J. Phys. Chem. 1983, 87, 1938-1941
1938
Reaction of NaCl(s) with S02(g) and O,(g) To Form Na,SO,(s). Reaction
A Charge-Transfer
Alfred B. Anderson' and N. C. Debnath Chemlstry Department,
Case Western Reserve Unlversity, Cleveland, Ohio
44 106 (Received: August 12, 1982)
-
We find, using an atom superposition and electron delocalization molecular orbital (ASED MO) theory, that the driving force for the reaction 02(g) + S02(g) + 2NaCl(s) Na2S04(s)+ Clz(g)is electron transfer from chloride ions to sulfate ions as they begin to form. The lowest energy pathway is catalyzed by SO3 which adds to an activated OZSO2molecular intermediate, yielding sulfate ions and SO3. Our results support the experimentally observed first-order dependence on SO3 pressure, and the low barrier for this exothermic reaction. We find that all gas-phase species bond weakly to the NaCl(s) surface and SOzbonds most strongly by donation from the 5al highest filled orbital to the empty Na 3s band.
Introduction The reaction of SOz and SO3 with sodium chloride at high temperatures in air leads to the formation of solid sodium sulfate, which in the molten state is a powerful corrosive agent.' Sodium sulfate attack of protective oxide layers on turbine blades can lead to premature failure by a phenomenon known as hot corrosion.2 This problem has motivated fundamental studies of the chemical kinetics of sodium sulfate f ~ r m a t i o n . ~One ~ ~ recent study4 employed the thermogravimetric technique over a 330-625 OC temperature range. By introduction of a platinum catalyst (which speeds attainment of equilibrium for the reaction SO2 + 1/202 + SO3),it was found that the formation of Na2S04is first order in SO3 concentration. An activation energy of 21 kJ/mol was determined for the overall reaction, which was believed to take the form 2NaCl(s)
+ SO&) + '/202(g)
-
NazSOa(s) + Cldg) (1)
The purpose of this paper is to examine the process given in reaction 1, as well as alternatives, from a quantum chemical viewpoint. We use an atom superposition and electron delocalization molecular orbital (ASED MO) t h e ~ r y . Our ~ reacting systems consist of small bulk-superimposable clusters of sodium chloride with 02,SOz, and SO3 reacting on their surfaces. We find that, overall, electron transfer from chloride ions to sulfate ions as they begin to form is the driving force for sulfate formation. Several conceivable overall processes for sodium sulfate formation are investigated and evidence is found for the importance of both S02(g)and SO,(g).
Method The ASED MO theory is a semiempirical technique based on an exact model. In the model the electronic charge in a molecule is separated into atomic parts and the electron delocalization bond part. The superposition of rigid atom electronic charge densities centered on the (1) G. C. Fryburg, F. J. Kohl, C. A. Stearns, and W. L. Fielder, J. Electrochem. SOC., 129, 571 (1982). (2) J. Stringer, Annu. Rev. Mater. Sci., 7, 477 (1977). (3) J. F. G. CondC, N. Birks, M. G. Hocking, and V. Vasantasree, 'Proceedings of the Fourth Conference on Gas Turbine Materials in a Marine Environment, Annapolis, MD, June 26-28,1979", Vol. 11, Naval Sea Systems Command, Washington, DC, 1979, p 386. (4) W. L. Fielder, C. A. Stearns, F. J. Kohl, and G. C. Fryburg, 'Formation of Na,SO,(c) from NaCl(c), SOz(g),and 02(g).",presented at the International Conference on High-temperature Corrosion, San Diego, CA, March 2-6, 1981. (5) A. B. Anderson, J. Chem. Phys., 62, 1187 (1975);60, 2477 (1974). 0022-365418312087-1938$01.50/0
TABLE I : Parameters Used in the Calculationsa
Na
S 0 C1
3 3 2 3
6.139 21.2 27.48 23.54
d
P
s
atom
0.836 2.220 2.046 2.356
3 2 3
11.36 12.62 11.97
1.927 2.027 2.039
3 3 3
5.0 1.9 2.0 2.0 1.0 2.0
a The values appearing under the s, p, and d headings are the principal quantum number, the ionization potential (electronvolts), and the orbital exponent (au), respectively, €or each atom.
atomic nuclei produces an easily calculated repulsive energy component called E& The electron delocalization density produces an energy component ED. The sum is the exact molecular binding energy, E: The ED component of the binding energy is difficult to obtain but has been found to be well approximated by a one-electron molecular orbital energy, EMO. The Hamiltonian that we use is similar to the extended Huckel Hamiltonian and the energy level and molecular orbital solutions are often qualitatively similar to those from traditional extended Huckel molecular orbital calculations. We pay particular attention to ionization potentials and Slater orbital exponents used in our determinations of EM, to produce accurate charge transfers and bond lengths for diatomic species. ER also depends on the orbital exponents. Thus determined, the same parameters are used in studying the structure and reactions of larger systems. The parameters used in this paper are in Table I. In a recent paper6 the reader will see that the bond lengths and angles produced by the theory for SO, SOz, SO,, and SO-: agree with experimentally determined values to within 0.03 A and 5 O .
Adsorption of 02,SOz, and SO3 on NaCl There is little covalency in the NaCl ionic bond. The C1 3p band is only 0.2 eV wide in a NagC19cluster measuring 3 X 3 X 2.' The C13s band is less than 0.1 eV wide. Consequently the adsorption bond is very localized and its strength is relatively independent of cluster size. This is different from adsorption t o a covalent material, such as a nickel surface.8 (6)A. B. Anderson, Chem. Phys. Lett., 93, 538 (1982). (7) Solid NaCl has simple cubic structure. The 3 X 3 X 2 cluster has two layers of nine atoms. An edge of one layer consists in NaClNa and for the other layer CINaC1. The 3 X 4 model is a single layer in thickness.
0 1983 American Chemical Society
The Journal of Physical Chemistry, Vol. 87, No. 11, 1983
Reaction of NaCl(s) with SO,(g) and Oz(g)
1939
TABLE 11: Structures and Binding Energies for 0,, SO,, and SO, on Na,Cl, molecule
site
orientation
0, 0, SO,
Na Na C1 C1 Na
SO, SO,
Na Na
SO,
Cl
SO,
SO,
C1 C1
SO,
Na C1
perpendicular parallel perpendicular parallel perpendicular, through S parallel, through S perpendicular, through 0 perpendicular, through S parallel, through S perpendicular, through 0 parallel parallel
0, 0,
SO,
height, adsorp A energy,"eV
1.8 2.1 3.30 3.0 2.2
0.46 (0.51) 0.44 0.03 0.11 0.76 (0.72)
2.4 2.5
0.69 0.44 (0.53)
3.4
0.07 (0.09)
2.5 2.6
0.40 0.23 (0.28)
2.6 2.6
0.65 (0.88) 0.53
so 3
a Several adsorption energies for a two-layer-thick Na,Cl, model are in parentheses.
No s band
Flguro 2. Molecular orbitals for SO3.
! /
n
n
-61
/
//
c
/
=n
x Figure 1. u donation from 5a, SOz orbital to Na s band.
All of the small molecules adsorb weakly to the NaCl surface because C1- and Na+ are closed-shell ions. All of the adsorption energies given in Table I1 are under 1 eV. Most of our studies were made with a 3 X 4 N+C& model,' but a few results for the two-layer-thick Na&$ model are included in Table 11, illustrating the slight dependence of adsorption energies on cluster size. These results indicate that these molecules will slide and tumble across the surface at reaction temperatures because the adsorption energies are comparable for a variety of orientations and for both sites. All of the surface-adsorbate bonds show stabilization of the lowest a-bonding framework adsorbate energy level, as already noted in other surface systems (ref 8). For the Na site this involves an orbital from the Na s band, which lies at about -4 eV. At the C1 site, the lowest adsorbate orbital stabilization involves the C1 d, p, and s bands which lie at -1, -12, -23.6 eV, respectively. The extra stabilization that SOz experiences at a Na site is due to donation from the highest occupied (3aJ orbital to the Na s band, as shown in Figure 1. This orbital has the proper symmetry to overlap and, at -11 eV, it is closer to the s band than the la;' SO3 orbital (Figure 2) with an energy of almost -17 eV. Other a symmetry orbital energy levels lie still deeper in SO3. Thus, SOz is a superior u donor and bonds more strongly to the NaCl surface. This capability is maintained by donation from the sulfur 3s component of the 3al orbital even when SOz is parallel to the surface (the 3p component is orthogonal to the sodium s orbital).
Formation of NatS04 The reacting system of interest contains NaCl(s), Oz(g), SOJg), and S03(g). No low-energy pathway was found for (8) A.
B. Anderson, J. Am. Chem. Soc., 100, 1153 (1978).
O2 so4
- 22
/
, /
- 26 241
Flgure 3. Bonding of structure.
/
'-fBf3 *
OS
-,,
O2 to SO2 forming SO, intermediate of
C,
sulfate ion formation by studying the dissociative reaction of O2with SO3: NaCl(s) + Oz(ads) + S03(ads) S042-(ads) OZ-(ads) 2Clz(g) + NaCl(s) (3)
+
-+ -
The C, symmetry SO4 adduct (see Figure 3) (4) Oz(g) + SO&) C,-SO,(g) studied previouslye shows the necessary activation. The O2bond is lengthened 0.3 A and (Figure 3) the bond order
1940
The Journal of Physical Chemisfry, Vol. 87,No. 11, 1983
Anderson and Debnath E=0 l43i
Ea=028eV
E=O02eV
I50i
I44i
CI -Na-CI-Na Figure 4. Transition state for first step in sulfate ion formation. -__--___ ___ 0
e :
-1 -
-
Gas ndsorbed
E a = O 61 eVI E a = O 68eV
-91.
>
.2 -10
-24-
-
I I
1
0 2 04 06 08 0 - 0 Stretch
(i)
Figure 5. 0,up' energy level position in SO, as it depends on 0,bond length. At the top of the figure the total energies are shown for gas phase and adsorbed SO,. They peak out when electron transfer occurs from the CI p band to the O2 up* orbital.
has been reduced to one due to mixing and filling of the 7r* orbitals, which are nearly degenerate in the adduct. Further, the O2 yp* orbital is stabilized by 3 eV, a result of bond lengthenmg. This becomes vide infra the acceptor orbital in sulfate ion formation. A number of pathways were explored in an attempt to rearrange C1-SO4into tetrahedral (t)SO, in the presence of NaCl NaCl(s) + Cl-SO,(ads) NaZSO4(g)+ C l M + NaCUs) ( 5 ) but all had high barriers. Once the tetrahedron forms, it will become a sulfate ion because its lowest unoccupied molecular orbital lies -0.4 eV below the top of the filled chloride p band in our calculations. Calculations verify the experimental exothermicity (AH E -201 kJ/mol) for the reaction 2S02(g) + OZk) 2SOdg) (6) Thus, SOz will not rearrange the SO, adduct according to the reaction NaCl(s) + C,-SO,(ads) + S02(ads) Na2S04(ads)+ S02(g)+ C12(g)+ NaCl(s) (7) because the weakly held O2 in the adduct will oxidize SO2. A successful reaction is the following: NaCl(s) + Cl-S04(ads)+ S03(g) SO,(ads) + t~-SO,~-(ads) + 2Na+(ads) + Clz(g) + NaCl(s) (8)
-
-
-
-
where t ~ - S 0 , ~ is -a trigonal-pyramidal sulfate ion. The transition state for reaction 8 is shown in Figure 4. The electron transfer begins at an 0-0 stretch of 0.6 8, from the C1-SO4value, as evident in Figure 5. With the 0.6-8, stretch the O2 up* level drops down to the top of the chloride p band and one electron transfers to tp-SO, which will settle down on an Na' ion, and accept a second electron to form NaS0,- which will pick up a sodium ion
T r i g o n a l Pyramid
Transition State
Tetrahedron
Figure 6, Conversion from trigonal-pyramidal NaSO,' to tetrahedral.
to form Na2S04. Overall, our results are consistent with the reaction
-
S03(g)+ S02(g) + OAg) + 2NaCl(s) NaZSO4(s)+ C12(g)+ S03(g) (9) The two activation energy curves in Figure 5 correspond to the formation of tp-S0,2- near and touching the surface. When it touches the surface, the barrier is 0.07 eV less, a minor result of covalent interactions with the surface. Our barrier of 59 kJ/mol overestimates the experimental determination of 21 kJ/mol, but it is clear that the calculated barrier is sensitive to the relative positions of the O2 ap.* level and the C1- p band (Figure 5), whose exact positions would require a self-consistent theoretical treatment. Once tp-NaS04- forms, it must undergo an umbrella distortion to the tetrahedral sulfate ion. We see from Figure 5 that, if the C1- p band moved up 1 eV or if the 0, up* level moved down 1 eV, the umbrella distortion of NaS0,- would be the overall transition state predicted by the theory. We have no basis for making either shift at present. The transition-state geometry (Figure 6) lies about halfway between the trigonal pyramid and tetrahedron. The calculated activation energy, 27 kJ/mol, is not far from the experimentally determined overall reaction energy (21 kJ/mol). According to the calculations, C1 atoms formed at adjacent sites as a result of sulfate formation will come together, forming ClP. The Clz molecules will desorb because of the weak physisorption bond, calculated to be only 11 kJ/mol.
Discussion The thermogravimetric studies indicated that sodium sulfate formation is first order in SO3 c~ncentration.~ Our postulated overall reaction based on theoretical analysis is consistent with first-order dependence in SO, concentration. However, SO3plays a catalytic role in reaction 9 and our overall reaction therefore differs from the one postulated in reaction 1. We expect that C1-SO4 exists briefly but for sufficient length of time to have SOzor SO, collide with it occasionally. When SO2 collides with ClSO4,two SO, molecules can form. When SO3collides with C1-S04,tp-Na2S04can form near the surface. The greater the concentration of SO3,the greater the rate of product formation by our mechanism. Recent experimental evidence suggests a possibility of pyrosulfate (Na2S207)formation below 500 O c a 9 A likely mechanism may be seen in Figure 4. As tp-S0,2- forms, it need only shift to the left to bond to SO3through the apical 0 atom. Umbrella distortions at both ends of the S2072-species would produce the pyrosulfate anion. Future studies will explore this and other possible mechanisms. We find that, once the electron transfer takes place, transfer of sodium ions to sulfate yields definite stability, about 200 kJ/mol in our approximate calculations. Dis(9) F. J. Kohl a n d W.
L. Fielder,
p r i v a t e communication.
J. Phys. Chem. 1983, 87, 1941-1951
cussion of the structure parameters of the Dzd-Na2S04 molecule is in ref 6. Our calculations omit Madelung energies in both sodium chloride and sodium sulfate crystals. On the basis of standard heats of formation, the heat released in the overall reaction 1is -250 kJ/mol. Assuming the Madelung contribution is qualitatively the same for the chloride and the sulfate, our calculated stability gain is in qualitative agreement with the expected value. This suggests that our C1- donor and t-SO, acceptor energy levels are in satisfactory relative positions because the
1941
electron stability gain is expected to dominate. Acknowledgment. We thank Drs. Fred J. Kohl and William L. Fielder of the NASA Lewis Research Center for their encouragement and interest. We also thank Miss Kelly Acker for calculational assistance. Our work was funded by a Select Research Opportunities Grant from the office of Naval Research. Registry No. NaCl, 7647-145; SO2, 7446-09-5;SO3, 7446-11-9; Na2S04,1757-82-6; 02,7782-44-7.
Diffusion-Limited Reactions in One Dimension David C. Torney and Harden M. McConneli' Stauffer Laboratory for phvsical Chemistry, Stanford University, Stanford, Callfornk 94305 (Received: September 10, 1982)
-
The irreversible diffusion-limitedreaction A + A P taking place on a ring is formulated exactly, as a stochastic process. Initially 2N particles are placed at random on a ring of length L. (We assume a dilute system and neglect the length of the particles.) Each diffuses with coefficientD until it collides with another particle which results in the removal of the pair. The expected fraction of the initial number of particles remaining at the dimensionless time { = Dt/L2in an ensemble of such rings is given for all systems containing initially an even number of particles. In the limit of an infinite number of particles put on an infinitely long ring with an initial density Ao,the survival fraction S ( r )is e8r erfc (8r)lI2. = AZDt.) The fluctuations about the mean number are of the order W I 2and the exact rate function is always larger than the Smoluchowski-Noyes rate function with the ratio increasing from 1 at the beginning of the reaction to ~ / at2 its completion. The exact survival fraction S ( r ) is smaller than the prediction of the Smoluchowski-Noyes theory for > 0.
(r
r
Introduction The shortcoming of standard diffusion-limited chemical reaction rate theory, the Smoluchowski-Noyes theory,' is that, although approximations are made, there is no estimate for the order of the error? Confidence in the theory is due to its agreement with experiment.' However, it is conceptually correct only in the limit of pseudo-first-order reaction^.^ With reactants of comparable mobility, finding the rate of a diffusion-limited reaction requires the consideration of a many-body problem with each body acting as a moving sink. Many theoretical treatments assume immobile sinks,4 but we are specifically interested in the reaction A + A P. In this paper the reaction occurring in one dimension is given an exact formulation with an eye to the development of a general method for calculating the rate of diffusion-limited chemical reactions. It is conventional to place the particles initially at random, which we have done. Initially there are 2N type A particles on a ring of length L, and we assume a dilute system with the aggregate length of the particles negligible compared with L. Thereafter each particle diffuses independently with diffusion coefficient D until it collides with another, resulting in the irreversible removal of the pair from the ring so that product molecules do not isolate the remaining reactive particles. (In two or three dimensions the presence of products does not physically block
-
(1) R. M. Noyes, Prog. React. Kinet., 1, 128-60 (1961). (2)D. C. Torney and H. M. McConnell, Proc. R. SOC.London, Ser. A,
the progress of the reaction. Also, the SmoluchowskiNoyes theory cannot be directly applied if unreacted particles do not diffuse freely.) The following method gives the probable number of particles surviving until time t. An ensemble of systems each with 2N particles with the same set of initial coordinates can be described as diffusing away from the initial position in 2N dimensions (2N D). There are 2N - 1 D surfaces of reaction where the coordinates of two particles are identical. In the extreme diffusion limit with all collisions resulting in reaction we can say that there is a vanishing boundary condition at each of the 2N surfaces around the 2N D prism containing the system's initial position. The point is that reaction is accounted for on fixed boundaries in 2N D. Random initial placement of the particles corresponds to uniform initial probability distribution inside each 2N D prism. In the mathematical calculations given in the next section, we begin by giving the survival fraction for two particles S2({). For an ensemble of rings initially containing two particles, &({) is the probability that a ring still contains two particles at the nondimensional time { = Dt/L2. As one can hold one of the particles fixed and consider relative diffusion, there is complete analogy with a previously solved problem; 58 the fraction of the initial heat remaining in an insulated rod which initially had a uniform temperature and its ends maintained at "zero" temperature is equal to the survival fraction S2({)when the parameters of the two problems are made equivalent. The initial probability distribution for the ensemble of
in . . . meas.
(3) E. W. Montroll, J. Chem. Phys., 14, 202-11 (1946). (4)B. U. Felderhof and J. M. Deutch, J. Chem. Phys., 64,4551-8
(1976). 0022-3654/83/2087-1941$01.50/0
(5)(a) H. S.Carslaw and J. C. Jaeger, 'Conduction of Heat in Solids", 2nd ed., Oxford University Press, Oxford, 1959,p 96;(b) ibid., p 97;(c) ibid., p 485;(d) ibid., p 58.
0 1983 American Chemlcal Society
