I
D. M. HIMMELBLAU' and A. L. BABB University of Washington, Seattle, Wash.
Reaction Rate Constants By Radioactive Tracer Techniques
CO,
- H,S - Na,CO, - NaHCO, - Na,S - H,O
Understanding this complex system i s essential to proper evaluation of processes for carbonating aqueous sodium carbonate-bicarbonate-sulfide solutions. Such processes might find suitable application, for example, in hydrogen sulfide recovery from natural and industrial gases. The reaction rate data are also valuable in studying chemical absorption phenomena. In the work reported here, the complex system was investigated in stepwise fashion. First, the principal reactions occurring in the system carbon dioxide-sodium carbonatesodium bicarbonate-water were investigated, followed by a study of the hydrogen sulfide-sodium bisulfide-water system in the same manner. Finally, the combined system was examined to verify the rate-controlling mechanisms previously found. A radioactive tracer technique simplified the experimental work and made it possible to measure both forward and reverse reaction rate constants of gas-liquid reactions without the influence of diffusion effects.
PREVIOCS
vapor-liquid equilibrium data have been reported for the carbonation reactions in this system over a wide range of temperatures, pressures, and sodium ion concentrations (79). This study is concerned with the kinetics of gas-liquid reacLions occurring among system components. The over-all reaction is: iYa3
+ 2C02 + 2H20
+
H2S 2NaHC03 (1)
T h e major problem in determining true reaction mechanisms and rate constants, rather than over-all mass transfer coefficients, is to distinguish between molecular and eddy diffusion processes and purely kinetic processes. I n studying the over-all carbonation reaction, the kinetics were investigated of the principal reactions occurring in the system carbon dioxide-sodium carbonate-sodium bicarbonate-water:
+ H20 e H + + HCOICO: + O H - = HCOB-
C02
(2)
(3)
Principal reactions in the system hydroPresent address, Department of Chemical Engineering, University of Texas, Austin, Tex.
gen sulfide-sodium bisulfide-water were then investigated :
HT H2S
+HS-e
HIS
+ OH- s HS- + H?O
(4) (5)
Finally. the over-all system carbon dioxide-hydrogen sulfide-sodium carbonate-sodium bicarbonate-sodium sulfide-water was examined to verify rate controlling mechanisms determined from the other two systems. Simplified theories of simultaneous diffusion and chemical reaction for a gasliquid system have been reviewed (30). Although no kinetic data have been discovered for the reaction of hydrogen sulfide with water hydroxyl ion, carbonate or other constituents in this system, work has been conducted on the reaction of carbon dioxide with water and bicarbonate salts (2-5, 7: 9, 70, 77, 20, 27, 22-29). T h e absorption of carbon dioxide in buffer solutions has also been discussed ( 6 ) . Some investigators determined the over-all mass transfer effect in a flow or batch process and then corrected for the effect of molecular and eddy diffusion, arriving a t a rate constant presumed to reprment solely the kinetic influence.
Others designed experiments so that diffusion influence was negligible and obtained a true kinetic rate constant. Because the latter approach is the simplest, it was used ir, this work. Mathematical Treatment
T h e essence of the procedure is to operate with the system a t chemical equilibrium but isotopic disequilibrium. By taking samples from a closed reaction vessel a t periodic intervals after initially injecting tracer in the form of carbon-14 labeled sodium bicarbonate or sulfur35 labeled sodium bisulfide (under controlled conditions of temperature and pH), reaction rates of Equations 2 and 3 or 4 and 5 were followed by determining the percentage of tracer in the solution in the form of labeled carbon dioxide or hydrogen sulfide. Mathematical analysis permits the use of a simple, first order, integrated rate eyuation in calculating reaction rate constants in the forward or reverse direction. A reaction rate can be formulated for the reversible reaction aA
VOL. 51, NO. 11
+ bB e rR + sS NOVEMBER 1959
(6)
1403
where uA of Equation 11 might be ( m ) (l/p) but preferably is the activity of species A in solution. This latter concept is theoretically more logical, for it interprets both sides of the equation as actual concentration corrected by a n activity coefficient to ideal behavior, or, in molal units
Thus, activity values are dimensionless, and activity coefficients have dimensions of reciprocal concentration. In this work, reaction rate was defined in terms of activity change per unit time so that true "thermodynamic rate constant;" could be measured experimentally. Precise calculation of a reaction rate would require knowledge of the activities of all species present, but these values may be approximated for engineering purposes by multiplying concentrations by activity coefficients from other data (72-74, 76).
H,S-H,O-NaHS-Na,S,
Assumed
reaction relationships are : H2S(g) e HZS
+
(1) 5 H T
-
+ HS-
OH-
S-
(13)
11
H-+S--
1, + H?O H2O
This study was designed to verify experimentally mechanisms by which these reactions occur. Individual reactions are : H2S & H'
HzS
Essential feature of the procedure is that the system is a t chemical equilibrium but isotopic disequilibrium After a tracer was iniected, samples were taken from the closed reaction vessel a t intervals and the percentage of tracer in the solution was determined
T h e concept that reaction rate (change of concentration per unit time per unit volume) depends on activities-i.e., for Equation 6 : r = -dc/dt
=
-
~ ( U A ) " ( ~ B ) ~k'(aR)r(Qs)"
(8)
r = dc/dt = k ( m ~ - / ~ ) 5 ( r n p /~)b k'(mRYR)'(mSYS)"
(9)
does not completely account for observed reaction velocities ( 8 ) . Denbigh suggested multiplying the right side of Equation 9 by p (a function of ionic concentraion) to satisfy thermodynamic requirements a n d yet provide a more
1404
O n e interpretation of p is that i t is the reciprocal activity of an intermediate complex in Reaction 5. However, if p is concentration dependent, reaction rate can be written as a "true thermodynamic" equation of ideal behavior : r = -d(aA)/dt
INDUSTRIAL AND ENGINEERING CHEMISTRY
d(aa)/dt =
~ ( Q A ) ~ ( u R -) k~ '
( a ~ ) ~ ( a ~ (11) )8
+ OH-=
+ HSHS- +
(14) H20
(15)
Other, more complicated reactions may occur; but results showed no need to consider them a t this stage of knowledge for this system. A wide range of p H and hydrogen sulfide and bisulfide concentrations was used to determine whether exchange occurs according to Reactions 14 or 15 or both. T h e reaction H & (g) e H2S (1) was not considered because it was a mass transfer process a t equilibrium during any run. Reactions 16 and 17 were presumed to be of ionic speed and thus in effective equilibrium when measurements were made to detect Reactions 14 and 15. Three possible situations are initially proposed for Reactions 14 and 15 because, in effect, they are competing for hydrogen sulfide in the solution : Reaction 14 takes place in a measurable time. Reactions 14 and 15 also occur simultaneously in a measurable time.
R E A C T I O N RATE C O N S T A N T S T h e isotopic exchange reaction was so rapid as to defy measurement by this technique a n d appeared to be in effective isotopic equilibrium when measured. Equilibrium concentration of labeled hydrogen sulfide in solution was so large that Reactions 14 and 15 could not be distinguished. Mathematical relations for the three possibilities show the experimental data needed. and resulting equations can be compared with corresponding rate equations developed for the carbon dioxide system ( 7 5 ) . REACTION: H2S F? H’ HS-. If reaction rate is expressed as
Equation 27 is obtained: which integrates into
which integrates into
+
k; (1 and the equilibrium constant for this reaction is
+ [:&E\)
+ C’
(f)
+
If Reactions 14 and 15 are simultaneous the over-all reaction rate is composed of the contributions of both Equations 27 and 33, or
(28)
+
REACTION: H,S OH- e HSH 2 0 . Reaction rate can be expressed as
-d-
then parallel rate equations may be written for the activity change with time for S35 and S 3 5 isotopes in the solution:
and the equilibrium constant is 1
+ [H?S3’] [HS2-]
(35)
which integrates to Parallel equations may again be written for activity change with time
If thr rate of Reaction 20 is zero (i.e.? a t equilibrium], then it may be substitutvd into Equation 21 to eliminate the reverse reaction rate constant:
By similar mathematical treatment a n integrated rate of isotopic exchange equation contains reverse reaction rate constants :
and these may be consolidated as before into
R U N 26 H,S - H 2 0 - N a H S - N a 2 S
6.00
Changing to concentration units (activity terms consist of ratios permitting cancellation of activity coefficients) :
SYSTEM
0°C.
co
9
5.00L
I
-
s,
4.00Also, by introducing the concept that a t isotopic equilibrium
A
v
L
Value a t t = Rb.
3.00
I
I
I
I
and utilizing a material balance o n the tracer VOL. 51, NO. 1 1
NOVEMBER 1959
1405
Table I.
Forward and Reverse Reaction Rate Constants CO?-HzO-NaHC03-Na?CO3
00 c.
100 C.
k ~ sec.-l , X lo3 k ~ sec.-l , X Kb = k F / k , X lo7 Kb, X 10' (74)
1 . 8 9 i ?yo 0 . 7 4 i. 1% 2.56 2.65
5 . 8 2 zt 10% 1 . 8 5 i 5y0 3.15 3.43
14.9 10% 3 . 3 2 f 570 4.53 4.14
COz O H - * HCOsk f , sec.-1 x 10-5 k,, set.-' x lo3 K* = k j / k , X K* x 10-7 (74)
1 . 4 4 =t5% 0.591 5y0 24.4 23.3
2 . 4 1 zt 10% 1 . 9 3 i 5% 12.5 11.4
2 . 7 7 i 20% 4.35 5oj, 6.37 6.09
+
Cog
Quantities to be measured were: [ H z S ~m~/ ][H2S3*]= ratio of S35 to S3* in counting gas a t isotopic equilibrium ( t = a). [H&335]j[H2S32] = ratio of S35 to S3? in counting gas at time t . [ H k P ] / [ H S 3 2 - ] = ratio of hydrogen sulfide in solution to bisulfide ion (negligible labeled quantities of both are present). Solution p H a n d activity coefficients of ions present also were measured : CO,-H,O-NaHCO3-Na2CO3. Mathematical treatment has been given elsewhere (75).
Experimental Experimental techniques were previously described (75). A 15-liter stainless steel reaction vessel was fitted with a stainless steel lid machined to hold a calomel p H electrode (Leeds and Northr u p Std. No. 1199-48), a glass electrode and pressure equalizing tube (Std. No. 1199-50), and a n automatic temperature compensator (Std. No. 199-47); conductance cell. stainless steel stirrer a n d impeller; thermometer well; and a n outlet to a n 80-cm. mercury manometer with one leg vented to the atmosphere. T h e steel lid was bolted to the reaction tank and made pressure tight with a 6.75-inch neoprene O-ring seal. T h e reaction vessel was fitted with nine sample outlets, seven in the front and two in the rear. All seven 3/g-inch toggle valves (Hoke No. RS492) in front were gang connected to open and close simultaneously. T h e 0.25-inch solenoid valves (General Controls K27) were electrically connected for the same reason. Each 100-ml. glass sample bulb was connected to the reaction vessel by saran fittings. Carbon dioxide and hydrogen sulfide samples were flashed from the sample bulb into the glass traps and vapor collection tube by opening the 0.25-inch stainless steel needle valve (Hoke No. Y344G 316) a t the bottom of the collection bulb. T h e labeled carbon dioxide or hydrogen sulfide content of vapor samples was determined using the proportional counting technique (7) as described (75). Voltage for carbon dioxide counting was 4350 volts, with 3700 volts for hydrogen sulfide counting.
1 406
Constant H20 HCOI-
+ H+
+
*
Results and Discussion
c.
*
*
6.2 to 8.2. Except for run 30, IHzS351/ .~ [ H z S ~ was ~ ] about constant with time within experimental error. For this ream - ( [H1S35]/ son, ( [H?S35]/[H&3L]) [HzS32]) had no significance, and it was not possible to calculate slopes and rate constants. No additional runs were ' C. results made a t 10' or 20' C . , as 0 showed that further work using this isotopic exchange technique would be futile. Two runs were made a t 0" C. in the combined systems (carbon dioxide, hydrogen sulfide, sodium bisulfide, and sodium bicarbonate present) to check the ~
Temperature range was restricted to 0 ' to 20' C. and p H range to 5.6 to 7.6. Below -4' C. the liquid solution froze, while above 20' C. the isotopic exchange rate was too rapid to obtain accurate data. 41so the latter passed through a minimum as p H changed from 5.6 to 7.5. Results (Table I) have been compared with previous work (75). Forward and reverse reaction rate constants for the hydrogen sulfide system (Table 11) could not be calculated because of the high rate of attaining isotopic equilibrium. T h e p H ranged from
Table II.
200
Experimental Data for H&-H20--NaHS--Na%S
Isotopic equilibrium was attained so rapidly that rote constants could not b e calculated
Temp.,
c.
Run 25
0.0
106
PH
Liter
Liter
HzS/ HS-
47 75 94 124 154 244 605
7.91 7.84 7.76 7.95 7.71 7.93 8.00 7.87
6.19
0.0204
0.125
6.11
4.15 4.20 4.20 4.14 4.24 4.25 4.40 4.25
7.25
0.0960
0.164
1.70
4.43 4.25 4.58 4.41 4.40 4.43
8.18
0.220
0.345
1.57
3.81 3.74 3.84 3.83 3.89
...
b
0.147
0.173
1.17
0.0672
0.0830
1.27
m
0.0
49 74 94 124 154 304 604
27
0.0
49 74 94 124 424
28
0.0
48 74 94 305
30
10.0
150 175 195 218 480 900
26
m
m
m
m a
INDUSTRIAL AND ENGINEERING CHEMISTRY
By titration.
HS- Concn., H:S Concn.," Mole/ Mole/
(R') X
Sec
t,
8.16 8.34 8.46 8.45 8.64 8.74 8.95
7.58
pH meter did not function properly.
R E A C T I O N RATE C O N S T A N T S Table I l l .
Experimental and Predicted Results Agreed Well for Combined Systems at 0" c.
+
k r ( a 0 ~1-I Diff., yo Pred.
[~F(~H,o)
Run
Exptl.
32 33
0.00208 0.00543
0.00197 0.00541
t5.6 +0.4
validity of carbon dioxide system results and to determine any influence of hydrogen sulfide and sodium bisulfide on reaction rate constants for the carbon dioxide system alone. T h e procedure was identical to that for the carbon dioxide system (75), except that carbon dioxide in the counting tubes was distinguished from hydrogen sulfide in the same tubes (which flashed out of solution when carbon dioxide samples were taken) by mass spectrographic analysis. Values for [kp(aH,o) k,(aoa-)l and ( k R [ H + ]- k, determined for these runs agreed hell with predicted values based on data from the carbon dioxide system alone (75). An additional assumption was that the effect of bisulfide and bicarbonate ions on activity coefficients of hydroxyl and hydrogen ions was the same as bicarbonate alone. The error thus introduced lies well within error limits of the activity coefficients themselves. Calculated and predicted values (Table 111) are \vel1 within possible experimental error. Carbon dioxide system reactions thus proceed independently from hydrogen sulfide system reactions, and addition of other components, except for applying dCliVity corrections, will not affect reaction rate of carbon dioxide or bicarbonate according to Equations 2 and 3. Division of Reaction 1 into separate parts was entirely sound. Rate of Attaining Isotopic Equilibrium. SYSTEM:CO2-H2O-NaHC03Na2C03. Half-times of over-all forward and reverse isotopic exchange reactions are :
+
t1/2
= 0.693/[k~(aa,o)
il
+ ~/(~OEI-)I
+ ([CO,l/[HCO3-1)1
(38)
Although half-time of the isotopic exchange reaction varies with p H and has a maximum value of about 100 seconds, a t the same temperature, 0' C., half-time of individual forward reactions was 367 seconds ior Equation 1 and 1.51 X 10-6 second for Equation 2. T h e isotopic exchange reaction is a complicated function of the sum of forward and reverse chemical exchange reactions. The technique can measure rate constants of
+
(LR[H+] k,/rH+) Exptl.
Pred.
Diff.,
0.0247 0.00123
0.0239 0.001 18
f3.3 +4.2
individual simultaneous reactions with half-times from 10-6 to lo3 seconds without any actual chemical reaction occurring, providing half-time of the over-all exchmge reaction is 10 seconds or greater.
SYSTEM:HZS - HzO - NaHS. There was no initial amount of labeled hydrogen sulfide in the reaction vessel, and the first measured sample of gas (about 45 seconds after tracer injection) nearly always showed approximately the same [H?S35]/[H2S3*]ratio as the sample a t 1 = a . Thus, either Reactions 14 or 15 or both must have forward and/or reverse rate constants of such high values that the isotopic exchange reaction was essentially complete within 45 seconds. Half-time of the over-all isotopic exchange reaction would have to be less than 10 seconds because changes in [H2S3j] [H?S3'] could be detected u p to four half-times with the experimental technique. Reaction 14 is known to occur, but it is difficult to tell whether Reaction 15 represents a real mechanism. Because the equilibrium concentration of labeled hydrogen sulfide resulting from the fast isotopic exchange was so high, it masked a n y concentration change which might have resulted from a slower reaction and made it impossible to measure such a reaction R u n 30 showed a slight change of [HzS3j]'[H2S3'] with time, and in two other runs values a t later times averaged above the early values of R', although well within the range of experimental error. Thus. both Reactions 14 and 15 may represent real mechanisms, but the results neither confirm nor negate this possibility. Later it is shown that Reaction 14 can be considered with or without Reaction 15 without affecting rate constant analysis. This analysis cannot determine exact values of reaction rate constants, but can indicate their relative magnitude and range. As the ionization constant a t 0' C. for Reaction 14 is about 10-7 and the reverse reaction is ionic, k: should a t least be as large as lOf/second a n d more likely 1010 to 1012/second. Therefore, the forward rate constant kF should range from l/second to as high as 105/second. This gives only a rough idea of actual rate constant magnitudes.
Similarly, for Reaction 15, the equilibrium constant a t 0' C. is about 108 so that K: = k;/ki = loa. The forward rate constant is several orders of magnitude larger than the reverse. By analogy to the carbon dioxide system, the forward rate constant for Reaction 15 could be about 1Ob/second (approximately that for reaction of carbon dioxide with hydroxyl ion). Then the reverse constant, k,', should be about 10-3/second. Again, this is primarily speculation, but based on this logic the magnitude of the reverse reaction rate constant (reaction of bisulfide with water) is about the same order of magnitude as the reaction of' carbon dioxide with water. If Reaction 15 does not occur, analysis of rate constant magnitudes for Reaction 14 is unaffected, as shown by equations for half-time of the isotopic exchange reaction in the hydrogen sulfide system (by analogy with Equations 38 and 39):
+
0.693/[kb k;(~oE-)] [I (~HzSl/[HS-l)l < 10 (40)
(I,,
+
or in terms of reverse reaction constants: t1/2
[1
=
0.693/(kk[H+]
+ [k:/-,H+])
+ (IHS-I/IH2Sl)l [ 7 / ~ + ? ~ ~ - / ~i ~ 10 ,~I
(41)
If kb i s greater than llsecond in Equation 40, the value of k;(aoa-) is unimportant. T h e value of 11 ([HzS]/ [HS-I)] ranges fron 2 to 7, so t l / Z would still be less than 10. Similarly, with respect to reverse rate constants in Equation 41, k: is so small that it has no effect on the magnitude of the denominator. Because [l ([HS-1,: [H2S])] [?'H+?HS-/~H,~] probably is about 0.5, kR[H+C]must only be greater than 0.2 for t u 2 to be less than 10. This seems likely as hydrogen ion concentration ranged from lo-' to lo-'* mole per lirer and k k [ H + ] would range from a n unlikely lo-? to a more probable lo2, to as high as lo6. Thus, half-time analysis of isotopic exchange reactions in the hydrogen sulfide system shows that rate constants for Reaction 14 can be approximated without considering Reaction 15. Exact values of forward and reverse reaction constants could not be ascertained by the isotopic exchange technique, but the approximate range of values is useful in comparison with the corresponding constants in the carbon dioxide system. Application of Thermodynamic Kinetic Data to Mass Transfer Calculations. From a mass transfer viewpoint, it is difficult to distinguish between physical absorption and that accompanied by chemical reaction. In absorption of carbon dioxide both processes occur, even in water, though chemical
+
+
VOL. 51.
NO. 11
NOVEMBER 1959
1407
reaction is not usually considered in mass transfer measurements. While a similar dual mechanism has not been definitely shown to exist for hydrogen sulfide, solution of hydrogen sulfide in water or dilute alkaline solutions by both processes has been confirmed recently ( 7 7 ) . Because of the complexity of over-all mass transfer processes, the turbulent mechanism of transfer cannot be considered in the following discussion, a n d analysis must be restricted to molecular diffusion and chemical reaction occurring simultaneously. Analytical Solutions to Mass Transfer Equations. Two general concepts of simultaneous molecular diffusion and chemical reaction have been developed. O n e describes a steady state process through a liquid film of negligible capacity but finite resistance with a rate proportional to a driving force or potential. T h e other assumes a n unsteady state process of molecular diffusion of solute into a whole of the liquid. Neither satisfactorily describes the true physical process. Because of simplifying assumptions with respect to chemical kinetics, real systems d o not usually conform to mathematical models and experimental verification of these models is difficult. I n the present work it is impossible to decide which concept is more nearly correct. I n both concepts, mass transfer rate equations have been based on rapid second-order, irreversible reactions or first-order. slow reactions ( 7 8 ) . Mathematical expressions for other mechanisms, particularly reversible reactions, are complex as to defy analytical solutions. Furthermore, they lead to solutions expressing concentrations in terms of contact time (or diffusion time) a n d diffusion distance (or film thickness). I n most cases mathematical solutions are so complicated, that their physical significance is not always obvious. An attempt to relate kinetic data to the over-all mass transfer coefficient (including turbulent flow effects), expressed by the rate equation
equipment, its geometry, and mass flows involved. Experimental studies similar to those recently reported (20, 37) on absorption rates of gases i n liquid jets, in which film thicknesses and contact times can be measured, offer the possibility of using the type of kinetic constants determined in this work in conjunction with analytical solutions to theoretical absorptionrate equations. T h e resulting theoretical absorption rates may be compared with experimentally determined rates for one geometry. Mathematical solution of absorption rate equations which account for simultaneous molecular diffusion a n d reversible chemical reactions is in progress a t this laboratory, and results are to be used in conjunction with experimental absorption studies using a jet technique. Nomenclature a
= activity of component indicated
C, C’
=
by subscript constants of integration = concentration of-component indicated by subscript = forward reaction rate constant for general bimolecular reversible reaction, set.-' = reverse reaction rate constant for general bimolecular reversible reaction, set.? = forward reaction rate constant for C O S OH- s H C O I - , sec. = forward reaction rate constant H20 e H’ for COz HCOs-, set.-' = reverse reaction rate constant OH- s HCOs-, for COz sec. -l = reverse reaction rate constant for COP HZO e Hf H C 0 3 - , set,.+ = forward reaction rate constant OH- e H S for H2S HzO. s e c . 3 = forward reaction rate constant H S - e H z S , set.? for H = reverse reaction rate constant OH- e H S for H2S
+
+
+
INDUSTRIAL AND ENGINEERING CHEMISTRY
+
+
+
+ +
+
= reverse reaction rate constant
= = =
= = = = = =
+
H S - g HzS, set? mass transfer coefficient, moles ’unit time-unit area-unit c equilibrium constant for CO:! H20 e H + f HCOsequilibrium constant for COSf OH- e H C 0 3 equilibrium constant for H + HS-$ H2S equilibrium constant for HzS OH-= HSH20 moles of A transferred per unit time per unit area in Equation 42 reaction rate, change of activity per unit time ratio of S35to in counting gas a t time t time, sec. activity coefficient of component indicated by subscript for H +
1408
+
HzO, set.-' = over-all
cannot be successful because a (interfacial area per unit volume) is not generally known. Kinetic data are usually expressed as change of concentration/unit time/unit volume basis, while mass transfer data are expressed as current through the surface of this volume, or on concentration change/unit timejunit area. Something must be known concerning the magnitude of the distance perpendicular to the transfer area to shift from one basis to the other. This has been the prime difficulty in attempting analytical treatment of gas absorption in a packed tower and why industrial equipment is designed on a n empirical basis with over-all transfer rate being a function of
+
+
+
+
+
Subscripts: 0 = initial condition m = a t infinite time Acknowledgment
T h e authors are grateful for financial support from the National Science Foundation. T. R. R e h m and M. C. Ward assisted with the manuscript preparation. literature Cited
(1) Bernstein, W., Ballentine, R . , Rec. Sci. Instr. 21, 158 (1950). (2) Booth, V. H., Roughton, F. J. W., Biochem. J . 40, 309 (1926). (3) Brinkman, R., Margaria, R., Roughton, F. J. W., Phil. Trans. Roy. SOC. London A232, 65 (1934). (4) Cullen, E. J., Davidson, J. F., Tranr. Faraday Soc. 53,113 (1957). (5) Cullen, E. J., Davidson, J. F., Trans. Znst. Chem. Engrs. (London) 35, 51 (1957). (6) Danckwerts, P. V., Appl. Sci. Research A3,385 [1951). (7) Danckwerts, P. V., Kennedy, A . M., Chem. Eng. Sci. 8 , 201 (1958). (8) Denbigh, K.? “The Principles of Chemical Equilibria,” Cambridge Univ. Press, Cambridge, 1955. ( 9 ) Eigen, M., Discussions Faraday Soc. 17, 194 (1954). (10) Faurholt, C., J . chim. phys. 21, 400 (1924). (11) Garner, F. H., Long, R., Pennell, A., J . Appl. Chem. (London) 8 , 325 (1958). (12) Harned, H. S., Bonner, F. T., J. Am. Chem. Soc. 6 7 , 1026 (1945). (13) Harned, H. S.: Davis, R., Zbid., 6 5 , 2030 (1943). (14) Harned, H. S., Owen, B. B., “The Physical Chemistry of Electrolytic Solutions,” 2nd ed., Reinhold, New York, 1950. (15) Himmelblau, D. M., Babb, A. L., A.Z.Ch.E. Journal 4, 143 (1958). (16) Kielland. J.: J . A m . Chem. Soc. 59, 1675 (1937). (17) Kiese, M.: Hastings, .A. B., J . Bzol. Chem. 132, 267 (1940). (18) Krevelen, D. W. van, Hoftyzer, P. J., Chem. Eng. Progr. 44,529 (1948). (19) Mai, K. L., Babb, A. L.. IND.ENG. CHEM.47, 1749 (1955). (20) Matsuyama, T., Mem. Fuc. Eng., Kyoto Univ. 15, 142 (1953). (21) Mills, G. A , ? Urey, H. C., J . .4m. Chem. SOL.62, 1019 (1940). (22) Moelwyn-Hughes, E. A . , “The Kinetics of Reactions in Solutions,” Oxford Univ. Press, Oxford, 1947. (23) Pinsent, B. R. W., Pearson, L. Roughton, F. J. W., Trans. Faraday Soc. 52, 1512 (1956). (24) Pinsent, B. R. W.. Roughton, F. J. W.: Ibid., 47, 263 (1951). (25) Roughton, F. J. W., J. Bid. Chem. 141, 129 (1941). (26) Roughton, F. J. W., Harcey Lectures 39, 96 (1943-44). (27) Roughton, F. J. W., Proc. Roy. SOC. (London) A104, 376 (1930). (28) Roughton, F. J. W., Booth, V. H., Biochem. J . 32, 2049 (1938). (29) Saal, R. N. J., Rec. trav. chim. 47, 264 (1928). (30) Sherwood, T. K., Pigford, R. L., “Absorption and Extraction,” McGrawHill, New York, 1952. (31) Vielstich, W., Chem. Zng.-Tech. 28, 543 (1956). RECEIVED for review August 21, 1958 ACCEPTEDJune I, 1959
