Soil Chemistry 51 CARBONATE EQUILIBRIA Carbonates are arguably the most important dissolved omponent of soil solutions and in alkaline soils this statement is even less disputable. Impliit in this statement is the relationship among dissolved arbonate speies whether or not they are in equilibrium with solid phase metal arbonates. The simplest example of arbonates is the ontrol whih dissolved arbon dioxide has on water ph and buffering. In this setion of the ourse we will onsider the effet of arbon dioxide on water ph, the influene of solid phase alium arbonate on solution omposition and the impliations of these reations. Several systems inluding arbon dioxide, solution and solid phase arbonates an be envisioned. Some of these inlude (after Garrels and Christ, 1967). 1. The solution ph of water in equilibrium with arbon dioxide and essentially devoid of other ontrolling speies.. The reation of alium arbonate saturated solutions with free aess to arbon dioxide. In essene this is the equilibrium of lime with air or soil air. Also referred to as an open system with solid phase present. This ase is of onsiderable interest as it represents the relationship of dilute natural waters in ontat with the atmosphere and system ontrol only by the arbonate equilibria.. The reations of alium arbonate with water where the gas phase is restrited or negligible. This ondition is not ommon, but in situations were the equilibrium system has little head spae and the mixing of air is restrited the attendant ph and slow rate of return to equilibrium is of interest. This is the famous "Turner Effet". Suh as system is also referred to as a losed system sine mass is not transferred or exhanged with the surroundings. 4. Equilibrium in systems with a fixed quantity of added alkalinity suh as the addition of strong base to a system open to the atmosphere. Other ases an be onsidered, however these serve to illustrate the great utility of being able to understand the equilbrium behavior of arbonate speies in soils and sediments. Setion 5 Carbonate Chemistry
Soil Chemistry 5 CASE 1 CO H O open system Aqueous arbon dioxide reats to form arboni aid via the following reation: CO HO H CO (aq) H CO CO.8 (aq) = 10 = 0.00159 The hydration of arbon dioxide is slow to attain equilibrium below ph 8 in pure systems. However, above ph 11, the hydration reation is relatively rapid as arbon dioxide reats diretly with hydroxide to form biarbonate. CO (aq) OH = HCO (1) In biologial systems the hydration of arbon dioxide is atalyzed by arboni anhydrase a Znontaining enzyme. Only a small portion of the aqueous arbon dioxide exists as arboni aid. However, in most H CO * (as defined below) is used to represent solution arbonates. H CO = H CO CO () * (aq) Where H CO * is the total dissolved abon inluding aqueous arbon dioxide. Carboni aid dissoiates into biarbonate and arbonate aording to the following equations: ( H ) ( HCO ) o 6.4 = K H CO = 10 ( ) H CO () where H CO = H CO CO (aq) = H CO * Note that K o HCO = *10 4 or pk =.69 if orreted for CO (aq) Setion 5 Carbonate Chemistry
Soil Chemistry 5 ( H ) ( CO ) o 10. = K = HCO 10 ( ) HCO (4) As for every aqueous reation the aid base relationship between the proton and hydroxide is an important relationship. ( ) ( ) H OH ( HO ) Starting with the eletrial neutrality expression: o = KW = 10 14 (5) [ H ] = [ OH ] [ HCO ] ] [ CO (6) The system is manipulated to ollet terms in the variables of interest hydrogen ion onentration and arbon dioxide partial pressure. Note that the eletrial neutrality expression is defined in terms of onentrations (mol/l). Therefore, in order to utilize values from the thermodynami equilibrium expressions, the onditional equilibrium onstants must be used whih relate onentrations rather than ativities to the ioni distributions at equilibrium. Defining in terms of yields: OH H [ ] = OH W K [ H ] (7) Solving for HCO gives: [ HCO ] = [ ] K H CO H CO [ H ] (8) Finding arbonate via biarbonate: K HCO [ CO ] = [ HCO ] [ H ] (9) Setion 5 Carbonate Chemistry
Soil Chemistry 54 Substituting for HCO : [ ] = K K [ HCO ] H CO HCO CO [ H ] The rewritten eletrial neutrality expression in terms of H and H CO is: (10) [ ] K K [ HCO ] [ H ] [ H ] [ H ] K W KH CO H CO H CO HCO [ H ] = (11) Rearranging, multiplying by (H ) and substituting k H * P CO for H CO yields: [ H ] K [ H ] K [ H ] k P K K k P = 0 W HCO H CO H CO HCO H CO ( ) [ H ] [ H ] K K k P K K k P = 0 w HCO H CO HCO HCO H CO (1) We are left with a polynomial equation in [H ]. In this formulation the only variables are (H ) and P CO. If a known and onstant value for the partial pressure of arbon dioxide is inserted into the above equation, (H ) an be found by a variety of numerial tehniques. Henry's Law Constant for this reation is strongly influened by temperature and slightly affeted by ioni strength. Salt Level k H @ 98K o Log k K 0. M NaCl 0.08 1.48 0.5 M NaCl 0.004 1.57 1.0 M NaCl 0.07 1.56 Temp o K k H @ 0. M NaCl Log k H @ 0. M NaCl 7 K o 0.07 1.1 78 K o 0.061 1.1 8 K o 0.051 1.9 88 K o 0.04 1.6 9 K o 0.07 1.47 98 K o 0.0 1.48 08 K o 0.06 1.58 Setion 5 Carbonate Chemistry
Soil Chemistry 55 Dissolved arbon is distributed among three speies H CO, HCO and CO as a funtion of ph. In soil systems where there may be an external (to the arbonate system) ontrol on ph it would be handy to know the distribution of the arbonate speies given ph. This distribution of arbonate speies an be derived from the HendersonHasslebah relationship knowing ph and pk s. A slightly different approah is shown next. Assume : C T = the sum of all arbonate speies onentration ativity oeffiients are negleted or equal to one. define z as: z = (H ) (H ) K HCO K HCO K HCO or more generally as: z = (H ) (H ) K 1 K 1 K where K 1 and K are the first and seond dissoiation onstants for the aid. then: [ ] [ ] = C Z HCO H T (1) [ ] [ ] = C Z HCO H K1 T (14) [ HCO ] C 1 T K K = (15) Z Figure 5.1. The distribution of arbonate speies as a fration of total dissolved arbonate in relation to solution ph. 1.00 a n = ( H n CO / C T ) 0.75 0.50 0.5 H CO HCO CO 0.00 pk 1 pk 4 6 8 10 1 14 Solution ph Setion 5 Carbonate Chemistry
Soil Chemistry 56 Figure 5.. The ativity of arbonate speies in relation to ph for a arbon dioxide level of 10.5 atmospheres. Log ativity of arbonate speies 0 4 6 8 10 P o = 10.5 H CO o HCO CO 4 6 8 10 1 ph Setion 5 Carbonate Chemistry
Soil Chemistry 57 Figure 5.. The ativity of arbonate speies in relation to ph for a arbon dioxide level of 1 atmosphere. Log ativity of arbonate speies 0 4 6 8 10 P o = 1 atm o H CO HCO CO 4 6 8 10 1 ph Equilibrium Reations in the CO H O system. Reation No. Equilibrium Reation Log K o 1. CO (g) H O W H CO o 1.46. H CO o W H HCO 6.6. HCO W H CO 10. 4. CO (g) H O W H HCO 7.8 5. CO (g) H O W H CO 18.15 Setion 5 Carbonate Chemistry
Soil Chemistry 58 Figure 5.4 Effet of arbon dioxide partial pressure on the solution onentration of arbonate speies in the CO water system. Log C 0 4 6 8 10 1 14 16 H H CO HCO HCO & H CO 10 8 6 4 0 Log P CO ALKALINITY Examination of Figure 5.4 indiates that as the arbon dioxide partial pressure goes to zero, the solution ph approahes 7 and inreasing pressures of arbon dioxide ause the system to be aidi. Therefore, in the CO water system there is never a net exess of base. The system an neutralize added base, but an not neutralize added aid. Another way of saying the same thing is to state that the system do not ontain any alkalinity. Alkalinity is an important onept in solution hemistry and relates to the aid neutralization apaity of solutions. The definition of alkalinity for the CO water system is: Alkalinity = [ OH ] [ HCO ] [ CO ] [ H ] (16) In the CO water system the eletrial neutrality ondition for the system is: [ H ] = [ OH ] [ HCO ] [ CO ] (17) Substituting the eletrial neutrality equation into the alkalinity definition : Setion 5 Carbonate Chemistry
Soil Chemistry 59 Alkalinity = [ OH ] [ HCO ] [ CO ] [ OH ] [ HCO ] [ CO ] (18) Therefore, there is no alkalinity in a CO water system, unless other soures of base are added. Case. CaCO CO H O system Calium arbonate in water with a fixed partial pressure of arbon dioxide. For the ase of a fixed partial pressure of arbon dioxide and alium arbonate dissolved in the aqueous phase one more equation is need to desribe the system. This is the solubility produt of alium arbonate: ( ) o = KCaCO Ca )( CO ( CaCO ) (19) The eletrial neutrality expression for this ase is: [Ca ] [H ] = [OH ] [HCO ] [CO ] (0) Substituting to produe an equation in H and H CO results in the following: CaCO K [ Ca ] = [ CO ] (1) As in the previous ase: [ Ca ] = K H CO K [ H K CaCO HCO [ H ] CO ] () [ HCO ] = [ ] KH CO H CO [ H ] () [ CO K K [ H CO ] H CO HCO ] = [ H ] (4) [ ] OH W = K [ H ] (5) Setion 5 Carbonate Chemistry
Soil Chemistry 510 These equations yield an expression in [H CO ] and [H ] whih are an be solved for [H ] at a given partial pressure of arbon. Knowing [H ] and P CO, Ca, HCO and CO an be found. Substituting into the eletrial neutrality expression yields: KCaCO W KH CO [ HCO ] [ ] K H = KH CO K [ HCO HCO ] [ H ] [ H ] [ H ] (6) K H CO K [ HCO H CO ] [ H ] Replaing [H CO ] with k H *P CO gives: K K k P CaCO KW H CO H CO [ ] H = KH COK HCO k H P CO [ ] [ ] H H [ H ] K K k P H CO HCO H CO [ H ] (7) Rearranging and multiplying by [H ] gives: 4 KCaCO [ H ] [ H ] KW [ H ] KH COk H PCO [ ] H KH COK HCO k H P CO (8) ( KH CO K k H PCO ) = 0 HCO This is a fourth power polynomial in (H ) whih an be solved by trial and error, graphial methods or numerial tehniques. Setion 5 Carbonate Chemistry
Soil Chemistry 511 Figure 5.5. Soluble Ca derived from alite, gypsum or Portlandite in relation to solution ph. A line depiting soil Ca at an arbitrary value of mmol/l is inlude, along with the solubility of Ca from gypsum. 8 portlandite Ca(OH) 6 alite (CaCO ) 4 Log (Ca ) 0 log Ca =. = ( K sp CaSO4 ) 0.5 4 Soil Ca =.5 6 8 4 5 6 7 8 9 10 ph Setion 5 Carbonate Chemistry
Soil Chemistry 51 Figure 5.6. Solution onentrations of Ca and arbonate speies derived from alite in relation to the partial pressure of arbon dioxide. 0 HCO OH Log onentration 4 6 8 10 CO Ca HCO H 1 OH 14 15 1 9 6 0 Log P CO Setion 5 Carbonate Chemistry
Soil Chemistry 51 Figure 5.7 Solution onentrations of Ca and arbonate speies derived from alite in relation to the partial pressure of arbon dioxide inluding ion pairs and omplexes.. 0 HCO Ca Log onentration 4 6 8 10 CaCO o HCO CaHCO CO CaOH OH 1 H 14 15 1 9 6 0 Log P CO Setion 5 Carbonate Chemistry
Soil Chemistry 514 CASE Calite in water without arbon dioxide additions In the ase where aess to arbon dioxide is restrited and there is little or no transfer of gases into or out of the solution, the ph an exeed the equilibrium ph of 8. for an airwateralium arbonate system. This is the ase of solid phase arbonate in a losed system and orresponds to dissolving pure alium arbonate in a gas free solution with no head spae. A pratial example might inlude irrigation water entering a dry soil ontaining arbonates. In this instane, the total arbonate speies are fixed by the solubility of the alite and do not depend on the partial pressure of the air as in the previous ase. Sine all of the arbonate speies are derived from dissolution of alite: [ Ca ] = [ H CO] [ HCO ] [ CO ] (9) Furthermore, the only harged speies in solution an be Ca, H, HCO, CO and OH whih yields the eletrial neutrality expression for this system. [ Ca ] [ H ] = [ OH ] [ HCO] [ CO ] (0) Solution of this ase follows on those already presented. In Case arbon dioxide (an aid) is not allowed to enter the system and the final ph of the solution is nearly 10. The alkalinity is generated by the hydrolysis of the arbonate ion. If the arbonate is more soluble, there is more arbonate ion and solution ph reahes a higher value. The reations generating the high ph are shown below. CaCO Ca CO (1) ( C) CO HOH HCO OH () Setion 5 Carbonate Chemistry
Soil Chemistry 515 Figure 5.8 is a graph of the hanges in ph generated when pure water is bubbled with arbon dioxide (Case 1) followed by the addition of alium arbonate to reah a ph of 9.9 (Case ). After the solution reahes the high ph assoiated with Case. As arbon dioxide is bubbled into the solution, CO reats with to form HCO and the solution ph returns to 8. or Case. When OH is present, it an reat with CO to form HCO. Figure 5.8. Changes in solution ph in relation to experimental manipulation of the arbon dioxidewater and arbondioxidealium oxide system. 11 10 Case ph = 9.9 Bubbled with air Solution ph 9 8 7 Addition of gypsum Case ph = 8.8 ph = 7.75 Addition of CaCO 6 5 Case 1 ph = 5.6 0 1 4 5 6 7 8 Time (hours) Case 4. The addition of NaOH to water open to the atmosphere. Case 4 is a situation where all of the added salt is soluble and the initial alkalinity is due to the addition of OH in diret relation to the onentration of added NaOH. As in Case, CO reats with OH to form HCO and over time NaOH beomes NaHCO. Alkalinity in this ase will be equal to the OH added to the system. This is the reason that strong bases are not used as primary standards for aidbase titrations, and a good reason to protet solutions of strong base from ontat with the atmosphere. Asarite is a good adsorbant of arbon dioxide and is often used as a trap for arbon dioxide on bottles of NaOH. Setion 5 Carbonate Chemistry
Soil Chemistry 516 The eletrial neutrality expression for Case 4 is: [ H ] [ Na ] = [ OH ] [ HCO ] [ CO ] () If Po is fixed at atmospheri values, then a set of simultaneous equations an be solved. The onentrations found for this ase are presented in Table 5.1. Table 5.1 is a summary of the various ases we examine in this setion on arbonate hemistry and alkalinity. Note that even thought the ph is higher in Case than in Case, the alkalinity is higher in Case. It is also interesting that the alkalinity in Case 4 is muh higher than in either Case or. The NaOH with an initial ph of nearly 1 has been titrated by the aid arbon dioxide to a ph of 9.77. Table 5.1 Solution onentration and properties for Cases 1,, and 4. Case 1 Case Case Case 4 ph 5.666 * 8.18 9.944 9.767 P CO (atm).0e4.0e4 6.1E7.0E4 H CO 1.04E5 1.04E5.19E8 1.04E5 HCO.16E6 9.87E4 8.59E5.48E CO 4.70E11 1.10E5.79E5.00E Ca 5.0E4 1.4E4 CaOH 1.8E8.0E7 CaHCO 5.50E6 1.8E7 o CaCO 5.7E6 5.7E6 Na 8.69E o NaHCO 1.04E NaCO 1.0E C T 1.6E5 1.0E 1.9E4 6.79E Ca T 5.14E4 1.9E4 Alkalinity 0 1.0E.59E4 0.100 I.S..16E6 0.0015 0.00041 0.1009 ( 1 0.998 0.957 0.977 0.781 ( 0.99 0.840 0.91 0.7 Conentrations are given in mol L_1., Case 1 is water in equilibrium with standard air. Case is water saturated with CaCO in equilibrium with standard air. Case is a losed aqueous system saturated with CaCO., Case 4 is 0.1 M NaOH brought to equilibrium with standard air., I.S. is ioni strength *Note: you an alulate ph more preisely than it an be measured. Setion 5 Carbonate Chemistry
Soil Chemistry 517 CARBONATE SUMMARY It is instrutive to note several features of the arbon dioxidewater and arbon dioxidewater alium oxide system. 1. Sine the eletrial neutrality expression and mass balane equations are speified in terms of onentrations and the aid dissoiation and solubility expressions are defined in terms of onentrations via onditional equilibrium expressions, the solutions presented here are general but relay on appropriate methods to determine ion ativities. For the limited solubility of the alite systems and the low ioni strength of the arbon dioxide water system, the assumption of unit ativity oeffiients is reasonable. However, in many waters and soil solutions there are indifferent eletrolytes that do not partiipate in the reations, but ontribute to ioni strength.. It should be evident that solving even these simplified equations is laborious and the general solution that iterates to determine the orret ioni strength is a monumental undertaking with out the aid of omputers. However, there are a few assumptions that an ease the labor. In the last ases presented, the final ph is alkaline and the ontribution of H to the eletrial neutrality expression is negligible. Similarly, the ontribution of H CO to the total arbonate speies is also very small and an be ignored with little loss in auray.. The final ph in the losed system depends on the solubility of the solid phase. For more soluble arbonates, solution alkalinity will be muh greater than in the alite ase. This setion has emphasized: 1. Simultaneous equilibria for quantitative desription of system behavior. a. Note that the "Lindsay" graphial tehnique is easy to apply beause it generally only onsiders a limited or speialized ase and it often makes drasti assumptions in order to develop a linear (loglog) relationship. But it is useful as a rough, quik and dirty guide to behavior. b. The quantitative approah requires a mathematial development inluding mass or harge balane plus the judiious seletion of the appliable equilibrium (thermodynami) expressions, iterative alulation of ioni strength and ativity orretions. It is pratiable only with a programmable alulator or a omputer. Setion 5 Carbonate Chemistry
Soil Chemistry 518. The quantitative approah an be extended from a speifi ase to a series of alulations overing a range of a key variable so that the results an be graphed. This is exemplified by the graphs on pages 51 and 51. This general type of graph is often alled a "Bjerrum" graph. Note how muh more information is given by the "Bjerrum" graph on pages 51 and 51 than by the Lindsay graph on page 511. Also note that the same equilibrium expressions (top of page 78) are used in onstruting both graphs.. Calium ion always ats as an aid.. Role of atmospheri (gas phase) CO in affeting solution phase omposition. a. In soils and sediments, partial pressure of CO an range widely; e.g., from 10 6 to one atmosphere. Note: At depth in water, hydrostati pressure inreases the total pressure on any gas phase (bubble). b. Sampling of soil water by vauum or sution tehniques an drastially affet equilibria in the water being extrated.. Bioaquati effets: Diurnal ph shifts. Lakes, marshes, estuaries, and rie paddies may show a maximum ph during the day (ph < 9.5) but a ph minimum during the night (ph > 7). The phenomenon is due to extration of CO from the alkalinity in the water during the day to arry on photosynthesis (algae, other phytoplankta). (Note that suh organisms are arrying on respiration during the day but CO uptake for photosynthesis is > > than CO release from respiration. During the night, the dominant proess is release of CO to the water by respiration. Nota Bene: During the day, ph annot go over 7 unless the water ontains alkalinity. The following program was written to alulate the distribution of arbonate speies in the arbon dioxide water system in relation to the partial pressure of arbon dioxide in the atmosphere. The results are tabulated and the data also is shown in Figure 5.4 on page 58. Setion 5 Carbonate Chemistry
Soil Chemistry 519 The Property of the Regents of the University of California February 1, 1989 100 REM Case speifi program to alulate ph and all arbonate speies SSC110 110 REM in pure water as a funtion of partial pressure of C0 R.G. Burau 10 CLS 10 DEFDBL AW 140 DIFF =.00001# 150 DEF FNDEL(A,B) = ABS(AB)* / ABS(AB) 160 OPEN "O",1,"B:SSPCO" 170 K1 = 10^6.5 : K = 10^10. : KW =.00000000000001# : KH = 10^1.46 180 RESTORE 190 190 DATA 1,1,lD7,0,0,0,0 00 READ G11, G1, H1, OH1, HCO1, C01, IS1 10 FOR ZPCO=10 TO 0 STEP.1 0 PCO = 10^ ZC0 0 REM...Top of NEWTONRAPHSON 40 KWC = KW/G11/G11 Z50 K1C = K1/G11/G11 60 KC = K/G1 70 F = H1 ^ (KWC K1C*KH*PCO)*H1 K1C *KC*KH*PCO 80 IF F = 0 THEN 570 90 F1 = *H1 ^ KWC K1C*KH*PCO 00 IF F1 <> 0 THEN 0 10 N1 = 1.0001*H1 0 GOTO 70 0 H = H1F/F1 40 HCO = K1C*KH*PCO/H 50 C0 = KC* HCO/H 60 OH = KWC/H 70 IS = *C0 (HCO OH H)/ 80 G = (SQ (ls)/(lsq(ls))).*is 9O G1 = 10^(.5*G) 400 G = 10^(*G) 410 SWAP H,H1 : SWAP HCO,HC01 : SWAP C0,C01 : SWAP OH,OH1 40 SWAP IS,IS1 : SWAP G1,G11 : SWAP G,G1 40 DEL = FNDEL(H1,H) 440 IF DEL > DIFF THEN 0 450 DEL = FNDEL(HCO,HCO1) 460 IF DEL > DIFF THEN 0 470 DEL = FNDEL(CO,CO1) 480 IF DEL > DIFF THEN 0 490 DEL = FNDEL(OH,OH1) 500 IF DEL > DIFF THEN 0 510 DEL = FNDEL(lS,IS1) 50 IF DEL > DIFF THEN 0 50 DEL = FNDEL(G1,G11) 540 IF DEL > DIFF THEN 0 550 DEL = FNDEL(G,G1) 560 IF DEL > DIFF THEN 0 570 ZPH = LOG(H1*Gll)/LOG(10) 580 ZHCO = LOG(KH*PCO)/LOG(10) 590 ZHCO = L0G(HCO1)/LOG(10) 600 ZC0 = LOG(CO1)/LOG(lO) 610 ZIS = IS1 60 ZG1 = Gll 60 PRINT ZPCO, ZPH, ZHCO, ZIS, ZG1 640 PRINT #1, ZPCO; ZPH; ZHCO; ZHCO; ZC0; ZIS; ZG1 650 NEXT ZPCO 660 CLOSE : END Setion 5 Carbonate Chemistry
Soil Chemistry 50 CARBONATE SPECIES AND ph AS A FUNCTION OF P CO Log onentration P CO H H C0 HCO G1 CO IS 10 6.999 11.46 10.81 14.19 1.0004 E07 0.9996 9.5 6.555 10.96 10.1 1.640 1.0006 E07 0.9996 9.1 6.999 10.56 9.910 1.40 1.0009 E07 0.9996 8.5 6.998 9.996 9.11 1.641 1.008 E07 0.9996 8.0 6.996 9.456 8.81 1.146 1.0081 E07 0.9996 7.5 6.989 8.956 8.0 11.660 1.046 E07 0.9996 7.0 6.968 8.460 7.841 11.0 1.075 E07 0.9996 6.5 6.91 7.960 7.96 10.81 1.14 E07 0.9996 6.0 6.796 7.460 7.01 10.546 1.5978 E07 0.9995 5.5 6.614 6.960 6.695 10.410.407 E07 0.9994 5.0 6.91 6.460 6.418 10.56 4.0644 E07 0.999 4.5 6.150 5.960 6.159 10.7 7.077 E07 0.9990 4.0 5.90 5.460 5.906 10.1 1.50 E06 0.9987.5 5.654 4.960 5.654 10.8.19 E06 0.998.0 5.404 4.460 5.404 10.6.9458 E06 0.9977.5 5.154.960 5.154 10.5 7.005 E06 0.9970.0 4.904.460 4.90 10. 1.456 E05 0.9960 1.5 4.654.960 4.65 10.1.51 E05 0.9946 1.0 4.404.460 4.40 10.18.969 E05 0.998 0.5 4.154 1.960 4.151 10.1 7.0658 E05 0.9905 0.1.954 1.560.950 10.09 1.16 E04 0.9880 R.G. Burau SSC10 February 1, 1989 COMPUTER SOLUTION OF SIMULTANEOUS EQUILIBRIA To illustrate the reation of a polynomial expression from a series of simultaneous equations whih an then be solved by a NewtonRaphson (NR) or other numerial proess, alulate equilibrium values for the system CaCO CaSO 4 H O Air. All the appropriate simultaneous equilibria are represented by: 4.47 *10 9 K 1 = [ Ca ][ CO ] = (1) γ.51*10 5 K = [ Ca ][ SO4 ] = () γ K K [ H ] [ HCO ] 4.45 * 10 7 = = [ HCO ] γ 1 11 [ H ] [ CO ] 4.69 * 10 4 = = [ HCO ] γ () (4) Setion 5 Carbonate Chemistry
Soil Chemistry 51 K 5 [ Ca ] [ ] = = 5. * 10 [ CaSO ] SO4 5 o 4 (5) HCO ) = kh P =1.05*10 (6) 5 ( CO K 10 14 w = [ H ] [ OH ] = γ 1 (7) The eletrial neutrality expression is: [ Ca ] [ H ] = [ CO ] [ SO4 ] [ HCO ] [ OH ] (8) Note that this last expression an be modified by elimination and substitution using (1) to (7) as neessary into a polynomial in 1 variable having exponents; 0, 1,,... m. Sine we are interested in ph, an expression in [H ] is onvenient. Substituting from above in (8): K [ H K4 K k K K ] = K K1 [ H K4 K [ H 1 ] 4 kh PCO kh [ PCO K kh PCO H H PCO ] ] [ H ] K [ H w ] (9) Multiplying both sides by [H ] and rearranging: [ ] K K K k P [ [ ] K K K P = 0 K K k P K 4 K1 H 4 H CO H ] ( khpco ) K Kw H 4 H CO 4 H CO 1 (1) Evaluating the oeffiients leads to the desired polynomial: 1 4 1 18 4.08*10 [H [H 4.68*10 [H ].46*10 = 0 = f ] ] Note that previous expressions all used onditional equilibrium onstants (K's). In a real omputer program, the ioni strength is estimated and from this estimate the ativity oeffiients are alulated allowing for an initial estimate of K. Subsequently, in eah iteration of the program new ioni strengths, K 's, ativity oeffiients and onentrations are estimated. These values are used in the NewtonRaphson proess (NR) whih onverges on the orret value for [H ]. In the first iteration all ativity oeffiients are set to 1. An initial guess for [H ] must 8.5 be entered for the NR proedure. Using [ H ] = i 10 for the initial guess, and finding the first derivative of the polynomial (ƒ'), whih is: 1.6 * 10 14 [ H ] [ H ] 4.68 * 10 1 = ƒ' If the urrent estimate of [ H ] is [ H ] i, then the next estimate by the NR tehnique is [ H ] i1 and the estimate is given by: [ H ] i1 = [ H ] i (ƒ / ƒ ) Setion 5 Carbonate Chemistry
Soil Chemistry 5 the new value of [ H ], [ H ] i1 is then used to alulate values for all other onstituents starting with equation () to alulate [HCO ] then proeeding to (4) to alulate [CO ],... and ending with () to alulate [SO 4 ]. These values are used to alulate a new estimate of ioni strength that is then proessed through the various equations to alulate new values of the ativity oeffiients using the Davies equation or other expressions. These "new" ativity oeffiients are then used to find new onditional K's and these in turn are used to find new values for the oeffiients in (10). Computer output during a NEWTONRAPHSON solution of polynomial for the system gypsumalitewaterair using ph = log [H ] = 8.5 as the initial estimate of ph. 1 4 5 6 7 8 9 10 11 1 1 14 15 16 17 18 19 0 1 4 5 6 7 ph 8.5000 5.8 5.408 5.54 5.6585 5.787 5.9088 6.09 6.1590 6.841 6.409 6.544 6.6595 6.7846 6.9097 7.048 7.1597 7.844 7.408 7.594 7.647 7.754 7.7881 7.8011 7.8017 7.8017 7.8017 Setion 5 Carbonate Chemistry
Soil Chemistry 5 Graphial solutions to the arbonate system an be very helpful to the understanding of arbonate hemistry. They an be rendered in either the log ativity form we are familiar with or in the log onentration form that follows from the equations presented above. Both endeavors rely on the availability of solubility onstants, dissoiation onstants and ion pair formation onstants for the appropriate speies. The following table taken from Lindsay (1979) presents the important onstants. Reation No. Equilibrium Reation log K o Carbonates 1. CaCO (alite) H = Ca CO (g) H O 9.74 14. CaCO (aragonite) H = Ca CO (g) H O 9.97 15. CaCO? 6 H O (ikaite) H = Ca CO (g) 7 H O 11.7 16. CaMg(CO ) (dolomite) 4 H = Ca Mg CO (g) H O 18.46 Soil, Oxides, Hydroxides, Ferrites 17. SoilCa = Ca.50* 18. CaO (lime) H =Ca H O.95 19. Ca(OH) (portlandite) H =Ca H O.80 0. CaFe O 4() 8 H = Ca Fe 4 H O 1.4 Sulfates 1. CaSO 4 (insoluble) = Ca SO 4 4.41. acaso 4 (soluble) = Ca SO 4.45. ßCaSO 4 (soluble) = Ca SO 4 1.75 4. CaSO 4? H 0 (gypsum) = Ca SO 4 H O 4.64 All of the above equations were taken from Lindsay 1979. Chemial Equilibria in Soils, Chapter 7. John Wiley and Sons, New York. Setion 5 Carbonate Chemistry
Soil Chemistry 54 NewtonRaphson Method of find Solutions for polynomials. The solution of polynomial equations relies on being able to find the point where ƒ(x) = 0. Normally the proedure would be to fator the equation into a series of terms suh as (xa) (xb) = 0 Then by inspetion, one an set x = a or b and the roots of the equation are found. In the ase of a quardati equation, this solution is: x b b 4a a x b b 4a a Multiplied out this is the more familiar ( ax bx = 0) While there are exat solutions for quadrati and ubi funtions, the general polynomial f x = a ax ax ax ( ) o 1... n n must be solved by numerial methods. If the funtion is ontinuous on over the interval of interest one may solve for the ondition ƒ (x) = 0 by several methods. One is the bisetion tehnique. In this method, two estimates of x are hosen so that ƒ(x 1 ) 0 and ƒ (x ) 0. By hoosing subsequent values of x that are between the limits of x 1 and x, the root of x an be approahed as losely as desired. This method will always work if the funtion is ontinuous on the interval, but may onverge slowly. A more elegant method is the NewtonRaphson Tehnique. In this method an initial guess of the root x 0 is used. Then the slope of the line at the point ƒ(x 0 ) is alulated and the point of intersetion with the y axis is used as the next estimate of the root. This is illustrated in below. In Newton s method the first guess (x o ) is the value of x used to evaluate ƒ(x o ) knowing the value of ƒ(x o ) the slope of the line tangent to the point ƒ(x o ),x o is: Slope = y x y x 0 1 0 1 where: y1 = 0, and y1 = f( x0) Setion 5 Carbonate Chemistry
Soil Chemistry 55 Illustration of the NewtonRaphson method for solution of polynomials. 0 18 16 14 1 f (x o ) 10 f (x) 8 6 4 0 x x 1 x 0 4 0 1 4 5 6 7 8 X Setion 5 Carbonate Chemistry
Soil Chemistry 56 Review Questions 1. What is the eletrial neutrality expression for the CO H O system? For the alitegypsum system?. Define alkalinity? What is the expression for alkalinity in a alitewater system? What assumptions if any have you made?. What are the ph limits in the alite water system in relation to arbon dioxide pressure? 4. What is the Turner Effet? 5. Fly ash ontains oxides of Ca, Na and K, whih dissolve in water to form hydroxides. The least soluble of these is Ca(OH). This hydroxide has a ph of approximately 1.5. However, the ph of alium hydroxide after a time dereases. Why? What will the equilibrium ph be? 6. What are the two system points in the Johnston Diagram in your syllabus? 7. In the CO H O System the program to alulate ph et. as a funtion of P CO ontains the following ode: 0 H = H1 F/F1 What is this line doing. What is F and F1? 8. What will the addition of gypsum do to the ph of a alite system in equilibrium with CO? 9. How does temperature and ioni strength affet CO water, and CO CaOwater systems? 10. When does Ca at as an aid in soil systems? Setion 5 Carbonate Chemistry
Soil Chemistry 47 11. The alium arbonate ion pairs are onstant over some ranges of P o in the Johnston diagram but not others. Why? Carbonate Chemistry