Standard Methods for Examination of Water & Wastewater_10 docx

11.1 CARBONATE EQUILIBRIA The carbonate equilibria is a function of the ionic strength of water, activity cient, and the effective concentrations of the ionic species.. Using calcium as

Water Stabilization

As mentioned in Chapter 10 on water softening, as long as the concentrations ofCaCO3 and Mg(OH)2 exceed their solubilities, the solids may continue to precipitate.This condition can cause scale to form, a solid that deposits due to precipitation ofions in solution To prevent scale formation, the water must be stabilized A water

is said to be stable when it neither dissolves nor deposits precipitates If the pH ishigh, stabilization may be accomplished using one of several acids or using CO2, aprocess called recarbonation If the pH is low, stabilization may be accomplishedusing lime or some other bases

Because of the universal presence of carbon dioxide, any water body is affected

by the reaction products of carbon dioxide and water The species produced from thisreaction form the carbonate system equilibria As discussed later, the stability orinstability of water can be gaged using these equilibria Thus, this chapter discussesthis concept It also discusses criteria for stability and the recarbonation processafter water softening

11.1 CARBONATE EQUILIBRIA

The carbonate equilibria is a function of the ionic strength of water, activity cient, and the effective concentrations of the ionic species The equilibrium coeffi-cients that are calculated from the species concentrations are a function of thetemperature This functionality of the coefficients can, in turn, be calculated usingthe Van’t Hoff equation, to be addressed later

coeffi-One of the major cations that can form scales as a result of the instability of water

is calcium Calcium plays an important role in the carbonate equilibria We willtherefore express the carbonate equilibria in terms of the interaction of the calciumion and the carbonate species which are the reaction products of carbon dioxide andwater In addition, since the equilibria occur in water, the dissociation of the watermolecule must also be involved Using calcium as the cation, the equilibrium equa-tions of the equilibria along with the respective equilibrium constants at 25°C are

514 Physical–Chemical Treatment of Water and Wastewater

to concentration, its value may be obtained using the value of the correspondingconcentration This relationship is expressed as follows:

(11.5)where sp represents any species involved in the equilibria such as Ca2+, , and so on The pair of brackets, [], is read as “the concentration of,” γ is the activitycoefficient

11.1.1 I ONIC S TRENGTH

As the particle ionizes, the number of particles increases Thus, it is not a surprisethat activity coefficient is a function of the number of particles in solution Thenumber of particles is characterized by the ionic strength µ This parameter wasdevised by Lewis and Randall (1980) to describe the electric field intensity of asolution:

(11.6)

i is the index for the particular species and z is its charge The concentrations are

in gmmols/L In terms of the ionic strength, the activity coefficient is given by the

DeBye-Huckel law as follows (Snoeyink and Jenkins, 1980; Rich, 1963):

1+1.14( µ) -

=

γ 10

0.5z i2( µ ) 1+1.14 ( µ ) - –

=

µ = 2.5 10( 5)TDS

mgmol/L If the pH of the solution is 7, what is the concentration of the bicarbonateion if the temperature is 25°C?

Solution:

mg/L, Ca2+= 80 mg/L, Mg2+ = 12.0 mg/L, Na+= 46.0 mg/L, ,and What is the ionic strength of the sample?

Solution:

in mg/L of the bicarbonate ion

HCO3−{ } = 8.93 10( 4) gmol/L=[HCO3−] Ans

11.1.2 E QUILIBRIUM C ONSTANT AS A F UNCTION OF T EMPERATURE

The equilibrium constants given previously were at 25°C To find the values of the

equilibrium constants at other temperatures, the Van’t Hoff equation is needed According to this equation, the equilibrium constant K (Ksp for the solubility product

constants) is related to temperature according to a derivative as follows:

(11.11)

T is the absolute temperature; ∆Ηο is the standard enthalpy change, where thestandard enthalpy change has been adopted as the change at 25°C at one atmosphere

of pressure; and R is the universal gas constant.

The value of R depends upon the unit used for the other variables Table 11.1

gives its various values and units, along with the units used for ∆Ηο and T By convention, the concentration units used in the calculation of K are in gmmols/L Enthalpy is heat released or absorbed in a chemical reaction at constant pressure.

Table 11.2 shows values of interest in water stabilization It is normally reported asenthalpy changes There is no such thing as An absolute value of an enthalpy does notexist, only a change in enthalpy Enthalpy is a heat exchange at constant pressure, soenthalpy changes are measured by allowing heat to transfer at constant pressure; theamount of heat measured during the process is the enthalpy change Also, the tableindicates enthalpy of formation This means that the values in the table are the heat

10

0.5 1 ( ) 0.023 ( ) 1+1.14 0.023 ( ) -=0.86 –

sp

{ } = γ sp[ ] = 0.86 0.0025( ) = 0.00215 mg/L Ans

dlnK dT

gmmol.K ° - gmmols

L

-measured when the particular substance was formed from its elements For example,when calcium carbonate solid was formed from its elements calcium, carbon andoxygen, −288.45 kcal of heat per gmmol of calcium carbonate was measured Thenegative sign means that the heat measured was released or liberated in the chemicalreaction.

Also, the state of the substance when it was formed is also indicated in the table.For example, the state when calcium carbonate is formed liberating heat in the amount

of −288.45 kcal/gmmol is solid, indicated as s The symbol l means that the state is liquid and the symbol aq means that the substance is being formed in water solution

Also, note the subscript and superscript They indicate that the values in thetable were obtained at standard temperature and pressure and one unit of activityfor the reactants and products The standard temperature is 25°C; thus the 298, which

is the Kelvin equivalent of 25°C The standard pressure is 1 atmosphere The script o symbolizes unit activity of the substances This means that the elementsfrom which the substances are formed were all at a unit of activity and the productsubstances formed are also all at a unit of activity

super-The enthalpy change is practically constant with temperature; thus ∆Hο may bereplaced by Doing this and integrating the Van’t Hoff equation from KT1 to

KT 2 for the equilibrium constant K and from T1 to T2 for the temperature,

(11.12)

This equation expresses the equilibrium constant as a function of temperature

TABLE 11.2 Enthalpies of Formation of Substances of Interest in Stabilization

=

11.1.3 ’ S FOR P ERTINENT C HEMICAL R EACTIONS OF THE

C ARBONATE E QUILIBRIA

Let us now derive the values of the of the various pertinent chemical reactions

in the carbonate equilibria as shown in Eqs (11.1) through (11.4) According to

Hess’s law, if the chemical reaction can be written in steps, the enthalpy changes

can be obtained as the sum of the steps

Thus, consider Equation (11.1) The corresponding reaction is

(11.13)

Writing in steps to conform to Hess’s law:

The values of the ’s are obtained from Table 11.2 Note that the values inthe table indicate of formation having negative values Thus, if the reaction

is not a formation but a breakup such as , the sign is positive.This reaction indicates that to break the water molecule into its constituent atoms+68.317 kcal/gmmol of energy is required The + sign indicates that the reaction isendothermic requiring energy for the reaction to occur For the ionization of thewater molecule as represented by and using Hess’s law

as shown previously, +13.36 kcal/gmol of HOH(l) is required

The Hess’s law steps for the rest of Eqs (11.1) through (11.4) are detailed asfollows:

Example 11.5 A softened municipal water supply enters a residence at 15°Cand is heated to 60°C in the water heater Compare the values of the equilibriumconstants for CaCO3 at these two temperatures If the water was at equilibrium at

25°C, determine if CaCO3 will deposit or not at these two temperatures

Solution:

Therefore,

HCO3 aq−( ) 1

2 -H2 C 6O2 ∆H298

Thus, the value of equilibrium constant is greater at 60°C than at 15°C Ans

The value of the equilibrium constant for calcium carbonate at 25°C is 4.8(10−9

)

At this condition, the ions Ca2+ and ions are given to be in equilibrium; thus,will neither deposit nor dissolve CaCO3 At the temperature of 15°C, the value ofthe equilibrium constant is 4.038(10−9) This value is less than 4.8(10−9) and willrequire less of the ionized ion; therefore at 15°C, the water is oversaturated and willdeposit CaCO3 Ans

At 60°C, the equilibrium constant is 8.10(10−9

), which is greater than that at

25°C Thus, at this temperature, the water is undersaturated and will not depositCaCO3 Ans

11.2 CRITERIA FOR WATER STABILITY

AT NORMAL CONDITIONS

In the preceding discussions, a criterion for stability was established using the

equilibrium constant called Ksp At normal conditions, as especially used in the water

works industry, specialized forms of water stability criteria have been developed These

are saturation pH, Langelier index, and the precipitation potential of a given water

11.2.1 S ATURATION pH AND THE L ANGELIER I NDEX

Because pH is easily determined by simply dipping a probe into a sample, nation of the saturation pH is a convenient method of determining the stability ofwater The concentrations of any species at equilibrium conditions are in equilibriumwith respect to each other Also, for solids, if the condition is at equilibrium noprecipitate or scale will form One of the concentration parameters of equilibrium

determi-is the hydrogen ion concentration, which can be ascertained by the value of the pH.Thus, if the pH of a sample is determined, this can be compared with the equilibrium

pH to see if the water is stable or not Therefore, we now proceed to derive the

equilibrium pH Equilibrium pH is also called saturation pH.

In natural systems, the value of the pH is strongly influenced by the carbonateequilibria reactions The species of these reactions will pair with a cation, thus

“guiding” the equilibrium reactions into a dead end by forming a precipitate For

example, the complete carbonate equilibria reactions are as follows:

(11.14)

(11.15)

K T 2at 60°K 4.8 10( 9)exp 2,950

1.987 298( ) 333( ) - 333( –298)

=8.10 10( 9) in gmol units

(11.17)

c is the charge of the cation that pairs with forming the precipitate Cation2(CO3)c(s) We call the formation of this precipitate as the dead end of thecarbonate equilibria, since the carbonate species in solution are diminished by theprecipitation

Let us digress for a moment from our discussion of the saturation pH in order

to find the dead end cation for the carbonate system equilibria Several of thesecations can possibly pair with the carbonate The pairing will be governed by the

value of the solubility product constant, Ksp A small value of the Ksp means that

only small values of the concentration of the constituent species are needed to form

a product equal to the Ksp This, in turn, means that solids with smaller Ksp’s will

easily form the solids Thus, of all the possible cations that can pair with the

carbonate, the one with the smallest Ksp value is the one that can form a dead end

for the carbonate equilibria reactions Mg forms MgCO3 with a Ksp of 10−5 Ca formsCaCO3 with a Ksp of 4.8(10−9) Table 11.3 shows other carbonate solids with therespective solubility product constants

From the previous table, the smallest of the Ksp’s is that for Hg2CO3 Thus,considering all of the possible candidates that we have written, Hg2CO3 is the onethat will form a dead end for the carbonate equilibria; however, of all the possiblecations, Ca2+ is the one that is found in great abundance in nature compared to the

rest Thus, although all the other cations have much more smaller Ksp’s than calcium,

TABLE 11.3 Solubility Product Constants of Solid

CO32−

they are of no use as dead ends if they do not exist The other cation that exists inabundance in natural waters is magnesium In fact, this is the other constituent

hardness ion in water Comparing the Ksp’s of the carbonate of these cations, however,

CaCO3 is the smaller Thus, as far as the carbonate equilibria reactions are concerned,the calcium ion is the one to be considered to form a dead end in the carbonate system

equilibria Cation2(CO3)c(s) is therefore CaCO3(s) For this reason, Equation (11.4) waswritten in terms of CaCO3 (See Table 11.4)

As will be shown later, the saturation pH may conveniently be expressed in

terms of total alkalinity and other parameters The alkalinity of water is defined as

its capacity to neutralize any acid added to it When an acid represented by H+ isadded to a hydroxide represented by OH−, the acid will be neutralized according tothe reaction H++ OH−
HOH Thus, the hydroxide is an alkaline substance Whenthe acid is added to a carbonate, the acid is also neutralized according to the reaction

Carbonate is therefore also an alkaline substance By writing

a similar reaction, the bicarbonate ion will also be shown to be an alkaline substance

As we know, these species are the components of the carbonate equilibria Theyalso represent as components of the total alkalinity of the carbonate system equilibria.They may be added together to produce the value of the total alkalinity To be additive,each of these component alkalinities should be expressed in terms of a common unit

A convenient common unit is the gram equivalent.

[OH−] is equal to {OH−}/γΟΗ, where γΟΗ is the activity coefficient of the hydroxylion {OH−} could be eliminated in terms of the ion product of water, Kw = {H+

}{OH−}

To establish the equivalence of the component alkalinities, they must all be referred

to a common end point when the acid H+ is added to the solution From generalchemistry, we learned that this is the methyl orange end point As far as the OH−ion is concerned, the end point for the reaction H++ OH−
HOH has already beenreached well before the methyl orange end point Thus, for the purpose of deter-mining equivalents, the reaction for the hydroxide alkalinity is simply H++ OH−
HOH and the equivalent mass of the hydroxide is OH/1 One gram equivalent ofthe hydroxide is then equal to one gram mole Therefore,

(11.18)

, where is the activity coefficient of the

Reaction of the acid H+ with the bicarbonate given by H++ ends exactly at the methyl orange end point From this reaction,the equivalent weight of the bicarbonate ion is HCO3/1; thus, one gram equivalent

is equal to one gram mole Therefore,

(11.19)

2H++CO32−
H2CO3

OH−[ ]geq [OH−] {OHγ −}

OH

- Kw

γOH { }H+ -

HCO3−

[ ] = {HCO3−}/γHCO3 γHCO3

HCO3−{ } { } COH+ 3

, where is the activity coefficient of the carbonate

Reaction of the acid H+ with the carbonate ion given by
also ends exactly at the methyl orange end point From this reaction, theequivalent mass of the carbonate ion is CO3/2; thus, one gram equivalent is equal to1(CO3/2)/CO3= gram mole Therefore,

(11.20)

Using the equation , Equation (11.19) becomes

(11.21)

The alkalinity of water has been defined as its capacity to react with any acid added

to it Thus, if any hydrogen ion is present, this must be subtracted to reflect the

overall alkaline capacity of the water Letting [A]geq represent the total alkalinity,

The Langlier Index (or Saturation Index) (LI) is the difference between the actual

pH and the saturation pH of a solution, thus

CO32−

{ } = K sp,CaCO3/ Ca{ 2+}HCO3−

+

{ }

γH

–

H+{ } –(D–A)+ (D–A)2–4CB

A positive value of the Langelier index indicates that the water is rated and will deposit CaCO3, whereas a negative value indicates that the water

supersatu-is undersaturated and will dsupersatu-issolve any CaCO3 that happens to exist at the particularmoment

140 mg/L, [Ca2+] = 0.7 mgmol/L, [Mg2+

] = 0.6 mgmol/L, [A]mgeq = 0.4 mgeq/L,temperature = 20°C, and pH = 6.7 Calculate the saturation pH and determine if thewater will deposit or dissolve CaCO3

Kw = 10–14 = { } OHH+ { −} at 25°C = K T 1

∆H298

o for Kw = +13.36 kcal/gmmol of HOH( )l = +13,360 cal/gmmol

0.94 - 7.24 10( –15)

D Ksp,CaCO3

2γCO3{Ca2+} -:

LI = pH − pHs = 6.7 − 8.8 = −2.1 and the water will not deposit CaCO3 but willdissolve it Ans

the concentration of the calcium ion in equilibrium with CaCO3 at this condition?Assume the rest of the data holds

Solution:

11.2.2 D ETERMINATION OF {Ca 2+}

The activity of the calcium ion is affected by its complexation with anions Ca2+forms complexes with the carbonate species, OH− and The complexation reac-tions are shown as follows:

–

2 2.44 10 [ ( 5) ] -

=

3.94 10 ( 4) 1.55 10 ( 7) 7.07 10 9

– +

2 2.44 10 [ ( 5) ] -

+

{ }

γH

–

=0.0004 6.80 10

15 –

0.94 10( 8) - 10

8

5.23 10( 9)

0.94 4.22 10[ ( –11)] Ca{ 2+} - 5.23 10

9

2 0.77( ) Ca{ 2+} - 10

8

0.94 -–

(11.30)

These complexes are weak enough that, in the complexometric titration mination using EDTA, they break and are included in the total calcium hardnessreported Thus, the total calcium hardness is composed of the “legitimate” cation,

deter-Ca2+, plus the complex ions as shown in the previous equations This total calciumhardness must be corrected by the concentrations of the complexes in order todetermine the correct activities of the calcium ions Let the total concentration of thecalcium species as determined by the EDTA titration be [CaT] Thus, the concentra-tion of the calcium ion is

(11.31)

Table 11.4 shows the equilibrium constants of the previous complexes at 25°C.For other temperatures, these values must be corrected using the Van’t Hoff equation.The use of this equation, however, requires the value of the standard heat of formation At present, none are available for , , CaOH+, and Research is therefore needed to find these values

Determination of the Calcium Complexes The whole thrust of the discussion

regarding water stability is the determination of whether CaCO3 precipitates at agiven solution condition The concentration of will therefore be determined

in relation to the solubility product of CaCO3 Applying Hess’s law,

TABLE 11.4 Equilibrium Constants for Various