AEROSOL CHEMICAL PROCESSES IN THE ENVIRONMENT - CHAPTER 6 docx

135

6

Reversible Chemical Reactions
in Aerosols

Mark Z. Jacobson

CONTENTS

Introduction 135
Definitions 136
Equilibrium Equations and Relations 136
Equilibrium Equations 136
Equilibrium Relations and Constants 139
Temperature Dependence of the Equilibrium Coefficient 142
Forms of Equilibrium Coefficient Equations 143
Mean Binary Activity Coefficients 144
Temperature Dependence of Mean Binary Activity Coefficients 146
Mean Mixed Activity Coefficients 147
The Water Equation 148
Method of Solving Equilibrium Equations 151
Solid Formation and Deliquescence Relative Humidity 153
Equilibrium Solver Results 154
Summary 155
References 155

INTRODUCTION

Aerosols in the atmosphere affect air quality, meteorology, and climate in several ways. Submicron-
sized aerosols (smaller than 1

µ

m in diameter) affect human health by directly penetrating to the
deepest part of human lungs. Aerosols between 0.2 and 1.0

µ

m in diameter that contain sulfate,
nitrate, and organic carbon, scatter light efficiently. Aerosols smaller than 1.0

µ

m that contain
elemental carbon, absorb efficiently. Aerosol absorption and scattering are important because they
affect radiative fluxes and, therefore, air temperatures and climate. Aerosols also serve as sites on
which chemical reactions take place and as sinks in which some gas-phase species are removed
from the atmosphere.
The change in size and composition of an aerosol depends on several processes, including
nucleation, emissions, coagulation, condensation, dissolution, reversible chemical reactions, irre-
versible chemical reactions, sedimentation, dry deposition, and advection. In this chapter, dissolu-
tion and reversible chemical reactions are discussed. These processes are important for determining
the ionic, solid, and liquid water content of aerosols.

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136

Aerosol Chemical Processes in the Environment

DEFINITIONS

Dissolution

is a process that occurs when a gas, suspended over a particle surface, adsorbs to and
dissolves in liquid on the surface. The liquid in which the gas dissolves is a

solvent

. A solvent
makes up the bulk of a solution, and in atmospheric particles, liquid water is most often the solvent.
In some cases, such as when sulfuric acid combines with water to form particles, the concentration
of sulfuric acid exceeds the concentration of liquid water, and sulfuric acid may be the solvent.
Here, liquid water is assumed to be the solvent in all cases.
A species, such as a gas or solid, that dissolves in solution is a

solute

. Together, solute and
solvent make up a

solution

, which is a homogeneous mixture of substances that can be separated
into individual components upon a change of state (e.g.,

freezing). A solution may contain many
solutes. Suspended material (e.g.,



solids) may also be mixed throughout a solution. Such material
is not considered part of a solution.
The ability of the gas to dissolve in water depends on the solubility of the gas in water.

Solubility

is the maximum amount of a gas that can dissolve in a given amount of solvent at a given
temperature. Solutions usually contain solute other than the dissolved gas. The solubility of a gas
depends strongly on the quantity of the other solutes because such solutes affect the thermodynamic
activity of the dissolved gas in solution. Thermodynamic activity is discussed shortly. If water is
saturated with a dissolved gas, and if the solubility of the gas changes due to a change in composition
of the solution, the dissolved gas can

evaporate

from the solution to the gas phase. Alternatively,
dissociation products of the dissolved gas can combine with other components in solution and

precipitate

as solids.
In solution, dissolved gases can dissociate and react chemically.

Dissociation

of a dissolved
molecule is the process by which the molecule breaks into simpler components, namely ions. This
process can be described by

reversible



chemical



reactions

, also called

chemical



equilibrium



reac-
tions

or

thermodynamic



equilibrium



reactions

. Such reactions are reversible, and their rates in the
forward and backward directions are generally fast. Dissociated ions and undissociated molecules
can further react reversibly or irreversibly with other ions or undissociated molecules in solution.

Irreversible



chemical



reactions

act only in the forward direction and are described by first-order
ordinary differential equations. When they occur in solution, irreversible reactions are called

aqueous

reactions

.

EQUILIBRIUM EQUATIONS AND RELATIONS

Reversible chemical reactions describe dissolution, dissociation, and precipitation processes. In this
section, different types of equilibrium equations are discussed and rate expressions, including
temperature dependence, are derived.

E

QUILIBRIUM

E

QUATIONS

An

equilibrium



equation

describes a reversible chemical reaction. A typical equation has the form
(6.1)
where

D

,

E

,

A

, and

B

are species and the

ν

’s are dimensionless

stoichiometric



coefficients

or number
of moles per species divided by the smallest number of moles of any reactant or product in the
reaction. Each reaction must conserve mass. Thus,

(6.2)
νν νν
DE AB
DE AB++…⇔ ++… ,
km
ii i
i
ν
∑
= 0,

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Reversible Chemical Reactions in Aerosols

137

where

m

i



is the molecular weight of each species and

k

i

= +1 for products and –1 for reactants.
The reactants and/or products of an equilibrium equation can be solids, liquids, ions, or gases.
Reversible dissolution reactions have the form
(6.3)
where (g) indicates a gas and (aq) indicates that the species is dissolved in solution. In this equation,
the gas phase and dissolved (solution) phase of species AB are assumed to be in equilibrium with
each other at the gas–solution interface. Thus, the number of molecules of AB transferring from
the gas to the solution equals the number of molecules transferring in the reverse direction. In the
atmosphere, gas–solution interfaces occur at the air–ocean, air–cloud drop, and air–aerosol inter-
faces. Examples of

dissolution



reactions



that



occur



at

these



interfaces

are
(6.4)
(6.5)
(6.6)
(6.7)
The reaction
(6.8)
is also a reversible dissolution reaction. In equilibrium, almost all sulfuric acid is partitioned to the
aqueous phase; thus, the relation is rarely used. Instead, sulfuric acid transfer to the aqueous phase
is treated as a diffusion-limited condensational growth process.
Once dissolved in solution, the species on the right sides of Equations 6.4 to 6.8 often dissociate
into ions. Substances that undergo partial or complete dissociation in solution are

electrolytes

. The
degree of dissociation of an electrolyte depends on the acidity of solution, the strength of the
electrolyte, the concentrations of other ions in solution, the temperature, and other conditions.
The

acidity

of a solution is a measure of the concentration of

hydrogen



ions

(

protons

or H

+

ions) in solution. Acidity is measured in terms of

pH

, defined as
(6.9)
where [H

+

] is the

molarity

of H

+

(moles H

+

L

–1

solution). The more acidic the solution, the higher
the molarity of protons and the lower the pH. Protons in solution are donated by acids that dissolve.
Examples of such acids are H

2

CO

3

(aq), HCl(aq), HNO

3

(aq), and H

2

SO

4

(aq). The abilities of acids
to dissociate into protons and anions vary. HCl(aq), HNO

3

(aq), and H

2

SO

4

(aq) dissociate readily,
while H

2

CO

3

(aq) does not. Thus, the former species are

strong

acids

and the latter species is a

weak



acid

. Because all acids are electrolytes, a strong acid is a

strong



electrolyte

(e.g.

,

it dissociates
significantly) and a weak acid is a

weak

electrolyte

. Hydrochloric acid is a strong acid and strong
electrolyte in water because it almost always dissociates completely by the reaction
(6.10)
AB(g) AB(aq),⇔
HCl(g) HCl(aq)⇔
HNO (g) HNO (aq)
33
⇔
CO (g) CO (aq)
22
⇔
NH (g) NH (aq)
33
⇔
H SO (g) H SO (aq)
24 24
⇔
pH H=−
[]
+
log ,
10
HCl(aq) H Cl⇔+
+−
.

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138

Aerosol Chemical Processes in the Environment

Sulfuric



acid

is also a strong acid and strong electrolyte and dissociates to bisulfate by
(6.11)
While HCl(aq) dissociates significantly at a pH above –6, H

2

SO

4

(aq) dissociates significantly at a
pH above –3. Another strong acid,

nitric

acid

, dissociates significantly at a pH above –1. Nitric
acid dissociates to nitrate by
(6.12)

Bisulfate

is also a strong acid and electrolyte because it dissociates significantly at a pH above
about +2. Bisulfate dissociation to

sulfate

is given by
(6.13)

Carbon



dioxide

is a weak acid and electrolyte because it dissociates significantly at a pH above
only +6. Carbon dioxide converts to

carbonic



acid

and dissociates to bicarbonate by
(6.14)
Dissociation of



bicarbonate

occurs at a pH above +10. This reaction is
(6.15)
While acids provide hydrogen ions,

bases

provide

hydroxide



ions

(OH

–

). Such ions react with
hydrogen ions to form neutral water via
(6.16)

An important base in the atmosphere is ammonia.

Ammonia

reacts with water to form

ammonium

and the hydroxide ion by
(6.17)
Since some strong electrolytes, such as HCl(aq) and HNO
3
(aq), dissociate completely in
atmospheric particles, the undissociated forms of these species are sometimes ignored in equilibrium
models. Instead, gas-ion equilibrium equations replace the combination of gas-liquid, liquid-ion
equations. For example, the equations
(6.18)
can replace Equations 6.4 and 6.10. Similarly,
(6.19)
can replace Equations 6.5 and 6.12.
H SO (aq) H HSO
24
⇔+
+−
4
.
HNO (aq) H NO
3
⇔+
+−

3
.
HSO H SO
44
2
−+
−
⇔+ .
CO (aq) H O(aq) H CO (aq) H HCO
22 23 3
+⇔ ⇔+
+
−
.
HCO H CO
33
2−
+
−
⇔+ .
H O(aq) H OH .
2
–
⇔+
+
NH (aq) + H O(aq) NH OH .
32 4
–
⇔+
+

HCl(g) H Cl
–
⇔+
+
HNO (g) H NO
33
–
⇔+
+
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© 2000 by CRC Press LLC
Reversible Chemical Reactions in Aerosols 139
Once in solution, ions can precipitate to form solid electrolytes if conditions are right. Alter-
natively, existing solid electrolytes can dissociate into ions if the particle water content increases
sufficiently. Examples of solid precipitation/dissociation reactions for ammonium-containing elec-
trolytes include
(6.20)
(6.21)
(6.22)
Examples of such reactions for sodium-containing electrolytes are
(6.23)
(6.24)
(6.25)
If the relative humidity is sufficiently low, a gas can react chemically with another adsorbed
gas on a particle surface to form a solid. Such reactions can be simulated with gas-solid equilibrium
reactions, such as
(6.26)
(6.27)
In sum, equilibrium relationships usually describe aqueous-ion, ion-ion, ion-solid, gas-solid,
or gas-ion reversible reactions. Relationships can be written for other interactions as well. Table

6.1 shows several equilibrium relationships of atmospheric importance.
EQUILIBRIUM RELATIONS AND CONSTANTS
Species concentrations in a reversible reaction, such as Equation 6.1, are interrelated by
(6.28)
where K
eq
(T) is a temperature-dependent equilibrium coefficient and {A}…, etc., are thermody-
namic activities. Thermodynamic activities measure the effective concentration or intensity of the
substance. The activity of a substance differs, depending on whether the substance is in the gas,
undissociated aqueous, ionic, or solid phases. The activity of a gas is its saturation vapor pressure
(atm). Thus,
(6.29)
NH Cl(s) NH Cl
44
–
⇔+
+
NH NO (s) NH NO
43 4 3
–
⇔+
+
NH SO (s) NH SO
4
2
444
2–
()
⇔+
+

2.
NaCl(s) Na Cl
–
⇔+
+
NaNO (s) Na NO
33
–
⇔+
+
Na SO (s) Na SO
24 4
2–
⇔+
+
2.
NH Cl(s) NH (g) HCl(g)
43
⇔+
NH NO (s) NH (g) HNO (g).
43 3 3
⇔+
AB
DE
KT
AB
DE
eq
{} {}
{} {}

=
()
νν
νν


,
Ap
sA
g
()
{}
=
,
.
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140

Aerosol Chemical Processes in the Environment

The activity of an ion in solution or an undissociated electrolyte is its

molality

(

m

A

) (moles
solute kg

–1

solvent) multiplied by its

activity



coefficient

(

γ

) (unitless). Thus,
(6.30)
(6.31)
respectively. An

activity



coefficient

accounts for the deviation from ideal behavior of a solution. It
is a dimensionless parameter by which the molality of a species in solution is multiplied to give
the species’ thermodynamic activity. In an ideal, infinitely dilute solution, the activity coefficient
of a species is unity. In a nonideal, concentrated solution, activity coefficients may be greater than

TABLE 6.1
Equilibrium Reactions, Coefficients, and Coefficient Units

No. Reaction A B C Units Ref.

a

1 HNO

3

(g)

⇔

HNO

3

(aq) 2.10

×

10

5

mol kg

–1

atm

–1

D
2NH

3

(g)

⇔

NH

3

(aq) 5.76

×

10

1

13.79 -5.39 mol kg

–1

atm

–1

A
3CO

2

(g)

⇔

CO

2

(aq) 3.41

×

10

–2

8.19 -28.93 mol kg

–1

atm

–1

A
4CO

2

(aq) + H

2

O(aq)

⇔

H

+

+ HCO

3
–

4.30

×

10

–7

–3.08 31.81 mol kg

–1

A
5NH

3

(aq) + H

2

O(aq)

⇔

NH

4+

+ OH

–

1.81

×

10

–5

–1.50 26.92 mol kg

–1

A
6 HNO

3

(aq)

⇔

H

+

+ NO

3
–

1.20

×

10

1

29.17 16.83 mol kg

–1

N
7 HCl(aq)

⇔

H

+

+ Cl

–

1.72

×

10

6

23.15 mol kg

–1

O
8H

2

O(aq)

⇔

H

+

+ OH

–

1.01

×

10

–14

–22.52 26.92 mol kg

–1

A
9H

2

SO

4

(aq)

⇔

H

+

+ HSO

4
–

1.00

×

10

3

mol kg

–1

R
10 HSO

4
–

⇔

H

+

+



SO

4
2–

1.02

×

10

–2

8.85 25.14 mol kg

–1

A
11 HCO

3
–

⇔

H

+

+ CO

3
2–

4.68

×

10

–11

–5.99 38.84 mol kg

–1

A
12 HNO

3

(g)

⇔

H

+

+ NO

3
–

2.51

×

10

6

29.17 16.83 mol

2

kg

–2

atm

–1

A
13 HCl (g)

⇔

H

+

+ Cl

–

1.97

×

10

6

30.19 19.91 mol

2

kg

–2

atm

–1

A
14 NH

3

(g) + H

+

⇔

NH

4
+

1.03

×

10

11

34.81 –5.39 atm

–1

A
15 NH

3

(g) + HNO

3

(g)

⇔

NH

4
+

+ NO

3
–

2.58

×

10

17

64.02 11.44 mol

2

kg

–2

atm

–2

A
16 NH

3

(g) + HCl(g)

⇔

NH

4
+

+ Cl

–

2.03

×

10

17

65.05 14.51 mol

2

kg

–2

atm

–2

A
17 NH

4

NO

3

(s)

⇔

NH

4
+

+ NO

3
–

1.49

×

10

1

–10.40 17.56 mol

2

kg

–2

A
18 NH

4

Cl(s)

⇔

NH

4
+

+ Cl

–

1.96

×

10

1

–6.13 16.92 mol

2

kg

–2

A
19 NH

4

HSO

4

(s)

⇔

NH

4
+

+ HSO

4
–

1.38

×

10

2

–2.87 15.83 mol

2

kg

–2

A
20 (NH

4

)

2

SO

4

(s)

⇔

2 NH

4
+

+ SO

4
2–

1.82 –2.65 38.57 mol

3

kg

–3

A
21 (NH

4

)

3

H(SO

4

)

2

(s)

⇔

3 NH

4
+

+HSO

4

+SO
4
2–
2.93 × 10
1
–5.19 54.40 mol
5
kg
–5
A
22 NaNO
3
(s) ⇔ Na
+
+ NO
3
–
1.20 × 10

1
–8.22 16.01 mol
2
kg
–2
A
23 NaCl(s) ⇔ Na
+
+ Cl
–
3.61 × 10
1
–1.61 16.90 mol
2
kg
–2
A
24 NaHSO
4
(s) ⇔ Na
+
+ HSO
4
–
2.84 × 10
2
–1.91 14.75 mol
2
kg
–2

A
25 Na
2
SO
4
(s) ⇔ 2 Na
+
+ SO
4
2–
4.80 × 10
–1
0.98 39.50 mol
3
kg
–3
A
Note: The equilibrium coefficient reads,
where T
0
= 298.15K and the remaining terms are defined in Equation 6.45.
a
A: Derived from data in Reference 21; D: From Reference 22; N: Derived from a combination of other rate coefficients
in the table; O, R: From Reference 23. With permission.
KT
T
T
T
T
T

T
eq
()














=−+−+AB C
000
exp ln ,11
A
AA
+
{}
=
++
m γ and
A
AA
aq

()
{}
= m γ ,
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© 2000 by CRC Press LLC
Reversible Chemical Reactions in Aerosols 141
or less than unity. Debye and Huckel showed that, in relatively dilute solutions, where ions are far
apart, the deviation of molality from thermodynamic activity is caused by Coulombic (electric)
forces of attraction and repulsion. At high concentrations, ions are closer together, and ion-ion
interactions affect activity coefficients more significantly than do Coulombic forces.
The activity of liquid water in an atmospheric particle is the ambient relative humidity (fraction).
Thus,
(6.32)
where a
w
denotes the activity of water and f
r
is the relative humidity, expressed as a fraction. Finally,
solids are not in solution, and their concentrations do not affect the molalities or activity coefficients
of electrolytes in solution. Thus, the activity of any solid is unity; that is,
(6.33)
Equation 6.28 is derived by minimizing the Gibbs free-energy change of a system. The Gibbs
free-energy change per mole (∆G) (J mole
–1
) is a measure of the maximum amount of useful work
per mole that may be obtained from a change in enthalpy or entropy in the system. The relationship
between the Gibbs free-energy change and the composition of a chemical system is
(6.34)
where µ
i

is the chemical potential of the species (J mole
–1
) and k
i
= +1 for products and –1 for
reactants. Chemical potential is a measure of the intensity of a chemical substance and is a function
of temperature and pressure. It is really a measure of the change in free energy per change in moles
of a substance, or the partial molar free energy. The chemical potential is
(6.35)
where µ
I
o
is the chemical potential at a reference temperature of 298.15K, and {a
i
} is the thermo-
dynamic activity of species i. The chemical potential can be substituted into Equation 6.34 to give
(6.36)
Rewriting this equation yields
(6.37)
where
(6.38)
HOaq
2
()
{}
==af
wr
,
A s
()

{}
= 1.
∆Gk
iii
i
=
∑
νµ,
µµ
ii i
RT a=+
{}
o*
ln ,
∆Gk RTka
iii
i
ii i
i
=
()
+
{}
()
∑∑
νµ ν
o*
ln .
∆∆GGRT a
i

k
i
i
i
=+
{}
∏
o *
ln ,
ν
∆Gk
iii
i
oo
=
()
∑
νµ
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© 2000 by CRC Press LLC
142 Aerosol Chemical Processes in the Environment
is the standard molal Gibbs free energy of formation (J mole
–1
) for the reaction. Equilibrium occurs
when ∆G = 0 at constant temperature and pressure. Under such conditions, Equation 6.37 becomes
(6.39)
The left side of Equation 6.39 is the equilibrium coefficient. Thus,
(6.40)
Substituting Equation 6.40 into Equation 6.39 and expanding the product term gives
(6.41)

which is the relationship shown in Equation 6.28.
TEMPERATURE DEPENDENCE OF THE EQUILIBRIUM COEFFICIENT
The temperature dependence of the equilibrium coefficient is calculated by solving the Van ’t Hoff
equation,
(6.42)
where
∆
H
T
o
is the change in total enthalpy (J mole
–1
) of the reaction. The change in enthalpy can
be approximated by
(6.43)
when the standard change in molal heat capacity of the reaction (∆c
P
o
) (J mole
–1
K
–1
) does not
depend on temperature. In this equation,

is the standard enthalpy change in the reaction (J
mole
–1
) at temperature T
o

= 298.15K. Combining Equations 6.42 and 6.43 and writing the result in
integral form gives
(6.44)
Integrating this equation yields the temperature-dependent equilibrium coefficient expression
(6.45)
exp .
*
−
()
[]
=
{}
∏
∆GRT a
i
i
i
o
ν
KT G RT
eq
()
=−
()
[]
exp .
*
∆
o
KT a

AB
DE
eq i
k
i
ii
AB
DE
()
=
{}
=
{} {}
{} {}
∏
ν
νν
νν


d
d
o
ln
,
*
KT
T
H
RT

eq
T
()
=
∆
2
∆∆∆HHcTT
TTp
ooo
o
o
≈+ −
()
∆H
T
o
o
dd
o
o
o
oo
o
ln .
*
KT
HcTT
RT
T
eq

T
T
Tp
T
T
()
=
+−
()








∫∫
∆∆
2
KT KT
H
RT
T
T
c
R
T
T
T

T
eq eq o
T
p
()
=
()
−−






−−+





















exp ln
**
∆
∆
o
o
o
o
o
oo
11
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© 2000 by CRC Press LLC
Reversible Chemical Reactions in Aerosols 143
where K
eq
(T
o
) is the equilibrium coefficient at temperature, T
o
. Values of

and ∆c
p
o

are measured
experimentally. Table 6.1 shows temperature-dependent parameters for several equilibrium
reactions.
FORMS OF EQUILIBRIUM COEFFICIENT EQUATIONS
Each reaction in Table 6.1 can be written in terms of thermodynamic activities and an equilibrium
coefficient. For example, an equilibrium coefficient equation for the reaction
(6.46)
is
(6.47)
where

is the saturation vapor pressure of nitric acid (atm), is the molality of
nitric acid in solution (moles kg
–1
), and

is the activity coefficient of dissolved, undissociated
nitric acid (unitless). The equilibrium coefficient has units of (moles kg
–1
atm
–1
).
When the equilibrium coefficient relates the saturation vapor pressure of a gas to the molality
(or molarity) of the dissolved gas in a dilute solution, the coefficient is called a Henry’s constant.
Henry’s constants (moles kg
–1
atm
–1
), like other equilibrium coefficients, are temperature and solvent
dependent. Henry’s law states that, for a dilute solution, the pressure exerted by a gas at the

gas–liquid interface is proportional to the molality of the dissolved gas in solution. For a dilute
solution,

= 1, and Equation 6.47 obeys Henry’s law.
A dissociation equation has the form
(6.48)
The equilibrium coefficient expression for this reaction is
(6.49)
where the equilibrium coefficient has units of (moles kg
–1
).
In Equation 6.49, the activity coefficients are determined by considering a mixture of all
dissociated and undissociated electrolytes in solution. Thus, the coefficients are termed mixed
activity coefficients. More specifically,

are single-ion mixed activity coefficients, and
is a mean (geometric mean) mixed activity coefficient. When H
+
, and NO
3
–
are alone in
solution,

are single-ion binary activity coefficients, and

is a mean (geometric
mean) binary activity coefficient. Activity coefficients for single ions are difficult to measure because
single ions cannot be isolated from a solution. Single-ion activity coefficients are easier to estimate
∆H

T
o
o
HNO (g) HNO (aq)
33
⇔
HNO aq
HNO g
HNO aq HNO aq
,HNO g
3
3
33
3
()
{}
()
{}
==
()
() ()
()
m γ
p
KT
s
eq
,
p
s,HNO (g)

3
m
HNO (aq)
3
γ
HNO (aq)
3
γ
HNO (aq)
3
HNO (aq) H NO
3
+
3
–
⇔+ .
HNO
HNO aq
3
-
H H NO NO
HNO aq HNO aq
HNOHNO
HNO aq HNO aq
3
-
3
-
3
-

3
-
+
() () () ()
{}{ }
()
{}
==
++ + +
3
2
33 33
mm
m
mm
m
γγ
γ
γ
γ
,
γγ
HNO
and
3
+
γ
HNO
3
-+

,
γγ
HNO
and
3
+
γ
HNO
3
-+
,
L829/frame/ch06 Page 143 Monday, January 31, 2000 2:59 PM
© 2000 by CRC Press LLC
144 Aerosol Chemical Processes in the Environment
mathematically. Mean binary activity coefficients are measured in the laboratory. Mean mixed
activity coefficients can be estimated from mean binary activity coefficient data through a mixing
rule.
A geometric mean activity coefficient is related to a single-ion activity coefficient by
(6.50)
where γ
±
is the mean activity coefficient, γ
+
and γ
–
are the activity coefficients of the single cation
and anion, respectively, and ν
+
and ν
–

are the stoichiometric coefficients of the cation and anion,
respectively. In Equation 6.48, ν
+
= 1 and ν
–
= 1.
Raising both sides of Equation 6.50 to the power ν
+
+ ν
−
gives
(6.51)
which is form of the mean activity coefficient used in Equation 6.49.
When ν
+
= 1 and ν
–
= 1, the electrolyte is univalent. When ν
+
> 1 or ν
–
> 1, the electrolyte is
multivalent. When ν
+
= ν
–
for a dissociated electrolyte, the electrolyte is symmetric; otherwise, it
is nonsymmetric. In all cases, a dissociation reaction must satisfy the charge balance requirement
(6.52)
where z

+
is the positive charge on the cation and z
-
is the negative charge on the anion.
MEAN BINARY ACTIVITY COEFFICIENTS
The mean binary activity coefficient of an electrolyte, which is primarily a function of molality
and temperature, can be determined from measurements or estimated from theory. Measurements
of binary activity coefficients for several species at 298.15K are available. Parameterizations have
also been developed to predict the mean binary activity coefficients. One parameterization is Pitzer’s
method,
1,2
which estimates the mean binary activity coefficient of an electrolyte at 298.15K with
(6.53)
where γ
0
12b
is the mean binary activity coefficient of electrolyte 1-2 (cation 1 plus anion 2) at the
reference temperature (298.15K), Z
1
and Z
2
are the absolute value of the charges of cation 1 and
anion 2, respectively, m
12
is the molality of electrolyte dissolved in solution, and ν
1
and ν
2
are the
stoichiometric coefficients of the dissociated ions (assumed positive here). In addition,

(6.54)
(6.55)
γγγ
νν
νν
±
+
()
=
()
+−
+−
+-
1
,
γγγ
νν
νν
±
+
()
+−
+−
=
+ –
,
zz
++ −−
+=νν0,
ln γ

νν
νν
νν
νν
γγ γ
12
0
12 12
12
12
12 12
2
12
32
12
1
2
2
2
b
fB C=+
+
+
()
+
ZZ mm
f
γ
=−
+

++
()






0 392
112
2
12
112
12
12
12
.

ln .
I
I
I
Be
12 12
1
12
2
212
2
2

4
1122
12
γ
β
β
=+ − +−
()
[]
()
()
−
I
II
I
,
L829/frame/ch06 Page 144 Monday, January 31, 2000 2:59 PM
© 2000 by CRC Press LLC

Reversible Chemical Reactions in Aerosols

145

where

I

is the ionic strength of the solution (moles kg

–1

). The

ionic



strength

is a measure of the
interionic effects resulting from attraction and repulsion among ions and is given by
(6.56)
In this equation,

N

C

is the number of different cations,

N

A

is the number of different anions in
solution, odd-numbered subscripts refer to cations, and even-numbered subscripts refer to anions.
In the case of one electrolyte, such as HCl(aq) alone in solution,

N

C

= 1 and

N

A

= 1. The quantites,
are empirical parameters derived from measurements. Pitzer parameters for
three electrolytes are shown in Table 6.2.
While Pitzer’s method accurately predicts mean binary activity coefficients at 298.15K from
physical principles, its limitation is that the coefficients are typically valid up to about 6 molal (m)
only. Figure 6.1 shows a comparison of activity coefficients predicted by Pitzer’s method to those
measured by Hamer and Wu.

3

The measured data are accurate to higher molalities.
Whether molality-dependent mean binary activity coefficients at 298.15K are determined from
measurements or theory, they can be parameterized with a

polynomial



fit

of the form
(6.57)

TABLE 6.2
Pitzer Parameters for Three Electrolytes

Electrolyte

ββ
ββ


11
11
22
22

(1 )

ββ
ββ


11
11
22
22

( 2)

C

12
((
((


γγ
γγ
))
))



HCl 0.17750 0.2945 0.0012
HNO

3

0.1119 0.3206 0.0015
NH

4

NO

3

–0.0154 0.112 –0.000045

Source:

From Reference 13. With permission

FIGURE 6.1

Comparison of binary activity coefficient data measured by Hamer and Wu

3

to those computed
using Pitzer’s method.

3

(From Reference 18. With permission.)
I =+








−−
==
∑∑
1
2
21 21
2

1
22
2
1
mm
ii
i
N
ii
i
N
C
A
ZZ
ββ
γ
12
1
12
2() ( )
, , and C
12
()
ln
,
γ
12
0
0112
12

212 312
32
b
BB B B=+ + + +…mmm

L829/frame/ch06 Page 145 Thursday, February 3, 2000 8:40 AM
© 2000 by CRC Press LLC
146 Aerosol Chemical Processes in the Environment
where B
0
, B
1
, are fitting coefficients. Coefficients for several electrolytes are given by Jacobson
et al.
4
Polynomial fits are used to simplify and speed up the use of binary activity coefficient data
in computer programs.
TEMPERATURE DEPENDENCE OF MEAN BINARY
ACTIVITY COEFFICIENTS
The temperature dependence of solute mean binary activity coefficients can be derived
5
from
thermodynamic principles as
(6.58)
where γ
12b
(T) is the binary activity coefficient of electrolyte 1-2 at temperature T, T
0
is the reference
temperature (298.15K), R* is the gas constant (J mole

–1
K
–1
), φ
L
is the relative apparent molal
enthalpy (J mole
–1
) of the species at molality m

(with subscript, 12, omitted),

is the apparent
molal heat capacity (J mole
–1
K
–1
) at molality m, and

is the apparent molal heat capacity at
infinite dilution. Further,
(6.59)
(6.60)
are temperature-dependent parameters.
The relative apparent molal enthalpy equals the negative of the heat of dilution, ∆H
D
. With
heat of dilution and apparent molal heat capacity data, polynomials of the form
(6.61)
and

(6.62)
can be constructed. Apparent relative molal enthalpy is defined as

, where φ
H
is the
apparent molal enthalpy and φ
0
H
is the apparent molal enthalpy at infinite dilution, which occurs
when m = 0. Equations 6.58, 6.61, and 6.61 can be combined to give temperature-dependent, mean
binary activity coefficient polynomials of the form
(6.63)
where F
0
= B
0
and
ln ln ,
**
o
γγ
νν
φ
∂φ
∂
νν
φ
∂φ
∂

φ
12 12
0
12 0 12
bb
L
L
L
C
c
c
c
T
T
RT
T
R
p
p
p
()
=+
+
()
+







+
+
()
+−








m
m
m
m
φ
L
=
φ
c
p
o
T
T
T
L
=−
0

1 and
T
T
T
T
T
C
=+






−1
00
ln
φ
L
UUU=+++
1
12
23
32
mmm
φφ
cc
pp
VVV=+ + + +
o

1
12
23
32
mmm
φφφ
LHH
=−
0
ln
,
γ
12
01
12
23
32
b
TFF F F
()
=+ + + +mmm
L829/frame/ch06 Page 146 Monday, January 31, 2000 2:59 PM
© 2000 by CRC Press LLC
Reversible Chemical Reactions in Aerosols 147
(6.64)
for each additional term, beginning with j = 1. In Equation 6.64,
(6.65)
(6.66)
With sufficient data, many temperature- and molality-dependent mean binary activity coeffi-

cients can be written in terms of Equations 6.63 and 6.64. Jacobson et al.
4
list B, G, and H values
for 10 electrolytes and the range of validity for data.
Determining the temperature-dependent binary activity coefficients of bisulfate and sulfate is
more difficult. They can be found by combining equations from the model of Clegg and
Brimblecombe
6
with Equations 6.72 and 6.73 of Stelson et al.
7
in a Newton-Raphson iteration.
Figure 6.2 shows results tabulated for the temperature range <201 to 328K and the molality range
0 to 40 m.
MEAN MIXED ACTIVITY COEFFICIENTS
The mean binary activity coefficients described by Equations 6.57, 6.58, and 6.63 were defined
under the assumption that an electrolyte dissociated alone in a solution. In atmospheric particles,
several electrolytes usually coexist in solution. For example, dissolved sulfuric acid, nitric acid,
hydrochloric acid, ammonia, and sodium chloride often exist together. In such cases, activity
coefficients are approximated with an empirical mixing rule that accounts for interactions among
ions. One such rule is Bromley’s method,
8
which gives the activity coefficient of electrolyte 1-2 in
a mixture as
(6.67)
FIGURE 6.2 Binary activity coefficients of sulfate and bisulfate, each alone in solution. Results are valid
for 0 to 40 m total H
2
SO
4
. (From Reference 18. With permission.)

FBGTHT
jjjL jC
=+ +
G
jU
RT
j
j
=
+
()
+
()
05 2
12 0
.
*
νν
and
H
jV
R
j
j
=
+
()
+
()
05 2

12
.
.
*
νν
log ,
10 12
12
12
12
12
12
1
1
2
2
1
γ
γm
TA
WW
()
=−
+
+
+
+







ZZ ZZ
ZZZ Z
I
I
m
m
L829/frame/ch06 Page 147 Monday, January 31, 2000 2:59 PM
© 2000 by CRC Press LLC
148 Aerosol Chemical Processes in the Environment
where A
γ
is the Debye-Huckel parameter (0.392 at 298K), Z
1
and Z
2
are the absolute values of
charge of cation 1 and anion 2, respectively, I
m
is the total ionic strength of the mixture, and W
1
and W
2
are functions of all electrolytes in solution. W
1
and W
2
are

(6.68)
(6.69)
where
(6.70)
(6.71)
Similar expressions are written for X
32
, X
52
…, etc. Y
41
, Y
61
…, etc. In these equations, γ
12b
(T), γ
14b
(T),
γ
32b
(T)…, etc. are temperature-dependent mean binary activity coefficients, and odd-numbered
subscripts refer to cations while even-numbered subscripts refer to anions. For example, m
1,m
and,
m
2,m
are molalities in the mixture of a cation and anion, respectively.
THE WATER EQUATION
Interaction between solvent and solute in solution is solvation. An example of solvation is when a
solvent bonds to a cation, anion, or nonelectrolyte (such as sucrose) in solution. When the solvent

is liquid water, the bonding is hydration. During hydration of a cation, the lone pair of electrons
on the oxygen atom of a water molecule bonds to the cation-end of the dipole. During hydration
of an anion, the water molecule attaches to the anion-end of the dipole via hydrogen bonding.
Several water molecules can hydrate to each ion.
When liquid water molecules bond to ions in solution, water vapor condenses to maintain
saturation over the solution surface, increasing the liquid water content. Liquid water content is a
unique function of electrolyte molality and sub-100% relative humidity. As the relative humidity
increases up to 100%, hydration increases the aerosol liquid water content. The liquid water content
also increases with increasing solute molality in solution. Above 100% relative humidity, particles
grow rapidly by condensation. When particles are large and dilute, the volume of water added to
them by hydration is small compared to the volume of water already present. Thus, hydration does
not affect water content much when the relative humidity exceeds 100%.
At ambient relative humidities below 100%, an important aspect of modeling aerosols is
determining their liquid water content as a function of electrolyte concentration. A convenient
parameterization of aerosol liquid water content is the Zdanovskii-Stokes-Robinson (ZSR) equa-
tion.
9
The equation can be applied to electrolytes or nonelectrolytes. The simplest form of the
equation, for two species x and y is
WY T A Y T A
bb
121
12
12
12
12
41
14
14
12

12
11
=
()
+
+






+
()
+
+






++log log γγ
γγ
ZZ ZZI
I
I
I
m
m

m
m
WX TA X TA
bb
212
12
12
12
12
32
32
32
12
12
11
=
()
+
+






+
()
+
+







++log log γγ
γγ
ZZ
ZZ
I
I
I
I
m
m
m
m
Y
21
12
2
2
2
=
+







ZZ
m
m
m
,
,
I
X
12
12
2
1
2
=
+






ZZ
m
m
m
,
.
I
L829/frame/ch06 Page 148 Monday, January 31, 2000 2:59 PM

© 2000 by CRC Press LLC
Reversible Chemical Reactions in Aerosols 149
(6.72)
where, m
x,a
and m
v,a
are the molalities of x and y, alone in solution at a given water activity, while
m
x.m
and m
y,m
are the molalities of x and y, when mixed together, at the same water activity. Water
activity is redefined as
(6.73)
where

is the saturation vapor pressure of water over a pure (dilute) liquid water surface,
and

is the saturation vapor pressure of water over a liquid water solution containing solute.
The latter term is always smaller than the former term because, when a solute hydrates, it binds
liquid water, requiring vapor to condense to replace the hydrated liquid water, reducing the vapor-
phase concentration of water. In the dilute solution case, the vapor-phase concentration of water is
not reduced.
The mixed and binary molalities in Equations 6.72 differ from each other because, in a mixture,
the quantity and type of ions differ from in a binary solution; thus, a different quantity of water is
hydrated in each case. Table 6.3 gives mixed and binary molalities of sucrose and mannitol alone
and mixed together in water. The table also shows that, when the molalities are applied to Equation
6.72, the equation is satisfied.

Equation 6.72 can be generalized for a mixture with any number of components by
(6.74)
where the summation is over all solutes in solution, m
k,m
is the molality of solute k in a solution
containing all solutes at the ambient water activity (moles kg
–1
), and m
k,a
is the molality of solute
k as if it were alone in solution at the ambient water activity (moles kg
–1
). For atmospheric aerosols,
this equation is rewritten as
TABLE 6.3
Demonstration of the ZSR Equation
Prediction Accuracy for a Sucrose
(Species a) — Mannitol (Species b)
Mixture at Two Different Water Activities
m
x,a
M
y,a
m
x,m
m
y,m
0.7751 0.8197 0.6227 0.1604 0.9990
0.9393 1.0046 0.1900 0.8014 1.0000
Source: From Stokes, R.H. and Robinson, R.A., J. Phys.

Chem., 70, 2126, 1966. With permission.
m
m
m
m
xm
xa
ym
ya
,
,
,
,
+
m
m
m
m
xm
xa
ym
ya
,
,
,
,
,+=1
HOaq
2
HO

HO
2
2
()
{}
===af
p
p
wr
sc
sd
,,
,,
,
p
sc,,HO
2
p
sc,,HO
2
m
m
km
ka
k
,
,
,
∑
= 1

L829/frame/ch06 Page 149 Monday, January 31, 2000 2:59 PM
© 2000 by CRC Press LLC
150 Aerosol Chemical Processes in the Environment
(6.75)
where c
w
is the liquid water content of particles in units of mole concentration (moles H
2
O(aq)
cm
–3
air), m
v
is the molecular weight of water (g mole
–1
), c
k,m
is the mole concentration (moles cm
–3
air) of solute k in a solution containing all solutes at the ambient water activity, and 1000 converts
g to kg.
Experimental data for water activity as a function of binary electrolyte molality are available
(e.g., see References 1, 10, and 11). Such data can also be fit to polynomial expressions of the form
(6.76)
where a
w
is the water activity (relative humidity expressed as a fraction), m
a
is the molality of an
electrolyte alone in solution, and the Y values are polynomial coefficients. Jacobson et al.

4
list sets
of Y values for 12 electrolytes.
In comparison to the temperature dependence of binary solute activity coefficients, the tem-
perature dependence of binary water activity coefficients under ambient surface conditions is
relatively small. The temperature dependence of water activity can be rewritten from Harned and
Owen
5
as
(6.77)
If the water activity at the reference temperature is expressed as
(6.78)
then Equations 6.77, 6.78, 6.61, and 6.62 can be combined to form
(6.79)
where
(6.80)
for each l greater than 2. Equation 6.79 shows that temperature affects the water-activity polynomial
beginning only in the fourth term. In Equation 6.79, temperature affected the solute activity
beginning with the second term of the polynomial. These equations indicate that the effect of
temperature on water activity is usually less than that on solute activity. At high molalities (above
10 m) and at ambient surface temperatures (273–310K), temperature affects water activity only
slightly. For example, at 16 m, HCl gives binary water activities of 0.09 at T = 273K and 0.11 at
310K. At lower molalities, temperature has even less of an effect.
In an atmospheric model containing mixed aerosols, the water equation is rearranged from
Equation 6.74 to
c
c
w
v
km

ka
k
=
∑
1000
mm
,
,
,
m
awww
YYa Ya Ya
12
01 2
2
3
3
=+ + + + ,
ln ln .
*
aT a
m
R
T
T
T
ww
v
LL
C

c
P
()
=− +






0
2
0
1000
m
mm
∂φ
∂
∂φ
∂
ln
,
aAA A A
w
0
01
12
23
32
=+ + + ++mmm

ln
,
aT A A A E E
w
()
=+ + + +
01
12
23
32
4
2
mmmm
EA
lm
R
T
T
UTV
ll
v
L
l
C
l
=−
−
()
+







−−
05 2
1000
0
22
.
*
L829/frame/ch06 Page 150 Monday, January 31, 2000 2:59 PM
© 2000 by CRC Press LLC
Reversible Chemical Reactions in Aerosols 151
(6.81)
where binary molalities of species alone in solution (m
a
) are obtained from Equation 6.76 at the
given relative humidity. In this equation, i,j is an electrolyte pair (where the odd/even subscripts
used previously are ignored), and c is the hypothetical mole concentration of the pair when mixed
in solution with all other components. In a model, hypothetical mole concentrations of electrolyte
pairs are not usually known; instead, mole concentrations of individual ions are. Thus, individual
ions must be combined into electrolyte pairs for Equation 6.81 to be solved.
METHOD OF SOLVING EQUILIBRIUM EQUATIONS
Equilibrium equations, activity coefficient equations, and the water equation are often solved together
in an atmospheric model to estimate particle composition, including liquid water content. One method
of solving these equations is with a Newton-Raphson iteration (e.g., see Reference 12). Other methods
are the bisectional-Newton method (e.g., see References 13 and 14) and a method
15,16

that minimizes
free energy. These methods require iteration and are mass and charge conserving.
Another method used to solve equilibrium problems is a mass flux iteration (MFI) method.
4,17-19
This method can converge thousands of equilibrium equations simultaneously, cannot produce neg-
ative concentrations, and is mass- and charge-conserving at all times. The only constraints are that
the equilibrium equations must be mass- and charge-conserving, and the system must start in charge
balance. For example, the equation HNO
3
(aq) = H
+
+ NO
3
–
conserves mass and charge. The charge
balance constraint allows initial charges to be distributed among all dissociated ions, but the initial
sum, over all species, of charge multiplied by molality must equal zero. The simplest way to initialize
charge is to set all ion molalities to zero. Initial mass in the system can be distributed arbitrarily,
subject to the charge balance constraint. If the total nitrate in the system is known to be, say, 20 µg m
–3
,
the nitrate can initially be distributed in any proportion among HNO
3
(aq), NO
3
–
, NH
4
NO
3

(s), etc.
The MFI method requires the solution of one equilibrium equation at a time by iteration. A
system of equations is solved by iterating all equations many times. Suppose a system consists of
a single aerosol size bin and 15 equations representing the equilibrium chemistry within that bin.
At the start, the first equation is iterated. When the first equation converges, the updated and other
initial concentrations are used as inputs into the second equation. This continues until the last
equation has converged. At that point, the first equation is no longer converged, because the
concentrations used in it have changed. The iteration sequence must be repeated over all equations
several times until the concentrations no longer change upon more iteration.
Equilibrium among multiple particle size bins and the gas phase is solved in a similar manner.
Suppose a system consists of several size bins, equations per bin, and gases that equilibrate with
dissolved molecules in each bin. Each gas’ saturation vapor pressure over a particle surface is
assumed to equal the gas’ partial pressure, which is a single value. In reality, the saturation vapor
pressure differs over every particle surface. In order to account for variations in saturation vapor
pressure over particle surfaces, nonequilibrium gas-aerosol transfer equations must solved.
Gas-particle equilibrium over multiple size bins is solved by iterating each equilibrium equation,
including gas-solution equations, starting with the first size bin. Updated gas concentrations from
the first bin affect the equilibrium distribution in subsequent bins. After the last size bin has been
iterated, the sequence is repeated in reverse order (to speed convergence), from the last to first size
bin. The marches back and forth among size bins continue until gas and aerosol concentrations do
not change upon more iteration.
To demonstrate the solution to one equilibrium equation, an example where two gases equili-
brate with two ions is shown. The sample equation has the form of Equation 6.1, with two gases
c
m
c
w
v
ijm
ija

j
N
i
N
A
C
=








==
∑∑
1000
11
,,
,,
,
m
L829/frame/ch06 Page 151 Monday, January 31, 2000 2:59 PM
© 2000 by CRC Press LLC
152 Aerosol Chemical Processes in the Environment
on the left side of the equation. The first step is to calculate Q
d
and Q
n

, the smallest ratio (MIN)
of mole concentration to moles among species appearing in the denominator and numerator,
respectively, of Equation 6.28. Thus,
(6.82)
(6.83)
where the subscript “1” refers to initial concentration. Initial concentrations can be selected arbi-
trarily with the requirement that mole concentrations (moles cm
–3
) of all individual species in a
mole-balance group must sum up to the total moles in the group. If an equilibrium equation contains
a solid, each solid’s concentration is included in Equation 6.82 or Equation 6.83.
Second, two parameters are initialized as z
1
= 0.5(Q
d
+ Q
n
) and ∆x
1
= Q
d
– z
1
. The iteration
begins by adding the mass flux factor (∆x, which may be positive or negative) to each mole
concentration in the numerator, or subtracting it from each mole concentration in the denominator
of the equilibrium equation. Thus,
(6.84)
(6.85)
respectively. Starting with Equation 6.84, iteration numbers are referred to by subscripts l and l +

1. If the equation contain solids, then the change in each solid’s concentration is calculated with
Equation 6.84 or 6.85 (solid, aqueous, and ionic mole concentrations are all identified with a c).
The above equations show that, during each iteration, mass and charge are transferred either from
reactants to products or vice versa. This transfer continues until ∆x = 0. Thus, the scheme conserves
mass and charge each iteration.
Third, the ratio of activities is compared to the equilibrium coefficient. The ratio is
(6.86)
To perform this calculation, mole concentrations are converted to units of either molality or
atmospheres. In the case of solids, the activities are unity; thus, none appears in Equation 6.86.
Further, mean mixed activity coefficients (e.g.,γ
AB,l+1
) are updated before each iteration sequence.
They converge after all iteration sequences are complete. Finally, the liquid water content (c
w
) is
updated either during or before each iteration sequence.
The fourth step in the process is to recalculate z for the next iteration. Thus,
(6.87)
Finally, convergence is checked with the convergence criterion:
Q
CC
d
D
D
E
E
=







MIN ,
,,11
νν
Q
cc
n
A
A
B
B
=






MIN , ,
,,11
νν
cc xcc x
Al Al
A
lBlBl
B
l,, ,,
, ,

++
=+ =+
11
νν∆∆
CC xCC x
Dl Dl
D
lElEl
E
l,, ,,
, ,
++
=− =−
11
νν∆∆
F
pp
KT
Al Bl ABl
Dl El
eq
AB AB
DE
=
()()()
()()









()
+++
+
++
mm
,,,
,,
.
111
11
νννν
νν
γ
zz
ll+
=
1
05.
L829/frame/ch06 Page 152 Monday, January 31, 2000 2:59 PM
© 2000 by CRC Press LLC
Reversible Chemical Reactions in Aerosols 153
(6.88)
Each nonconvergence, ∆x is updated, the iteration number is advanced, and the code returns to
(84). Ultimately, all molalities converge to positive numbers.
SOLID FORMATION AND DELIQUESCENCE RELATIVE HUMIDITY
Insoluble solids can form within a particle by precipitation or on its surface by chemical reaction.

Precipitation is defined as the formation of an insoluble compound from solution and can be
simulated as a reversible equilibrium process, such as
(6.89)
where the equilibrium coefficient for this reaction is called the solubility product. A solid precipitates
from solution when the product of its reactant ion concentrations and mean activity coefficient
exceeds its solubility product. In other words, precipitation occurs when
(6.90)
Similarly, gas deposition and solid-forming reaction on a surface can be simulated with a
reaction such as
(6.91)
In this case, the solid can form when one gas adsorbs to a surface and the other gas collides and
reacts with the adsorbed gas. Alternatively, both gases can adsorb to a surface and then diffuse on
the surface until a collision and reaction occur. In Equation 6.91, a solid is assumed to form on
the surface when
(6.92)
In either of the above two cases, solid formation is accounted for with the MFI equilibrium
solution method, described above. When a solid forms, F from Equation 6.88 converges to 1.0 and
solid, ion, and/or gas concentrations are updated with Equations 6.84 and 6.85. When a solid does
not form, F does not converge, and Equations 6.84 and 6.85 predict no net change in concentrations.
The process by which an initially dry particle lowers its saturation vapor pressure and takes
up liquid water is deliquescence. If a particle consists of an initially solid electrolyte at a given
relative humidity, and the relative humidity increases, the electrolyte does not take on liquid water
by hydration until the deliquescence relative humidity (DRH) is reached. At the DRH, water rapidly
hydrates with the electrolyte, dissolving the solid, and increasing the liquid water content of the
particle. Above the DRH, the solid phase no longer exists, and the particle takes up additional
liquid water to maintain equilibrium.
F
x
x
ll

ll
=
>→ =−
<→ =+
=→







++
++
1
1
1
11
11
∆
∆
z
z
Convergence
.
NH NO (s) NH NO
43 4 3
–
⇔+
+

,
mm
NH NO NH NO
4343
2
+−+−
>
()
γ
,
.KT
eq
NH NO (s) NH (g) HNO (g).
43 3 3
⇔+
pp KT
ss eq,,
.
NH HNO
33
>
()
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© 2000 by CRC Press LLC
154 Aerosol Chemical Processes in the Environment
If a particle consists of an initially aqueous electrolyte, and the relative humidity decreases
below the DRH, water evaporates, but dissolved ions in solution do not necessarily precipitate
(crystallize) immediately. Instead, the solution is supersaturated and remains so until solid nucle-
ation occurs. The relative humidity at which nucleation occurs and an initially aqueous electrolyte
becomes crystalline is the crystallization relative humidity (CRH). The CRH is always less than

or equal to the DRH. Table 6.4 shows the DRHs and CRHs of several electrolytes at 298K. Some
electrolytes, such as NH
3
, HNO
3
, HCl, and H
2
SO
4
, do not have a solid phase at room temperature.
These substances, therefore, do not have a DRH or a CRH. In a mixed solution, the DRH of a
solid in equilibrium with the solution is lower than the DRH of the solid alone.
16,20
EQUILIBRIUM SOLVER RESULTS
Graphical results from the equilibrium solution method discussed in the section “Method of Solving
Equilibrium Equations” are shown here for two cases. Figure 6.3 shows the change in composition
of a bulk particle solution as a function of sodium chloride mole concentration. The figure shows
TABLE 6.4
DRHs and CRHs of Several Electrolytes at 298K
Electrolyte
DRH
(percent)
CRH
(percent) Electrolyte
DRH
(percent)
CRH
(percent)
NaCl 75.28
a

47
c
(NH
4
)
2
SO
4
79.97
a
37–40
b
Na
2
SO
4
84.2
b
57–59
b
NH
4
HSO
4
40.0
b
0.05–22
b
NaHSO
4

52.0
d
<0.05
d
NH
4
NO
3
61.83
a
25–32
d
NaNO
3
74.5
d
0.05-30
b
(NH
4
)
3
H(SO4)
2
69
b
35–44
b
NH
4

Cl 77.1
a
47
e
KCl 84.26
a
62
c
a
From Reference 24. With permission.
b
From Reference 20. With permission.
c
From Reference 25. With permission.
d
From Reference 26. With permission.
e
From Reference 10. With permission.
FIGURE 6.3 Aerosol composition vs. NaCl concentration when the relative humidity was 90%. Other initial
conditions were H
2
SO
4
(aq) = 10 µg m
–3
-air, HCl(g) = 0 µg m
–3
, NH
3
(g) = 10 µg m

–3
, HNO
3
(g) = 30 µg m
–3
,
and T = 298K. NaCl dissolves and dissociates completely at this relative humidity. (From Reference 4. With
permission.)
L829/frame/ch06 Page 154 Monday, January 31, 2000 2:59 PM
© 2000 by CRC Press LLC
Reversible Chemical Reactions in Aerosols 155
that an increase in sodium chloride caused water to condense and hydrate, increasing the rate of
dissolution and dissociation of nitric acid.
Figure 6.4 shows a model simulation of the change in aerosol composition as a function of
relative humidity. As humidity decreased down from 100%, water, chlorine, nitrate, and ammonium
decreased steadily. At about 62% relative humidity, which is near the DRH of ammonium nitrate,
both ammonium nitrate and ammonium sulfate precipitated. Ammonium sulfate did not precipitate
at relative humidities of 62 to 80%, although its DRH is about 80% because it was undersaturated
at those humidities. When the relative humidity is decreasing, ammonium sulfate can remain in
the aqueous phase until the relative humidity reaches 40% (Table 6.4).
SUMMARY
In this chapter, chemical equilibrium equations were discussed. When equilibrium equations are
solved in a model, mean mixed activity coefficient and water content equations are also needed. A
method of calculating the water content and expressions for temperature-dependent mean binary
activity coefficients were given. Mean mixed activity coefficients were calculated from mean binary
activity coefficients with a mixing rule. A method of iterating equilibrium, activity coefficient, and
water content equations together was also given.
REFERENCES AND FURTHER READING
1. Pitzer, K.S. and Mayorga, G., Thermodynamics of electrolytes. II. Activity and osmotic coefficients
for strong electrolytes with one or both ions univalent, J. Phys. Chem., 77, 2300, 1973.

2. Hamer, W.J. and Wu, Y C., Osmotic coefficients and mean activity coefficients of uni-univalent
electrolytes in water at 25°C, J. Phys. Chem. Ref. Data, 1, 1047, 1972.
3. Pitzer, K.S., Ion interaction approach: theory and data correlation, Activity Coefficients in Electrolyte
Solutions, 2nd ed., edited by Pitzer K.S., CRC Press, Boca Raton, FL, 1991.
4. Jacobson, M.Z., Tabazadeh, A., and Turco, R.P., Simulating equilibrium within aerosols and nonequi-
librium between gases and aerosols, J. Geophys. Res., 101, 9079, 1996.
5. Harned, H.S. and Owen, B.B., The Physical Chemistry of Electrolyte Solutions, Chap., 8, Reinhold,
New York, 1958.
6. Clegg, S.L. and Brimblecombe, P., Application of a multicomponent thermodynamic model to activ-
ities and thermal properties of 0–40 mol kg
–1
aqueous sulfuric acid from <200K to 328K, J. Chem.
Eng. Data 40, 43, 1995.
FIGURE 6.4 Aerosol composition vs. relative humidity. Initial conditions were H
2
SO
4
(aq) = 10 µg m
–3
,
HCl(g) = 0 µg m
–3
, NH
3
(g) = 10 µg m
–3
, HNO
3
(g) = 30 µg m
–3

, and T = 298K. (From Reference 4. With
permission.)
L829/frame/ch06 Page 155 Monday, January 31, 2000 2:59 PM
© 2000 by CRC Press LLC
156 Aerosol Chemical Processes in the Environment
7. Stelson, A.W., Bassett, M.E., and Seinfeld, J.H., Thermodynamic equilibrium properties of aqueous
solutions of nitrate, sulfate and ammonium, Chemistry of Particles, Fogs and Rain, edited by Durham,
J.L., Ann Arbor Publication, Ann Arbor, MI, 1984.
8. Bromley, L.A., Thermodynamic properties of strong electrolytes in aqueous solutions, AIChEJ 19,
313, 1973.
9. Stokes, R.H. and Robinson, R.A., Interactions in aqueous nonelectrolyte solutions.I. Solute-solvent
equilibria, J. Phys. Chem., 70, 2126, 1966.
10. Cohen, M.D., Flagan, R.C., and Seinfeld, J.H., Studies of concentrated electrolyte solutions using the
electrodynamic balance., 1, Water activities for single-electrolyte solutions, J. Phys. Chem., 91, 4563,
1987.
11. Cohen, M.D., Flagan, R.C., and Seinfeld, J.H, Studies of concentrated electrolyte solutions using the
electrodynamic balance. 2. Water activities for mixed-electrolyte solutions, J. Phys. Chem., 91, 4575,
1987.
12. Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T., Numerical Recipes: The Art of
Scientific Computing, Cambridge University Press, Cambridge, 1992.
13. Pilinis, C. and Seinfeld, J.H., Continued development of a general equilibrium model for inorganic
multicomponent atmospheric aerosols, Atmos. Environ., 21, 2453, 1987.
14. Kim, Y.P., Seinfeld, J.H., and Saxena, P., Atmospheric gas-aerosol equilibrium I. Thermodynamic
model, Aerosol Sci. Technol., 19, 157, 1993.
15. Bassett, M.E. and Seinfeld, J.H., Atmospheric equilibrium model of sulfate and nitrate aerosol, Atmos.
Environ., 17, 2237, 1983.
16. Wexler, A.S. and Seinfeld, J.H., The distribution of ammonium salts among a size and composition
dispersed aerosol, Atmos. Environ., 24A, 1231, 1990.
17. Jacobson, M.Z., Developing, Coupling, and Applying a Gas, Aerosol, Transport, and Radiation Model
to Study Urban and Regional Air Pollution, Ph.D. thesis, Dept. of Atmospheric Sciences, University

of California, Los Angeles, 1994.
18. Jacobson, M.Z., Fundamentals of Atmospheric Modeling, Cambridge University Press, New York,
656, 1999.
19. Villars, D.S., A method of successive approximations for computing combustion equilibria on a high
speed digital computer, J. Phys. Chem., 63, 521, 1958.
20. Tang, I.N. and Munkelwitz, H.R., Composition and temperature dependence of the deliquescence
properties of hygroscopic aerosols, Atmos. Environ., 27A, 467, 1993.
21. Wagman, D.D., Evans, W.H., Parker, V.B., Schumm, R.H., Halow, I., Bailey, S.M., Churney, K.L.,
and Nuttall, R.L., The NBS tables of chemical thermodynamic properties: selected values for inorganic
and C
1
and C
2
organic substances in SI units, J. Phys. Chem. Ref. Data, 11, Suppl., 2, 1982.
22. Schwartz, S.E., Gas- and aqueous-phase chemistry of HO
2
in liquid water clouds, J. Geophys. Res.,
89, 589, 1984.
23. Perrin, D.D., Ionization Constants of Inorganic Acids and Bases in Aqueous Solution, 2nd ed.,
Pergamon, New York, 1982.
24. Robinson, R.A. and Stokes, R.H., Electrolyte Solutions, Academic Press, New York, 1955.
25. Tang, I.N., Thermodynamic and optical properties of mixed-salt aerosols of atmospheric importance,
J. Geophys. Res., 102, 1883, 1997.
26. Tang, I.N., Chemical and size effects of hygroscopic aerosols on light scattering coefficients, J.
Geophys. Res., 101, 245, 1996.
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© 2000 by CRC Press LLC