4936
J. Phys. Chem. 1983, 87, 4936-4939
-
- +
does producing secondary and radical reactions as follows: CINH,
+0
CH=C-CH2-NH2
+co
+O
+O
CHEC-NH~ CH* CN* + H2O +CO + Hz Of course, a detailed mechanism of these complex reactions would perhaps require not only more experimental work, but also a different experimental technique mainly (8)Nicovich, J. M.; Gump, C. A,; Ravishankara,A. R. J.Phys. Chem. 1982,86,1684.
directed toward identifying those intermediates or precursors present in our system. A t the moment our main goal was to report the main experimental findings and to relate them with the basic chemical structure of the reactant aromatic rings. In this direction work is in progress with thiophene and other heterocyclics in our laboratory. Acknowledgment. This work received financial support from the Comision Asesora of Spain. Registry No. Pyridine, 110-86-1;atomic oxygen, 17778-80-2.
Gas Absorption with Instantaneous Chemical Reaction. Absorption of Sulfur Dioxide by Aqueous Sodium Bisulfite D. G. Lealst Depertment of Chemlsby, University of Western Ontario, London, Ontario, Canada N6A 567 (Received December 9, 1982)
Equations are derived for estimating rates of absorption of gas by ternary absorbents. For semiinfinite, convection-freeabsorbents, the rate of absorption equals Acl(D,/nt)1/2where Acl is the quantity of gas required to saturate a unit volume of bulk absorbent to the partial pressure of gas at the surface of the absorbent, D, is an apparent diffusion coefficient defined in terms of ternary diffusion coefficients in the absorbent, and t is the time. This result applies to absorption by chemically inert systems, and to absorption with simultaneous chemical reaction provided the reaction rates are sufficiently rapid to effectively guarantee local chemical equilibrium. For purposes of illustration, absorption of sulfur dioxide by aqueous solutions of sodium bisulfite is discussed. In this system ternary diffusion of the partially hydrolyzed dissolved gas yields a concentration-dependent apparent diffusion coefficient that varies between 1.5 X and 8.5 X m2 s-l at 20 OC.
Introduction Efficient gas absorption often depends on chemical reaction of the dissolved gas with an involatile component in the absorbent. Reaction of oxygen with hemoglobin, for example, plays a key role in respiratory processes. In the chemical industry, aqueous solutions of acids or bases are widely used to absorb gases such as ammonia, carbon dioxide, hydrogen sulfide, and sulfur dioxide. So that the rate of absorption with reaction can be estimated,'+ it is common practice to assume that diffusion in the absorbent is pseudobinary. In real systems, however, the solute fluxes are coupled by nonideal thermodynamic behavior and by chemical reaction.'^^ With absorption processes involving electrolytes, there is the additional complication that fluxes of ions are coupled to each other by the electric field they produce, even a t very low concentrati~ns.~J~ Very large (1)Danckwerta, P. V. Trans. Faraday SOC.1951,47,1014. (2)Danckwerta, P. V. Ind. Eng. Chem. 1951,43,1460. (3)Sherwood, T. K.; Pigford, R. L. 'Absorption and Extraction"; McGraw-Hill: New York, 1952;p 332 and Chapter 9. (4) Crank, J. "The Mathematics of Diffusion"; Oxford University Press: London, 1956;p 124. (5)Bird, R. B.; Stewart, W. E.; Liahtfoot, E. N. "Transport Phenomena"; Wiley: New York, 1960; p 599. (6)Danckwerta, P. V. Chem. Eng. Sci. 1968,23,1045. (7)Vitagliano, V. J. Phys. Chem. 1970,74,2949. (8)Wendt, R. P. J.Phys. Chem. 1965,69,1227. (9)Leaist, D. G.; Lyons, P. A. Aust. J. Chem. 1980,33,1869. 0022-365418312087-4936~0~ .5010
departures from pseudobinary behavior are likely whenever diffusivities of the various species are widely different.+14 How is gas absorption influenced by multicomponent diffusion in the absorbent? To answer this question, we derive equations to describe absorption with ternary diffusion in the absorbent. For purposes of illustration, absorption of sulfur dioxide by aqueous sodium bisulfite is discussed in detail. This system was chosen for several reasons. First, are available to make reliable estimates of the appropriate ternary diffusion coefficients. Second, absorption of sulfur dioxide by aqueous solutions is of importance to pollution studies. And finally, the recent reportz4that the ternary diffusivity of acetic acid (10)Leaist, D. G. J. Chem. SOC.,Faraday Trans. I 1982,78,3069. (11)Revzin, A. J . Phys. Chem. 1972,76,3419. Reinfelds, G.;Costing, L. J. J.Phys. Chem. 1973,77,934. (12)Kim, H.; (13)Leaist, D. G.; Lyons, P. A. J. Phys. Chem. 1982,86,564. (14)Leaist, D. G. Chem. Eng. Sci., submitted for publication. (15)Campbell, W. B.; Maass, 0. Can. J.Res. 1930,2,42. (16)Morgan, 0.M.; Maass, 0. Can. J. Res. 1931,5, 162. (17)Eriksen, T. Chem. Eng. Sci. 1967,22,727. (18)Robinson, R. A.; Stokes, R. H. "Electrolyte Solutions", 2nd ed; Academic Press: New York, 1959;Appendix 6.2. (19)Whitney, R. P.; Vivian, J. E. Chem. Eng. Progr. 1949,45,323. (20)Groothius, H.;Kramers, H. Chem. Eng. Sci. 1955,4,17. (21)Lynn, S.;Straatemeier, J. R., Kramers, H. Chem. Eng. Sci. 1955, 4, 49. (22)Toor, H.L.; Chiang, S. H. AIChE J. 1959,5,339. (23)Johnstone, H. F.;Leppla, P. W. J.Am. Chem. SOC.1934,56,2233. (24)Leaist, D. G.; Lyons, P. A. J. Phys. Chem. 1981,85,1756.
0 1983 American Chemical Society
The Journal of Physical Chemistv, Vol. 87, No. 24, 1983 4937
Gas Absorption with Instantaneous Chemical Reaction
in aqueous sodium acetate is substantially larger than its binary diffusivity in water led us to inquire whether aqueous sulfur dioxide (sulfurous acid) diffuses more rapidly in bisulfite buffers. The addition of a substance to the absorbent that increases the diffusivity of the dissolved gas component would improve the performance of gas absorption equipment.
Theory Consider isothermal absorption of a single gaseous component by a semiinfinite, convection-free absorbent. Let it be assumed that the absorbent contains N + 1 distinguishable chemical species (including solvent and dissolved gas species) with R independent,%.%homogenous chemical reactions. To simplify matters, it is assumed that the rates of the reactions are rapid compared to diffusi~n.~’ Hence diffusion is constrained by R local chemical equilibria and by the Gibbs-Duhem equation. Under these conditions, diffusion in the absorbent is described in terms of N - R linearly independent fluxes of components. If the absorbent contains ions, the requirement that the electric current must vanish imposes an additional constraint, and only N - R - 1 fluxes are independent.’O In the following, consideration is limited to the case of ternary diffusion (two independent component fluxes) in the absorbent of dissolved gas and one involatile solute. The ternary diffusion equations Ji(z,t) = -Dil(dci/az) - D12(ac,/az)
(1)
J z ( ~ , t )= -D2,(aci/az) - Dzz(ac2/dz)
(2)
relate total molar fluxes (reacted plus unreacted forms) of dissolved gas (component 1) and involatile solute (component 2) to gradients in volume-formal concentrations. Dll and Dzzare the main diffusion coefficients. The cross-coefficientsD12and D21describe interactions between fluxes of the components. At time t = 0 the partial pressure of the gas in contact with saturated absorbent is raised from p1to p1+ Apl. At the same instant the concentration of the dissolved gas at the surface of the absorbent (z = 0) is raised from its bulk value cl0 to c: + Acl, and subsequently held at this value. The object is to determine Jl(O,t),the rate of absorption of gas expressed in moles per unit time per unit surface area of the absorbent. Provided Acl is not too large, the absorption process is adequately described by time-dependent diffusion equations with constant coefficients
Boundary conditions with z = 0 and t
>0
and initial conditions with z > 0 and t = 0
tration profiles are obtained Cl(Z,t)
cl0
=
+ Acl[All erfc(z/2(Dlt)1/2)+ A12 e r f ~ ( z / 2 ( D ~ t ) l / ~ ) ] (9)
c2(z,t) = c20
+ Acl[Azl erf~(z/2(D,t)’/~) + A22e r f c ( ~ / 2 ( D ~ t ) ’ / ~ ) ]
(10) where erfc(y) is the complementary error function, Di denote eigenvalues of the diffusion coefficient matrix Di = [(Oil + D22) + (Dii - Dz2) X [1 + 4D1@21/(D11 - D ~ z ) ~ I ” (11) ~I/~ D2 = [(Oil + D22) - 0 1 1 - 4 2 ) X + 4DnD21/(Dil - D22)211/21/2(12) and parameters Aik are functions of the diffusion coefficients A11 = (Dl - D22)2D21/2/B (13)
A12 = DlzD21D11/2B
(14)
-421
= (Dl - D z z ) D ~ ~ D ~ ’ / ~ / B
(15)
-422
= (D22 - D I ) D ~ ~ D ~ ~ / ~ / B (16)
B = (DZZ - D1)2D21/2+ D12D21D11/2
(17)
Differention of eq 9 and 10 provides (acl/az)lz=o= - [ ( A l l / ( ~ D l t ) ” ~+) ( A I Z / ( T D ~ ~ ) ~ / ~ ) I A ~ ~ (18) (acZ/az)l,=o
= -[(A2i/(TDit)li2)
+ (Azz/(~D2t)’/~)lAc,
(19) It is interesting to note that although the flux of (involatile) component 2 is zero at the surface of the absorbent, the gradient in concentration of that component at the interface does not vanish (eq 19). Also, ternary diffusion of the gas into the absorbent changes the surface concentration of component 2 from its bulk value c20 to cz(O,t) =
~ 2 ’
+ (A21 + A ~ J A c ~
(20)
With eq 1, 18, and 19, the following expression is obtained for the rate of absorption of the gas Jl(0,t)= ACl(D,/Tt)’/2 (21) where D,, the apparent diffusion coefficient describing absorption of the gas, is a complicated function of the ternary diffusion coefficients
Should D12 and/or Dzl be zero, then D, = Dll and the following well-known equations are retrieved for absorption followed by pseudobinary diffusion in the absorbent:
c ~ ( z , ~ ; D ~=~cl0 D+ ~ ~Acl= e~r )f ~ ( z / 2 ( D , ~ t ) ” ~ ) (23) J1(0,t;D12D2,=0) = AC1(Dll/xt)’/’
(24)
It is worth emphasizing that rates of absorption with ternary diffusion (eq 21) and with pseudobinary diffusion (eq 24) are both proportional to t-lI2 apply. Equations 3 and 4 may be integrated by standard method^.^^,^^ The following expressions for the concen(25) Aris, R.; Mah,R. H.S. Ind. Eng. Chem. Fundam. 1963,2, 90. (26) Denbigh, K. “The Principles of Chemical Equilibrium”, 3rd ed; Cambridge University Press: London, 1971; p 169. (27) Stockmayer, W. H. J. Chem. Phys. 1960, 33, 129.
Results The general equations developed in the preceding section are now applied to the particular case of absorption ~
(28) Toor, H. L. AIChE J. 1964, 10, 448. (29) Stewart,W. E.; Prober, R. Ind. Eng. Chem. Fundam. 1964,3,224.
4938
The Journal of Physical Chemistry, Vol. 87, No. 24, 1983
Leaist
1
t
1 100
0.
0 75’
t
050’
~
0 25
-2-
0’0: ’
0 10 ,
,
1.o 10 (C,tC2)/10-3mol drn”
+ +
100
+
Flgure 1. Ternary diffusion coefficients for SO2 (c ,) NaHSO, (c,) H 2 0 at 20 OC plotted against total solute concentrations c c2 for several values of the ratio f , = c,/(c, c2).
+
+
+
t
1.01
099 LL
1
r
cn
w
rnE
-O -0
4F 01
.
6 100
10 ( C ~ + C ~ ) / l O - ~dm” rnol 10
+
Flgure 2. Ternary diffusion coefficients for SO, (c,) NaHSO, (c,) -tH,O at 20 OC plotted against total solute concentration c 1 c2 for several values of the ratio f , = c,/(c, c,).
+
+
of sulfur dioxide gas by sulfur dioxide (cl) + sodium bisulfite (c2) + H 2 0 absorbents at 20 “C.In this system the dissolved gas is subject to rapidm hydrolysis. Bisulfite and hydrogen ions are produced
SOz + HzO
l l l 100 I J
Flgure 3. Apparent diffusion coefficient D, describing absorption of sulfur dioxide gas by SO, (c,) NaHSO, (c,) H20 absorbents at 20 OC plotted against total solute concentration c, c 2 for several values of the ratio f , = c,/(c, c,).
000
0.1
I ’ l 10 l I I l l 1 ’0 ’ (C, +C2)/10-3moldrn”
HS03- + H+
withz3K = 0.0146 mol dm-3. In addition to solvent mol(30) Eigen, M.; Kustin, K.; Maas, G. Z . Phys. Chem. (Frankfurt a m M a m ) 1961, 30, 130.
+
+
ecules, there are four31 species: r n ~ l e c u l a r ~SO ~ - ~H+ ~ Na+, and HS03- with respective d i f f ~ s i v i t i e s l ~of- ~1.45 ~ X 8.40 x 1.17 X and 1.21 X m2 s-l. Since N = 4 and R = 1, the fluxes of sulfur dioxide and sodium bisulfite are independent. Ternary diffusion coefficients of the components were estimated by the extended N e r n ~ t - H a r t l e f l ’ ~method ~ ~ ~ used p r e v i o u ~ l y . ~ ~ The results are plotted in Figures 1 and 2 against total solute concentration c1 + c2 for several values of the ratio fl = cl/(cl c2). The concentration dependence of the diffusional properties of sulfur dioxide + sodium bisulfite buffers is striking. As anticipated, diffusion in this system is very similar to diffusion in acetic acid + sodium acetate buffers (discussed in detail in ref 24). Values of the apparent diffusion coefficient D, describing absorption of sulfur dioxide gas by aqueous SO2+ NaHS03mixtures are plotted against total solute concentration in Figure 3. In Figure 1-3, the curves labeled 1.00 refer to solutions of aqueous sulfur dioxide without added sodium bisulfite. In this case the apparent diffusion coefficient describing absorption of the gas is simply the binary diffusion coefficient of aqueous sulfur dioxide. Because of hydrolysis, diffusion of the dissolved gas closely resembles diffusion of a 1:l weak acid. Neglecting activity coefficient terms, we haveW7-39 27
9
+
~
D,(f1=1) = Dll(fl=l) = 2 - 8 [(l-P)D,+
1
2
(31) At extremely low concentrations (less than about mol dm-9 dissociation of bisulfite ion to sulfite and hydrogen ions (pK, = 6.9)17and self-ionization of water32@must also be taken into account. (32) Woolf, L. A. J . Phys. Chem. 1972, 76,1166.. (33) Passiniemi, P.; Liukkonen, S.;Noszticzius, A. J. Chem. Soc., Faraday Trans. 1 1980, 76, 2552. (34) Falk, M.; GiguBre, P. A. Can. J . Chem. 1958, 36,1124. (35) Cotton, F. A.;Wilkinson, G. “Advanced Inorganic Chemistry”,3rd ed; Interscience: New York, 1972; p 447. (36) Small quantities of the hydrate species H,S03 may also be present,17,34i35 in which case the experimental value for the diffusivity of molecular sulfur dioxide is the number-weighted average of the diffusivities of unhydrated and hydrated molecular sulfur dioxide.37Because these species are present in constant proportion, separate values of their diffusivities are not required for our purposes. (37) Leaist, D.G.Can. J. Chem. 1983, 61, 1494.
Gas Absorption with Instantaneous Chemical Reaction
The Journal of Physical Chemistry, Vol. 87, No. 24, 1983 4939
where p is the degree of hydrolysis
P=
[H+I
(26) [H+l + [SO21 Do is the diffusivity of molecular sulfur dioxide (1.45 X 10-9 m2 s-l) and D, is the Nernst diffusion coefficient (2.12 X m2 s-l)
for fully hydrolyzed ( p = 1) sulfur dioxide. The addition of sodium bisulfite to aqueous sulfur dioxide produces a remarkable increase in both the diffusivity of the dissolved gas component (Dll) and its apparent diffusivity (D,) during the absorption process, especially at low total solute concentrations. This is interpreted as follows.24 In binary SO2 + H 2 0 mixtures, the hydrogen ions and bisulfite ions produced by hydrolysis diffuse at the same rate in order to maintain zero electric current. In SO2 + NaHS03 + H20 mixtures, however, H+, HSOf, and Na+ are present. In general, the three ions diffuse at different rates. The highly mobile hydrogen ion, no longer required to diffuse at the same rate as the slower disulfte ion, diffuses rapidly into the bulk of the absorbent at the expense of countercurrent flux of sodium ion to the surface of the absorbent. (Note that values of &-the coefficient determining the coupled flux of sodium bisulfite produced by the flux of sulfur dioxide into the absorbent-are large and negative.) In the limiting case of absorption of trace amounts of sulfur dioxide by aqueous sodium bisulfite (fl = 0), D, and Dll are equal, and are number-weightedaverages of the diffusivities of molecular sulfur dioxide and hydrogen D,(fi=O) = Dll(fi=O) = (1 - P)Do
+ PDu+
(28)
At low sodium bisulfite concentrations where hydrolysis is extensive, the apparent diffusion coefficient describing absorption of trace amounts of gas approaches the diffum2 d). sivity of the hydrogen ion (8.45 X The mechanism described here for the increase in the diffusivity of the sulfur dioxide component in aqueous sodium bisulfite will increase the multicomponent diffusivity of sulfur dioxide in other supporting electr01ytes.l~ Therefore, the rate of absorption of sulfur dioxide by (38) Muller, G. T. A.; Stokes, R. A. Trans. Faraday SOC.1957,53,642. (39) Holt, E. L.; Lyons, P. A. J.Phys. Chern. 1965,69, 2341.
aqueous salt solutions (such as seawater) will be substantially larger than the rate of absorption by water alone, especially at low partial pressures of the gas. The exceedingly large diffusivity of the aqueous hydrogen ion is responsible for this unexpected behavior. The effects of multicomponent diffusion on absorption will also be significant for aqueous alkaline absorbents containing the highly mobile hydroxide ion (diffusivity 5.2 X m2 s-l at 25 "C).
-
Discussion Because several approximations were made, the expressions developed in this paper best describe absorption processes with low rates of absorption. At high mass transfer rates, the following points should be considered: (a) Enthalpies of solution and reaction of the gas may cause departures from isothermal conditions. (b) If transport in the gas phase is a rate-limiting factor, the surface of the absorbent may not be fully saturated with dissolved gas. (c) In cases where the flux of dissolved gas is large, the concentrations and hence diffusion coefficients may vary appreciably along the diffusion path. (d) Large concentration gradients may produce density inversions and subsequent c o n v e ~ t i o n . ~ ~ The expressions derived in this study refer to absorption by a convection-free absorbent. In practical absorption processes, however, the absorbent is well-agitated in order to increase the mass transfer rate. To describe absorption with turbulent motion in the liquid phase, our result for absorption by a quiescent, ternary absorbent may be combined with the surface renewal theory developed by Danckwerts1s2to obtain
Jl(O,t) = Acl(sD,)'/* (29) where s is the fractional rate of surface renewal-an empirical measure of the degree of turbulence. If chemical reactions accompany absorption of the gas, the reaction rates must be rapid enough to ensure local chemical equilibrium, otherwise eq 21 and 29 do not apply. Although discussion has been confined to absorption processes, the equations that have been derived can of course be used to describe desorption processes (Acl < 0), such as those used to regenerate spent absorbent. Also, the analysis can be extended to more complicated geometries, and to quaternary and higher-order diffusion. Registry No. SOz, 7446-09-5; NaHS03, 7631-90-5. (40) Wendt, R. P. J.Phys. Chern. 1962, 66, 1740.
