AEROSOL CHEMICAL PROCESSES IN THE ENVIRONMENT - CHAPTER 6 docx

Jacobson CONTENTS Introduction ...135 Definitions ...136 Equilibrium Equations and Relations ...136 Equilibrium Equations...136 Equilibrium Relations and Constants ...139 Temperature Dep

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 µm in diameter) affect human health by directly penetrating to the deepest part of human lungs Aerosols between 0.2 and 1.0 µm in diameter that contain sulfate, nitrate, and organic carbon, scatter light efficiently Aerosols smaller than 1.0 µm 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 reacdissolu-tions are discussed These processes are important for determining the ionic, solid, and liquid water content of aerosols

136 Aerosol Chemical Processes in the Environment

DEFINITIONS

Dissolution is a process that occurs when a gas, suspended over a particle surface, adsorbs to anddissolves in liquid on the surface The liquid in which the gas dissolves is a solvent A solventmakes 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 andsolvent make up a solution, which is a homogeneous mixture of substances that can be separatedinto individual components upon a change of state (e.g.,freezing) A solution may contain manysolutes 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 giventemperature Solutions usually contain solute other than the dissolved gas The solubility of a gasdepends strongly on the quantity of the other solutes because such solutes affect the thermodynamicactivity of the dissolved gas in solution Thermodynamic activity is discussed shortly If water issaturated 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 dissolvedmolecule is the process by which the molecule breaks into simpler components, namely ions Thisprocess 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 theforward and backward directions are generally fast Dissociated ions and undissociated moleculescan 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-orderordinary 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 thissection, different types of equilibrium equations are discussed and rate expressions, includingtemperature 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 thereaction Each reaction must conserve mass Thus,

Reversible Chemical Reactions in Aerosols 137

where m iis 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 witheach other at the gas–solution interface Thus, the number of molecules of AB transferring fromthe gas to the solution equals the number of molecules transferring in the reverse direction In theatmosphere, 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 theaqueous 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 dissociateinto ions Substances that undergo partial or complete dissociation in solution are electrolytes Thedegree of dissociation of an electrolyte depends on the acidity of solution, the strength of theelectrolyte, 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 higherthe molarity of protons and the lower the pH Protons in solution are donated by acids that dissolve.Examples of such acids are H2CO3(aq), HCl(aq), HNO3(aq), and H2SO4(aq) The abilities of acids

to dissociate into protons and anions vary HCl(aq), HNO3(aq), and H2SO4(aq) dissociate readily,while H2CO3(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 dissociatessignificantly) and a weak acid is a weak electrolyte Hydrochloric acid is a strong acid and strongelectrolyte in water because it almost always dissociates completely by the reaction

(6.10)

AB(g)⇔AB(aq),

HCl(g)⇔HCl(aq)HNO (g)3 ⇔HNO (aq)3

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, H2SO4(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 ofbicarbonate 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 HNO3(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

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 Table6.1 shows several equilibrium relationships of atmospheric importance

E QUILIBRIUM R ELATIONS AND C ONSTANTS

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 Thermodythermody-namic 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

140 Aerosol Chemical Processes in the Environment

The activity of an ion in solution or an undissociated electrolyte is its molality (mA) (molessolute 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 givethe 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

1 HNO3(g) ⇔ HNO3(aq) 2.10 × 10 5 mol kg –1 atm –1 D

2 NH 3 (g) ⇔ NH 3 (aq) 5.76 × 10 1 13.79 -5.39 mol kg –1 atm –1 A

3 CO2(g) ⇔ CO2(aq) 3.41 × 10 –2 8.19 -28.93 mol kg –1 atm –1 A

4 CO 2 (aq) + H 2 O(aq) ⇔ H + + HCO 3 4.30 × 10 –7 –3.08 31.81 mol kg –1 A

5 NH3(aq) + H2O(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

Note: The equilibrium coefficient reads,

where T0 = 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.

T

T T

T T

or less than unity Debye and Huckel showed that, in relatively dilute solutions, where ions are farapart, 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-ioninteractions 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,

(6.34)

where µi 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 µI 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

= o+ * ln∏{ } ν ,

i

o=∑ ( ν µo)

142 Aerosol Chemical Processes in the Environment

is the standard molal Gibbs free energy of formation (J mole–1) for the reaction Equilibrium occurswhen ∆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

oln

c R

T T

T T

o

where K eq (T o ) is the equilibrium coefficient at temperature, To Values of and ∆c p are measured

reactions

FORMS OF EQUILIBRIUM COEFFICIENT EQUATIONS

Each reaction in Table 6.1 can be written in terms of thermodynamic activities and an equilibriumcoefficient For example, an equilibrium coefficient equation for the reaction

(6.46)is

(6.47)

nitric acid in solution (moles kg–1), and is the activity coefficient of dissolved, undissociatednitric 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

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 NO3 are alone in

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 To

o

HNO (g)3 ⇔HNO (aq)3

HNO aqHNO g

HNO aq HNO aq

,HNO g 3

HNO aq

3 -

HNO aq HNO aq

HNO aq HNO aq 3

3 -

-3 - 3 - +

γ

γγ,

γH and γNO

3 +

3 - +,

144 Aerosol Chemical Processes in the Environment

mathematically Mean binary activity coefficients are measured in the laboratory Mean mixedactivity coefficients can be estimated from mean binary activity coefficient data through a mixingrule

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 cationand 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 molalityand 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

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 theinterionic 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 insolution, 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 forthree electrolytes are shown in Table 6.2

While Pitzer’s method accurately predicts mean binary activity coefficients at 298.15K fromphysical 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 thosemeasured 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 frommeasurements or theory, they can be parameterized with a polynomial fit of the form

(6.57)

TABLE 6.2 Pitzer Parameters for Three Electrolytes

Electrolyte ββββ (1 ) 11112222 ββββ ( 2) 11112222 C 12(((( γγγγ))))

HCl 0.17750 0.2945 0.0012 HNO 3 0.1119 0.3206 0.0015

NH4NO3 –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