiometry that expresses quantitative relationships between reactants
and products participating in a given reaction. An introduction to the types
of chemical reactions and reaction stoichiometry used in water treatment
processes is presented below.

Types
of Reactions

Chemical reactions commonly used in water treatment processes can be
described 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 reactants
are consumed and products are formed until one of the reactants is totally
consumed. Irreversible reactions are signified with an arrow in the chemical
equation, pointing from the reactants to the products. Symbols commonly
used in chemical equations are described in Table 5-1. In the following
reaction, reactants A and B react to form products C and D:
A+B→C+D

(5-1)

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

Table 5-1
Symbols used in chemical equations
Symbol
→

Description

Comments

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 calcium
carbonate, CaCO3 (s)

(l)

Liquid phase

Used to designate chemical component present
in liquid phase, e.g., liquid water, H2 O(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)

Catalysis

Chemical species, represented by x, catalyzes

reaction, e.g., cobalt (Co) is the catalyst in the

x

→

reaction SO3

2−

Co

+ 12 O2 −→ SO4

2−

↑

Volatilization

Arrow directed up following a component is
used to show volatilization of given component,
2−
CO2 (g) ↑ +H2 O
e.g., CO3 + 2H+

↓

Precipitation

Arrow directed down following a component is
used to show precipitation of given component,
2+
2−
CaCO3 (s) ↓
e.g., Ca + CO3

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:
C+D→A+B

(5-2)

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

C+D

(5-3)

229


230

5 Principles of Chemical Reactions

Theoretically, all reactions are reversible given the appropriate conditions;
however, under the limited range of conditions typically experienced in
water treatment processes, some reactions may be classified as irreversible
for 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 dissolved
organic 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. Reactions
that require the use of a solid-phase catalyst may also be considered
heterogeneous.
Reaction
Sequence

An understanding of the sequence of reaction steps is needed for engineering and control of reactions in water treatment reactors. Chemical
reactions in water treatment can occur via a single reaction step or multiple
steps in a sequential manner. In addition, reactions may occur in series or
parallel or in a combination of series and parallel reactions. Due to the
diverse chemistry of water originating from surface and subsurface sources,
many reactions occur during water treatment processes.
SERIES REACTIONS

The conversion of a reactant to a product through a stepwise process of
individual reactions is known as a series reaction. For example, reactant A
forms product B, which in turn reacts to form product C:
A→B→C

(5-4)

For example, the two-step conversion of carbonic acid (H2 CO3 ) to carbonate (CO32− ) takes place in water according to the following series reaction:
H2 CO3
HCO3

−

HCO3− + H+
CO3

2−

+H

+

(5-5)
(5-6)

The extent and rate of the reactions shown in Eqs. 5-5 and 5-6 are determined by the water pH, temperature, and other properties, as discussed
later in this chapter.


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:
A→B

(5-7)

A→C

(5-8)

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
moles of desired product formed, [B]
(5-9)
S=
moles of all products formed, [B] + [C]
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:
A+B→C

(5-10)

A+C→D

(5-11)

For example, the reaction of ozone (O3 ) with bromide ions (Br− ) in
groundwater occurs by the following three-step process:
O3 + Br− → OBr−
OBr− + O3 → BrO2
−

BrO2 + O3 → BrO3

−
−

(5-12)
(5-13)
(5-14)

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.
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 •). Intermediate 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.

Reaction
Mechanisms


232

5 Principles of Chemical Reactions
ELEMENTARY REACTIONS

Reaction mechanisms involving an individual reaction step are known as
elementary reactions. Elementary reactions are used to describe what is
happening 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 and
Smith, 1974):
(5-15)
O3 + H2 O → HO3+ + OH−
HO3+ + OH− → 2HO2
O3 + HO2 → HO • + 2O2
HO • + HO2 → H2 O + O2

(5-16)
(5-17)
(5-18)

In this series of elementary reactions, ozone reacts with water to form,
among other compounds, HO • (hydroxyl radical) and HO2 (superoxide),
which are very reactive and sometimes used for the destruction of organic
compounds.
OVERALL REACTIONS

A series of elementary reactions may be combined to yield an overall
reaction. The overall reaction is determined by summing the elementary
reactions and canceling out the compounds that occur on both sides of
the reaction. For the elementary reactions shown in Eqs. 5-15 to 5-18, the
overall reaction may be written as
2O3 → 3O2

(5-19)

The specific reaction mechanism and intermediate products that are
formed cannot be determined from the overall reaction sequence. In many
cases the elementary reaction mechanisms are not known and empirical
expressions must be developed to describe the reaction kinetics.
Reaction
Catalysis

A catalyst speeds up a chemical reaction, but it is neither consumed nor
produced by the reaction. For a reaction between two molecules to occur,
the molecules must collide with the proper orientation. However, molecules
have a tendency to move in ways that make the proper orientation less likely.

For example, molecules move about their axis in two directions (called a
rotation and a translation) and they vibrate. Adsorption and reaction on a
catalyst surface reduce this motion and increase the local concentration of
reactant.
Catalysts may be homogeneous or heterogeneous in nature. Homogeneous catalysts are dissolved in solution and speed up homogeneous
reactions. For example, cobalt, a homogeneous catalyst, is known to speed
up the following reaction, which is used to deoxygenate water for oxygen
transfer studies (Pye, 1947):
Co

SO32− + 12 O2 −→ SO42−

(5-20)


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 determine the overall reaction with the initial product CO2 and the final product of
2−
CO3 :
CO2 (g)
CO2 (aq)
CO2 (aq) + H2 O

H2 CO3
−

HCO3 + H+

H2 CO3
−

2−

HCO3

CO3

+ H+

Solution
1. Eliminate species that occur on both sides of the elementary reaction
equations:
CO2 (aq)
CO2 (g)
CO2 (aq) + H2 O
H2 CO3
−

HCO3

H2 CO3
−

HCO3 + H+
2−

CO3

+ H+

2. Determine the overall reaction by combining the remaining species
from step 1:
CO2 (g) + H2 O

2H+ + CO3

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.
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:
aA + bB

cC + dD

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

(5-21)

Reaction
Stoichiometry


234

5 Principles of Chemical Reactions

Using the stoichiometry of a reaction and the molecular weight of the
chemical species, it is possible to predict the theoretical mass of reactants
and 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) ↓ +2H2 O

(5-22)

As shown in Eq. 5-22, 1 mole of Ca(HCO3 )2 and 1 mole of Ca(OH)2 react
to form 2 moles of CaCO3 (s) and 2 moles of H2 O. The molecular weights
can be used to determine the theoretical mass of calcium hydroxide needed
to react with a specified mass of calcium bicarbonate and the amount of
calcium 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) that
will 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 the
quantity 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 the
reactants and products involved in the reaction. The molecular weights
are written below each species in the reaction.
Ca(HCO3 )2 + Ca(OH)2
74

162

2CaCO3 (s) ↓ + 2H2 O
2×100

2×18

2. Determine the molar relationship for the disappearance of Ca(HCO3 )2
and formation of CaCO3 (s):
2 mol CaCO3 (s)
1 mol Ca(HCO3 )2
= 1.23

100 g CaCO3 (s)
mol CaCO3 (s)

1 mol Ca(HCO3 )2
162 g Ca(HCO3 )2

g CaCO3 (s)
g Ca(HCO3 )2

Therefore, for each gram of Ca(HCO3 )2 removed, 1.23 g of CaCO3 (s)
will be produced.
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 )2 removed = (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]
3

× [1.23 g CaCO3 (s)]/[g Ca(HCO3 )2 ](1 kg/10 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.

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
B
a

c
d
C+ D
a
a

(5-23)

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 concentration of A, assuming there is no volume change upon reaction (which is
valid for most water treatment problems):
XA =
where

NA0 − NA
moles of A reacted
=
moles of A present initially
NA0

(5-24)

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
CA0 − CA
(5-25)
XA =
CA0

Reactant
Conversion


236

5 Principles of Chemical Reactions

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 CA are written in terms
of the conversion, the following expressions are obtained:
NA = NA0 (1 − XA )

(5-26)

CA = CA0 (1 − XA )

(5-27)

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 are
determined by subtracting the product of moles of A reacted and the
stoichiometric 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 in
terms of moles is shown below.
b
NB = NB0 − XA NA0
(5-28)
a
where

NB = final amount of reactant B, mol
NB0 = initial amount of reactant B, mol

The final amount of B in terms of concentration is
NB
b
= CB0 − XA CA0
CB =
V
a
b CA0 − CA

b
= CB0 −
CA0 = CB0 − (CA0 − CA )
a CA0
a
where CB = final concentration of reactant B, mol/L
V = solution volume, L
CB0 = initial concentration of reactant B, mol/L

(5-29)

Similarly, for reactants C and D,
c
(CA0 − CA )
(5-30)
a
d
CD = CD0 + (CA0 − CA )
(5-31)
a
where
CC , CD = final concentration of reactants C and D, mol/L
CC0 , CD0 = initial concentration of reactants C and D, mol/L
CC = CC0 +

The final concentration of the various species are related to one another
and to the conversion, as summarized in Table 5-2. As illustrated on
Fig. 5-1, the addition of a catalyst (or other change in the reaction conditions) may improve the selectivity and the reaction conversion for a given
time. The conversion from reactant to product can eventually reach the
thermodynamic limit of the reaction, as discussed in the following section.

5-2 Equilibrium Reactions

Table 5-2
Final concentration of various species related to one another and
to conversion
Initial Amount
Present, mol

NA0
NB0
NC0
ND0

Change in
Initial Amount,
mol

Final Amount
Present, mol

Final
Concentration,
mol/L

−NA0 XA

NA = NA0 (1 − XA )

CA = CA0 (1 − XA )

b
− XA NA0
a
c
XA NA0
a

b
NB = NB0 − XA NA0
a
c
NC = NC0 XA NA0
a

b
CB = CB0 − (CA0 − CA )
a
c
CC = CC0 + (CA0 − CA )
a

d
XA NA0
a

d
ND = ND0 + XA NA0
a

d
CD = CD0 + (CA0 − CA )
a

100

Reactant
conversion, %

Thermodynamic limit to conversion

Conversion
with catalysis
Conversion
without catalysis
0
Time

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

237


238

5 Principles of Chemical Reactions

system, the equilibrium will shift to form more A. In a chemical system, the
difference between the actual state and the equilibrium state is the driving
force 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 of
the reactants all raised to the power of the corresponding stoichiometric
coefficients is known as the equilibrium constant (Kc ) and, for the reaction
shown in Eq. 5-21, is written as
[C]c [D]d
= Kc
[A]a [B]b
where

(5-32)

Kc = 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,
respectively

For example, the ionization of carbonic acid, given previously as Eq. 5-5, is
shown as
HCO3− + H+
H2 CO3
The equilibrium constant at 25◦ C (neglecting nonidealities) for the reaction
shown above may be written as
[H+ ][HCO3− ]
= Kc = 5.0 × 10−7
[H2 CO3 ]

(5-33)

The value of equilibrium constants and reactant and product concentrations are typically small and, therefore, are often reported in the literature
using the operand ‘‘p,’’ which is defined as
p[i] = −log10 [i]
where

(5-34)

[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

(5-35)
pK = −log10 K
Therefore, the Kc reported in Eq. 5-33 may be written as
pKc = −log10 Kc = −log10 (5.0 × 10−7 ) = 6.3
Ionic Strength

(5-36)

In dilute solutions, the ions present behave independently of each other.
However, as the concentration of ions in solution increases, the activity of
the ions decreases because of ionic interaction. The ionic strength may be


5-2 Equilibrium Reactions

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 constant Kc for the dissociation of HOCl into OCl− and H+
(also known as an acid dissociation constant and typically reported as Ka ) is
10−7.5 (pKa = 7.5).
Solution
1. Write the equilibrium relationship for the equation provided in the

problem statement
−

[H+ ][OCl ]
= Ka = 10−7.5
[HOCl]
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 ]
= 10−7.5
[HOCl]

−

[OCl ]
= 10−0.5 = 0.32
[HOCl]

b. At pH 8.0, the hydrogen concentration [H + ] is equal to 10−8 and
the equilibrium relationship is written as
−

(10−8 )[OCl ]
= 10−7.5
[HOCl]

−

[OCl ]
= 100.5 = 3.2
[HOCl]

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.

239


240

5 Principles of Chemical Reactions

determined using the equation (Lewis and Randall, 1921)
I =
where

1
2

i

Ci Zi2

(5-37)

I = ionic strength of solution, mol/L(M)
Ci = concentration of species i, mol/L(M)
Zi = 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 strength
may be estimated from the total dissolved solids concentration using the
correlation (Stumm and Morgan, 1996)
I = (2.5 × 10−5 )(TDS)

(5-38)

where TDS = total dissolved solids, mg/L
To account for nonideal conditions encountered due to ion–ion interactions (e.g., at high ionic strength), an effective concentration term called
‘‘activity’’ is used.
Activity
and Activity
Coefficients

The activity of a substance is defined by the standard state conditions of

the substance and is based on commonly used standard conditions. The
standard reference conditions for zero free energy are defined as 1 atm
of pressure a temperature of 298.15 K (25◦ C), elements in their lowest
energy 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 chemical
references use 1 bar rather than 1 atm as the standard state, but the
difference is small (1 atm = 1.01325 bar). Nonetheless, when looking up
values for free energy in reference tables, note whether 1 atm or 1 bar is
used for the standard state. The activity coefficient of a chemical in water
may be determined as discussed below.
For ions and molecules in solution,
{i} = γi [i]
where

(5-39)

{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 for
electrolytes. As the solution becomes dilute (applicable to most applications in water treatment), γi approaches 1 and {i} approaches [i]. In the
dilute aqueous solutions normally encountered in water treatment, activity
coefficients are assumed to be equal to 1.


5-2 Equilibrium Reactions

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

{i} = γi Pi
where

(5-40)

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

(5-41)

xi = 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–Huckel
limiting law (Debye and Huckel,
1923):

log10 γi = −AZi2 I 1/2
where

(5-42)

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 modification of the Debye–H¨uckel equation, known as the Davies equation, can be
applied with acceptable error (Davies, 1967):
log10 γi = −AZi2
where

I 1/2
− 0.3I
1 + I 1/2

(5-43)

A = constant (see Eq. 5-44)

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)
√
2
6
A = 1.29 × 10
(5-44)
(Dε T )1.5

where

T = absolute temperature, K (273 + ◦ C)
Dε = dielectric constant (see Eq. 5-45)

241


242

5 Principles of Chemical Reactions

The dielectric constant may be determined using the equation (Harned
and Owen, 1958)
Dε ∼
= 78.54{1 − [0.004579(T − 298)] + [11.9 × 10−6 (T − 298)2 ]
+ [28 × 10−9 (T − 298)3 ]}
where

(5-45)

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
{C}c {D}d
=

=K
(γa [A])a (γb [B])b
{A}a {B}b
where

(5-46)

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
=K
{A}{B}b/a

(5-47)

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. The
calculation 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+ , Ca
of 0.001, 0.005, and 0.01 M at 25◦ C.

2+

, and Al3+ at ionic strengths

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):
log10 γNa+ = −0.51(1)2 0.001 = −1.61 × 10−2 ∴ γNa+ = 0.96
log10 γCa2+ = −0.51(2)2

0.001 = −6.45×10−2

log10 γAl3+ = −0.51(3)2 0.001 = −0.14

∴ γCa2+ = 0.86
∴ γAl3+ = 0.72


5-3 Thermodynamics of Chemical Reactions

2. Determine the activity coefficients for an ionic strength of 0.005 M at
25◦ C using the Debye–H¨uckel limiting law:
log10 γNa+ = −0.51(1)2 0.005 = −3.61 × 10−2 ∴ γNa+ = 0.92
log10 γCa2+ = −0.51(2)2 0.005 = −0.14

∴ γCa2+ = 0.72

log10 γAl3+ = −0.51(3)2 0.005 = −0.32

∴ γAl3+ = 0.48

3. Determine the activity coefficients for an ionic strength of 0.01 M at
25◦ C using the Davies equation (Eq. 5-43):

√
0.01
2
− 0.3(0.01)
log10 γNa+ = −0.51(1)
√
1 + 0.01
= −4.48 × 10−2
log10 γCa2+ = −0.51(2)2

∴ γNa+ = 0.90
√
0.01
√
1 + 0.01 − 0.3(0.01)

∴ γCa2+ = 0.66

= −0.18

√

2

log10 γAl3+ = −0.51(3)
= −0.40

0.01
− 0.3(0.01)
√

1 + 0.01

∴ γAl3+ = 0.40

Comment
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 determining 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

243


244

5 Principles of Chemical Reactions

many processes and reactions. Equilibrium thermodynamics can be used
to determine whether the treatment process is feasible, and the reaction
kinetics, described in the following sections, will provide a basis for the
treatment device size.
To determine whether a reaction will proceed (i.e., is thermodynamically
favorable), 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 zero
for a reaction to proceed. When evaluating chemical reactions, the entropy
requirement is typically satisfied, especially when heat is produced by
the reaction and, therefore, is not considered further in this text. The
second thermodynamic criterion necessary for a reaction to proceed is the
requirement that the change in free energy (final energy state minus initial
energy state) of the reaction must be less than zero.
To understand how the free energy of reaction changes as a reaction
proceeds, it is useful to examine the total free energy of reaction as a
function of the reaction extent, as shown on Fig. 5-2. Because the absolute
free energy of reaction cannot be determined easily, it is most common to
determine the change in free energy of a reaction. The free energy of the
reaction curve shown on Fig. 5-2 is compared to a convenient set of standard
conditions. 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 ion
concentration; and (3) a specified temperature, usually 25◦ C. For most
water treatment applications, the molar concentration is essentially equal
to the molal concentration and a 1 M solution is 1 mole per 1000 g of
solvent.

Total free energy, GT

Reference
Conditions

Slope of tangent line = ΔGRxn
Equilibrium
ΔGRxn = 0

Convenient
standard
conditions
Figure 5-2
Total free energy as function of the extent of the reaction.

Extent of reaction


5-3 Thermodynamics of Chemical Reactions

245

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

Free Energy
of Formation

GF ,i =
where

◦

GF ,i + RT ln{i}

(5-48)

GF ,i = free-energy change of formation of species i, kJ/mol
GF◦,i = free-energy change of formation per mole of i at standard
conditions, kJ/mol
R = universal gas law constant, 8.314 × 10−3 kJ/mol · K
T = absolute temperature, K(273 + ◦ C)
{i} = activity of species i

Thermodynamic constants may be found in various reference books, including Stumm and Morgan (1996) and Lange’s Handbook (Dean, 1992).
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:
c
d
b
C+ D
A+ B
a
a
a
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:
b
c
d

GF ,B +
GF ,C +
GF ,D
(5-49)
GRxn,A = − GF ,A −
a
a
a
GRxn,A = free-energy change of reaction per mole of A, kJ/mol
where
GF ,A = change in free energy of reactant A, kJ/mol
GF ,B = change in free energy of reactant B, kJ/mol
GF ,C = change in free energy of product C, kJ/mol
GF ,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
expression
c
d
◦
◦
GF ,C + RT ln{C}c/a +
GF ,D + RT ln{D}d/a
GRxn,A =
a
a
(5-50)
b

◦
◦
GF ,B − RT ln{B}b/a
− GF ,A − RT ln{A} −
a

Free Energy
of Reaction


246

5 Principles of Chemical Reactions

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/L

The free-energy change of the reaction per mole of A at standard conditions
◦
, can be written as
(25◦ C and 1 atm pressure), GRxn,A
b
c
d
◦

◦
◦
GF ,B +
GF ,C +
GF ,D
(5-51)
a
a
a
Equation 5-50 can be further simplified by substituting in the relationship
shown in Eq. 5-51:
{C}c/a {D}d/a
◦
(5-52)
GRxn,A = GRxn,A + RT ln
{A}{B}b/a
The logarithmic term in Eq. 5-52 is called the reaction quotient Q:
◦

◦

GRxn,A = − GF ,A −

{C}c/a {D}d/a
(5-53)
{A}{B}b/a
If the stoichiometric coefficient a had not been factored out of Eq. 5-21,
then Eq. 5-52 would be written per a moles of A as
Q =

{C}c {D}d
(5-54)
{A}a {B}b
When examining thermodynamic data, it is important to make certain that
the free energy that is reported is per mole of A. Finally, the thermodynamic
criterion that must be met for a reaction to proceed as written from the
initial state toward the final state may be expressed as
◦

a GRxn,A = a GRxn,A + RT ln

GRxn,A must be < 0

(5-55)

While a reaction is thermodynamically feasible when GRxn,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 energy
than the reactants. Alternately, if GRxn,A > 0, the reverse reaction would
be thermodynamically feasible.
Free Energy
at Equilibrium

Another useful relationship, known as the equilibrium state, is obtained
when GRxn,A = 0. When GRxn,A = 0 in Eq. 5-52, the reaction quotient is
equal to the equilibrium constant K as shown below:
{C}c/a {D}d/a
=0
(5-56)
{A}{B}b/a

If the relationship shown in Eq. 5-47 for the equilibrium constant is
substituted for the reaction quotient, the following expression is obtained:
GRxn,A =

◦

◦

GRxn,A + RT ln

GRxn,A = −RT ln

{C}c/a {D}d/a
{A}{B}b/a

= −RT ln [K]

(5-57)


5-3 Thermodynamics of Chemical Reactions

247

Rearranging Eq. 5-57 and solving for the equilibrium constant result in the
expression
K = e−

◦
GRxn,A /RT

(5-58)

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.
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, GF◦,A,aq , is needed to calculate GRxn,A , and the free energy of
formation may be reported for the gas phase, GF◦,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,
GF◦,A,aq , based on the free energy of formation in the gas phase, GF◦,A,gas :
◦

GVol,A =
where

◦

◦

GF ,A,gas −

GF ,A,aq = −RT ln HPC

Calculation
of Free Energy

of Formation
Using Henry’s
Constant

(5-59)

◦
GVol,A
= free-energy change of volatilization per mole of A at
standard conditions, kJ/mol
GF◦,A,gas = free-energy change of formation per mole of A in gas
phase at standard conditions, kJ/mol
GF◦,A,aq = free-energy change of formation per mole of A in
aqueous phase at standard conditions, kJ/mol
HPC = 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:
◦

◦

GF ,A,aq =

GF ,A,gas + RT ln HPC

(5-60)

Consequently, GF◦,A,aq can be calculated from GF◦,A,gas if HPC is known.
Henry’s law is presented and discussed in detail in Chap. 14.

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

where

T
T =298 K

=

T
T =298 K

−

◦
HRxn
(T )
dT
T2

◦
HRxn
(T ) = standard enthalpy of reaction that depends
on temperature

(5-61)

Temperature
Dependence
of Free-Energy
Change


248

5 Principles of Chemical Reactions

The temperature-dependent standard enthalpy of reaction is defined as
◦

HRxn (T ) =
where

T
T =298 K

◦

Cp,Rxn dT +

HRxn,298 K

(5-62)

Cp,Rxn = change in heat capacity for the reaction, kJ/mol

◦
HRxn,298K
= standard enthalpy at 298 K

The heat capacity term may be calculated using the equation (Poling et al.,
2001)
Cp,i = A + BT + CT 2 + DT 3
(5-63)
where A, B, C, D = constants
Cp,i = isobaric (constant-pressure) heat capacity
for compound i
To calculate Cp,Rxn , the difference of each constant (A, B, C, and D)
between products and reactants needs to be calculated:
Cp,Rxn =

A+

BT +

CT 2 +

DT 3

(5-64)

For the reaction shown in Eq. 5-23, the terms in Eq. 5-64 are given by the
expressions
c
b
d

AD + AC − AB − AA
a
a
a
d
c
b
C = CD + CC − CB − CA
a
a
a
A=

d
c
b
BD + BC − BB − BA
a
a
a
(5-65)
d
c
b
D = D D + DC − DB − DA
a
a
a
B=

Substituting the relationships shown in Eq. 5-64 into Eq. 5-62 and subsequently into Eq. 5-61, the following expression is obtained:
◦
GRxn
T

T

⎡

T =298 K
T

=−

T =298 K

⎢
⎢
⎣

⎤
A(T − 298)
B(T 2 − 2982 )
+
+
⎥
T2
T2
2T 2
⎥ dT

3
3
⎦
C T − 298
D T 4 − 2984
+
+
2
2
3T
4T
(5-66)

◦
HRxn,298
K

◦
◦
For the case where HRxn
does not depend on temperature ( HRxn
is constant), Eq. 5-66 can be simplified as
◦
GRxn
T

T
T =298 K

=

◦
GRxn,T

T

−

◦
GRxn,298
K

298 K

=

◦

HRxn,298 K

1
1
−
T
298 K
(5-67)

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

5-3 Thermodynamics of Chemical Reactions

constant (Keq ) at different temperatures:
lnK |TT =298 K =
ln
where

KT
K298 K

=

◦
HRxn,298
K

T

T =298 K
◦
HRxn,298
K

RT 2

dT

1
1

−
298 K
T

R

(5-68)

KT = equilibrium constant at temperature T , K (273 + ◦ C)
K298 K = equilibrium constant at 298 K

Using Eq. 5-68, the linear relationship between ln K and 1/T can be
determined by plotting the function
ln K = −

◦
HRxn,298
K

RT

+ const

(5-69)

◦
For most reactions occurring in water treatment processes, HRxn
can be
assumed to be constant because HRxn does 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
H2 O
are as follows:
GF◦,H O = −237.18 kJ/mol
2

GF◦,H+
G◦ −
F ,OH

= 0 kJ/mol
= −157.29 kJ/mol

−

H+ + OH

HF◦,H

2O
◦
HF ,H+
H◦ −
F ,OH

= −285.83 kJ/mol

= 0 kJ/mol
= −230.0 kJ/mol

Calculate the pH of neutrality and free-energy change of the reaction at
◦
10◦ C. Assume that HRxn
does not change with temperature.
Solution
1. Calculate the equilibrium constant, using Eq. 5-58, for water at 25◦ C.
◦
using Eq. 5-51:
a. Calculate GRxn,H
O
2

◦

GRxn,H

2O

=

◦

GF ,OH− +

◦

GF ,H+ −

◦

GF ,H

2O

= −157.29 + 0 − (−237.18 K) = 79.89 kJ/mol

249