J . Phys. Chem. 1986, 90, 144-152
144
Contributlon to the Theory of Electrolyte Mixtures at Equllibrium T. K. Lim, Department of Chemistry, University of Malaya, Kuala Lumpur, Malaysia
En Ci Zhong, and Harold L. Friedman* Department of Chemistry, State University of New York, Stony Brook, New York 11794 (Received: July 17, 1985; I n Final Form: September 10, 1985)
The electrolyte mixing coefficient gl(I)measures mainly the contributionsof three-ion clusters to the change in Gibbs function when two electrolytes with a common ion are mixed at fixed ionic strength I. Earlier incomplete information indicated that, for a mixture of unlike charge types, g l ( I ) has a remarkable shape. Here the HNC integral equation approximation is applied to a simple model for aqueous MnC12-LiCI mixtures to calculate g l ( I ) down to I = 5 X 10” m, revealing a strong maximum near m, where the long-range forces that give a limiting law (gl(I) = Ill2 as I 0) are compensated by the short-range forces which become more important at higher I. The model calculation shows that some pair correlation functions, e.g., gMn,Li(r), as functions of I deviate maximally from the corresponding Debye-HUckel approximation near the I value at which g l ( I ) has a maximum. These novel phenomena may be relevant to various other mixture data, notably to the I dependence of the rates of activation-controlled reactions between like-charged ions.
-
1. Introduction
The properties of charged systems such as ionic solutions at low concentration continue to be of considerable interest.’ The long range of the Coulombic interaction leads to peculiar behavior such as the well-known Debye-Hiickel limiting laws in which the partial molar excess functions of the electrolyte become proportional to the square root of the electrolyte concentration at low concentrations. Less well-known are the additional limiting laws for the coefficients which characterize electrolyte m i x t ~ r e s . ~ - ’ ~ In the work reported here our objective is to locate a remarkable peak in the ionic strength dependence of a particular mixing coefficient as it departs from limiting law behavior in extremely dilute solutions. When solutions of electrolytes A (ion species 1 and 3) and B (ion species 2 and 3)15 with the same molal ionic strength I are mixed at fixed pressure and temperature, the change in the excess Gibbs function Gexper unit of solution may be written in the form
A,GeXb,I)
E
GeXb,I)- yGeX(1,I) - (1 - y)Gex(O,I)
= ~ R T Y -Y)Zgn(I)(l ( ~ -2~)” n=O
(1.1) (1) P. J. Rossky and H. L. Friedman, J . Chem. Phys., 72, 5694 (1980). (2) H. L. Friedman, J . Chem. Phys., 32, 1 134 (1960). (3) H. L. Friedman, “Ionic Solution Theory”, Wiley-Interscience, New York, 1962. (4) R. H. Wood and R. W. Smith, J . Phys. Chem., 69, 2974 (1965). (5) R. H. Wood and H. L. Anderson, J. Phys. Chem., 70, 992 (1966). (6) H. L. Friedman and P. S. Ramanathan, J . Phys. Chem., 74, 3756 (1970). (7) R. A. Robinson, R. H. Wood, and P. J. Reilly, J . Chem. Thermodyn., 3, 461 (1971). (8) H. L. Friedman, A. Smitherman, and R. DeSantis, J . Solution Chem., 2, 59 (1973). (9) R. B. Cassel and R. H. Wood, J . Phys. Chem., 78, 1924 (1974). (10) H. L. Friedman and C. V. Krishnan, J. Phys. Chem., 78, 1927 (1974). (1 1) (a) K . S. Pitzer, J . Solution Chem., 4, 249 (1975); (b) K. S. Pitzer, J . Phys. Chem., 87, 2360 (1983). (12) H. L. Friedman, J . Solution Chem., 9,525 (1980). There are misprints in eq 4 in this paper which can be corrected by replacing R T by 2RT in each case. (13) T. K. Lim, Int. J . Quantum Chem., Quantum Chem. Symp., 16,247 (1982). The notation Ig, in his report should be identified with the present g”.
(14) (a) R. N. Roy, J. J. Gibbons, J. C. Peiper, and K. S. Pitzer, J. Phys. Chem., 87, 2365 (1983); (b) J. Vanhees, J. P. Francois, J. Mullens, Y. Yperman, and L. C. Van Poucke, J . Phys. Chem., 89, 2661 (1985). (15) We assign the labels A and B to the two electrolytes in such a way that 1z1z31b 1z2z31so that gl(r)/11’2is not negative in the I = 0 limit. Many authors use other conventions, which may change Y = (1 - 2y) to -Y. Since 4y( 1 - y ) = 1 - p,this change does not affect the sign of goand its derivatives, but it does change the sign of gl and other g, with odd n.
0022-3654/86/2090-0144$01.50/0
where y is the fraction of I coming from the ions of electrolyte A and g, is the nth-order mixing coefficient. By convention lzl/z,l 2 1; then it may be shown that g , is positive at small I for unsymmetrical mixt~res.’~ The unit of solution here is the quantity of solution containing a kilogram of solvent. Of course, A&‘” is the generating function for the changes in other thermodynamic variables which are derivatives of G (section 2). For example, the expansion for the enthalpy of mixing has the same form as eq 1.1 except that the free energy mixing coefficients gn(I) are replaced by the corresponding enthalpy mixing coefficients It follows that the theoreticalgJOresult g l ( 0 ) = 0 forces hl(0) = 0 too. Therefore, the experimental h,(I) data shown in Figure 1 imply that this function has a strong maximum at small I. To elucidate this feature, model calculations of gl(I) have been made,1° with the results shown in Figure 1 together with the limiting law for g,. The calculated gl(I) is qualitatively consistent with the h l ( I ) data (except for the sign, see Appendix), but the techniques for solving the H N C equation then available were not powerful enough to give the accurate results at small I needed to locate the extremum and the approach to limiting law behavior. The zeroth-order mixing coefficient go is closely related to the and aBA, which familiar first-order Harned coefficients12.16aAB also characterize common-ion mixtures and also depend on I but not on the mixing ratio y . However, go is more reduced in the sense that the Harned coefficients depend both on the difference in properties of the solutions of pure A and pure B and on the changes in interactions when A and B are mixed at fixed I,while go and the higher g, depend only on the latter effects. T h e ~ r y ~ , ~shows % ~ - that * go is governed by one limiting law 1n [go(I)/go(o)] = (Z12X312/2a)(N~v/V,)’/21’i2 (1.2) if electrolytes A and B have the same charge type, and another go(l) =
(ZI
- Z2)*(X3/96a2)(N~v/V~w) 1x1 I
(1.3)
if z1 # z2. In these equations X 4?re2/ckBT,where e is the electronic charge, t the solvent dielectric constant, kBBoltzmann’s constant, and T the temperature. Thus, X is 857 times the Bjerrum length for a 1-1 electrolyte.I6 Also, N A V is Avogadro’s number and V, is the volume, in the same units as X3, occupied by a kilogram of solvent in the same P,T state as the solution. These limiting laws are distinct from the familiar Debye-Huckel limiting law for Gexand its thermodynamic derivatives for a single ~~
~~~
(16) H S. Harned and B. B. Owen, ‘The Physical Chemistry of Electrolyte Solutions”, 3rd ed , Reinhold, New York, 1958
0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 1, 1986
Electrolyte Mixtures at Equilibrium O.O2It
0' 0
I
0.3
I
0.6
Jr
Figure 1. Experimental and calculated first-order mixing coefficients as functions of molal ionic strength. Experimental data of Cassel and Wood:9 A, 4 / 2 0 for Na2S0,-NaC1 mixtures; B, h,/20 for BaC12-NaCl mixtures. Theoretical results for MnC12-LiCI mixture: C, g, from H N C approximation; D, limiting law for g, from cluster expansion. This figure was adapted from ref 10. The scale factors (20) for A and B are merely to bring the h , data into the range of the computed g, values in the figure.
such effect may be expected for the ion triple with the strongest + - interactions. This picture is analogous to the molecular interpretation of the minimum in [$(I) - l]/Z'/2 in aqueous solutions of single electrolytes of the CuS04 charge type','8a (4 is the osmotic coefficient). It also is akin to the explanation advanced by RosseinskyIsb for the remarkable I dependence of the activation entropy for electron-transfer reactions between highly charged metal ions in aqueous solution. Here we report accurate numerical solutions of the H N C integral equation for unsymmetrical electrolyte mixtures at low ionic strength. There the Debye shielding is so weak that the various ion-ion correlation functions such as hab(r)have to be followed out to extremely large values of r in order to obtain sufficiently accurate solutions of the H N C equation. Therefore, we adapt the numerical technique that proved successful in the case of a single electrolyte at very low concentration.' The thermodynamic theory of the mixing coefficients is collected and completed in section 2, the limiting law results are given in section 3, and in section 4 the model and the H N C approximation are described. In section 5 we describe the computational techniques employed to get satisfactory results in the present case. The new results for mixing coefficients are given in section 6 and for pair correlation functions in section 7, while a review of the conclusions is given in section 8. 2. Thermodynamics of the Mixing Coefficients
In view of d(G/ T ) / d ( 1/ T ) = H it follows that eq 1 . 1 implies the following equation for the change in enthalpy on mixing at fixed I, P, and T
electrolyte. To explain, we consider the appropriate cluster exp a n ~ i o n ~ sfor ~ , the ' ~ extensive excess Helmholtz free energy AeX in a McMillan-Mayer state (specified temperature, volume V, solvent activity, and numbers N , , N2, ..., N,, ... of ions of various species) of the solution
-AeX/VkT = K3/127r
+ CCP,PfBSl(K) + C C C P S ~ , ~ U ~ S l , (+K )." (1.4) S
l
S
f
U
Here ps = N s / V ,K = (Xx,psz>)'/2, and Bsfu...(~) is the K-dependent nth-order virial coefficient resulting from the Mayer resummation of chains of Coulombic interaction (Mayer renormalization). In terms of eq 1.4, the ordinary limiting laws for the osmotic coefficient and electrolyte activity coefficients derive from the K3 term while the go limiting laws derive from the &(K) terms. The available experimental data for go(Z) and its thermodynamic derivatives are consistent with the t h e ~ r y . ~ - ~ ~ ~ - " The theory also shows2q3that go at low I is made up of a weighted sum of contributions &(K) from pairs of ions and that the limiting law behavior derives from the Debye-like shielding of the pairwise ion-ion interaction by all of the other ions. If A and B are of the same charge type (symmetrical mixture), then there are contributions to go only from close ion pairs with the common ion excluded; indeed, eq 1.2 has also been derived by Robinson et al. from a chemical model under the Debye-Hiickel theoryS7 In the other case, distant pairs of all charge types determine g0.2J The coefficient g , derives mostly from the coefficient B s l u ( ~ ) , thus from clusters of three ions, again under Debye-like shielding due to all of the other ions. On this basis a molecular explanation for the strong maximum (Figure 1 ) in g, in dilute solutions may be tentatively suggested: Among the triple ions which may be found in a mixture of electrolytes A and B are some with mixed charges, say two positive charges and a negative charge. Such an ion triple, presumably mostly in a linear + - + configuration, is stabilized by reducing I , thus reducing the Debye shielding of the two - interactions. However, as I is reduced even more, we reach a regime in which the secular (law of mass action explicit concentration factors in eq 1.4) concentration dependence is the dominant factor in the stability of the complex. The biggest
+
(17) J. E. Mayer, J . Chem. Phys., 18, 1426 (1950).
-
145
(2.1) where h, = -T dgn/dT. The corresponding equation for the volume change A,V(y,I) is equally evident. All of the above equations except (1.4)are in the so-called Lewis and Randall thermodynamic ~,~ system of independent variables ( T , P , and m ~ l a l i t i e s )which are most directly applicable to analyzing the thermodynamic data. On the other hand, the statistical-mechanical calculations are made in the framework of the McMillan-Mayer t h e ~ r y 'and ~ * are ~ ~ most simply based on the change in Helmholtz free energy when A and B are mixed at fixed Debye K (hence fixed molar ionic strength), fixed temperature, and fixed solvent activity. The subtle aspects of the conversion from the resulting McMillan-Mayer system to the Lewis-Randall system are well-known2' but are of little consequence at low I. Therefore, we here identify A e x / Vwith Gex/V , and we convert particle number densities p s to molalities m, simply by converting the units, thus neglecting complications due to the compressibility of the solvent and the volumes of the ions. Among the free energy functions that characterize the common-ion electrolyte mixture, the following are the most straightforward to determine, whether by experiment or by model $(y,I) is the osmotic coefficient, given by calculations:3~6~10~lZ
( 1 - $)m,RT = [ ~ ( G e x / m t ) / ~ ( l / m , ) l y
(2.2)
where m, is the total molality of ions m, = ml
+ m2 + m3 = 211v/lz1~3l+ ( 1 - Y ) / I z ~ z ~ I I
DEL(y,I) is t h e generalized compressibility, given by22
DEL(y,Z) = (8 In ~ + / dIn
Dy
(2.3)
(18) (a) H.L. Friedman and B. Larsen, J . Chem. Phys., 70, 92 (1979); (b) D. R. Rosseinsky, private communication. (19) W. G . McMillan and J. E. Mayer, J . Chem. Phys., 13,276 (1945). (20) H. L. Friedman and W. D. T. Dale in "Statistical Mechanics", Part A, B. J. Berne, Ed., Plenum Press, New York, 1977. (21) H. L. Friedman, J . Solution Chem., 1, 387, 418 (1972).
Lim et a].
146 The Journal of Physical Chemistry, Vol. 90, No. I , 1986
where y+ is the usual mean ionic activity coefficient given by R T In y+ = (dCex/dm,),
(2.4)
g,, and gB, are generalized Harned coefficients, given by2z,23
g*yb,l) = (&?*/dY)l
(2.5)
where g, is the reduced mean ionic excess chemical potential R T in y* of electrolyte A in the mixture g*b,l) = 2PY*b~l)/1Z1~31
(2.6)
and finally gByb,l) = (dgB/dy)l
(2.7)
gBb,l) = 2P%b,l)/1z2z31
(2.8)
with
T h e y dependence of each of these functions can be derived from eq 1.1, giving the following equations in terms of y and Y (1 - 2Y) gAyb,I) = - I R T C
n20
c p ( n + l)(bn -
n20
DELb,I)] = y ( l -y)IRT(do
+ ...I ...I
+ d l Y + ...)
(2.10)
(2.12)
+ (1 -y)IRT(ao + uIY+ ...) (2.13) (2.14) = gB(0,I) + yIRT(bo + b , Y + ...)
All of the mixing coefficients g,, a,, b,, d,, and w, depend on I but not on y . The g, mixing coefficients are interrelated in a way that can be precisely described if all of them with n > N vanish. Then we would havez4 dgN/ar = NgN/r (2.15) N
[ak + (-)"+kbklfn.k
(2.16)
k=n
where we define fn,k =
k+ 1
I-1
j=n+l
m=n
c (j!,)(-y+k-Io. + 1)-l c (32j-m-I
g,'
(2.11)
= gA(1,I)
gn =
+ l)(n + 2)(gn+2 - gn)
n
= -2ko - g2) - 6(g1 - g3)Y - .'.
(2.9)
which define the mixing coefficients w, and d,, while a, and b, are defined as modified Harned coefficients:'*
gBb,l)
(gAy - gBy)/IRT =
bn+l)
A,,,{mtb,l)[+bJ) - 111 = ~ ( -YV~(WO 1 + wiY+
g,b,l)
satisfied. It follows that the precise truncation of the expansion in eq 1.1 is not realistic. One cannot do more than neglect the terms for which n > N where N is the highest order for which a mixing coefficient can be derived from experiment or model calculation. Thermodynamic consistency leads to various relations among the mixing coefficients of given order n.
(2.18)
+ a, + 2(a, + a 2 ) Y + ...I
= ZRT[bo - 61 + 2(bl - b2)Y
A,[m,(yJ)
4b]/~~V.
W n + I)(an + an+,)
= -ZRT[ao g B y b ? I ) = IRT
3 40 4b Figure 2. Cluster integrals that contribute to the g, limiting law. In these graphs a bond from circle 1 to 2 represents the function e x p ( - ~ r ~ ~ ) / 4 ~ r ~ ~ and the circles are black circles of unit weight. The functions in eq 3.1 are defined as dimensionless combinations in the thermodynamic limit , = [graph 4 a ] / ~ ~ V and , idb = [graph ( V - m). i, = [graph ~ ] / K Vila
(2.17)
For a mixture of nonionic solutes eq 2.15 is exact even if N is replaced by any n because in that case g, (concentration)". In an ionic system the long-range Coulomb interactions cause every quotient g,/P to be dependent on the Debye shielding and hence not to have a power series expansion in I. (Cf. the modified cluster integrals in eq 1.4.) Therefore, eq 2.15, which is required to make the truncation of the expansion consistent, cannot be exactly
d, = d(Pw,)/dZ
(2.19)
wn = a(Ign)/ d l
(2.20)
dg,/dI = (a, g, = a,
+ b, - 2g,)/z
(2.21)
+ b, - w,
(2.22)
These relations are not all independent; for each n there are three thermodynamic consistency tests, analogous to the familiar virial-compressibility interdependence in the theory of simple fluids.6J0 It also may be noticed that gA,, gB,,, 4, and DEL, unlike AmCex,are easily calculated6 once the equilibrium pair correlation functions have been calculated for a given state of a model for an electrolyte mixture. To extract the mixing coefficients g,, a,, etc., one needs the ion-ion pair correlation functions for a sequence of states differing only in y .
3. Limiting Laws for go and g l The limiting laws for the mixing coefficients can be derived from the cluster expansion of AeXfor common-ion mixtures of general charge type zl, z2, z3 to provide a wider context for understanding the present results. Using the method already reported,I0 we obtain the results already given in eq 1.2 and 1.3 together with the following.25 gl(l) = X9/2(NAv/2Vw)3/2[2A334i3 - A3333(i4a+ 2i4b/3)]I,/* (3.1) Here
A334 !h(zl - zd3(z1+ z2 + z 3 )
(3.2)
- zd3(z1+ zz + 2 4
(3.3)
A33332 (z,
are coefficients appearing in the following changes in products of the moments (3.4)
P , = CPsZsn 5
(22) It is traditional to regard In y+ as an experimental quantity, but in the present context there is some disadvantage in so doing because In y+ only is measured up to an additive constant to be determined by a procedure based on extrapolation of experimental data to I = 0. In the present case such extrapolation cannot be tolerated because it is precisely' the subtlety in the small-I dependence which is being investigated. Fortunately, there is no penalty except the break with tradition to regard the derivatives DEL,gAY, and gByas the experimental variables rather than respectively In y* for the entire electrolyte, In yt for electrolyte A, and In y+ for electrolyte B. The derivatives can be quite direct1 obtained from the experiments using wellknown numerical or graphical2 Y techniques to convert finite differences into - . derivatives. (23) T. F. Young and 0. G. Vogel, J . Am. Chem. SOC.,54, 3030 (1932). (24) T. IC. Lim, C. Y. Chan, and K. H. Khoo, J . Solution Chem., 9, 507 (1980).
where ps is the particle number density of species s, which appear in the asymptotic estimates of the cluster integrals at small I . Am(P,'P4)/b'(l
-Y)P131 = -(zI - ZJ2[(zi2+ Z 3 2 + z1z3) + '/z(z, + z2 + z3)(3z* + 423 + Z d l + A334Y ( 3 . 5 )
Am(P~,~)/b'(l-Y)P241 = -(ZI - Z2I2[(7/4) X (zI - 22)' + 6 ( ~ 2+ Z ~ ) ( Z+I 2311 + A3333Y-
(ZI
- z2I4p/4
(3.6) (25) In the case z, = z2 this formula for gl evidently vanishes, but the residual term, analogous to eq 1.2, has yet to be derived.
The Journal of Physical Chemistry, Vol. 90, No. 1, 1986 147
Electrolyte Mixtures at Equilibrium In eq 3.5 and 3.6 it is only the terms in Y that contribute to g,. The terms in contribute to goand the terms in contribute to g2, but they are of no consequence for the gl limiting law; they are included here to avoid confusion since they are indeed generated by the respective A,,, operations. We also have (cf. Figure 2) i3 = 2.5g0 x
i4a = 3.25, x
iqb
= 7.295 x (3.7)
The last term26should be omitted in comparison with the H N C approximation because it is missing from the latter. Since i4bis so compared to i3,its contribution is not significant.
4. Model and HNC Approximation Beginning with a Hamiltonian model for an electrolyte mixture, we estimate the set of model ion-ion pair correlation functions gab(r)from which the free energy functions 4, DEL, gAy, and gBy are calculated. We use one of the models for a LiC1-MnCl2 mixture in water previously studied by Friedman and Krishnan.lo The solvent-averaged ion-ion pair potentials have the form fiab(r)
= Dab/r9 + GURab(r) f
Z,Zbe2/€r
ps is
the number density of solute species s, defines the
(26) M. Dixon and P. Hutchinson, J. Phys. Chem., 79, 1820 (1975). (27) J. D. Talman, J . Comput. Phys., 29, 35 (1978). (28) P. J. Rossky and W. D. T. Dale, J . Chem. Phys., 73, 2457 (1980). (29) The pair potential notation of Rossky and Friedman' is slightly changed here to specify the solvent-averaged potential (overbar) because, in work to be reported elsewhere, we see whether the limiting law for g, is affected by the nonpairwise additive part of the McMillan-Mayer solventaveraged N-point potential U p
0.325 913 3247 1834 2.89 3.91
0.032 682 k"lA-' 324.7 ,,?rb 1603 1 0 5 ~ ~ ~2.95 105~,d 5.43 k'1A-I r 'b
3.247 1143 32474 2064 2.91 4.92
5.84 41.8
"All of these examples are for y = 0.5. They are typical of the results for all y values. b k ' = exd-10.24 + 0.01nl: k " = exd-10.24 +
effective direct correlation function cab for which the H N C approximation is Cab
=
hab
- In (1
+ hab)
- Pa,b
(4.3)
where = l/kBT and where, in the context of the McMillanMayer t h e ~ r y ,we l ~use ~ ~the ~ solvent-averaged potential, denoted by an overbar2*(cf. eq 4.1). The potential is separated into a Coulombic part
(4.1)
where the f 9term represents the core repulsion, where the "Gurney" term, which incorporates an adjustable parameter Aab, has a short range and a modest effect but enables us to fit the model to one set of data and then apply it to another, and where we also have the Coulomb term with the ionic charges z,e and zbeand the dielectric constant t of the pure solvent. Further details are given as model A in ref 10. The earlier effortlo to apply the H N C approximation to a model to calculate the first-order mixing coefficients for a very dilute unsymmetrical common-ion mixture gave results which were unsatisfactory; the gl-level thermodynamic consistency tests were extensively violated. It was not possible then to tell whether the problem was due to the limitations of the H N C approximation method (since those correlation functions that are exactly derived under H N C need not exhibit thermodynamic consistency) or whether it was due to failure to get sufficiently accurate solutions of the H N C equation. In the work reported here we bet on the latter interpretation; we apply a modification of the technique for numerical Fourier transforms which enables us to get accurate results for the H N C equation even when the ion concentrations are so low that the Debye shielding is so weak that the r-space correlation functions have to be followed out many thousands of angstroms. Thus, we apply the algorithm for fast Fourier transforms with exponential spacing of the sample points (EXPFFT), due to Tal~nan,~' which has been adapted' for integral equation calculations of correlation functions over an extremely large range of r. We solve the H N C equation computationally using the algorithm of Rossky and Dale2* which we need to describe here in order to specify some minor modifications which prove to be essential to success in the present work. In the notation of Rossky and Friedman' (eq 1-14) the H N C equation and the algorithm for solving it may be described as follows in terms of hab(r) = gab(r)- 1, where g a b is the usual pair correlation function (or radial distribution function) connecting a particle of species a with one of species b at a separation r. The Ornstein-Zernike equation
where
TABLE 1: Examples of k' and k" Parameters and Moment Defects Plm 10-5 10-4 I 0-3 10-2
@tia< =
zazbX/4*r
(4.4)
and a short-range remainder ti,,,* iiab - iia{; Mayer's resummation" of the long-range part generates Mayer " q bonds" qab(r) -/3iia 0.01 m the above algorithm, employed with FFT with N = 21° uniformly spaced sampling points, was satisfactory, as before,1° giving zeroth-moment defects A. in the range from 10" to and second-moment defects Az (30) J. C. Rasaiah and H. L. Friedman, J . Chem. Phys., 48, 2742 (1968).
148 The Journal of Physical Chemistry, Vol. 90, No. I , 1986
in the range from to (cf. Table I). While the larger values are rather high, we did not try to do better at this time by increasing N o r the number of iterations because of our overriding interest in the behavior at small I. In the present work, the solutions of the H N C equation in the I < 0.01 m regime, even using EXPFFT and the above algorithm, were found to be insufficiently accurate for our purpose. The basic problem seems to be intrinsic in the EXPFFT calculation when very high accuracy is required, as in the calculation of a higher order mixing coefficient like g,. To explain, we note that the vth EXPFFT sampling point in r space, expressed in angstroms, is given by’ rv = exp(p, v A p ) , v = 0, 1, ..., N - 1 (5.1)
+
With the parameters’ N = 21°, p, = -N/ 100, and Ap = 0.04, we A; thus, ruvaries from the have ro = 10-4.458,and rN-, = 1013.32 nuclear to the geographic range. The important range around a few angstroms is found for Y = 300 while for v > 717 we have r greater than 1 cm. Thus, we have over 500 v values for which r is macroscopic. In this range the computed values of the correlation functions are expected to be determined by numerical noise which gets folded into the smaller r range in subsequent Fourier transformations. It is not possible to cure this fault by a different choice of the parameters in eq 5.1 because of the intrinsic qualities of the EXPFFT.’ In the applications of EXPFFT reported here we actually employed N = 212, pm = -N/400, and A p = 0.01 to get better accuracy, so the above problem is considerably aggravated. Moreover, the k-space equation has the same form, namely, for k in A-1 k , = exp(c, v A K ) , v = 0, 1, ..., N - 1 (5.2)
+
and various constraints’ lead to the choices K , = pm and AK = Ap so we have the same problem in k space. Thus, in both r space and k space we find cases in which numerical inaccuracy can be traced to functions of large v which become important in some circumstances. Moreover, the range of numerical values involved here is so enormous that simply changing the entire computation from 32 bit (single precision) to 64 bit word length scarcely improves the results. After a long study of this problem, a four-part modification of the basic procedure was found sufficient to obtain the high accuracy at low concentration needed for calculating the first-order mixing coefficients. (a) One problem was traced to the matrix inversion in eq 4.1 1, which has to be done with extreme accuracy when k is so large that the matrix elements of 7-H are all so small ( k > k’the matrix inverse is evaluated as the three terms on the right side of eq 5.3, while for k > k”the inverse is evaluated as the first two terms on the right of eq 5.3. In each case the H matrix is evaluated an2lytically and, if k < k’, the remaining operations in computing C are done in double precision, but the result is reduced to single precision before going to the next step in the iteration loop. To determine the parameters k‘and k”, we observe that in the H N C approximation Yab(r)varies like hab(r)2at large r while for dilute ionic systems hab(r)at large r and small I is approximately qab(r);at small r the term -qab(r) dominates Tab(r).It follows that at large k we have
Lim et al. MabbTZ&) I(@H)abI(CPs/Pb) S
I(@H)abl
a 1(4’H)abI
(5.4)
while the function giving the bound is easily evaluated. Mab(y,I,k) = (Cp,)(ZaZblAk2/(kZ + K2)*
(5.5)
S
For each solution state y,Z we choose k’as the smallest k larger for all ab and k”as than 0.003 A-’ for which Mab(y,Z,k)< the smallest k larger than k’for which Mab(y,I,k)< IO-* for all ab. When k < k’then 1(T’H),bl is at most 2-3 orders of magnitude smaller than Mab, so it is large enough so that double precision is adequate. This procedure has been found to be adequate for our purpose. (b) We use the analytical ijab(k)rather than the numerically computed form. The latter is, in the abstract, surely more consistent with the use of numerical Fourier transforms throughout the computation, but EXPFFT is not well suited to functions which, like qab(r),are extremely strong functions of r at small r, no doubt because ro = lo-’ A, as noted above. (c) Expressions of the form 8 - 1 are approximated by power series expansions when x is very small, in analogy to the procedure described under (a). (d) At very large r the function hab(r)tends to be inaccurate in the sense that the relative error is large. While the difference between 1O-Io and is unimportant for many purposes, such differences have a substantial effect when the computed hab(r) is used to estimate integrals of the form
Even for the virial integral, where one hasfir) = 1 / r at large r, the effect of errors in the correlation function at large r is significant; the problem is most serious, of course, in the integral r2. representing the second-moment defect, for which f ( r ) Problems of this kind are more important with EXPFFT than simple FFT because of the enormous size of rlv-l in the former case. To suppress this source of error, we locate the large-r range for each species pair ab in which Ihab(r)ldamps down to values between lo-’ and lo-* and locate a place rMin this range in which is less than 0.01. Then the integration in eq 11 - (hab(r)/qab(r))l 5.6 is divided into two ranges: for r < rMthe H N C hab(r)is used while for r > rM it is replaced by qab(r). With the EXPFFT employed in the context of these special techniques, for I < 0.01 m and with 212 sampling points as described above, the solutions of the H N C equation initiated with ~~ in less than ten iterations to the g(A) a p p r o ~ i m a t i o nconverged give moment defects & < 3 X and A2 < 7 X IO”, which seem to be satisfactory for our purpose, judging from the results in the next section. The computational requirements are considerable, mainly because of the core storage requirements, namely, 6 X 6 X 4096 words for the principal functions (hab(r)and five more in the program), and the fact that it takes two ordinary FFT steps to accomplish one EXPFFT step.’ A computation to determine the set of hab(r)for a single y J state takes about 1 min on a Univac 1100/82 computer. The results reported next cover more than 100 such states, but they are only a fraction of the calculations used in program development.
-
6. Results for go and g , For each selected y,Z state the H N C equation was solved to determine the pair correlation functions hab(r)for the six species pairs 1-1, 1-2, 1-3, 2-2, 2-3, 3-3. For each state the coefficients 4, gAy,gByr and the zeroth- and second-moment defects were calculated6 from the pair correlation functions as described in connection with eq 5.6. For the I < 0.01 m range typical values of the computing parameters k’ and k ” and the zeroth- and second-moment defects are given in Table I. For most values of I we studied five values of y , namely y = 0 and y = 1 and three values near y = 0.5 as shown for typical data in Figure 3. The solutions for pure A and pure B are needed to evaluate A,(m,(+ - 1)) for the intermediate y value^.^' Then
The Journal of Physical Chemistry, Vol. 90, No. I, 1986 149
Electrolyte Mixtures at Equilibrium
TABLE III: First-Order Mixing Coefficients Calculated from the Model' I/m
-0.01
0.01
0
-0.01
0
Y I
I
1
I
I
I
(dl
(C)
-',4619tiJ -0.01
0.01
Y
Y 0
. 595 . 3 -
0 Y
-0.01
0.01
Figure 3. Extraction of the mixing coefficients from the computed data (circles). The functions plotted on the ordinate scales are (a) gA,/IRT, (b) gBy/IRT, (C) A,[mt(4 .- 1)1/Y(l - V)p,and (d) (gAy each as a function of y at given I , here 0.0001 m . The lines determined by least-squares fit are gAy/IRT= 2.777 + 0.236Y, gBy/IRT = -0.813 + 0.29823 A,,,[mt(@- l ) ] / y ( l - y ) p = -1.46187 -k o . o ~ o o land ~ , (gAy - gBy)/IRT = -3.590 + 0.062Y. TABLE 11: Zeroth-Order Mixing Coefficients Calculated from the Model' Ilm 0.000005 0.00001 0.00005 0.0001 0.0003 0.0006 0.001 0.002 0.0025 0.0025 0.003 0.004 0.007 0.01 0.02 0.03 0.05 0.07 0.1 0.15 0.3 0.5 1 .o
1000a0, e~ 2.9 -8939 -5772 -3275 -2658 -1855 -1492 -1260 -988 -91 1 (-9 15) -851 -76 1 -603 -556 -372 -285 -198 -147 -96 -46 23 64 110
1000b0,
eg 2.10 3917 2384 -4852 -663 -629 -55 1 -48 1 -345 -307 (-287) -275 -228 -132 -34 -37 2 2 -7 -18 -37 -89 -129 -181
1ooow,, eg 2.11 -2640 -2352 -1718 -1461 -1080 -859 -707 -520 -465 (-466) -422 -358 -249 -183 -100 -67 -39 -28 -22 -20 -23 -28 -37
1ooog,,
1ooog,,
eg 2.22
eg 2.18
-2382 -1035 -2042 -1861 -1403 -1 185 -1033 -814 -753 (-736) -704 -63 1 -486 -407 -309 -215 -156 -126 -92 -63 -43 -36 -33
-2 145 -2296 -20 18 -1794 -1416 -1 183 -1018 -805 -74 1 (-737) -690 -612 -472 -385 -261 -199 -138 -107 -82 -6 1 -4 1 -35 -34
"The data in parentheses are explained in the text. the zeroth- and first-order mixing coefficients were calculated by least-squares fit of these data to eq 2.9-2.1 1 and 2.18 but neglecting all coefficients of second and higher order, as illustrated in Figure 3. The computed zeroth- and first-order mixing coefficients are given in Tables I1 and 111, respectively. In Table I1 the various columns for go are obtained from the data in various ways as indicated; the degree of their agreement shows the thermodynamic consistency, at this level, of the H N C pair correlation functions. The w I data in Table I11 and Figure 4 show the expected (in view of Figure 1 and eq 2.20) peak in w,(Z) to be rather sharply located at 0.003 m. By integration of the w I data according to eq 2.20 (with the help of a cubic spline fit routine IMSL ICSICU), we obtain the g, results also shown in Figure 4 as well as in Table 111. The degree of agreement of g,(Z) calculated in diverse ways (31) To avoid possible inconsistencies, we use the HNC program for mixtures to calculate pair correlation functions and osmotic coefficients for pure A and B. To do this for A, we set y = 0.5 with ion 1 2 ion 2 = Mn2* while to do this for B we set y = 0.5 with 1 2 = Li'.
0.000005 0.00001 0.00005 0.0001 0.0003 0.0006 0.001 0.002 0.0025 0.0025 0.003 0.004 0.007 0.01 0.02 0.03 0.05 0.07 0.1 0.15 0.3 0.5 1.o
1000al, 1000bl, lOOOw,, lOOOg,, lOOOg,, lOOOg,, eq 2.9 eq 2.10 eq 2.11 eq 2.22 eq 2.18 eq 2.20 398 -309 -134 -117 -157 -141 -131 -1 17 -111 (-1 10) -106 -98 -84 -67 -45 -43 -29 -2 1 -16 -1 1 -3 0 2
-336 894 141 149 191 180 162 159 152 (173) 146 135 124 111 67 73 51 38 32 28 13 5 0
2.3 6.7 7.4 10.1 12.2 13.9 14.6 15.0 15.1 (15.1) 15.1 14.7 13.6 12.2 9.3 7.1 5.1 3.8 2.4 1.6 0.3 -0.2 -0.8
60 577 -1 21 21 24 16 27 26 (49) 24 21 26 31 11 21 16 13 12 15 8 5 2
21 195 2 10 11 13 10 14 14 (14) 13 12 13 14 7 10 7 5 5 5 3 2 1
3.7 8.4 10.3 11.7 12.7 13.8 14.1 14.2 14.4 14.3 13.9 12.2 10.9 9.1 7.8 6.3 4.9 2.9 1.7 0.6
"The data in parentheses are explained in the text.
JI Figure 4. First-order mixing coefficients as functions of ionic strength: (a), w, data point; (-), to fit w1 data points; (---), gl calculated by
numerical integration of w,(l). (Table 111) provides a more subtle thermodynamic self-consistency test than the test in Table 11; it was at the first order that the earlier computations failed.I0 We see that g, calculated from w , by integrating eq 2.20 is in agreement with that calculated by eq 2.22 but not eq 2.18. The assumption that the coefficients with n 2 2 can be neglected is probably responsible for the discrepancy; it seems that the contribution of higher order (n > 1) mixing coefficients would need to be taken into account to get accurate values of g, from eq 2.18. To investigate this point, we calculated more y states with Z = 0.0025 m so the coefficients up through second order can be determined by a least-squares fit (Figure 5). The results for the zeroth- and first-order mixing coefficients are shown in parentheses in Tables I1 and 111, respectively. From both the theory and the graphs we can see that wo and w 1 are almost unaffected by omission of the higher order coefficients, while the effects of such omission on a , and bl are larger than on g, obtained according to eq 2.18. Also, the w, coefficients are the most stable of the coefficients with given order. This is probably due to the fact that the virial equation (unlike the compressibilityequation) has a direct contribution from the pair potentials while the zeroth-moment integrals that determine6 gAuand gBydepend for their accuracy
Lim et al.
150 The Journal of Physical Chemistry, Vol. 90, No. 1, 1986
Y 0.02 0.03
0.01
0
0.04
Ji Figure 6. w l ( f )in the small-f region, showing the approach to the limiting law. The change in the limiting law due to the inclusion of the i4b contribution is too small to show on the scale of this figure. - 0 45-
i’
-050‘
I 0
- 0.1
I
1
0.1
Y
-0.465
-0.466
-0.467 -0 1
0
0.1
Y
on the pair correlation functions alone. Figure 6 is a magnified part of Figure 5 to show that w1calculated by the HNC approximation approaches the limiting law at small I quite convincingly. In preliminary studies the mixing coefficients do and d l also were calculated from the correlation functions, but it was found that they are even less accurate than the corresponding a,, and b,, coefficients so they do not add significantly to our knowledge of the system.
7. Pair Interactions While we have verified the sharp peak in g l ( l ) at small I,which seems per se quite remarkable, we have not found a fully satisfactory molecular picture of this phenomenon. The limiting law for g, results from clusters of three or four ions interacting at great distances, as indicated by the fact that the cluster integrals in Figure 2 become infinite for K = 0; only the Debye shielding, weak though it is at small I,makes the integrals converge. On the other hand, the peak in g, can be thought of as being generated by deviation from the limiting law, and here short-range interactions, which are easier to visualize, come into play. To test this idea, we have studied the changes in w l ( r ) ,near its maximum, caused by changes in the model Gurney parameters. If a given such parameter A& is made more positive (or less negative) by O.lkBT, the peak in w1does change, more so if ab is a pair for which the other parameters are such that the change in &, gives a larger change in iiab(r),e.g., for pairs in which the mutual volume factor is larger. This is just the response expected if the deviations from the limiting law are indeed associated with short-range interactions in clusters of three (or four) ions. Another interesting question is whether the peak in g, may be associated with other dilute solution phenomena, even kinetic, in electrolyte mixtures. So we have examined the behavior of the pair correlation functions gab(r? as functions of I in the dilute range. For this purpose r’is chosen a little outside of the core repulsion region, namely r’ = 0.4 A. Because large effects are not expected in such dilute solutions, we focus on the reduced pair correlation function, at y = 0.5 and given I gab(r?
-0 1
0.1
0
Y
Figure 5. As in Figure 3, except that here f = 0.0025 m. The lines determined by least squares are (a) gAy/IRT = 1.024 + 0.224Y +
+
+
O.O07Y-?,(b) f n , / f R T = -0.460 0.305Y O.O63F, (c) A,[m,(@I)]/y(l - y ) p = - 0 . 4 6 5 7 O.O151Y+ O.O002p, and (d) (gAy--gB,)/ IRT = 1.484 - 0.081Y - 0.056p.
+
E
gab(r?/exp[-@aab*(r? + qab(r?l
(7.1)
which is the ratio of the full correlation function (obtained by HNC in this case) to the correlation function calculated by a Debye-Huckel approximation for the same pair potential. Some typical results displayed in Figure 7a,b show that gabmay have a maximum a t I values near 0.005 m, whether the maximum is found (or how large it is, if found) being very sensitive to the model pair potentials. While a systematic study of gab(r? has not been made, the maximum has only been found for cation-cation pairs in this system, and it changes only gradually if r’is changed over a moderate range.
The Journal of Physical Chemistry, Vol. 90, No. 1, 1986 151
Electrolyte Mixtures at Equilibrium
0.85' 0
.A
I 0.1
-
OO
Oil
Ji
Figure 7. (a, b) Ionic strength dependence of reduced pair correlation functions (see eq 5.1). The ordinate is gij for the Li', Mn2+ pair in (a) and for the Mn2+, Mn2+ pair in (b). In both cases curve I is for the model fitted to thermodynamicdatai0and used to calculate the data in Figures 3-6 while curve I1 is for the same model except that the Gurney parameter AMn,CI is increased by O.1RT. (c) The ordinate is gijfor the pair Ru(NH3),bpy2+, Ru(NH,),bpy3+ calculated from the electron-exchangerate data of Brown and S ~ t i n For . ~ ~curve I the medium is aqueous HC10, while for curve I1 it is aqueous CF,S03H, both at 25 "C.
It is interesting to see whether similar behavior may be found in some experimental systems. As an example, we choose the R U ( N H ~ ) ~ ( ~ electron-exchange ~ ~ ) ~ + / ~ + kinetics for which the I dependence of the rate constant, here denoted kz3,has been studied as a function of ionic strength in aqueous perchlorate and trifluoromethyl sulfonate media.32 The connection to the data in Figure 7a,b may be made by reference to the equation
where i23(r) is the local rate constant,33a strongly decreasing function of r which is insensitive to I . Also, the subscripts here and in the remainder of this section refer to the charges of the species rather than to the conventional species indices defined in reference to eq 1.1. Thus, gijis a pair correlation function connecting a i+ ion to a j+ ion. Since gz3(r)is a strongly increasing function of r in the relevant range, the integrand tends to be sharply peaked,33say at r = R , so we have34
where 6R is a measure of the width of the peak of the integrand in eq 7.2. Now, using a superscript to indicate the ionic strength, we may write
since the other factors are insensitive to ionic strength, at least in the low-I range of interest here. In view of eq 7.4 the experimental value of the reduced pair correlation function defined in eq 7.1 can be expressed as follows
(32) G . Brown and N. Sutin, J . Am. Chem. SOC.,101, 883 (1979) (Table 11). (33) B. L. Tembe, H. L. Friedman, and M. D. Newton, J . Chem. Phys., 76, 1490 (1982). (34) M. D. Newton and N. Sutin, Annu. Reu. Phys. Chem., 35, 437 (1984).
since z2z3= 6 in the present case. It may be recognized that this derivation is an alternative to the traditional theory of the Bransted-Livingston diagram of the kinetic salt effect.35 We apply it here to estimate k23(R)from the rate constant data of Brown and S ~ t i to n get ~ ~the results in Figure 7c. This figure, like their Figure 1 for the Idependence of the rate constants per se, shows the remarkable sensitivity of k23to whether the anion is C104- or CF3S03-. The qualitative similarity of part c to parts a and b (Figure 7) is striking. The change in k23(R) due to changing the anion from C104- to CF3S03-is qualitatively similar to the changes in i l 2 ( r 3 (for the Li', Mn2+ pair) and &+22(r')for the Mn2+, Mn2+ pair) when the model interaction of the C1- with the Mn2+is made a little less attractive (or more repulsive). Figure 7a,b also indicates that the sensitivity of k&') to the close interaction of the anion with a cation is strongly dependent on the charges i+ and j+ in a way that determines the trend in height of the maxima from a to b to c (Figure 7). At the very least these results provide a stimulus to the further study of a common-ion mixture of charge types z , = 3, z2 = 2, and z3 = -1 in the small-I range, possible with a third electrolyte, the common-ion acid, also present to suppress hydrolysis in the relevant experimental systems. The feasibility of model calculations using the H N C approximation applied to mixtures with four ionic species already was d e m ~ n s t r a t e d . ~It~ might be mentioned, however, that there are few if any experimental thermodynamic data for such mixtures with the charge types that are relevant here.
8. Conclusions The feasibility of using the H N C approximation to study models for mixed electrolytes at very low concentrations has been demonstrated. The H N C results for the zeroth- and first-order mixing m are accurate enough to be coefficients for I as small as useful, judging by thermodynamic self-consistency tests. The particular model studied here exhibits the remarkable maximum in the first-order mixing coefficient gl(Z) at low I which had been predicted on the basis of the thermodynamic data of Cassel and Wood.g While it is not easy to find a satisfactory molecular picture for the phenomena measured by the mixing coefficients, as others also have noticed,'J'J3 the maximum in gl(Z) is sensitive to (35) V. K. LeMer, Chem. Reu., 10, 179 (1932).
J . Phys. Chem. 1986, 90, 152-156
152
short-range interactions, as shown by direct “experiments” with the model calculation. Also, the Li+-Li+, Li+-Mn2+, and Mn2+-Mn2+pair correlation functions, which presumably strongly reflect three-particle configurations involving two cations and an anion, show markedly non-Debye-Hiickel ionic strength dependence. These results provide a tentative explanation for certain data for the rate constant of an electron-transfer reaction which otherwise would suggest complexes involving perchlorate ions in very dilute solutions. Perhaps equally important, this study provides specific examples that contradict the tempting assumption that the ionic strength dependence of a mixing coefficient can be represented by a power series in I .
Acknowledgment. The support of this research by the National Science Foundation is gratefully acknowledged, as is the computing facility made available by the computing center in Kuala Lumpur for a portion of this research. T.K.L. thanks Prof. Friedman for his warm hospitality during his sabbatical leave in Stony Brook. Appendix Concerning the h , data reported by Cassel and Wood,9 there is a sign discrepancy. According to theory,I0 h, and g, have opposite signs in the limiting law region, but according to Figure 1 the calculated g, and the experimental hl are both positive unless the latter function changes sign below the experimental I range.
This contradiction has not been resolved. Because the present study involves only the free energy coefficients,it is more important to know whether there is evidence of a corresponding sign discrepancy involving calculated and experimental g,. To investigate this question, we can use the exact equationI2
b, = g , + g2 + Y 2 k l l
+ Y2gd
(A. 1)
which, combined with the limiting laws for g , and go and the presumed negligibility of g2 compared to g , in the small-Z region, 1 as I 0. leads to the conclusion that b , / g , In ref 12 the model results for b , ( l ) are compared with experimental data for36CoC12-HC1 mixtures; qualitative agreement is found, not only for the sign of bl but also its Z dependence. Moreover, the same qualitative features are shown by bl measured3’ for A1Cl3-HC1 mixtures, as one can see by using the relation foundI2 between the coefficients reported by Khoo and co-workers and the coefficients employed here. Finally, LimI3 has applied a different data reduction method to the emf data for several HBr-MBr2 mixtures and concluded that they give positive g , in the small-Z region.
- -
(36) K. H. Khoo, T. K. Lim, and C. Y . Chan, J . Solufion Chem., 10,683 (1978), (37) K. H. Khoo, T. K. Lim, and C. Y . Chan, Trans. Faraday SOC.,74, 2037 (1981).
CHEMICAL KINETICS Fourier Transform Infrared Spectroscopic Study of the Thermal Stability of Peroxyacetyl Nitrate G. I. Senum,* R. Fajer, and J. S. Gaffneyt Environmental Chemistry Division, Department of Applied Science, Brookhaven National Laboratory, Upton. New York 1 1 973 (Received: December 14, 1984: I n Final Form: September 10, 1985)
The unimolecular decomposition of peroxyacetyl nitrate (PAN) to form methyl nitrate and carbon dioxide has been studied over the temperature range 298-338 K by Fourier transform infrared spectroscopy (FTIR). Pure PAN samples (>98%; 2-30 torr) were found to decompose thermally, nearly quantitatively, to these products, with the only other observed product being nitromethane (< 10%). Both PAN decomposition and methyl nitrate formation rates yielded the Arrhenius equation, k = (2.1 X 10”) exp(-24.800 1800 cal/(K mol/RT)) s-’. Perdeuterio-PAN was synthesized and observed to decompose to perdeuteriomethyl nitrate and carbon dioxide with no apparent isotope effect. Experiments adding nitric oxide to the PAN are described which confirm the existence of two unimolecular pathways for the decomposition of peroxyacyl nitrates. One pathway is a concerted reaction most likely proceeding via a cyclic intermediate, and the other the previously identified equilibrium between PAN and the peroxyacetyl radical and nitrogen dioxide. The rate of the PAN decomposition in a large excess of NO to form the peroxyacetyl radical and nitrogen dioxide was determined to be 2.2 X s-l at 298 K. This rate is somewhat slower than previously reported for this pathway. The results are discussed with respect to the atmospheric lifetime of PAN and the potential atmospheric production of methyl nitrate.
*
Introduction Peroxyacyl nitrates (PANS) are known to be produced during the photochemical oxidation of organic molecules in the troposphere, and the gas-phase chemistry of peroxyacetyl nitrate (PAN) has been investigated by a number of workers.14 Measurements of PAN in the troposphere have found PAN to have a widespread occurrence in the clean troposphere as well as in polluted air and indicate that P A N can be an important reservoir for NO, and ‘Present address: Los Alamos National Laboratory, Group INC-7, MSJ519, Los Alamos, N M 87545.
0022-3654/86/2090-0152$01.50/0
a measure of the photochemical age of an air parceL5s6 To better evaluate the impacts of this known phytotoxin on ecosystems and (1) C. T. Pate, R. Atkinson, and J. N. Pitts, Jr., Enuiron. Sci. HealthEnuiron. Sci. Eng., A l l , 19-31 (1976). (2) R. A. Cox and M. J. Roffey, Enuiron. Sci. Technol., 11, 900-906
(1977). (3) D. G. Hendry and R. A. Kenley, J . Am. Chem. SOC.,99, 3198-3199 (1977). (4) U. Schurath and V. Wipprecht, “Proceeding of the First European Symposium on Physico-Chemical Behavior of Atmospheric Pollutants”, Ispra, Italy, Oct 1979, B. Versino and H. Ott, Ed., EUR 6621, 1980, pp 157-166.
0 1986 American Chemical Society
