Ind. Eng. Chem. Res. 1987,26, 1291-1296 1291

Absorption of SO2 in Citric Acid-Sodium Citrate Buffer Solutions

Binay K. Dutta,* Ranjan K. Basu, Amit Pandit,+ and Parthasarathi Ray Department of Chemical Engineering, Calcutta University, Calcutta 700009, India

Absorption of SO2 in citric acid-sodium citrate buffer solutions has been carried out by using a continuous flow stirred cell with a flat gas-liquid interface. The effects of gas concentration, liquid concentration, pH, and stirring speed on the absorption rate have been investigated. Theoretical models for the absorption process have been developed based on the film theory and on an ap- proximate surface renewal theory. It has been observed that the absorption data could be satis- factorily interpreted using the model based on film theory of mass transfer.

Removal of sulfur dioxide from stack gases is extremely important in the control of atmospheric pollution. The most widely used processes for this purpose are dry ab- sorption in lime and limestone injected into the hot flue gas and the wet process where the flue gas is scrubbed in lime slurries, sometimes with additives that increase the rate of absorption. These processes, besides being non- regenerative in nature, create the problem of disposal of large amounts of waste.

One regenerative process of sulfur dioxide removal in- volving absorption in citric acid-sodium citrate buffer solution has received considerable attention in recent times. The process was developed by Erga (1975) and later on by Farrington and Bengtsson (1979). The pH range used in this process is 3-5 so that the dissolved SO2 can subsequently be recovered from the loaded solution by steam stripping. Although some aspects of the process and the nature of solution equilibrium have been discussed by Erga (1980), the details are not available in open literature. Recently some studies on this absorption process have been reported by Dutta et al. (1986). The present work is in- tended to investigate the absorption characteristics of sulfur dioxide gas in sodium citrate-citric acid buffer so- lution using a continuous flow stirred vessel with a flat gas-liquid interface.

Experimental Setup and Procedure The absorption cell was made of a 10-cm section of

7.5-cm-i.d. Corning glass pipe with the ends closed by rubber corks. I t was fitted with stirrers for both gas and liquid phases. The stirrer shafts were concentric but driven by different motors provided with speed changing devices. The rpm of the stirrers could thus be changed independ- ently so that the effect of rpm in both gas and liquid phases could be studied individually. The liquid holdup in the absorption cell was 137 mL, and the liquid level was maintained constant by a liquid seal. The gas-phase stirrer was placed about half an inch over the liquid surface. Liquid samples for analysis were collected in two sampling tubes of known volumes provided with stopcocks and placed alternately in the exit liquid line. The stirred cell with the sampling device is shown schematically in Figure 1.

The gas phase was a mixture of SO2 and air, prepared by mixing these gases a t desired proportions. SO2 was available in cylinder, and air was supplied by a compressor. The individual gas flow rates were measured by calibrated capillary flow meters. The absorbent liquid was taken in a large Morriotte bottle, and the flow rate was measured

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by another capillary flow meter. The flow rates of the various streams were adjusted by fine-control needle valves.

The system was continuous with respect to both gas and liquid phases. The gas flow was started after the liquid level could be adjusted and maintained at the desired position. The attainment of steady state was ascertained by taking liquid samples at different times during a run and analyzing the SO2 content in the samples. I t usually required about 15 min for attainment of steady state. However, in all the runs, absorption rates were found out by taking liquid samples after the system operated under uniform conditions for more than 25 min. Liquid flow rate was 45 mL/min and gas flow rate was maintained at 8 L/min. I t was found that a maximum of only 2% of the SO2 entering was absorbed in the liquid at steady operating conditions. I t was also found by analysis of several gas samples that the inlet and exit gas concentrations were virtually equal. Absorption rates were measured in water as well as in sodium citrate-citric acid buffer solutions at different gas and liquid concentrations and pH. Effect of stirrer speed in both of the phases was also studied inde- pendently. All the runs were taken at a constant tem- perature of 35 f 0.5 OC and essentially atmospheric pressure. Fluctuations in flow rates and gas concentration during any run were below 2 % . Several runs were repeated to ensure reproducibility of the experimental data col- lected. Oxidation of SOz in solution could not be detected.

The concentration of dissolved gas was estimated by iodometric titration. pH values of fresh and loaded buffer solutions were measured by Phillips Model PM-9045 di- gital pH meter.

Theoretical Considerations When SO2 is absorbed in a polyfunctional buffer solution

such as sodium citrate-citric acid buffer, the following ionization reactions occur.

SOp + HzO + H+ + HS03- (1)

HS03- 6 H+ + S032- (2)

H,Ci + H+ + H2Ci- (3)

H2Ci- + H+ + HCi2- (4) HCi2- 6 H+ + Cis- (5)

Where H3Ci, H2Ci-, HCi2-, and Ci3- represent citric acid and corresponding anions. Reactions 1 and 2 only occur if the absorbent liquid is pure water. The hydrolysis of dissolved SO2 (eq 1) is known to be very fast (Eigen et al., 1961), the proton-transfer reactions 2-5 being even faster. Thus, all the above reactions may be regarded as instan- taneous.

0 1987 American Chemical Society

1292 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987

- Srirred Cell 2 , 3 - G a s inlet 8 outlet 4, 5 - L i q u i d inlet a outlet 6 - Thermometer 7,8- Liquid a G a s stirrers 9 - Static head meter 10,Il- Sampl ing tubes

Figure 1. Experimental setup-stirred cell.

In order to apply the surface renewal theory, the un- steady-state diffusion equations for this system may be expressed as follows. For SO2 and corresponding anions

8x2 - - d2[HS03-] d2[ so32-]

+ Dso32- ax2

at at at

+ DHSOj d2[S021

Dso* ~ ax* a[SO,] + d[HS03-] d[S03*-] +---- (6)

For H3Ci and corresponding anions d2 [ H3Ci] a2[HzCi-]

DH,Ci 7 + D b C I a x L +

a[H2Ci-] d[HCi2-] d[Ci3-] +-+- ( 7 )

In addition, the following equation must be satisfied for at at at

electrical neutrality of the solution.

- a2[H+] a*[HSO3-] a 2 1 ~ 0 ~ 2 - 1

D H + - - DHSOj - 2 D S 0 , J ax2 ax2

- d2[H2Ci-] a* [ HCiL-]

d2[Ci3-] d[H+] d[HS03-] d[S03*-] dX2

DH,Ci - 2 D H C ~ 2 dX'

- ~- - 2 - - 3DCl ____ -

d[H,Ci-] d[HCi2-] a[CiJ-] ax 2 at at at

2 ~ - 3 - (8)

The appropriate initial and boundary conditions are

~- at at at

t = 0 [SI = [SI, (9)

where [SI is the concentration of any species in eq 6-8. Also

x = 0 [SO,] = [SO,], (10)

a [ H3Ci] dX

a [ H2Ci-] dX

a[HCi2-] a[Ci3-]

d[HSO3-] a [so32-] dX ax

+ DH2C~ ~ + DH?C~ ~

+ Dc,3 - - - 0 (11) DHC~2- 7 dX

- 2Dso32 ~ - DHS03- aw+1

ax DH+ - -

d[H2Ci-] a [ HCi2-] d[Ci3-] 2 D ~ c ~ 2 ___ - 3% 7 = o

D H z C ~ 7 - ax (12)

Boundary conditions 11 and 12 indicate that there is no net flow of the citric acid or citrate ions and of the charge across the gas-liquid interface.

Also, as x - S = [SI, (13)

In addition, the concentrations of the individual species in eq 6-8 are interrelated by the equilibrium relations which, for eq 1-5, may be expressed as

K,' = = K 1 ~ H + ~ H S O , - Yso2 Yso2

%02 YH+YHSO~- Y H+Y HSO~-

aH+aS032- YHS03- YHS03- K2' = ___ = K2

QHS03- YH+YS032- YH+YS032-

aH+aH2Ci7 YH3Ci YH&i = K3 K,' = ~

aH3Ci YH'YH2Ci- YH+YH2Ci-

aH+aHCi2- YHzCi- YH2Ci- = K4 K4r = ~

uH2Ci- YH+YHCi2- YH+YHCi2-

(14)

where K,-K6 are the respective thermodynamic equilib- rium constants and K,'-K,' are the effective equilibrium constants in concentration units.

It is not possible to obtain analytical solution for the above coupled set of partial differential equations (eq 6-8) subject to prescribed conditions. Numerical solution would also be quite involved. The corresponding film theory equations will, however, be much simpler and would admit of closed-form solutions. The film theory equations are

YHCi2- aH+aci3- YHCi2- K5' = ___ ~ = K5-

uHCiZ- YH'YCi" YH+YCi3-

d2[S032-] = 0 (15)

dx2 Dso32-

d2[H3Ci] d2[H2Ci-] + dx2 + DH2Ci? dx2 DH3Ci

d2 [ HCi*-] d2[Ci3-]

dx2 dx + Dci3- ___ = 0 (16) DHCi2-

and

- d2[H+] d2[HS03-] d2[S032-]

- 2DSo32- dx2 dx2 D H + - - DHS03- dx2

- d2[H2Ci-] d2[HCi2-]

dx2 dx2 - 2D~ci2- DH2C~

d2[ Ci3-]

dx 3Dci3- ~ = 0 (17)

The above equations assume that there is no potential gradient across the film. The boundary conditions at x = 0 (gas-liquid interface) remain the same as eq 10-12. The boundary condition 13 in this case reduces to

x = 6 s = [SI, (18) Concentration distribution of all the species concerned

may be obtained by solving eq 15-17 subject to boundary conditions 10-12 and 18 together with the equilibrium relations 14. However, calculation of the interfacial con- centrations of HS03- and SO3- are particularly necessary for finding out the theoretical enhancement factor for absorption. The interfacial concentration of hydrogen ion, [H+Ii, is required for this purpose. It can be shown that

Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1293

I- Liquid i t l r re r r . p . m . o

c 0 4. 0

2 - x

[H+Ii may be calculated by solving the sixth degree alge- braic equation (more details are given in the Appendix) [H+]: + CU~[H+]: + a2[H+]t + C Y ~ [ H + ] ~ + a4[H+]i2 +

aS[H+]i + (Ye = 0 (19)

where Dd(3/- b3

= (D7K4 - b2D3 - b3Ds)Ki - D1K1’ [SOJi as = (D&,’ - 2b2D4 - b3D7)KiK; -

(DlD&i + 2D2K2/ )KI’ [SO2li = -(DiD7K: + 2020&2/)K1’Ki[S02]i -

(3b2D5 + b3DB)KiK:K5/ a 5 = -(D1D$(5/ + 2D,D,K,’)K,’K3’K: [SOzli

“6 = -2D2DsK1’K2/K3’K4‘K5/[SO2]i (20) b2 and b3 are given by

b2 = [H3Ci], + D6[H2Ci-I0 + D7[HCi2-Io + DB[Ci3-l0

b3 = [H+l0 - D1[HSO3-lo - 2D2[S032-]o - D3[H2Ci-Io - 2D4[HCi2-Io - 3[Ci3-], (21)

The enhancement factor for the absorption process may now be readily calculated by using the film theory.

Ecf = 1 + (DHs~,-([HSO~-]~ - [HSO,-]o) + Dso3~([S032-Ii - [S032-Io))/lD~~,([S021i - [SO2IO)l (22) I t was pointed out by Olander (1960) that for single

equilibrium reaction the film and surface renewal theory expressions for enhancement factor are identical if the diffusivities of the species are equal. In other cases, an approximate expression for the enhancement factor was suggested by Chang and Rochelle (1981,1982a) which is the same as the film theory expression except that the diffusivity ratios are replaced by the corresponding square roots. For the present system, the enhancement factor for approximate surface renewal theory would be E,, = 1 + (D~so,-’/~([HSo,-]i - [HSO,-]o) +

D s o ~ z - ” ~ ( [ S O ~ ~ - I ~ - [S032-l~))/(D~,,l/2([sO~li - [SOZIO)) (23)

For the case of absorption in pure water, the expression for enhancement factor for the film theory given by Olander (1960) may be used.

Temp: 3 5 ‘ ~ Cos stirrer r ,p.m. : 266 Liquid stirrrr r.p.m.:

a90 V 7 0 A 5 0

The same expression for the surface renewal theory may be approximated by (Chang and Rochelle, 1981)

where the second ionization of the dissolved SO2 is ne- glected, as it occurs to a negligible extent in solution in pure water.

Equations 22-25 may be used to test the chemical ab- sorption mechanism of SO2.

Experimental Results Absorption rates were measured in sodium citratecitric

acid buffer solution as also in pure water at different gas and liquid stirrer speeds, gas concentrations, liquid con- centrations, and pH. The fraction of the feed gas absorbed at steady state was very small, and the gas concentration could thus be calculated from observed flow rates of air and SO2.

9 c

a I -.-

0 200 400 600 8 0 0 1000 Go8 phose r t i n e r r , p . m .

Figure 2. Effect of gas-phase stirrer rpm on absorption rate.

0 0 h I O I 2 14 16 1.8 2 0

Gar c o n c ( v o l % S o 2 1

Figure 3. Effect of gas concentration on absorption rate in water.

Test for Gas-Phase Resistance. For studying gas absorption with chemical reaction, it is convenient to eliminate gas-phase resistance to mass transfer. The sig- nificance of gas-phase resistance for the system under study was tested by taking several runs with different gas-phase stirrer speeds while keeping all other system variables fixed for SO2-buffer and SO2-water systems in- dividually. No effect of increasing stirrer rpm from 255 to 997 on the rate of absorption was observed (Figure 2). Thus, even at a speed of 255, the gas-phase stirrer could produce sufficient turbulence in the medium so that the resistance to mass transfer in that phase was negligible.

Effect of Different System Variables on Absorption Rate. The specific absorption rates of SO2 in both water and buffer solutions were found to increase linearly with gas concentration (Figures 3 and 4). This is because of increased solubility of the gas (and hence of interfacial concentrations of bisulfite and sulfite ions) with increased partial pressure of SO2 in the gas phase. The effect of pH on absorption rate is shown in Figure 5. It is found that the absorption rate increases with pH to some extent at each stirrer speed. The effect of concentration of the feed solution was studied by preparing solutions with different amounts of sodium citrate and citric acid so that the pH remained constant. Figure 6 shows that the absorption rate is nearly independent of feed concentration. This is because at low concentrations of gas used and at the in- terfacial area per unit volume of liquid available, the ob- served concentration of the dissolved gas was too small to change the pH of the absorbent substantially even for the case of lowest solution concentration (3.65%). However, if the amount of dissolved sulfur dioxide in the liquid is large, the absorption rate is likely to be more for solutions of higher buffer concentration because of correspondingly larger buffer capacity.

1294 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987

Temp : 35.C 00s stlrrrr r . p . m . : 2 5 5

Buffer ccnc: 0.178(M)

o f f e e d solution: 4.42

Liquid stirrer r . p . m.:

e 90

001 I I 1 I

0.0 0.5 1.0 1.5 2.0 Gos conc. ( V O I .% so, )

Figure 4. Effect of gas concentration on absorption rate in buffer solution.

Temp: 35.C Goa conc: 1.70% SO2 &ps sh'rrer r . p . m . 2 5 5 Liquid stirrer r . p m .

* -

L a I

0.0 ' I 3.0 3.5 4.0 4.5 5.0

P" of feed Solution

Figure 5. Effect of pH on absorption rate.

Temp: 35'c Goa conc :I.70%SO2 Gas stirrer r.p.m.: 255

Liquid stirrer r. p.m. of fo rd solullon: 4.42

I 7 0

Y--A * a

4 8 12 16 20 0. oo Feed liquid conc. ('/e by W t . )

Figure 6. Effect of feed liquid concentration on absorption rate.

Interpretation of Absorption Data The theoretical enhancement factor (E) for absorption

in both SO2-water and SO2-buffer systems may be cal- culated by using eq 22 and 24 for the film theory and eq 23 and 25 for the approximate surface renewal theory, respectively. By comparison of the specific rates of ab- sorption in S02-buffer (kbuffer) and in SOz-water (kwater)

(26) kbuffer Ecf

kwater E w f for film theory -- - -

kbuffer

kwater E w s for surface renewal theory (27) - - - -

Equations 26 and 27 provide simple but effective ways of testing the validity of the theories for the system studied. Different physical properties and data required for this purpose have been taken from literature or esti- mated by using suitable correlations.

2.5

2.0 -

W

0.0 0.5 1.0 1.5 2.0 2.5

(E, /E, )exp Figure 7. Theoretical and experimental enhancement ratio plot- film theory. SO2 concentration, 1.05-2.00% ; pH, 3.58-4.80; solution concentration, 3.65-12.16%; liquid rpm, 50-90.

Physical Solubility of SO2. Johnstone and Leppla (1934) reported the ionization constant and solubility in SOz-water system. Rabe and Harris (1963) reported a detailed investigation of the equilibrium and solubility for this system. They recommended the following expression for the Henry's law constant as a function of temperature:

H = exp[2851.1/T - 9.37951 (28) The physical solubility, however, is likely to be influ-

enced by other species present in the solution, and a me- thod of estimation of the magnitude of such effect was suggested by van Krevelen and Hoftijzer (1948). As the salting out parameters for the different ionic species present were not available, eq 28 was used to estimate the physical solubility of SO2.

Diffusivity of SO2 and Other Species. Diffusivity of molecular SOz in water was reported to be 2.00 X cmz/s a t 30 "C (Peaceman, 1951). The diffusivities of H+, HS03-, and SO:- ions were estimated by using the Nernst equation (Robinson and Stokes, 1959)

RTXo D=- (29)

where 3 is the Faraday number and vj is the charge of the species. The infinite dilution ionic conductivity, Xo, for the above species was taken to be 350, 50, and 72 cm2/ (ohm-g-mol), respectively (Landolt-Bornstein, 1960).

Some experimental data are available on diffusivity of H3Ci and H2Ci-. Muller and Stokes (1957) reported these values a t 25 "C and at different acid concentrations. No data are available on the diffusivities of HCi2- and Ci3-. The values for H&i- and HCi2- were, however, assumed to be equal following Albery et al. (1967) who observed that the diffusivities of dibasic acid anions were only slightly affected by the ionic charge. The same value was also taken for Ci3-, as the third ionization of H3Ci occurs to a very little extent in the pH range used in this study, and even an appreciable error in the diffusivity of Cis will not have any significant effect on the computed results.

The diffusivity values were corrected for change in temperature and viscosity wherever necessary by using the Stokes-Einstein relation. The values for H3Ci and HzCi- used were 0.839 X and 1.035 X cm2/s, respectively.

Ionization Equilibrium Constants. The thermody- namic equilibrium constant for the ionization reaction 1 was reported by Rabe and Harris (1963).

K1 = exp[1972.5/T - 10.9670] (30) The equilibrium constant for reaction 2 was reported by Tarter and Garretson (1941) to be K z = 6.2 x at 25

v j 3 2

Ind. Eng. Chem. Res., Vol. 26, No. 7 , 1987 1295

pH and the higher the degree of dissociation attainable. The behavior is also in conformity with the experimental observations.

Conclusions Absorption of SOz in low concentrations using citric

acid-sodium citrate buffer solutions exhibits considerable enhancement over absorption in water. Absorption rate is weakly dependent on pH in the range 3-5 but is inde- pendent of the concentration of the feed solution. The rate, however, increases linearly with the gas concentration. Absorption data could be better interpreted by the film theory of mass transfer with multiple equilibrium reactions than by the approximate surface renewal theory.

Nomenclature u1-a3, b,-b3 = integration constants, eq 34-36 a = activity D = liquid-phase diffusivity DI = Dnso3-ID~+

D3 = DH2Ci-/DH+ Dz = DSo32-/DH+

D4 = DHCiz-/DH+ D, = DCp/DH+ DE = DH2Ci-/DH3Ci 01 = DHCi2-/DH3Ci D, = DcPIDH~c~ E = enhancement factor H = Henry's law constant, eq 28 HnCi3-" = citric acid and corresponding anions (n = 0-3) K = thermodynamic equilibrium constant K' = effective equilibrium constant in concentration units kbuffer = rate of absorption in SOz-buffer system k,,,, = rate of absorption in SOz-water system R = universal gas constant S = species in solution T = absolute temperature t = time x = coordinate Greek Symbols y = activity coefficient 6 = film thickness Subscripts c = citric acid-sodium citrate buffer solution f = film theory i = gas-liquid interface 0 = bulk solution s = refers to surface renewal theory w = water

Appendix

15-17 results in Derivation of Equations 19 and 20. Integration of eq

/ Temp: 35'c

c , .*. ' '

a - 0 1 2 3 4 5

( E c / h ) e \ p

Figure 8. Theoretical and experimental enhancement ratio plot- surface renewal theory. SO2 concentration, 1.05-2.00%; pH, 3.58- 4.80; solutioa concentration, 3.65-12.1670; liquid rpm, 50-90.

OC. As the second ionization of dissolved SO2 was very small in the pH range of this study, this value was used in the calculations. The equilibrium constants for reactions 3-5 are available as functions of temperature (Bates and Pinching, 1949).

PK3 = - 1255'6 - 4.5635 + 0.01167T T

PK4 = - 1585*2 - 5.4460 + 0.01640T T

PK5 = - 1814'9 - 6.3664 + 0.02239T (31)

Activity Coefficients of the Species in Solution. Activity coefficient of unionized H3Ci was obtained from the data provided by Muller and Stokes (1957) at different acid concentrations, and those of citrate ions were given by Conway (1952) at different total ionic concentrations. Activity coefficient of hydrated SOz was estimated by the relation (Harned and Owen, 1958)

log yi = 0.0761 (32) where I is the ionic strength of solution. The ionic activity coefficients of H+, HS03-, and were estimated by using the modified Debye-Huckel limiting law (Lowell et al., 1970).

T

c

The values of CBi and Cqi available a t 25 OC were used (Chang and Rochelle, 1981).

Comparison of Theoretical and Experimental En- hancement Factor Ratios. The theoretical enhancement factors for both SOz-buffer and SOz-water systems can be calculated by using the above equations and data available. The experimental value of the enhancement ratio is the ratio of the rates of absorption in SOz-buffer and SOz-water systems, respectively, under identical conditions.

The theoretical enhancement ratios using both film theory and approximate surface renewal theory are plotted against the experimental values in Figures 7 and 8, re- spectively. I t appears from these parity plots that the experimental data could be better represented by the film theory of mass transfer with multiple equilibrium reactions than by the approximate surface renewal theory. Com- puted results also indicate higher enhancement at lower feed gas concentration, because the less the interfacial concentration of dissolved sulfur dioxide, the more is the

[HCiz-] + [H3Ci] + - [HzCi-] + - DH2Ci- DHCi2-

DH3Ci DHBCi DCi3- ---[Ci3-] DH3Ci = a2x + b2 (35)

Dci3-

DH+ 3-[Ci3-] = u3x + b, (36)

1296 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987

Differentiating eq 35 and 36 with respect to x and com- paring with eq 11 and 12, a2 = 0 = a,. Then,

[HCi2-] + [H3Ci] + - [H2Ci-] + - DH&!- DHCi2-

DH4Ci DH9Ci Dci3-

DH3Ci -[Ci3-] = b2 (37)

DHSOL Dso32- [H+] - - [HS03-] - 2-[S032-] -

DH+ DH+ Dci3-

DH+ DH+ DH+ [Ci3-] = b, (38) -[H2Ci-] - 2-[HCi2-] - 3-

DH2Ci- DHCi2-

Thus, the quantities on the left-hand side in eq 37 and 38 are invariant over the thickness of the film, and it may be noted at this point that b2 and b3 can thus be calculated from the bulk concentrations of the different species in- volved.

At the gas-liquid interface, eq 34-36 may be written as

Dso32-

Dso* [HSOa-1, + -[SO~2-ll

[HCi2-], + [H3Ci], + - [H,Ci-], + - DHzC~- DHCi2-

D H X ~ DHICi Dci3- - [ Ci3-] DH8Ci

and

DHS03- Dso32- DH’ DH+

DH+ DH+ DH+

[H+]i - - [HSO,-]i - 2-[S03’-]i -

DH2Ci- DHCi2- Dci3- -[H2Ci-Ii - 2-[HCi2-Ii - 3-[Ci3-Ii

= b, (39)

= b, (40)

= b3 (41)

Now the interfacial concentrations of the various species in solution may be related by using the effective equilib- rium constants [eq 141 as

K3’ [H3CiIi [H2Ci-Ii =

[H+Ii

K3’K4’ [H3Ci]i [HCi2-Ii = . [H+Ii2

K3K4’K5‘ [H3CiIi [ci3-li = (42)

On substitution of eq 42 in eq 40 and on simplification,

W+l?

Dei K3’K4’Kj’ DH3Ci3- ----) [ H’] i 3

(43)

Also, from eq 41 and eq 42,

[H+]i - DHSO~- - K1’[S02]i - 2- Dso32- KI’K2’[S02ji - DH+ [H+]i DH+ [H+li2

DH2Ci- K,’[H3CiIi DHCiz- K3’K4’ [HaCiIi -- .. - 2-

Dcia- K3’K,’K,’ [H3Cili DH+ [ H+] i3

DH+ [H+]i DH+ [H+li2

3-- = b3 (44)

On substituting eq 43 in eq 44 and simplifying, we get eq 19 with a’s given by eq 20. The principles underlying the theoretical development are similar to those mentioned by Chang and Rochelle (1982b).

Registry No. SOz, 7446-09-5; citric acid, 77-92-9; sodium citrate, 994-36-5.

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Bates, R. G.; Pinching, G. D. J . Am. Chem. Soc. 1949, 71, 1274. Chang, C. S.; Rochelle, G. T . AZChE J . 1961, 27, 292. Chang, C. S.; Rochelle, G. T. Ind. Eng. Chem. Fundam. 1962a, 21.

Chang, C. S.; Rochelle, G. T . AZChE J . 1962b, 28, 261. Conway, B. E. Electrochemical Data; Elsevier: Amsterdam, 1952. Dutta, B. K.; Basu, R. K.; Pandit, Amit; Ray, P., Paper presented

a t the World Congress I11 of Chemical Engineering, Tokyo, Sept 1986.

Eigen, M.; Kustin, K.; Maas, G. 2. Phys. Chem. (Frankfurt) 1961, 30, 130.

Erga, 0. U.S. Patent 3 886060, 1975. Erga, 0. Chem. Eng. Sci. 1980, 35, 162. Farrington, J . F.; Bengtsson, S. 108th AIChE Meeting, 1979, New

Harned, H. S.; Owen, B. B. The Physical Chemistry o j Nonelectro-

Johnstone, H. P.; Leppla, P. W. J . Am. Chem. Soc. 1934, 56, 2233. Londolt-Bornstein Physihalish-Chemische Tabr lkn , Bd 11-7,

Springer-Verlag: Berlin, 1960. Lowell, P. S.; Ottmers, D. M.; Schwitzgebel, K.; Strange, T. I.; De-

berry, D. w . A.P.T.D. 1287, PB 193-029, 1970; USEPA, Wash- ington, DC.

Muller, G. T. A,; Stokes, R. H. Trans. Faraday due. 1987, 58, 64%. Olander, D. R. AZChE J . 1960, 6 , 233. Peaceman, D. W. Sc.D. Thesis, MIT, Cambridge, 1951. Rabe, A. E.; Harris, J. F. J . Chem. Eng. Data 1963, 8, 333. Robinson, R. A.; Stokes, R. H. Electrolytic Solutions; Butterworths:

Tarter, H. V.; Garretson, H. H. J . Am. Chem. Soc. 1941, 63, 808. Van Krevelen, D. W.; Hoftijzer, P. J. “Chimie et Industrie, Numero

Speciale du XXI e”, Presented a t the Congress International de chiemie Industrielle, Bruxelles, Sept 1948.

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lytic Solutions; Reinhold: New York, 1958.

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Received for reuieu June 3, 1985 Revised manuscript received July 17, 1986

Accepted February 24, 1987