4238
J. Phys. Chem. 1982, 86, 4238-4244
Cesium-I 33 Nuclear Magnetic Resonance and Electrical Conductance Study of Ion Association of Cesium Salts in Methylamine Sadegh Khazaell, Alexander I. Popov, and James L. Dye' Department of Chemistry. Michigan State Un/versity,East Lansing, Michigan 48824 (Received: March 7 7, 1982; In final form: June 22, 7982)
Ion-ion and ion-solvent interactions of cesium salts in methylamine solutions were investigated by cesium-133 NMR and by electrical conductance measurements. The concentration and temperature dependence of the 133Cschemical shift for cesium salts in methylamine was fitted by a model involving ion pairs and triple ions. The average formation constants for ion pairing, Kip,at 25 "C calculated from the NMR data are (2.6 f 0.4) x 105 and (1.3 f 0.4) X lo4 M-' for cesium iodide and cesium tetraphenylborate, respectively;the corresponding enthalpies of association are 3.6 f 0.4 and 4.0 f 0.9 kcal mol-'. The value of Kip for CsI from conductance measurements depended on the conductance equation used. The Onsager limiting law gave the best fit with (Kip)cBI = (3.8 f 0.5) X lo4 M-' at -15.7 "C which, however, is only 35% of the value obtained from NMR data at this temperature.
Introduction The concept of ion-pair formation was introduced by Bjerrum in 1926.' Since that time, various properties of ion pairs have been studied, primarily in solvents of high or medium dielectric constant, but ion association in solvents with dielectric constants lower than 15 is complicated by the formation of higher aggregates. The formation of triple ions in low-dielectric media was invoked by Fuoss and Kraus many years ago? but only in the last few years has much attention been given to the nature of triple ions and higher ion clusters in s o l u t i ~ n . ~ - ~ We present here a study of ion association of cesium salts in methylamine by 133CsNMR spectroscopy. The sensitivity of 133CsNMR is relatively high, compared to other alkali nuclei, and the natural line widths are small (6l l
d ,
126 0
04
1.2
0.8
[Cs'] Flgure 1. Conductance cell,
wide-band, variable-temperature probe. A small probe which contained paramagnetically doped water was placed 1.5-2.5 cm from the sample, to serve as an external l o ~ k . ' ~ J ~ A Nicolet 1080 computer, coupled to a Diablo magnetic disk system, was used to carry out spectral averaging and Fourier transformation of the data. The chemical shifts were initially referred to a 0.7 M aqueous cesium bromide solution which was sealed in a 5-mm NMR tube and coaxially mounted in a 10-mm tube. The space between the two tubes was evacuated and vacuum sealed.1° The reference chemical shift was then corrected to infinitely dilute aqueous solutions. Chemical shifts were also corrected for differences in the bulk diamagnetic susceptibilities of the nonaqueous solvent and water'& as appropriate for Fourier transform NMR.'4b A downfield (paramagnetic or deshielding) chemical shift from the reference is positive. Resistance measurements were made at 400, 600, lK, 2K, and 4K Hz with a conductance bridge and oscilloscope null dete~tor.'~For resistances greater than 90 000 Q,the cell was connected in parallel with a 90000-Q standard precision resistor. The Erlenmeyer-type conductance cell (Figure 1) was thermostated within f0.04 "C in an ethylene glycol bath. The cell constant of 1.0213 f 0.0004 cm-' was determined in the resistance range of the samples at 25.0 "C by using aqueous potassium chloride solutions and applying the conductance equation of Barthel et al.16 The measured resistances were corrected for both irreversibility at the electrodes and the capacitance bypass effect by using the equation R,,, = Ro af2 + b / f f 2in which R,,, and Ro are the measured and corrected resistances, f is the frequency, and a and b are adjustable parameters. All adjustable parameters described in this paper were obtained by fitting the appropriate equations to the data with the weighted nonlinear least-squares program K I N F I T ~ ~ 'on a CDC-6500 or CDC-7501 computer. Listed uncertainties are marginal standard deviation estimates and therefore include effects due to "coupling" of the parameters.
+
Results and Discussion I. Cesium-133NMR Data for Cesium Salts in Methylamine. The concentration and temperature dependence (12) Traficanta, D. D.; Simms, J. A.; Mulcay, M. J.Magn. Reson. 1974, 15,484. (13) Wright, D. Ph.D. Thesis, Michigan State University, East Lansing, MI, 1974. (14) (a) Live, D. H.; Chan, S.I. A w l . Chem. 1970,42,791. (b) Martin, M. L.; Delpuech, J.-J.; Martin, C. J. "Practical NMR Spectroscopy"; Heyden: London, 1980. (15) Thompson, H. B.; Rogers, M. T. Reu. Sci. Instrum. 1966,27,1079. (16) Barthel, J.; Feuerlien, F.; Neueder, R.; Wachter, R. J. Solution Chem. 1980,9, 209. (17) Nicely, V. A.; Dye, J. L. J. Chem. Educ. 1971, 48, 443.
, \ ,
x
25O0c
1.6 (M)
2.0
1
2.4
Flgure 2. Concentration dependence of the 133Cschemical shift of cesium iodide in methylamine at 25.0 and -15.7 "C. The s o l i curves were calculated according to assumption I11 of Table I .
-10-
250%
-5-
13 2% 5 Boc
0-
5-29oc
30
15
d
:9 50 4
20
o
p
/
04
06
:
25 60 00
301 3
00
02
[C"]
02
04
5
0.6
08
X IO2
(MI
1.0
1.2
1
[Cs+]x IO2 (M)
Flgure 3. Concentration dependence of the 133Cschemical shift of cesium tetraphenylborate in methylamine at various temperatures. The points containing error bars were obtained by extrapolation of mole ratio studies of the complexation of CsBPh, by 18-crown-6 in methylamine; the solid curves were calculated according to assumption 111 of Table I . Inset: Concentration dependence of the '%s chemical shift of cesium tetraphenylborate in 90% v/v methyiamine in dimethyl sulfoxide at 25 "C.
of the cesium-133 chemical shifts of cesium iodide and cesium tetraphenylborate was examined and the results are given in Figures 2 and 3. For cesium iodide solutions, in addition to the data shown in Figure 2, chemical shifts at 13.0, 6.2, -2.0, -9.5, and -32.0 "C were also measured over the concentration range 2.8 X 10-3-2.4 X M. A total of 73 data points were used to calculate the thermodynamic parameters for this salt. For cesium iodide solutions at all temperatures an increase in the concentration of the salt resulted in a paramagnetic shift, indicating that the cesium cation interacts more strongly with the iodide ion than with the solvent. Conversely, a diamagnetic shift occurs for cesium tetraphenylborate. Because of the limited solubility of cesium tetraphenylborate in methylamine, the chemical shift of this salt at high concentrations was determined by adding 18-crown-6 to the salt solutions. Since the formation constant of the resulting 1:l complex is larger than 104,18 ~~
~
(la) Khazaeli, S.;Popov, A. I.; Dye, J. L. J. Phys. Chem., in press.
4240
Khazaeli et al.
The Journal of Physical Chemistfy, Vol. 86,No. 21, 1982 --I 62
ao
-
68 -
sat.
csuo, CsSCN
70 72 -20
-10
0
IO
20
t("c)
Figure 4. Temperature dependence of the 133Cschemical shift of cesium thiocyanate in methylamine. Each point Is the average of six to eight determinations over the concentration range 9.4 X 104-1.8 x 10-2 M.
a plot of the chemical shift vs. the mole ratio (18-crown6)/(CsPh4B)is linear in the range 0 < mole ratio < 1. This permitted the chemical shift of the uncomplexed salt at concentrations above the saturation limit to be obtained by extrapolation of the data in the presence of 18-crown-6 to zero mole ratio. For solutions of cesium thiocyanate the chemical shift is independent of concentration at each temperature over M). the concentration range studied (9.4 X 104-1.8 X The plot of the average chemical shift at each temperature (6,) vs. temperature ( t )is linear, as shown in Figure 4, and follows the equation 6, = 60.73-0.224(t - 25.0) (1) To permit comparison of the results for various salts, we have plotted the concentration dependences at 25 "C in Figure 5. The chemical shifts of cesium bromide, cesium perchlorate, and cesium nitrate shown in Figure 5 could only be studied at saturation because of the low solubilities of these salts in methylamine. The chemical shift of the solvated cesium cation in methylamine at infinite dilution must be independent of the counterion. It is obvious from Figure 5 that substantial ion association occurs even at the lowest concentration studied. Data accumulation times of 8-10 h per sample were required at these concentrations so that it was not feasible to study more dilute solutions. From the concentration dependence of the 133Cschemical shifts of cesium iodide and cesium tetraphenylborate and the requirement that these salts and cesium thiocyanate have a common intercept, it was possible to determine the ion-pair formation constants of the former two salts. However, the long extrapolation of these data to infinite dilution made numerical calculations very difficult, and a number of attempts were made to obtain the chemical shift of the solvated cesium cation independently. Although these attempts did not give the chemical shift of the free cation directly, the data contain some information about ion association which will be discussed in section 11. It was impossible to obtain the ion association constant for cesium thiocyanate from the NMR data because the chemical shift was found to be independent of the salt concentration. Three possibilities exist for this salt: (1) Cesium thiocyanate may not form ion pairs even at the highest concentration. The behavior of the other salts in this solvent makes this highly unlikely. In addition, the '=Cs line widths are larger for this salt than for the other salts, which is indicative of ion-ion interactions. (2)
Flgure 5. Concentration dependence of the 133Cschemical shift of cesium salts in methylamine at 25 OC; the dashed lines are the curves calculated from the simple ion-pair model for CsBPh, and CsI. S o l i curves are calculated from the full model.
Cesium thiocyanate may be completely ion paired even at very low concentrations. If this is the case, the chemical shift might change at concentrations which are too low to be studied. (3) The chemical shifts of the solvated cesium cation and cesium thiocyanate ion pairs might be nearly the same. This is plausible since both thiocyanate and methylamine are nitrogen donors. Because of the concentration independence of the chemical shift of CsSCN at all temperatures and its behavior upon complexation by 18-crow1-6,~~ the third assumption formed the basis for our choice of the chemical shift of Cs+ at infinite dilution. Various models were used in an attempt to analyze the concentration dependence of the 133Cschemical shifts of cesium iodide and cesium tetraphenylborate in methylamine solutions. We first tried to fit the chemical shift data with a simple model in which only ion pairs form according to the following equations: Kip
c s + + x- P cs+.x-
(
y+ = exp -
(2)
)
(4.19764 X 106)(Ca)'/2 (4) ( ~ ~ ) 3 / 2 [+ 1 50.299(Cc~)~~~/(Dnl/~i bobsd
= XCS+6CS+ + xcs+.x-6cs+.x'
(5)
in which Cs+ and X- are the solvated cation and the solvated anion, respectively, Cs+-X-is the ion pair, the terms in parentheses are the molar concentrations of various species, K . is the ion-pair association constant, CY is the degree of a)issociation of the ion pair, y* is the mean activity coefficient of the salt in solution, C is the analytical concentration of the salt, D is the dielectric constant of methylamine: T i s the temperature in Kelvin, 9 = 5.3 8, is the distance of the closest approach, doM is the measured chemical shift, bCs+ and 6cst.x- are the chemical shifts of the free solvated cation and the ion-paired cation, respectively, and Xcs+ and Xcs+.x-are the relative mole fractions of the free and ion-paired cations, respectively. The distance of the closest approach was chosen as the average of rcs++ rr-and rcs++ rI- + rs in which rcs+= 1.69 A and rr- = 2.16 8, are the Pauling crystal ionic radii of the
Ion Association of Cesium Salts in Methylamine
The Journal of Physical Chemistty, Vol. 86, No. 21, 1982 4241
cesium cation and the iodide anion, respectively, and re = 2.87 A is the van der Waals radius of the methylamine molecule. However, the exact choice of the distance parameter does not affect the results significantly. The cesium iodide and cesium tetraphenylborate NMR data at 25.0 "C were fitted simultaneously by the above equations. The calculated chemical shifts according to this model are shown as the dashed lines in Figure 5. This simple ion-pair model requires a sharp change in the chemical shift at low concentrations and a flat portion at high concentrations. Even the "best fit" of the data is very poor at all temperatures. Close examination of the experimental chemical shift-concentration plots (Figures 2 and 3) shows the existence of rapid changes of the chemical shift with concentration at low concentrations and slower but continuing changes at higher concentrations. These results indicate that other interactions are important in the solution and that a simple ion-pair model cannot describe the behavior of cesium salts in methylamine. It must be emphasized, however, that most of the changes in chemical shift with concentration can be accounted for on the basis of ion-pair formation. This means that the remaining experimental information needed to improve the model is relatively minor and we cannot expect to extract many additional equilibrium constants and limiting chemical shifts from the data. It should be noted that in solvents such as methanol, plots of 133Csand 87Rbchemical shifts of CsI and RbI vs. concentration are not simple; 19p20 i.e., the chemical shift also changes rapidly at lower concentrations and more slowly at high concentration. The origin of this behavior is not known. The next logical step invokes triple-ion formation. According to theory,21the maximum concentration, C, for which triple-ion formation by a univalent electrolyte in a solvent of dielectric constant D may be safely neglected is given by C, i= (1.19 X 10-14)(DT)3.In our case C, N 5 X 10-4M therefore, the formation of triple ions cannot be ignored even at the lowest temperatures and concentrations studied. Therefore, we analyzed the NMR data by considering the following equilibria (charges are omitted for simplicity):
-
cs + x
x K t a x.cs.x + cs.x
Kip
where X-Cs-X and Cs.X.Cs are anionic and cationic triple ions and Kt,and K , are formation constants for the anionic and cationic triple ions, respectively. The calculated concentrations of the various species were then used in the chemical shift equation: Jobsd
=
cxisj i
(7)
where Xi's and &'s are the mole fractions and chemical shifts of the species which contain the cesium cation. A complete solution to the above equations for both salts at various temperatures would require the adjustment of 19 parameters (six association constants, six enthalpies of (19) Li, Z., private communication. (20) Khazaeli, S.;Dye, J. L.; Popov, A. I.; Spectrochim. Acta, in press. (21) Fuoss, R. M.;Accascinia, F. 'Electrolyte Conductance-; WileyInterscience: New York, 1959.
formation, and seven chemical shifts) even if the temperature dependence of the chemical shifts were ignored. In our attempts to fit the data to much more restricted models, it became apparent that those parameters which are highly correlated with each other, and/or are characteristic of minor species in the solution, cannot be determined simultaneously. Therefore, assumptions had to be made in order to obtain any quantitative information about the system. Fortunately the ion-pair association constants, enthalpies of association, and chemical shifts of the ion pairs proved to be rather insensitive to the assumptions used to describe triple-ion formation. After a large number of trials22the following set of assumptions was used. (1) The concentration-independent chemical shift of cesium thiocyanate suggested that the chemical shifts of the solvated cation, the ion-paired cation, and triple ions are the same for this salt. The effect of complexation by 18-crown-6 in solutions of CsSCN in methylamine is in agreement with the assumption that Csf and Cs+.SCN- species have the same chemical shift.'* It should be mentioned, however, that the choice of 6cs+is important, since ion association constants are significantly affected by the choice of the chemical shift of the free cation. For example, a change in sa+ from 80 to 40 causes the ion-pair formation constant of cesium iodide to decrease by a factor of 4 while the ion-pair formation constant of cesium tetraphenylborate increases by the same factor. In summary, our choice of the chemical shift of the free cesium cation at a given temperature is given by eq 1. (2) The formation constants for cationic and anionic triple ions are the same. It was not possible to determine individual values for Kt,and K , since they are highly correlated with each other and with the ion-pair formation constant. Less than 10% of the cesium cations are present in triple ions even at the highest concentrations studied. Fortunately, the values of ion-pair formation constants, enthalpies of their formation, and chemical shifts of the ion pairs are not very sensitive to the choice of formation constants for triple ions. Therefore, the assumption that the two triple-ion formation constants are the same is acceptable even though the degrees of solvation of the cation and the anion are undoubtedly different. (3) The formation constants for triple ions were not independently adjusted. Since triple ions are minor species and the ion-pair parameters do not change significantly with changes in the triple-ion formation constants, it is possible to fix the formation constants of the triple ions at "reasonable" values with the understanding that the resulting chemical shift parameters for the triple ions will depend upon this choice. Two procedures were used: (a) the formation constants of triple ions at different temperatures were calculated from the Fuoss equation2 where a3 = 5.3 A was used as the distance of closest approach of the ion pair to the other ion, and b3 = e2/ (aJIkT), in which e is the charge on an electron, and k is the Boltzmann constant; (b) independent of the model used, the value of the ion-pair formation constant for cesium iodide at 25.0 OC was almost 5 times larger than the Fuoss value,23while it was nearly 5 times smaller than the Fuoss value for cesium tetraphenylborate. If we assume that the ratio of the ion-pair and triple-ion formation (22) Khazaeli, S. Ph.D. Thesis, Michigan State University, East Lansing, MI, 1982. (23) Fuoss, R. M.; Kraus, C. A. J. Am. Chem. SOC.1933,55, 1919.
4242
The Journal of Physical Chemistty, Vol. 86, No. 27, 1982
Khazaeli et al.
TABLE I : Thermodynamic Parameters for Ion Association of Cesium Iodide and Cesium Tetraphenylborate in Methylamine at 25 "C Obtained from NMR Data according to Various Assumptions assumptions
AH"ip, kcal mol-'
Kip, M-'
salt
(I) 1, 2, 3a, 4
CsI CsBPh, (11) 1, 2, 3b, qb CsI CsBPh, (111) 1, 2, 3a, 4, 5 CsI CsBPh,
(2.54 f 0.31)X l o 5 (1.41 i: 0.30)X lo4 (2.68 f 0 . 1 9 ) ~l o 5 (1.16 f 0.15)X l o 4 (2.50 i: 0.22) x l o 5 (1.21 i 0 . 2 1 ) ~l o 4
3.86 f 4.71 f 3.75 i: 4.09 i: 3.15 f 3.31 i
a Average standard deviation of chemical shifts. and was used as a constant.
0.28 0.40 0.11 0.15 0.21 0.79
37.67 f 0.97 34.8 i: 1.4 37.41 i: 0.39 32.31 f 0.56 35.26 f 0.73 29.8 i 2.7
6csx
8x.cSx
bcsx,
ppm deg-l
124.19 f 0.31 458 t 24 0.39 -13.1 i 4.2 -298 i 42 124.05 f 0.20 299 t. 1.2 0.30 -17.8 i: 3.2 - 9 0 6 i 51 124.47 f 0.23 469 i 1 9 0.020 i 0.002 0.27 -16.1 i 3.7 - 2 8 0 f 27 -0.16 0.09
The ratio A = Kt/Kip = 6.8 X
constants calculated from the Fuoss equation is correct, we can write Kt = AKip where A = (Kt/Kip)Fuoes. This assumption suggests that any effects which are missing in the calculation of the ion-pair formation constant according to the Fuoss theory are also missing in the calculation of the triple-ion formation constant. Again, it should be emphasized that some such assumption is forced upon us by the inability of the chemical shift data to define all of the unknown parameters. (4) The chemical shift of the cationic triple ion and the ion pair are the same. The nearly complete correlation between the chemical shifts of the two kinds of triple ions prevents their simultaneous determination. Because they are formed in equal concentrations, the chemical shift of one could go down by an arbitrary amount provided that it is balanced by a corresponding increase in the chemical shift of the other triple ion. We must conclude that the chemical shifts of the triple ions cannot be determined from the NMR data. However, fixing the chemical shift of one of the triple ions at an arbitrary value leads to a reasonably well-determined value for the chemical shift of the other triple ion. (5) The chemical shifts of the ion pairs are linearly dependent upon temperature. Unfortunately, there is no quantitative theory for the temperature dependence of the chemical shift. The CsSCN data show linear dependence of the chemical shift on temperature (eq 1). Therefore, it was assumed that the chemical shifts of the solvated cesium cation and the ion pairs are also linearly dependent on temperature (6, = 6 b(t - 25)). Including this assumption gave a somewhat better fit of the data but had little effect on the ion association parameters. The calculated thermodynamic parameters for the ion association of cesium iodide and cesium tetraphenylborate obtained according to the above-mentioned assumptions are given in Table I. Inspection of Table I shows that the parameters for the formation of ion pairs are well determined and they are independent (within their standard deviations) of the model used. One obvious deficiency is that the chemical shifta obtained for the anionic triple ions are very different from the values that one expects on the basis of a simple overlap model. The addition of a second anion has a greater effect on the chemical shift of Cs+ than does the first. Changing the chemical shift of Cs+.X-Cs+ from that of the ion pair toward that of Cs+ only makes matters worse. In view of the arbitrariness involved in the adjustment of triple-ion formation constants and chemical shifts, speculation about the origin of this effect does not seem warranted. It should be noted, however, that the chemical shift of a 0.02 M cesium iodide solution in methylamine in the presence of saturated tetraphenylphosphonium iodide at this concentration is 129.38 ppm compared to a value of 127.06 ppm in the absence of added iodide. Similarly the addition of tetraphenylarsonium tetraphenylborate to a 0.0032 M cesium tetraphenylborate solution in methylamine resulted in a 0.38-ppm upfield
+
ASoip, eu
*
was calculated from Fuoss equations
shift at 25.0 "C. These chemical shift changes are in the direction expected for the formation of additional triple ions. The chemical shift of a 0.02 M cesium thiocyanate solution in methylamine which is saturated with tetraphenylphosphonium thiocyanate was not different from that of pure cesium thiocyanate. However, the results are inconclusive because of the very limited solubility of tetraphenylphosphenium thiocyanate in methylamine. It should be emphasized that not any collection of parameters can fit the model; the model chosen must fit the concentration dependence of the chemical shifts of CsI and CsBPh4 not only at a given temperature but at all temperatures studied. While the data do not permit an unambiguous choice of all possible parameters, the model used is a reasonable one-theory predicts that ion pairs and triple ions should form in this solvent. The model chosen is compatible with the data, provides well-determined thermodynamic parameters and chemical shifts for ion pairs, and at least provides a description of the behavior at higher concentrations. According to our model, at the lowest concentrations studied ( ~ X 5lo4 M), 90% and 55%, respectively, of the cesium cations in cesium iodide and cesium tetraphenylborate solutions are ion paired at room temperature. At the highest concentrations (-0.01-0.02 M, where the complexation of cesium salts by macrocyclic ligands is usually studied) only 3% and 25% of the cesium cations in cesium iodide and cesium tetraphenylborate, respectively, are free. Therefore, in methylamine solutions, ion association of cesium salts cannot be ignored in the study of the thermodynamics of the complexation of these salts by macrocyclic ligands. II. Cesium-133 NMR Data for Mixed Solvents and Mixtures of Salts. Since a long extrapolation of the chemical shift data of cesium salts in methylamine is required to obtain the chemical shift of the free solvated cations, attempts were made to extract this information independently. Even though these attempts failed, they provided some information which is in accord with our proposed model. (1)It was hoped that we could extrapolate the chemical shift-concentration data for cesium salts in mixed solvents (in which one of the solvents is methylamine) to infinite dilution. The chemical shifta of the cesium cation obtained in this way at various solvent compositions could then be extrapolated to obtain the chemical shift in pure methylamine. The chemical shift of cesium tetraphenylborate in 90% v/v methylamine in dimethyl sulfoxide (Me2SO) as a function of concentration at 25 "C is shown in the inset to Figure 3. Cesium tetraphenylborate does not form contact ion pairs in MezSO and the chemical shift of the cesium cation in this solvent is 68 The inset to Figure 3 shows that cesium tetraphenylborate remains (24) Dewitte, W. J.;Liu, L.; Mei, E.;Dye, J. L.; Popov, A. I. J.Solution Chem. 1977, 6,337.
-
Ion Association of Cesium Salts in Methylamine
The Journal of Physical Chemistry, Vol. 86,No. 21, 1982 4243
TABLE 11: Equivalent Conductance of Cesium Iodide Solutions in Methylamine at - 1 5 . 7 "C
M
A , a-1 cm-I equiv-I
U,A,~
M
cm-' equiv-I
3.522 9.022 15.42 22.87 31.02
99.2 70.7 59.2 52.3 47.3
1.7 1.4 0.97 0.65 0.38
49.28 73.34 103.3 142.0 204.5
39.23 33.48 29.137 25.715 22.189
105. (concn),
105. (concn),
A,
a-1 UAO
0.18 0.10
0.058 0.048 0.027
Calculated standard deviation in equivalent conductance. 0
0.2 0 4
06
08
IO
5Flgure 6. Chemical shift variation in mixtures of CsBPh, and CsI at total cesium concentrations of 0.0007 (0)and 0.003 (A)M. Solid lines - gCalcdwhere 6, is given by eq 9. Dashed lines are plots of, 6 have been corrected for ion-pair formation.
highly associated in this mixed solvent and that an extrapolation to infinite dilution would be difficult. Since the chemical shifts are about 40 ppm downfield from the corresponding values in pure methylamine solutions, they are dominated by the chemical shift of the cesium cation in Me2S0, indicating preferential solvation by Me2S0. (2) The chemical shift-concentration data shown in Figure 5 might suggest that cesium iodide and cesium tetraphenylborate form ion pairs in methylamine, while cesium thiocyanate does not. To examine this possibility, we measured chemical shifts of mixtures of cesium iodide and cesium thiocyanate in methylamine at a constant total concentration of the cesium cation but at various mole ratios of the two salts at 25 "C. If CsSCN were completely dissociated, the reaction Cs+ + I- s Cs+.I- would be shifted to the right and the observed chemical shift in mixtures would be downfield from that predicted by the equation (9) &bad = XC&CN6C~SCN + xCs16CsI in which 6csIis the chemical shift of pure cesium iodide solutions at the same concentrations as in the mixture. If, on the other hand, CsSCN is more strongly associated than CsI, eq 9 will be obeyed. The values of &bsd XC~CN~C - XCs16~eI, ~ S C Nat XcSI= 0.26,0.52, and 0.76 and at a total cesium salt concentration of 0.0054 M, are 1.04, -0.24, and 0.38, respectively. These differences are within experimental error of zero and indicate that CsSCN is ion paired to at least the same extent as CsI. (3) The chemical shift data shown in Figure 5 and the simultaneous fit of the data for cesium tetraphenylborate and cesium iodide to equations which describe ion-pair and triple-ion formation indicate that ion pairing is more pronounced in CsI solutions than in CsBPh, solutions. This suggests that the observed chemical shift in mixtures of these two salts would be downfield from that predicted by an equation similar to eq 9. Figure 6 shows plots of bo^ - XCsBp~6CsBph4- Xcs16csI vs. XcsIat two total cesium salt concentrations. The downfield curvature is clearly evident. The introduction of the ion-pair parameters corrects for more than 50% of the deviation. However, the exact correction would require the introduction of triple-ion parameters and consideration of complicated equilibria in which mixed triple ions also form. III. Electrical Conductance Measurements of Cesium Iodide i n Methylamine. If the ion-pair formation constants of cesium salts in methylamine could be independently determined, it might be possible to determine the thermodynamic parameters of the minor species from the NMR data. Ion-pair association of cesium salts (picrate, tetraphenylborate, and iodide) cannot be studied by
TABLE 111: Values of the Association Constant and Limiting Equivalent Conductance of Cesium Iodide in Methylamine at - 1 5 . 7 "C Obtained by Various Conductance Equations or from NMR Data Ao,
equation
K A ,M-'
limiting lawz6 (3.43 t 0.20) X l o 4 Pitts28+29 (2.4 t 1 . 0 ) X l o 4 F ~ o s s - H s a i ~ ~ (2.6 * ~ ' f 1.9)X lo4 Fuoss-Hsai(8.7 t 0.49) x l o 3 Chen30*32 Justice33.34 (1.4 * 2.4) x lo6 NMR 9.9 x 104
cm' equiv-'
7~~
161.0 t 4 . 2 148 i 25 159 t 44 97.6 i 3.1
0.076 0.35 0.55 0.40
922
0.90
t
890
a Average standard deviation in the equivalent conductance.
UV spectroscopy since the anion absorption band is either weak or insensitive to ion pairing with the cesium cation.% A usual approach is to obtain ion-pair formation constants by electrical conductivity measurements. To test the feasibility of this approach, we measured the conductivity of solutions of cesium iodide in methylamine at -15.7 "C. The conductance data are given in Table 11. The data were analyzed according to various conductance equations. The Onsager limiting law for weak electrolytes is expressed as26 A = .[A0 - S(CCY)~/'] (10) with KA = (1 - C Y ) / ( C C Y ~ ~Extended +~). conductance equations which add higher-order terms to the conductance equation have the general linearized formz7 A = A0 - S(CCY)'/~ + EC log (CCY)+ J1(Ca) KAh(Ca)7+'- J'(CCYY)~/~ (11) in which S is the coefficient of the limiting law, and E depends only on the properties of the solvent and the charge on the ions, while J1and J2 depend on the same parameters and also on the distance of closest approach of ions. The coefficients E , J1,and J2 have different values according to the particular theory employed. Among the conductance equations are Pitt's equation,28the FuossHsia equation30 (both linearized by Fernandez-PriniBbl) , the Fuoss-Hsia equation corrected by Chen (FHC),32and the Justice e q ~ a t i o n .The ~ ~ Justice ~ ~ ~ equation consists essentially of setting the distance parameter to the Bjerrum (25) Gilkerson, W. R., private communication. (26) Onsager,L. Phys. 2. 1927, 28, 277. (27) Fuoss, R. M.; Onsager, L. J.Phys. Chem. 1957, 61, 668. (28) Pith, E. Proc. R. SOC.London, Ser. A 1953, 217, 43. (29) Fernhdez-Prini, R.; Prue, J. E. Z. Phys. Chem. (Leipzig) 1965, 228, 373. (30) Fuoss, R. M.; Hsai, K. L. Proc. Natl. Acad. Sci., U.S.A. 1967,57, 1550.
(31) Fernhdez-Prini, R. Trans. Faraday SOC.1969, 65, 3311. (32) Chen, M. J . Phys. Chem. 1977,81, 2022. (33) Justice, J. C. J. Chim. Phys. Phys.-Chim. B i d . 1968, 65, 353. (34) Justice, J. C. Electrochim. Acta 1971, 16, 701.
J. Phys. Chem. 1982, 86, 4244-4256
4244
distance in the FHC equation. The results of the analysis of the conductance data according to the above equations are given in Table 111. For all calculations a value of 5.3 8, was selected as the distance parameter except for the Justice method in which the Bjerrum distance of 33.0 8, was used. Also the formation of triple ions was ignored since the mobilities of triple ions are not known. It is clear that the ion association constants obtained from these conductance measurements depend strongly on the conductance equation used and are different from the value obtained from the NMR data. Similar behavior was observed by G i l k e r ~ o nfor ~ ~ion association of cesium tetraphenylborate in acetonitrile, and of silver nitrate and lithium picrate in 2-butanone. The apparent reason for the differences in the ion association constants obtained from various conductance equations is that both the higher-order term in the conductance equation and the terms caused by association have the same concentration dependence in the first a p p r o ~ i m a t i o n .An ~ ~increase ~~~ in the sum ECa log (Ca)+ JICa can be compensated for by a decrease in the ion association constant. Equations which introduce higher-order terms into the conductivity equation produce smaller ion association constants for a given value of the distance parameter. An increase in the distance parameter causes J1to decrease and J2to increase. Both such changes are compensated for by an increase in the ion association constant. One expects association constants obtained from NMR and from conductance measurements to be essentially the same. The equilibria in the solutions are (35) Gilkerson, W. R.; Roberta, A. M. J. Am. Chem. Soc. 1980, 102, 5181. (36) Karl, D. J.; Dye, J. L. J. Phys. Chem. 1962, 66, 477. (37) Kay, R. L.; Dye, J . L. R o c . Natl. Acad. Sci. U.S.A. 1963,49, 5.
M+
+ X- --f K [M+.S.X-] & K M+.X-
(12)
in which M+ and X- are the solvated cation and anion, respectively, and M+.S.X- and M+.X- are solvent-separated and contact ion pairs. Since electrical conductance measures the fraction of uncharged species, then
The second equilibrium is unaffected by changes in concentration except via changes in the relative activity coefficients of the two types of ion pairs. Thus, the observed chemical shift can be written as =
+ 8[XM+SX- + XM+X-]
(14) in which 8 = (6M+.s.x-+ K26M+.X-)/(K2 + 1) is the population-averaged chemical shift of contact and solvent-separated ion pairs. Since the exchange between the two types of ion pairs is fast on the NMR time scale, the relative concentrations of these two species cannot be determined by NMR methods. This leads to 6obsd
6M+xM+
KNMR
= Kcond
Table 111 shows that none of the values of the ion association constant from the conductance measurement is comparable with the NMR value. The discrepancy between the values obtained by the two methods is probably due to the inadequacy of the conductance equations and the inability to measure the conductance of extremely dilute solutions which leads to an uncertainty in A,,. Acknowledgment. We gratefully acknowledge the support of this work by National Science Foundation Grants CHE-80-10808 (A.I.P.) and DMR-79-21979 (J.L.D.)
Dielectric Friction and Molecular Reorientation Paul Madden and Danlel Klvelson’ Department of Theoretical Chemlstry, Cambrklge University, Cambrklge, England and Department of Chemistry, University of California, Los Angeles, California 90024 (Received: Aprll 1, 1982; I n Final Form: April 5, 1982)
We have developed a molecular theory for the dielectric friction on a rotating molecule in a polar medium. The theory accounts for numerous important effects, which are not included in the existing continuum theories, such as molecular translations and the anisotropic relaxation of the polarization induced in the surrounding medium.
1. Introduction
The rate of reorientational motion of a molecule in a liquid is thought to be principally dependent upon the molecular shape and size.’ In liquids of polar molecules appreciable additional contributions might be expected from the slowly relaxing long-range dipolar interactions; for example, it has been observed that the in-plane-reorientation rate of polar heterocycles is slower than that of nonpolar ones.2 Two effects resulting from dipolar interactions are the enhanced two-particle equilibrium orientational correlations and the dielectric friction which
arises from the slow relaxation of the reaction field or, alternatively, of the dipolar torques on a given molecule. The first of these effects is of greatest importance in dielectric relaxation3 where a collective dipole relaxation is observed, but the second should be significant in all observations of the reorientation of polar molecules. Discussions of reorientation of polar molecules have usually been carried out within the framework of dielectric experiments, and the relevant calculations have been based upon models in which single dipoles are introduced into ~
(1) Kivelson, D.; Madden, P. A. Annu. Reo. Phys. Chem. 1980,31,523.
( 2 ) Pedersen, E. J.; Vold, R. R.; Vold, R. L. Mol. Phys. 1978,35,997. QQ22-3654182/2Q86-4244$0 1.2510
(3) See, for example, Brot, C. “Dielectric and Related Molecular Processes”; Davies, M., Ed.; The Chemical Society: London, 1975; Vol. 2.
0 1982 American Chemical Society
