Monte Carlo Quasi-Classical Trajectory Study of CI
+ HBr
Corporation, by National Science Foundation Grant MPS7344656A01, and by the University of Toledo Research Award program.
References and Notes (1) M. Diem and D. F. Burow, J. Chem. Phys., 64, 5179 (1976). (2) B. A. Hudgens and D. F. Burow, Thirty-First Symposlum on Molecular Spectroscopy and Structure, Columbus, Ohio, 1976, Paper WD-9, to be published. (3) M. 2. ECSabban and B. Zwolinski, J. Mol Specfrosc.,19, 231 (1966). (4) H. C. Urey and C. A. Bradley, Phys. Rev., 38, 1969 (1931).
479 (5) T. Slmanoutl, J. Chem. Phys., 17, 245 (1949). (6) See, for example, E. B. Wilson, J. C. Declus, and P. C. Cross, “Molecular Vibrations”, McGraw-Hill, New York, N.Y., 1955, pp 169-17 1. (7) For theoretical justlflcatlon of this assumption, see, for example, L. S. Bartell and K. Kuchitsu, J. Chem. Phys., 37, 69 (1962). (8) J. H. SchacMschnelder, “Vibratlonal Analysis of Polyatomlc Molecules V and VI”, Report No. 231-64, Shell Development Co.,Emeryville, Callf. 94608, 1964. (9) J. Overend and J. R. Scherer, J. Chem. Phys., 33, 446 (1960). (10) L. H. Ngal and R. H. Mann, J. Mol Specfrosc., 38, 322 (1971). (11) J. Aldous and I. M. Mills,
[email protected], 19, 1567 (1963). (12) T. Simanouti, J Chem. Phys., 17, 848 (1949).
Monte Carlo Quasi-Classical Trajectory Study of CI on Reaction Rate and Product Energy
+ HBr.
Effect of Reactant Vibration
James Corgan Brown,+ H. E. Bass, Department of Physics and Astronomy, The University of Mlssissippi, University, Misslsslppi 38677
and Donald L. Thompson* Departments of Physics and Chemistry, The University of Mississippi, University, Mississippi 38677 and University of Californla, Los Alamos ScientificLaboratory, Los Alamos, New Mexico 87545 (Received October 4, 1976) Publication costs assisted by the University of Callfornia
Monte Carlo quasi-classicaltrajectories have been used to study the reactions C1+ HBr P HC1+ Br at 300 and 1000 K. A semiempirical valence-bond (London equation) formulism was used to represent the potential-energy surface. The surface was not adjusted for this reaction system, but was computed using parameters adjusted with kinetic information for related systems. The distributions of product molecules among vibrational and rotational states were computed and compared with experimentalresults (at 300 K); the agreement is good. The computed thermal rate coefficient is in good accord with the reported measured value. The effect of reactant vibrational energy on product distributions and reaction rate was investigated. The principle of detailed balancing in quasi-classical trajectories was also examined for a limited set of conditions for this system.
I. Introduction We present here the results of a quasi-classicaltrajectory study of the forward and reverse reactions C1+ HBr 7r! HC1+ Br. The main purposes of this study are to examine the influence of reactant, vibration, and translational energy on the reaction rate and product energy partitioning, and to test the principle of detailed balancing applied to quasi-classicaltrajectories. The potential-energy surface (described in detail in section 11) is of the semiempirical valence-bond formulism of Raff; Stivers, Porter, Thompson, and Sim5.l Since the surface is obtained from atomic parameters adjusted to give agreement of the energy barriers with the activation energies of other HBr + H and H2 + C1- HC1+ reactions2 (H2 + Br H), the first step in this work was to compute results that can be compared with experimental data to check the validity of the surface. Atom-diatomic molecule reactions involving hydrogen and halogen atoms have been studied extensively. Polanyi and ~ o - w o r k e r s ,using ~ infrared chemiluminescence techniques, have obtained extensive detailed information about the distribution of energy among internal states of products in exothermic reactions and about the effects of reactant vibration and rotation on endothermic reactions
-
Current address: Department of Chemistry, University of Toronto, Toronto, Ontario M5SlA1, Canada. On leave from the Los Alamos Scientific Laboratory, 1975-1976.
by application of the principle of microscopic reversibility. There have been several experimental studies of the reaction considered in this work. Maylotte, Polanyi, and W o ~ d a l have l ~ ~ reported results of infrared chemiluminescence measurements of reaction product energy distributions and reaction rates for this and similar reactions. Airey4has studied this system in relation to the C1+ HBr pulsed chemical laser. Wodarczyk and Moore6 have measured the reaction rate for this system when the reaction is initiated by the laser photolysis of C12 in a flowing C12/HX gas mixture. Bergmann and Moore6have studied the isotope effect on this reaction rate for the substitution of deuterium; they found that DBr reacts more slowly than HBr. There has also been a great deal of experimental data reported for related reactions. Cowley, Horne, and P01anyi~~ studied C1+ HI HC1+ I at enhanced collision energies using infrared chemiluminescence, and Polanyi and co-workers3efhave measured the effect of reactant vibration on the rate and product energy partitioning in F + HC1- HF + C1. Polanyi and ~ o - w o r k e r shave ~ ~ ? made ~ use of the principle of microscopic reversibility to extract information about endothermic reactions from experimental chemiluminescence data of the reverse, exothermic reactions. It is reasonable to question whether the results of quasiclassical trajectories obey the principle of detailed balancing since the forward and reverse processes are not the same in the calculations. This was the subject of a detailed
-
The Journal of Physlcai Chemistry, Vol 81, No. 5, 1977
480
J.
-
three-dimensional trajectory study for the case of F + H2 HF + H; the results indicated that microscopic reversibility would yield dependable information concerning the endothermic reaction, if the exothermic data referred to room temperature reagent^.^ Kuppermann and cow o r k e r ~ found * ~ ~ discrepancies between the forward and reverse reaction probabilities computed from collinear quasiclassical trajectories for F Hz(v = 0) P HF(v = 2,3) + H and H C Hz(v = 0) + H2(v= 1) H. Because of the limitations of available computer time we were not able to carry out extensive calculations to check detailed balance; we have, however, included in this work some examples to illustrate the degree to which the principle is obeyed for this reaction system.
+
+
C. Brown, H. E. Bass, and D. L. Thompson
TABLE I: Potential-Energy Surface Parameters Atom pair Param- eter HBra HCla BrCl ID, eV 3.91825 4.61534 2.275b au-I 0.95878 0.95848 1.0373' 3 D , eV 1.3668678 1.34956 1.51172 p, au-l 0.8360 0.79 0.32402 R e , au 2.673 2.41 3.117 20d C, eV au 12312.0 17634.0 12434.72 A, au - 2.9774 - 2.88770 - 2.83518 a, au-I 2.366 2.561 2.196 R,, au 3.7 3.6 3.4 From ref 2. b From ref 11. C Com uted from data reported by Clyne and Coxon, ref 12. 2 From ref 13. 01,
I , ,
11. Potential-Energy Surface The potential-energy surface was obtained from the semiempirical formulation of Raff et al.' This nonionic, valence-bond formalism, which treats the systems as consisting of three electrons moving in the potential of a proton and the nonpolarizable Br+ and C1+ cores, contains atomic parameters for each of the halogens. We make the assumption that these atomic parameters are the same for the atoms regardless of the chemical environment in which they are considered. The analytical form of the potential is the London equation:
V = QAB +
+
(JBC
+ QAC - JAB - JBc)' - JAC)2 + (JAC - J A B ) 2 1 )"'
=
+
I
3
5
7
9
II
Flgure 1. Contour map of the potential-energy surface for collinear collision paths of CIHBr. Contour energies are in units of kcal/mol.
(3)
'0,{1- exp[-aij(Rij -
- 'Dij
(4)
and the triplet-state energies are given by the Pedersen-PorterlO functions:
30,{1 + exp[-@,(R, - Re,)] }2 - 3D, for Rij 5 RCijand =
3E, = C,(R,
+ A,) exp(-a,Rij)
(5)
'
= we ( 2r CE.1BrCl/Deh)
'
(6)
(7)
where pBrCl is the reduced mass of BrCI, c is the speed of light, De is the dissociation energy (from ref 15), h is Planck's constant, and we (= 440 cm-l) is the fundamental vibrational frequency reported by Clyne and Coxon.12 The The Journal of Physical Chemistv, Vol. 8 I, No. 5, 1977
equilibrium internuclear separation Re was taken from the semiempirical calculations of Pohl and Raff.13 The remaining BrCl parameters were computed by the method described by Raff et al.' In this formulation the potential-energy surface is parameterized through the electron core integral where 4xis the appropriate atomic orbital. The value of the integral fx, the adjustable parameter in this formulation, was obtained by adjusting the barrier in the minimum potential-energy reaction pathway of X t H, - H X + H
for Rij > Rcij. The values of the atom-pair potential parameters are listed in Table I. The parameters for HC1 and HBr were taken from ref 2. The singlet-state Parameters for BrCl were obtained as follows: lD was taken from the compilation given by Herzberg.ll The Morse exponential constant a was computed from (Y
I
l / ~ [ l ~ i j3 ~ , 1
Overlap terms are neglected in eq 1-3. The singlet-state energies are given by the Morse function
3&,
, -106.08
RHCS hu.)
J, = v 2[ ' E , ~- 3 ~ , 1
=
Br
(1)
and
'E,
H
QBC
where Qij and Jijare the Coulomb and exchange integrals, respectively. They are defined in terms of the singlet- (lEij) and triplet-state (3Eij)energies using the Heitler-London equations:
Q,
CL
(X=ClorBr)
to be in slight excess of the experimental activation energyS2 These atomic parameters are assumed to be transferable for descriptions of the forces in other reactions. The atomic parameter (Cx) values used are those computed by Porter et a1.2 Figure 1is a contour map of the potential-energysurface for collinear configurations of ClHBr. There is a slight well in the C1-HBr reactant valley. This well increases in depth slightly as the ClHBr angle goes to 90' as can be seen by examining the contour map for this configuration in Figure 2. 111. Computational Methods The basic methodology and equations for the classical mechanical problem have been described in detail by
Monte Carlo Quasi-Classical Trajectory Study of CI II
+ HBr b
=
[1(1 t l)]"'(h/2n~Vp)
The vibrational phase was selected using the method of Porter, Raff, and Miller.l8 The orientation angles B,4, and y, and the internal states (when averaged over) were selected in the usual way.17 The value of the reaction-shell radius R, was fixed at 10 au; at this separation the interaction of the atom with the diatomic molecule is negligible. The vibrational state u was held constant in most of the calculations and the rotational state J was selected by playing the Monte Carlo game.17 The rotational energy of the product molecule is computed using
9
-7 ?
0
& I:
u5
E,' = L"/2,ure2 3
-9224
\
- 69 IS
(12) L', the angular momentum of the diatom, can be computed exactly. In eq 12 p and re are the reduced mass and equilibrium separation, respectively, of the diatom. The final state rotational quantum number J' is given by the closest integer solution of
E,
= B[J'(J'
+ l)]- D [ J ' ( J ' + l)] '
(13)
where B and D are constants.'l The product vibrational energy E,/ is given approximately by
E,'
= E,,'
- E,'
(14)
where E,' is the total internal energy of diatom; E,' can be computed exactly. In using eq 14 it is assumed that the vibrational-rotational coupling energy is small relative to the vibrational energy. The product vibrational quantum state u' is then the closest integer solution of
+
E,' =
(VI
where
we and wex,
1h)ue
-
(u'
+
(15)
1/2)2aexe
are ~0nstants.l~
IV. Results and Discussion We have carried out a series of quasi-classical trajectory calculations for the reactions C1+ HBr + Br + HC1. One of the objectives of this study was to compare the relative effectiveness of reactant energy in different vibrational states for promoting reaction. Also of interest is the testing of detailed balancing in quasi-classical trajectory calculations; these comparisons are made at 300 and 1000 K with the HC1 or HBr molecule initially in the u = 1,2, 3, or 4 vibrational state. The distribution of energy among the various product molecule modes was also examined. The reaction rates and product energy distributions are compared with experiment. The thermal rate coefficient k for C1+ HBr HCl + Br has been measured by Wodarczyk and Moore5 at 295 K by the laser photolysis of C12in a flowing C12/HBr gas mixture; they obtained k = 4.5 X 10l2cm3/molecules. The value obtained in these trajectory calculation is 30.2 X 10l2 cm3/molecule s. This comparison indicates that the potential-energy surface used in this work is a reasonable one. It should be noted that this surface was not calibrated using kinetic data for this reaction. Arnoldi, Kaufmann, and Wolfrumlghave measured the effect of reactant vibration on Br + HC1 HBr C1. They report that at room temperature that exciting HCl(u = 0) to u = 2 gives a rate enhancement of loll. We did not compute the u = 0 rate because of the very low reaction probability (based on the Arnoldi et al.19 results a rate coefficient of about 10 cm3/molecule s is estimated). We can, however, compare our computed rate coefficient of (5.6 f 3.2) X lo1' cm3/molecule s at 300 K for HCl(u = 2) with the experimental valuelgfor u = 2 of (8.8 f 3.1) X loll
-
-
+
The Journal of Physlcal Chemlstw, Vol. 8 1, No. 5, 1977
482
J. C. Brown, H. E. Bass, and D. L. Thompson
v’= 2
6O C
I
4 0 1
3 0.5
0.2
O
u O
2
l
3
I
V’
-
I
1
1
-1I
+
Figure 3. Computed distribution of final vibrational states for CI HBr HCI(v’) Br at 300 and 1000 K; initial vibrational and rotational states thermally distributed.
+
c . 4
-I
n
->
4-
a 0.4
W
z
0.5
0.3 ‘0
U
5
113
/rl
i0
j5
30
J‘
I z0.2
Figure 5. Computed distribution of final rotational states for CI 4- HBr HCI(v’, J’) 4- Br at 300 K; initial vibrational and rotational states thermally distributed. The points are the computed results: the solid line is a smooth curve fltted to the points.
0.I
+
Flgure 4. Computed distribution of flnal Vibrational states for CI 4- HBr(v = 1) HCI(v’) Br at 300 K; initial rotational states thermally distributed. -+
. .
+
cm3/molecule s. The agreement is good. The distribution of product vibrational states has been HC1 Br by measured for the reaction C1 HBr Maylotte, Polanyi, and W ~ o d a l using l ~ ~ an “arrested relaxation” variant of the infrared chemiluminescence technique. They found N(u’ = 2)/N(u’ = 1) = 0.4 with a reactant gas temperature of approximately 300 K. [Here, N(u’ = i) is the number of product molecules in the ith vibrational state.] Our calculations gave N(u’ = 2)/N(u’ = 1) = 0.38 at 300 K. This close agreement again supports our choice of the potential-energy surface. The overall distributions of HC1 molecules among the final vibrational states are shown in Figure 3 at 300 and lo00 K, with initial u and J selected from thermal distributions; N(u? is the number of molecules formed in the u’ vibrational state. There is a definite temperature effect on the product vibrational distributions. Particularly noticeable is the shift in the relative populations of the u’ = 0 and u’ = 2 states; a t both temperatures approximately half of the product molecules are in the u’ = 1 state. The effect of initial vibration on the product vibrational distribution is shown in Figure 4 for the case of C1+ HBr(u = 1) HCl(u9 + Br at 300 K. For these conditions the
+
-
-
The Journal of Physical Chemistty. Vol. 8 1, No. 5, 1077
+
final vibrational states 1, 2, and 3 are about equally populated, in contrast to the peaking that occurs under thermal conditions as shown in Figure 3. The rotational states of the product molecules were computed by solving eq 13 for J. The distribution of products among rotational states at 300 K is shown in Figure 5 for final vibrational states (u’) equal to 0, 1, and 2. Due to the large number of product rotational states available, the statistical fluctuations of the data points in Figure 5 are large. A solid line was drawn through the points by taking the average between the value for two successiveJ states and then visually drawing a line through the points so obtained. The curves should be considered only qualitatively correct. Maylotte, Polanyi, and WoodallMmeasured the most probable rotational quantum for u’ = 1 to be 13 f 1 compared to 12-13 number (3’) found here and for u’ = 2 to be 3 f 1compared to 9 found here. Figure 6 shows the effect of increasing reactant temperature on the rotational state distribution. The increased temperature shifts the rotational distribution to higher J’values but the most dramatic effect is the decreased breadth of the J’distribution. In Figure 7 comparisons of the partitioning of the energy available to products ET among rotation ER,vibration E”, and translation ETrare illustrated for various conditions. The reaction probability is also given in Figure 7 for each set of conditions. There is not a significant change in the partitioning of the available energy for the processes C1
Monte Carlo Quasi-Classical Trajectory Study of CI I
I
I
+ HBr
483
TABLE 11: Computed Rate Coefficients for the Reaction Br + HCl(u) -+ HBr + C1 for 300 and 1000 K
I
I (C)
0.8
v’.
2
T.K
u
300
1 2 3 4 0
1000
1 2 3 4
r
a
0.56 * 0.32 2.12 0.94 2.96 * 0.92 b 0.56 * 0.56 4.34 i 1.52 3.88 i 1.88 1.06 i 1.74
a Only 150 trajectories were computed for u = 1, T = 300 K ; no reactions occurred. No reactions were observed out of the 250 trajectories computed for v = 0,
1 v‘.
k(1013 cm’/molecule s)
T = 1000 K.
I
1000 O
I .o
K
a 0.6
3z
0
-
I
I
. I
I
I
3.0x 16‘-
(a) 0
v’9
2.0
16’-
1.0 x
5
0
IO
20
IS
J’
:I 25
m
O
+
1 , 2
3
V
+
30
+
J - I r
bi-44-i---
ET ET ET
ET ET ET
ET ET ET
ET ET ET
Flgure 7. Computed partitioning of the energy available to the products in CI HBr HCI Br for various condfiions. The reaction probability P, is given for each case: (a) 300 K, vand Jthermally distributed; (b) 300 K, v = 1, and Jthermally distrlbuted; (c) 1000 K, vand Jthermally distributed; and (d) = 0.65 eV, v and J thermally distributed.
+
kL
O O
+
12-r
--+
3
-+
0.2
+
2
Figure 8. Computed product vibrational state distributions as a function of initial vibrational state at 300 and 1000 K for the reaction Br HCI(v) CI -I- HBr( v’).
0.3
u E’ E I E
1
VI
Figure 6. Computed distribution of final rotational states for CI HBr HCI(v! J’) Br at 1000 K; initial vibrational and rotational states thermallv distributed. The Doint 3re the computed results: the solid line is a-smoc curve fitted to e points. +
0
HBr (u and J thermal) and C1 + HBr (u = 1 and J thermal) as can be seen by comparing Figures 7a and 7b, however, the reaction probability is doubled. Changing the temperature from 300 to lo00 K produces a change in the partitioning, see Figures 7a and 7c, and about a 30% increase in the reaction probability. A dramatic change in energy partitioning is brought about by fixing the relative translational energy at 0.65 eV (which corresponds to approximately the energy difference in the u = 0 and u = 1 states of HBr) as illustrated in Figure 7d in which the internal states are thermally distributed for 300 K. It is interesting to note that the increased translational energy causes a decrease in the reaction probability. The rate coefficient and the distribution of energy among various product modes with HC1 in the vibrational states u = 0, 1,2,3, and 4 at 300 and 1000 K were computed for the reaction Br + HC1. The calculated rate coefficients are given in Table 11. The distributions of product vibrational states for different initial reactant vibrational states are shown in Figure 8 for T = 300 and 1000 K. In these calculations, increasing the initial vibrational energy shifts the distribution of product molecules t~ higher vibrational states. The fraction of the total available energy in the vibrational modes of the products also increases with u at 1000 K but does not at 300 K. The Journal of fhyslcal Chemistry, Vol 81, No. 5, 1977
404
J. C. Brown, H. E. Bass, and D. L. Thompson 1000 'K
IL
'r
I .o
'I
300' K
CP + H B r
1
- E: E,;
E ';
ET
ET
ET
v=3
v=2
V'I
v=4
i
300 O K
'i
'1 v =2
-
v.3
v:4
Figure 9. Computed partitioning of available energy among product modes as a function of initial vibrational state at 300 and 1000 K for the reaction Br HCI( v ) CI HBr(v').
+
+
The partitioning of the available energy from the Br + HCl(u) reactions for u = 1, 2 , 3 , and 4 is shown in Figure 9 for the temperatures 300 and 1000 K. At both temperatures, changes in u cause significant changes in energy partitioning. At 300 K, vibration is the dominant mode for deposition of the available energy for u = 2 and 3 with translation becoming more important for u = 4. However, at 1000 K vibration is more important than rotation and translation for only u = 4, being essentially comparable to translation for u = 3. Perhaps, the most interesting facet of these results is the dramatic shift in the relative importance of rotation and translation as u goes from 2 to 4. Because of the combination of quantum mechanical rules (for assigning initial states) and classical mechanics (for computing the motion and thus assigning the final states) it is reasonable to question the validity of detailed balancing in quasi-classical trajectory calculations. Since we have computed rate coefficients for the forward and reverse reactions
0
-
where AG = AH - TAS is Gibb's free energy. If we make the assumption that the entropy change AS is negligible then we can write
-
Let us consider the process C1+ HBr(u = 0) Br + HCl(u = 1). The heat of reaction is -16.5 kcal/mol for C1 + HBr(u = 0) Br + HCl(u = 0), and thus for the process we are considering the equilibrium constant computed from K = exp(-AH/RT) gives 54.8 at 1000 K. The trajectory computed forward rate coefficient is 5.12 X 1013 cm3/molecule s and the reverse 5.6 X 10l2cm3/molecule The Journal of Physical Chemistry, Vol 81, No. 5, 1977
1
~
~
'
'
'
~
1
IOOO" K CP + H B r
0.6
i
0.5 LT
a
0.3
o'21
71
0. I
we have the information to check detailed balancing in this case, A completely unequivocal statement on detailed balancing in quasi-classical trajectories would, however, require results of greater statistical accuracy than the computer time available for the present work permits. We should be able to detect a gross violation of the principle with the present results. The equilibrium constant is
-
120 144 168
96
+
~
kr
kr exp(-AHIRT)
72
+
kf
=
48
Figure 10. Computed differenital scattering cross section for CI HBr HCI Br at 300 K. The angle 8 is the angle between the final molecule velocity and the initial atom velocity.
C1+ HBr t Br t HCl
kf
24
SCATTERING ANGLE ( 8 / DEGREES
"
0
-
24 48 72 96 120 144 168 SCATTERING ANGLE ( 8 ) / DEGREES
Flgure 11. Computed differential scattering cross section for CI 4- HBr HCI Br at 1000 K. The angle 8 is the angle between the final molecule velocity and the initial atom velocity.
+
s. We obtain from these results = 9.2. Product angular scattering distributions were also computed for thermal distributions at 300 (see Figure 10) and lo00 K (Figure 11)for C1+ HBr HC1+ Br. Plotted in Figures 10 and 11 are the differential scattering cross sections (I(e)/sin e), where 0 is the angle made by the product molecule velocity and the initial reactant atom velocity. Therefore small 0 values correspond to scattering into the forward hemisphere. There are, to our knowledge, no published results for the scattering distributions for this reaction. There are results (unpublished) by McDonald and HershbachZ0for scattering in C1 HI HC1 I, which strongly peaked in the forward direction for the two sets of conditions studied relative translational energies of about 5 kcal/mol and thermal collision energies for 400 K. As can be seen in Figures 10 and 11 the trajectory results predict definite forward scattering for C1 HBr
-
+
-
+
+
~
1
~
-
Tracer and Mutual Diffusion in Electrolyte Solutions
HC1+ Br at 300 and 1000 K. We are not certain how close an analogy there is between the C1+ HBr and C1+ HI systems, though it does seem reasonable to expect qualitative similarities in the scattering. Trajectory calculations on C1+ HI HC1+ I using a surface of the same formulation as used here gave forward scattering a t temperatures of 1000 and 2000 K, but isotropic to backward scattering at 300 K.21 Acknowledgment. D.L.T. wishes to express his graditude to the Departments of Physics and Chemistry at Ole Miss for their hospitality during the course of this work. This work was supported in part by the US. Energy Research and Development Administration.
-
References and Notes (1) L. M. Raff, L. Stlvers, R. N. Porter, D. L. Thompson, and L. B. Sims, J. Chem. Phys., 52, 3449 (1970). (2) R. N. Porter, L. 8. Slms, D. L. Thompson, and L. M. Raff, J. Chem. Phys., 58, 2855 (1973). (3) See, for example, (a) K. G. Anlauf, J. C. Polanyl, W. H. Wong, and K. B. Woodall, J. Chem. Phys., 49,5189 (1988);(b) K. 0. Anlauf, D. H. Maylotte, J. C. Pdanyl, and R. 8. Bemstein, bkl,51,5716(1969); (c) L. T. Cowley, D. S.Horne, and J. C. Polanyl, Chem. Phys. Lett, 12,144 (1971);(d) D. H. Maylotte, J. C. Polanyl, and K. B. Woodall, J. Chem. Phys., 57,1547 (1972);(e) L. J. Kirsch and J. C. Polanyi, ibM., 57, 4498 (1972);(f) A. M. G. Ding, L. J. Kirsch, D. S.Perry, J. C. Polanyi, and J. L. Schreiber, Faraday Discuss., Chem. Soc., 55, 252 (1973). (4) J. R. Airey, J Chem. Phys., 52, 156 (1970).
485 (5) F. J. Wodarczyk and C. B. Moore, Chem Phys Lett, 26,484(1974). (6) K. Bergmann and C. B. Moore, J. Chem. Phys., 63,843 (1975). (7)D. S.Perry, J. C. Polanyi, and C. W. Wilson, Jr., Chem. Phys. Lett., 24,484 (1974). (8) J. M. Bowman and A. Kuppermann, Chem Phys Lett, 19,166 (1973). (9) J. M. Bowman, G. C. Schatz, and A. Kuppermann, Chem Phys. Lett, 24,378 (1974). (10) L. Pedersen and R. N. Porter, J. Chem. Phys., 47,4751 (1967). (11) 0. Herzberg, “Molecular Specira and Molecular Structure. I. Spectra of Diatomic Molecules”, 2nd ed, D. Van Nostrand, New York, N.Y., 1950. (12) M. A. A. Clyne and J. A. Coxon, Roc. R. Soc (London),Sec A, 298, 424 (1967). (13) H. A. Pohl and L. M. Raff, Inf. J. Quanfum Chem., 1, 577 (1967). (14) M. Karplus, R. N. Porter, and R. D. Sharma, J. Chem Phys, 43,3259 (1965). (15) S.Gill, Proc. Cambridge Phi/. Soc., 47,96 (1951). (16) H. H. Suzukawa, Jr., D. L. Thompson, V. B. Chang, and M. Wdfsberg, J. Chem. Phys., 59,4000 (1973). (17) L. M. Raff, D. L. Thompson, L. B. Sims, and R. N. Porter, J. Chem. Phys., 56, 5998 (1972).For a comprehensive description of Monte
Carlo methods for quasi-classicaltrajectorles, see R. N. Porter and L. M. Raff, “Classlcal Trajectory Methods in Molecular Collisions”, in “Modem Theoretical Chemistry, Volume 111. Dynamics of Molecular Collisions”, W. H. Miller, Ed., Plenum Press, New York, N.Y., 1970. (18)R. N. Porter, L. M. Raff, and W. H. Miller, J. Chem. Phys, 63,2214
(1975). (19) D. Arnoldi, K. Kaufmann, and J. Wolfrum, Phys. Rev. Lett., 34,1597 (1975). (20) J. D. McDonald and D. R. Hershbach, “Molecular Beam Kinetlcs: Reaction of Chlorine Atoms with Hydrogen Iodide”, unpublished; J. D. McDonald, Ph.D. Thesis, Harvard University, 1971. (21) H. E. Bass, L. S. Kenton, and D. L. Thompson, Chem. Phys. Lett.,
44, 452 (1976).
Coupling between Tracer and Mutual Diffusion in Electrolyte Solutions Pierre Turq,” Marius Chemla, Habib Latrous, and Jaiel M’haliat Laboratoire d Elecfrochimle, ERA 3 IO, Universik Pierre et Marle Curie, 4, Place Jusslerc 75230 Paris Cedex 05, France (Received December 1, 1975; Revised Manuscript Received November 2, 1976) Publication costs assisted by Universik Pierre et Marie Curie
The coupling between tracer and mutual diffusion in electrolyte solutions is experimentally realized and theoretically analyzed. The essential coupling terms are the electrical diffusion potential gradient, and the activity coefficient gradient, arising from the individual excess chemical potential of each ion.
I. Introduction Self-diffusion processes are among the most simple transport phenomena occurring in a given medium and are generally measured by special techniques using isotopically or spin-labeled compounds. On the other hand, the phenomenon of mutual or chemical diffusion, which can be easily observed experimentally, has been studied by many classical techniques. However its complete analysis involves a detailed consideration of several complicated effects. The relationship between the chemical diffusion coefficient of an electrolyte and the separate ionic self-diffusion coefficients is simple at infinite dilution (Nernst-Hartley relation) but becomes more complex for finite values of the concentration. The equation describing these two phenomena requires different polynomes, and the terms beyond the limiting laws in C1fzwere calculated only a few years a g ~ . l - ~ The problem considered in the present study is the interdependence of tracer diffusion and of mutual diffusion in electrolyte solutions when these processes occur simultaneously.
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W i t h the technical coworkship of Yvette Roumegous.
Self- and mutual-diffusion processes are closely related in the following ways: (a) They are both consequences of the thermal motion of the particles and are expressed by the square-average displacement by unit time. (b) In an electrolyte solution, the value of the self-diffusion coefficient depends essentially on the ion-solvent interaction; other interactions give only a small contribution when the solution concentration is below 1 M. Therefore, the tracer diffusion coefficient in the presence of a concentration gradient of electrolyte will depend not only on the local and instantaneous values of the concentration in the solution, but also on the fields of forces involved in the electrolyte diffusion. In this case, we shall have together a tracer and an electrolyte flow which can occur in different directions. We shall assume that a steady state will be locally established after a very short time; this means that the potentials and the corresponding fields take well-defined values at any place and any time. Also, in each elementary volume of the solution, an electrolyteflow and a tracer flow (of same or opposite direction) occur simultaneously. The tracer diffusion process, which has only a small dependence on the electrolyte concentration, occurs even in the presence of a countercurrent flow of electrolyte. In fact, The Journal of Physical Chemistry, Vol. 8 1, NO. 5, 1977
