128
GORDON ATKINSON AND S. K. KOR
The Kinetics of Ion Association in Manganese Sulfate Solutions. I. Results in Water, Dioxane-Water Mixtures, and Methanol-Water Mixtures at 25”
by Gordon Atkinson and S. K. Kor Department of Chemistry, University of Maryland, College Park, Maryland
(Received June 23, 1964)
Ultrasonic absorption has been measured in water, dioxane-water, and methanol-water solutions of MnS04 a t 25”. The three-peak relaxation spectrum is interpreted by the three-step association between Mn+2(aq) and S04-z(aq) proposed by Eigen. All six rate constants for the three-step process are calculated by a straightforward method. Step I is diffusion controlled with a forward rate constant of 4.2 X 10’0 M-’set.-' in water while the step I11 forward rate (4.8X lo’ sec.-’) seems controlled by the rate of exchange of solvent on the Mn+2 ion. The over-all equilibrium constant including the three ionpair states is shown to be the same as the KA determined by conductance both in water and the mixed solvents.
The classic model for electrolyte solution theories has been the continuum solvent model of Debye and Onsager. The great power of this approach in giving the low concentration equilibrium and nonequilibrium properties of solutions cannot be overlooked. For associated electrolytes, the continuum solvent models of Bjerruni and Fuoss have given much insight into electrolyte behavior. Yet in recent years, it has become more and more apparent that the model was inadequate to describe many elect>rolyte systems. Corrections and additions that take a more specific and molecular view of the solvent have been either mathematically intractable or of very limited service. In some recent articles, FUOSS’ has pointed up some cases where a molecular approach to the solvent seemed necessary. A few years ago Gilkerson2 developed a more general approach to association to explain specific solvent effects. Ramsey3 and Hyne4 have also emphasized specific solvent effects in conductance behavior. In a very careful analysis of KCl conductance data, PetruccP has demonstrated that a simple Stokes law approach is inadequate even for this uncomplicated salt. And in a series of papers6 we have pointed out the great inadequacies of the simple model for 2-2 salts such as MnS04 and manganese mbenzenedisulfonate. The Journal of Phyeical Chemistry
Besides these analyses from the viewpoint of classical solution measurements, other approaches and techniques have demonstrated the limitations of the continuum model. Frank and his co-workers have organized much data using the “structure-making” and “structure-breaking” concept.? The coordination chemists have consistently looked a t many ions as having definite numbers of solvent molecules around them in solution and have developed ways of measuring the numbers in a t least some ca~es.8.~ I n our recent work on the conductance of MnSOl and manganese m-benzenedisulfonate (MnBDS) in various
(1) R. M. FUOBS and A. D’Aprano, J. Phya. Chem., 67, 1704, 1722 (1963). (2) W. R. Gilkerson, J . Chem. Phys., 25, 1200 (1956). (3) J. B. Ramsey and H. K. Bodenseh, J. Phys. Chem., 67, 140 (1963). (4) J. B.Hyne, J . Am. Chem. Soc., 85, 304 (1963). (5) 5. Petrucci, Acta Chem. Scond., 16, 760 (1962). (6) G.Atkinson, et al., J. Am. Chem. Soc., 83, 3759 (1961); 84, 721 (1962);86,7 (1964). (7) H. S. Frank and W. Wen, Discusswns Faradag Soc., 24, 133 (1957). (8) H. Taube and J. P. Hunt, J. Chem. Phys., 19, 602 (1951). (9) R. E. Connick, “Advances in the Chemistry of the Coordination Compounds,” 5. Kirschner, Ed., Macmillan and Co.. New York. N. Y., 1961,p. 15.
KINETICS OF IONASSOCIATION I N RlnSOp SOLUTIONS
129
solvent mixtures, very specific solvent effects showed up the pulse technique which has proven to be the most in a number of ways. The classic log KA us. [1/D] plot accurate in the megacycle range.’S l9 The basic frequency range of the main apparatus is was not a single straight line for a given salt in any solvent but depended on the specific solvent mixture. The 1-60 Rlc. using a cell with a maximum path length of 1 , with “mean distance of closest approach,” a ~changed 111. For the range 60-250 Mc. a cell with a iiiaxiniuni solvent coniposit ion in sonie cases. Finally, the Walden path length of 5.08 cni. is used. The temperature of product, AOvo, exhibited specific dependence on solvent the solution being measured can be controlled to +0.05’ over the range 0-65’. The purification of 1lnSO4 composition. This led us to explore other techniques that would and the solvents have been described in the papers on give us a niore detailed view of ion-ion interaction in conductance previously referenced. solution. The most promising development in many The absorption coefficient o is defined by the equayears has been the array of relaxation methods detion I , = I , exp(-22az) where l o = intensity of sound veloped and utilized by Eigen and his co-workers.1° beam a t 0, I , = intensity of sound beam a t x, and x = Of the various methods used by Eigen, the most versacentimeters of path length. The relaxation spectra tile for electrolyte work seenied to be the ultrasonic are usefully plotted as [ah]*,the absorption per wave absorption technique. The excellent analyses of this length in the solution niinus the absorption per wave technique presented elsewhere” l 2 niake it unnecessary length in the solvent, us. frequency. to go into it in detail here. Figure 1 shows the relaxation spectra of MnS04 in The particular system that we chose to examine was water a t three different salt concentrations. It is quite MnS04 ion association. I n this system we had acapparent that the peak a t approximately 30 l l c . curate conductance data in water and in a variety of reported by SL exists. Figure 2 shows the effect of niixed solvent systems. Furthermore, its ultrasonic increasing dioxane content on the relaxation spectra of relaxation spectrum was measurable with a single type of apparatus. Three previous groups of workers “I had reported various detailed experiments in ultrasonic Mn SO, absorption in 11nS04 solutions. Eigen, Kurtze, and Water as Solvent I O I M Sol Tanim (EKT) had measured absorption over a wide 2 005M Sol frequency range as part of a very extensive pr~graiii.’~ Sniithson and Litovitz14 (SL) had made some very I accurate measurements in H20, D20,and HZO-CH3OH mixtures over a more limited frequency range. Vernia -1 and Kori5 (VK) had niade measurements in H 2 0 and 2 water-dioxane iiiixtures over a still more limited frequency range. There were sonie curious inconsistencies in the three sets of data. E I i T report relaxation peaks a t approxiniately 3 and 200 1Ic. in 0.1 M aqueous RInSO4 while SL also report a distinct peak at about 30 IO IS 20 30 40 SO 100 IIc. The effect of methanol noted by SL seemed quite - - ” I 1 1 . 5 2 3 4 5 Frequency Mcs different from the effect of dioxane nieasured by VK. Figure 1. Furthermore, the effect of temperature on relaxation frequency seemed very different for SL and for VI(. I n view of these inconsistencies we decided to nieas(10) M. Eigen and L. De Maeyer, “Technique of Organic Chemistry,” Vol. VIII, Interscience Publishers, New York, N. Y.,Part 2 , ure ultrasonic absorption as a function of concentration Chapter 18. in aqueous MnS04and in dioxane-water and niethanol(11) G. S. l‘erma, Rec. Mod. Phys., 31, 1052 (1959). water mixtures. The best theoretical analysis of ultra(12) It. T. Beyer. et al., ibid., 23, 353 (1951). sonic absorption in electrolytes is that given by Eigen (13) 11. Eigen, G. Kurtze, and K. Tarnm, 2. Elektrochem., 5 7 , 103 (1953). and and we have based our interpretation of (14) J. li. Sniithson and T. A. Litovitr, J . A r o i d . Soc. Am., 28, 462 data on their general approach. (1956).
.J
Experimental The equipment used for the ultrasorlic l~~easurements has been described in detail elsewhere.” It’is based on
(15) S. K. Kor and G. S. Verma, .I. Chem. Phvs.. 29, 9 (1958). (18) 11.Eigen aiid K. Tamm, Z . Elektrochem.. 6 6 , 93, 107 (1962). (17) G. Atkinson, S. K. Kor, aiid It. L. Jones, Rer. Sci. Instr., in press. (18) J. It. I’ellam and K. J. Galt, J . Acoitst. Soc. Am.. 18,251 (1948). (19) J. M . M .I’inkerton, Proc. P h p . SOC. ( L o d m , ~ 6 2 , 2 8 (1949). 8
Voliime 69. .Ytrmhrr 1 J a n u a r y 1965
GORDON ATKISSOX9 S D s. K. KOR
130
I
I
I
I
~
I
1
I l l
I
I
l
l
l
l
l
ABSORPTION OF MnSO, I N D I O X A N E - WATER 0 0 5 MOLAR
2o
t
1
association process as postulated by Eigen and Tanim. Eigen has shown that other equilibria leading to relaxation peaks can be eliminated from consideration experimentally or theoretically. Various hydrolysis equilibria
Table I : Ultrasonic Absorption of MnSOc Solutions a t 250a
3
15 %
Dioxane
0
25% 35%
II
a
1
,
,
,
2
3
4
ii
II
, , I I , l 5 6
8
1
FREQUENCY
I
20
IO
Mc
30
1
YIII*?
Mc.
Mo.
0.01 0.05 0.10
32 34 35
Water 2.7 3.1 3.3
0.01 0.05
I
I t :b 1
1
1
'
1
I 1
I
I
I
I
I
I
I
I
1
2o
0 IO
0 3 4
104
15% (w./w.) dioxane 3.5 48.0 58,8 4.2 4.5 160.0
26.0 73.0 140.0
0.10
34 36 37
25% (w./w.) dioxane 3.6 55.0 4.4 127,O 4.8 187.0
26.5 73.7 145.0
0.01 0.05 0.10
35 37 38
35% (w./w.) dioxane 3.9 73.5 4.65 132.0 5.1 218.0
26.5 75.3 160.0
0.01
157, (w./w.) methanol 34 2.4 36.0 37.5 3.2 103.0
29.3 202.0
0.01
25% (w./w.) methanol
Water
i
25% M a O H 35% M e O H
-i
2 t
1 1 1 2 3 4
x
33 35 36
0.10
\
[ax1111*
28.0 87.5 160.0
0.05
ABSORPTlON OF MnSO, IN METHANOL-WATER 005 MOLAR
[aAl~i* x 104
31.2 42.5 100.0
0.10
1
t
VII*>
l l l l l 70 90
40 50
Figure 2.
30
Conon., M
1
I 1 1 1 1 1
5 6
8 1 0
20
1
I
30 40
I
I 60
I I I I
80
FREQUENCY ( Mc
Figure 3.
one concentration of JlnSOd. Although the 30-Mc. peak is lower in amplitude than the 3-1112. peak in water, it rapidly becomes higher as dioxane is added. Figure 3 shows the analogous effect upon adding methanol. Table I suiiiniarizes the relaxation frequencies and absorption niaxinia measured in this work. The frequencies were calculated from the measured excess absorption values using the Jlikhailov technique.20
Data Treatment Since measurements of the 200-Me. peak were not available to us until very late in the work we have used the data of Kurtze and Tanin121for this peak. The existence of three peaks (approximately 3, 30, and 200 )IC,) is very strong evidence for a three-step The Joztrnal of Physical Chemistry
0.05 0.10 0.01
0.05
38.5
74.2 120.0
110.0 202.0
357, (w./w.) methanol 37 1.9 38.7 39.5 2.5 105,O
35.0 137.0
40
2.6 2.9
' Y X I * = frequency of 2nd absorption maximum; y I I I * = frequency of 3rd absorption maximum; ( a X ) r I * = absorption per wave length a t 2nd maximum (excess); ( ~ ) I I I *= absorption per wave length a t 3rd maximum (excess).
are easily eliminated by examination of pH dependence. Structural relaxation of the solvent itself must occur at a frequency much higher than 200 ;\IC.,and ion-atmosphere relaxation is too small to account for the peaks in this frequency range. Therefore, we have assumed a three-step associat,ion equilibrium between ;\Inf2 and SO4-* as a working hypothesis. (20) I. G . Mikhailov, Dokl. Akad. .Vauk SSSR, 89, 991 (1953). (21) G. Kurtse and K. Tamm, Acustica, 4 , 380 (1953).
KINETICSOF IONASSOCIATION IN MnS04 SOLUTIONS
0
I
State @
+ S04-2(solvated) E [Mn+2S04-21 kll
Mnf2(solvated)
kai
Ikn
131
3. Activity coefficients can be ignored for states 2, 3, and 4. With these assumpt’ions we can define a quantity 8 [ C ] as
11
ksr
[Mn+2SO4-2]
[Mn+2S04-2] krs
0
and write
0
I11
Step I involves the approach of two completely solvated ions. Steps I1 and I11 are most probably the successive removal of solvent molecules from the region between the ions giving, finally, a contact ion pair. For such a three-step process Eigen has shown that the relationships between the relaxation times and the rate constants are given by the equation
where
~/TII
=
27~vm11=
[ + ] kl2’
k32
k12’
k23
= k32
k,l
k43
+
k34’
+
k23’
(4)
where CY = degree of association in step I a t C; yk = mean activity coefficient of ions in state 1 a t concentration C; C = equilibrium concentration of MnS04; v m ( = frequency of maximum absorption for concentration C; T~ = relaxation time for step i. The more complete mathematical treatment using transformation matrices did not significantly alter the numerical results of this analysis. For data analysis we have made the following assumptions. 1. Step I can be described using the Fuoss-Bjerrum association model. 22 An equilibrium constant can be calculated for a given solvent system with the Bjerrum equation and a distance parameter equal to the sum of the ionic radii plus the diameter of two solvent molecules (the calculations are quite insensitive to the distance chosen). 2. The activity coefficient, yi.,can be calculated using the extended Debye-Huckel equation. Brubakerz3 has shown that this is quite good for an unassociated 2-2 salt. The measured activity coefficients for MnS04 include a contribution from association and cannot be used.
(7)
A small computer program is used to calculate 8 [C] for a given K l zand solvent. Plots of 2rvmtus. the quantities in brackets in eq. 6, 7, and 8 are straight lines yielding the rate constants for the given step as slope and intercept. The graphical method is very useful for smoothing the vm data. Water Results. Table I1 summarizes the rate constant results in water. The most significant features can be summarized as follows. (1) k12 agrees very well with that predicted for a diffusion-controlled rate.24 (2) k34 is very close to Connick’s best estimate of the rate of exchange of water in the first coordination sphere of R ! ~ I ( I I ) . This ~ ~ encourages our belief that the ratedetermining step in the association is the replacement of a solvent molecule in the first coordination sphere by the ligand. (3) All three “ion-pair” states exist in reasonable quantities but state 3 is definitely less stable. It is fruitful to compare the over-all constant for association
Kr.
=
CZ
+ C3 + C4 C12Y*2
=
139
with the association constant determined by conductK A = 133. This very close ance agreement, as well as the results noted in points 1 and 2 above, give us confidence in the general model chosen and in the method of data treatment. Mixed Solvent Results. The change of step I rates (k12 and k21) is what is expected from consideration of (22) E . g . , R. M.Fuoss and C . A. Kraus, J. Am. Chem. Soc., 7 9 , 3301 (1957). (23) C. H . Brubaker, Jr., and P. G . Rasmussen, J . Phya. Chem., 67, 330 (1963). (24) See ref. 10, pp. 1032, 1033. (25) See ref. 9, p. 17.
Volume 69, Number 1
Sanuarg 1966
GORDON ATKINSON AND S. K. KOR
132
A '
Table 11: Rate Constants for MnS04 Association in Water a t 25" k l n = 4.2 kzl = S.0
K,2 =
x x
1Olo C-I see.-' 108 see.-' 0.0192 l./mole
kzs = 6.9 X lo7 sec.-l kaz = 1.9 X lo8 see.-'
Kz3
CZ
= - =
1
RATES
1
A
A
I
the Debye equation for diff usion-controlled reactions. This reinforces our belief that the continuum model is perfectly adequate, as long as we are dealing with ions separated by two or more solvent molecules. However, in steps I1 and I11 we start seeing the individual behavior of the different solvent mixtures. Figure 4 shows the change of the step I1 rate constants with added methanol or added dioxane. It is apparent that methanol has no effect on the rates implying that the ions do not distinguish between water molecules and methanol molecules in this step. The dioxane drastically increases the forward rate and decreases the reverse. Figure 5 shows the effect of the added organic solvent on step 111. Here both solvents have markedly different effects on the rate constants. With results on just one salt, molecular interpretation seems premature. There is no particular reason for plotting the constants against mole fraction organic in Figures 4 and 5 . However, in step I11 if we consider k13 as a pseudo-first-order constant, X.,, = / C ~ ~ ~ [ H and ~ Ocalculate ], kr30 we find that the dioxane line and the methanol line coincide. This implitis a decided preference for water over the organic component by one (or both) of the ions in the pair. This reinforces King's recent results on the preference of Cr(II1) for water over methanol.26 In Figure 6 we have plotted the K , calculated from the ultrasonic data on the same plot as the K A values obtained froin conductance work. The two sets of data agree very well. The difference between the methanol line and the dioxane line noted in the conductance work is found to result from the very different effects of the two organic solvents on steps I1 arid I11 of the association process. We niight also point out that, at least in the case of -MnSO4,conductance includes all three ionpair states in the nonconducting group.
i
0 0 dioxane A A methonol
L . - 1 - '
c-3 = 0.29
Physical Chemistru
E
STEP
IO
2.8
C4
of
i
zW3
kS4 = 4.8 x l o 7 sec.-l k43 = 1.4 X lo7 sec.-l
T h e Journal
1
i
L\
I
c 3
K~~ =
I
~
05
x.
IO
20
15
Figure 4. i
I
I
STEP
I
IU
I
i
0 A A
RATES
I
dioxane methanol
-,
10
I
!
I
!
x.
IO
05
-20
15
Figure 5.
40
I
i
log K,
log ( I/KI) Comparison
methanol
3.
K,(A)
30
=
13 3
39
0
A
Ultrosoiics
I
0 A Conductance 15
20
25
(IOO/D)
Figure 6. ~
~~~~~~~
~
~
(26) E. L. King, et a!., Proceedings of the 7th International Coordination Chemistry Conference, Paper 6B1, 1962.
KINETICS OF IONASSOCIATION IN MnS04 SOLUTIONS
Summary The Eigen three-step mechanism for 2-2 salt association has been demonstrated to hold very accurately for MnS04, both in water and in dioxane-water and methanol-water mixtures. The first and fastest step is the diffusion-controlled interaction of ions with complete solvation spheres. The second and third steps most probably involve the successive removal of solvent molecules from between the ions, giving a contact ion pair in the final state. Only the first step can be adequately described using the classical continuum model. A theoretical description of the second and third steps niust await a theoretical model recognizing the discrete structure of the solvent. I n the final step the rate seeins determined by the rate of exchange of solvent molecules on the cation. The dependence of the reverse rate constant of this final step (k4J on the actual water concentration in the low organic content solvent mixtures examined implies a definite preference for a H20 molecule on the part of the Mn(I1) ion.
133
The over-all association constants calculated from the ultrasonic data agree very well with those previously obtained by conductance work. This means that for this salt, a t least, none of the three ion-pair states contributes to the electrical conductance. Furthermore, the individual effects of different added organic solvents, so inexplicable by classical electrolyte theories, are the result of distinctly different solvent effects on the individual steps of the association reaction. I n a succeeding paper we shall report the effect of temperature on the relaxation spectra and examine the energetics of the individual steps of the association process. Acknowledgment. The authors wish to acknowledge the generous support given this research by the Army Research Office (Durham) under Grant DAARO(D)-31124-G175. They also wish to acknowledge the aid of Dr. Hiroyuki Tsubota in the preparation, purification, and analysis of the materials used, and the help of A h . D. W. Ebdon in the computer programming.
Volume 69. Number 1 January 1965
