J . Phys. Chem. 1985,89, 5849-5855 TABLE V Mulliken Pooulation of Reactants
HO
CH3
C1 17.1 1
8.89 10.00 9.29
8.71
9.98 9.90 9.98 9.92 9.95 9.88 9.94 9.90 9.94 9.86
8.72 8.56 8.67 8.55 8.74 8.52 8.70 8.53 8.78 8.49
(-0.08)b (-0.06) (-0.07) (-0.04) (-0.08)
(-0.16)
(-0.12) (-0.22) (-0.17) (-0.29)
18.00 17.30 17.55 17.37 17.54 17.27 17.59 17.34 17.59 17.24 17.64
(0.25)
(0.17) (0.32) (0.25) (0.40)
systema CH3CI HOCH3OH C11 2 3 4 5 6
7 8 9 10
#See footnote a of Table IV. bChanges from reactant compexes to transition states are shown in parentheses. hydrogen-bonded H and the nonbond H. In all cases in Table IV, the total population on a water molecule is not far from 10.0, indicating the net charge transfer from/to a solute molecule to/from a solvent water molecule is small. The Mulliken populations on HO, CH3, and C1 for reactants, reactant complexes, transition states, and products are summarized in Table V. Upon going from the reactant complexes to the transition states, only a small charge transfer from OH- to CH3 and C1 occurs, and a dominant charge polarization takes place between C H 3 and C1. This feature is consistent with the notion of an “early” transition state for reactions A-E. The amount of increased charge polarization going from reactant complexes to transition states changes consistently with the number of water molecules, as was discussed with Table 111. Increased charge polarization changes in the order 10 > 6 > 2 8 > 4, the same order as the lateness of transition states. Reaction D, where one water molecule has hydrated each side of the reacting system, may appear to be a reasonable model for a reaction in solution and has been used as such in some theoretical studies.14 As discussed above, its barrier, 2.3 kcal/mol is, however,
-
5849
not much different from that of reaction A, 2.8 kcal/mol, without hydration. One of the most distinct features often attributed to the solution phase reaction of the present SN2reaction’ is the high overall barrier without a potential minimum corresponding to the reactant complex. Reaction D hardly satisfies this criterion. Our preliminary calculations suggest that one needs several water molecules to complete the first hydration shell before such a solutionlike potential energy profile is obtained.I6
Conclusions (OH)-(H,O), CH3C1 gas-phase reactions are investigated for n = 0, 1, and 2 by ab initio calculation using a flexible basis set. A large activation energy in the dihydrated reaction E is consistent with the results of the experimental rate constants, and this reaction is on the threshold of a positive temperature dependence of the reaction rate. A negative temperature dependency is expected for nonhydrated and monohydrated reactions. Though the water transfer process has been known to be a part of the rate-determining step in the symmetric reaction 2, it takes place after the transition state of the rate-determining step in the reaction 1. In general, in highly exothermic reactions such as reaction 1, water transfer is not likely to be the rate-determining step, because that transition state is early and crossing of potential profiles having different ( n , m ) values does not take place before the transition state of CH3 inversion. In slightly exothermic or thermoneutral reactions such as reaction 2, water transfer may be involved in the rate-determining step, depending on the relative solvation energy of the reactants, transition states, and products. Dependency of geometrical structures and charge distributions on the number of water molecules ( n , m ) is consistent with the dependency of the exothermicity and lateness of the transition states.
+
Acknowledgment. We thank Dr. S. Sakai for stimulating discussions. Numerical calculations were carried out at the Computer Center of IMS. The research was supported in part by a Grant-in-Aid from the Ministry of Education of Japan to K.O. Registry No. C H Q , 74-87-3. (16) Sakai, S.; Morokuma, K., unpublished results.
Conductivity and Dielectric Relaxation in Calcium Nitrate Tetrahydrate and Sodium Thiosulfate Pentahydrate near T , D. R. MacFarlane* and D. K. Y. Wong Department of Chemistry, Monash University, Clayton, Victoria, Australia 3168 (Received: May 16, 1985; In Final Form: August 15, 1985)
Admittance data for the liquids Ca(N03)2.4H20and Na2S203-5H20 at temperatures in the vicinity of Tgare reported. The imaginary part of the dielectric modulus is found to be well described, as a function of frequency, by the Davidson-Cole relaxation function. Average conductivity relaxation times as a function of temperature are found to indicate a varying degree of return to Arrhenius behavior in the two liquids. Na2S203.5H20also shows a distinct secondary relaxation at low temperatures probably associated with the anion or anion bound water molecules.
Introduction The ionic conductivity of a liquid is one of the most sensitive probes available for the observation of the structural changes that are taking place as the liquid is supercooled beyond its normal equilibrium melting point, ~~~~~d the glass transition temperature, Tg,the conductivity, along with the other transport properties, can change by an order of magnitude in a temperature region in which spectroscopic or thermodynamic properties are 0022-3654/85/2089-5849$01.50/0
virtually constant. The transport properties of the equilibrium liquid above Tgcan be related to a thermodynamic transition to an amorphous ground state which must take place Tg if a thermodynamically catastrophic crossover of the liquidand solid-state entropy surfaces is to be avoided.’ Such relationships, whether based on a continuity of entropy at the transition (1) C. A. Angel1 and J. Donella, J . Chem. Phys., 67,4560 (1977).
0 1985 American Chemical Society
5850 The Journal of Physical Chemistry, Vol. 89, No. 26, 1985
or of volume, uniformly produce a transport equation, known as the VTF equation which describes the temperature dependence of the transport properties, P( T ) : P ( T ) = A exp(-B/(T-
To))
Conductivity and Dielectric Rela~ation’,~ At temperatures well above Tgit is normally possible to reach a plateau region in the ac conductivity at the lower end of the frequency spectrum and the assumption is then made that this plateau conductivity is representative of the true dc conductivity. As the temperature is lowered toward Tgthe intrinsic relaxation time of the conduction process becomes of the order of the period of the applied ac and the plateau region is invariably replaced by a steadily rising conductivity with frequency. Appropriate (see below) data manipulation techniques must then be used to derive the average conductivity relaxation time, this being related to the dc conductivity. The complex permittivity t of a system is defined as t’
proximation, ionic conduction can be considered to be frequency independent and hence make a contribution only to the dielectric loss component of the permittivity, i.e. e:’
(1)
Here A and B are only weakly temperature dependent compared with the modified Arrhenius temperature dependence predicted by the exponential term, and To is the thermodynamic transition temperature which marks the absolute end of the liquid state. In other words, the liquidlike transport properties are dependent on the acquisition of some liquidlike thermodynamic property; in one theory2 this property is identified as being free volume, in the other3 configl! 5onal entropy, this quantity disappearing as the temperat-re approaches To. A considerable amount of effort4 has been expended over the past two decades investigating the validity of eq 1 and the underlying assumptions in the theories from which it is prodwed. From high-pressure measurements5 has come an indication that free volume is not the controlling property, while at atmcu ,‘iericpressure conductivity and viscosity data in a variety of liqui , have yielded Tovalues in satisfactory agreement with those predicted by calorimetric measurements, Le. the temperature where the configurational entropy of the liquid disappears. While eq 1 has been found to give a good description of the transport properties at temperatures well above Tg,in much of this work a disturbing trend was observed in which the data were found to require increasingly smaller values of To in order to provide a reasonable fit as the fitted temperature range was lowered.6 This seemed to indicate a return to Arrhenius behavior ( T o = 0) at temperatures in the vicinity of Tg,suggesting that the models which predict eq 1 were in need of modification in order to satisfactorily describe the transport properties in this low-fluidity regime. Unfortunately this trend toward Arrhenius behavior has been rather difficult to characterize because of the difficulties associated with the measurement of transport properties in this region of temperature. One of the most easily and precisely measureable transport properties is the ionic conductivity although even these measurements become more complex at temperatures in the vicinity of Tgdue to the rapidly falling relaxation time of the ion in the alternating electric field. This we proceed to discuss in more detail.
= e / - it’[ =
MacFarlane and Wong
- [(./weo)
= uo/weo
where u,, is the limiting low-frequency, or dc, conductivity. Any remaining part of the loss and all of the dielectric constant are presumed to arise from polarization mechanisms not involved in the long-range transport of charge by ionic migration, for example, dipole reorientation. The admittance data can then be treated in terms of the dielectric modulus defined by M = 1/ e =
+ P2))+ i(t”/(e’2 + P2))= M‘+
(t’/(~’~
iM”
which, on insertion of the relationship between u and e” and setting T , = eots/uo, where cs is the limiting high-frequency dielectric constant, can be written
M‘=
(1/4(wTu)2/(1
+ (w7J2)
A plot of M”vs. frequency will thus exhibit a maximum at frequency f(max) = 1/2aT, while the M’plot will show an S-shaped wave. If relaxations other than that due to the conductivity are present these will give rise to separate maxima in the modulus plot, unless their characteristic relaxation times are similar in which case overlapping may occur. If the relaxation is primarily due to conductivity the relaxation time, T, (or its average), will be calculable from the limiting low-frequency conductivity. On the other hand, if the relaxation is dielectric in nature the relaxation time will not correlate with the zero frequency conductivity at all. In the absence of electrode polarization the low-frequency behavior of M’ is also different for the two types of relaxation. For a dielectric relaxation M’e,,-1 as w 0 where e< is the relative permittivity, compared with 0. the conductivity relaxation case where M’- 0 as w One of the main advantages of the modulus spectrum plot is that it tends to suppress circuit elements which contain large capacitance^.^^'^ Thus the large rise in impedance expected at low frequencies because of the purely capacitive nature of the electrodes (Le. “blocking” electrodes) is almost completely quenched in the modulus plot. The total contribution to the modulus due to the electrode double layer capacitance can be shown to be an extra term cO/cdl,in the real part of the modulus, where Cois the vacuum capacitance of the cell and cd,the double layer capacitance, the imaginary part being unchanged. Where the value of the double layer capacitance (usually of the order of to F cm-2 of electrode area) is much larger than that of the sample capacitance (in this work of the order of 10-100 pF) then this double layer term becomes negligible. If the conductivity and dielectric constant are found to be frequency dependent and the dependence cannot be attributed to any dielectric relaxation, then it becomes necessary to describe the conductivity in terms of a distribution of relaxation times and to modify the above modulus equations to read8
-
-
where t’ is the relative permittivity or dielectric constant, e” the dielectric loss, u the measured conductivity, w the angular frequency, and eo the permittivity of free space. As a first ap(2) M. H. Cohen and D. Turnbull, J . Chem. Phys., 31, 1164 (1959). (3) G. Adam and J. M. Gibbs, J . Chem. Phys., 43, 139 (1965).
(4). For a review see S. I. Smedley, “The Interpretation of Ionic Conduc-
tivity
in Liquids”, Plenum Press, New York, 1980. ( 5 ) A. Jobling and A. C. S.Lawrence, Proc. R. SOC.London, Ser. A, 206, 257 (1951). (6) J . H. Ambrus, C. T. Moynihan, and P. B. Macedo, J . Electrochem. SOC.,119, 192 (1972). (7) C . T . Moynihan, R.D. Bressel, and C. A. Angell, J . Chem. Phys., 55, 4414 (1971). (8) J. H . Ambrus, C. T. Moynihan, and P. B. Macedo, J . Phys. Chem., 76, 3281 (1972).
where g(r,) is the distribution function of the relaxation times and M , is defined below. In this case the low- and high-frequency limits of the conductivity and permittivity become 60
=
t’(w-m)
~ S ( T ; ) / ( T , ) ~
=
t,
= l/Ms
(9) I. M. Hodge and C. A. Angell, J . Chem. Phys., 67, 1647 (1977). ( I O ) I. M. Hodge and C. A. Angell, J . Chem. Phys., 68, 1363 (1978).
Ca(N03)2.4H20 and Na2S2O3*5H20 Admittance Data
where
Cf(7,))
a(w-0)
= uo = e o t , / ( 7 , )
a(w---)
= ( e o 4 ( 1/70)
The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 5851
is defined by cf(7,))
=
J37J
g(7,) d7u
Previous studies6tt1of the conductivity relaxation process in hydrate solutions have shown that the magnitude of the primary relaxation in the modulus spectrum is too great to be explicable in terms of a simple relaxation process characterized by a relaxation time, 70, in the usual relaxation function E(?) = E ( 0 ) exp(-t/TO) where E(t) is the electric field vector at time t. In order to develop an understanding of this discrepancy a distribution of these relaxation times has been postulated, this modifying the modulus spectrum as described above. This distribution function can take various, largely empirical, forms. Macedo, Moynihan, and coworkers have found8J1-I6that the Williams-Watts function
E(t) =
exp(-(t/70)@)
where 0 < p C 1, provides a useful description of conductivity relaxation in alkali aluminosilicate glasses and in K+/Ca2+ nitrate glasses. In concentrated aqueous solution studies such as the present, M” has also been fitted by using a logarithmic Gaussian distribution of relaxation time^,^^^ this function only accurately describing the data on the low-frequency side of the modulus peak. Hodge and Angel19 have suggested that the use of the Davidson-Cole function.17 Here the real and imaginary components of the modulus are given by
$)?I - cos (74)(cos $)?I
M” = M,[sin (y$)(cos M’ = M,[1
4 = arctan
(W70)
y being a measure of the half-width of the modulus peak and y = 1 indicating single relaxation time behavior. This was foundI7 to provide a better description of the data than did the Williams-Watts function, especially at frequencies in the region of the modulus peak. The same authors proceed to show that the Davidson-Cole functions appears to have quite general applicability in a variety of vitreous materials, including the silicate and nitrate glasses, as well as the supercooled liquids mentioned previously. Thus with the use of the most appropriate distribution function conductivity relaxation time data can be extracted from the raw experimental data, this yielding the transport property information at temperatures in the region of Tg that is of interest. In the present work attention has been focused on two hydrate liquids, Ca(N03)2.4H20and Na2S203-5H20,which have been extensively studied at temperatures well above Tg,18the current measurements being extended to temperatures as much as 6 K below T g . This is possible, of course, because Tg is determined normally by an experiment carried out at some arbitrary heating rate. If sufficient time is allowed for full equilibration, liquid-state equilibrium can be obtained over a short range of temperature below this Tg,thus (1 1) P. B. Macedo, C. T. Moynihan, and R. Bose, Phys. Chem. Glasses, 13, 171 (1972).
(12) V. Provenzano, L. P. Boesch, V. Volterra, C. T. Moynihan, and P.
B. Macedo, J. Am. Ceram. SOC.,55, 492 (1972).
(1 3) T. J. Higgins, L. P. Boesch, and N. L. Laberge, Phys. Chem. Glasses, 14. 122 (1973). (14) L. P. ‘Boesch and C. T. Moynihan, J . Non-Cryst. Solids, 17, 44
(1975). (15) F. S.Howell, C. T. Moynihan, and P. B. Macedo, Bull Chem. SOC. Jpn., 57, 652 (1984). (16) F. S. Howell, R. A. Bose, P. B. Macedo, and C. T. Moynihan, J . Phys. Chem., 78, 639 (1974). (17) D. W. Davidson and R. H. Cole, J. Chem. Phys., IS, 1417 (1951). (18) C. T. Moynihan, J . Phys. Chem., 70, 3399 (1966).
000
1000
2000
low
4000
5000
Log I f / H z i
Figure 1. Conductivity and permittivity data as a function of frequency over a range of temperatures for Ca(N03)2.4H20.
extending the measurements to the (experimentally) absolute limit of the liquid state.
Experimental Section The low-temperature dielectric cell and temperature control equipment have been described previo~sly.’~To summarize some of the more important details, the cell was of an all metal construction with large stainless steel electrodes maintained parallel and separated by a 1-mm-thick polyethylene spacer, which also served to contain the solution. At frequencies below 500 H z a modified Berberian-Cole bridgeZowas used. For frequencies in the audio range a Wayne-Kerr B331 bridge was used in conjunction with a GRl309-A oscillator and GR1232-A tuned amplifier and detector. Samples of the hydrate liquids were obtained by melting J. T. Baker analyzed reagents in a sealed container in an oven maintained just above their respective melting points. On complete fusion the cell was filled, quenched into liquid nitrogen, and then mounted into the controlled temperature block. Data as a function of frequency was obtained isothermally over a range of temperatures around Tg. At temperatures below the 10 K min-’ Tg the attainment of liguid-state equilibrium was always monitored via the trend toward a steady-state value of the conductivity and the frequency scan only begun once this steady-state had been achieved. Both of the liquids studied here have a tendency to crystallize at temperatures above Tgand for this reason data were obtained at a rate of 3 frequencies/decade in order to minimize the measurement time. Compositions of the fused hydrate samples were checked by drying under vacuum and were found to be, within the error limits of the determination, the same as the stoichiometric hydrates. As received hydrates often have slightly higher than stoichiometric water contents, it is probable that some loss of water took place during this fusion. The fitting of the experimental data to the various response functions investigated, and of the derived average relaxation times to the VTF equation, was carried out using the nonlinear leastsquares fitting technique.21 This has the advantage, over those (19) C. A. Angel1 and D. R. MacFarlane in “Advance in Ceramics”, Vol. IV, Ohio University Press, Athens, OH, 1984, pp 66-76. (20) J. G. Berberian and R. H. Cole, Rev. Sci. Instrum., 40,8 11 (1969).
5852 The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 2 000 -
MacFarlane and Wong
%
M”
\
0 2 l 2 6l( 0 211 2 K 211 9 K V 2152K A 2IS9K 217 2 K P I 0 7K A 215 L K V 221 1 K 0 221 IK 5 2 5 01( X 228 5 1
Figure 3. Imaginary part of the dielectric modulus as a function of frequency over a range of temperatures for Ca(N0,)2.4H20.
0 211 0
K OZlSZK OZl74K VZl99K
A ~ Z Z L K
-13 0 0 ’ -I
000
I
OOC
I
200c
IO00
1
3 000
J LOO0
Log [ f/Hz) Figure 2. Conductivity and permittivity data as a function of frequency over a range of temperatures for Na2S203.5H20.
used previously, that meaningful standard deviation data is obtainable for all of the fitted parameters. Also parameter covariance can be analyzed to indicate whether or not the chosen functional form is properly descriptive of the data.
Results and Discussion Conductivities and relative permittivities were calculated from the measured conductance, G, and capacitance, C, via u = e o G / C o and
e’ = C/Co
and the resultant data are shown in Figures 1 and 2 for Ca(N03)2.4H20and Na2S2O3-5H20.In both cases the upper limit in temperature was imposed by the sudden crystallization of the liquid and it was for this reason that measurements were not pursued into the experimentally more time-consuming low-frequency region at the higher temperatures. At these temperatures in the vicinity of Tg( T g = 218 K for Ca(N03)2.4H20 and 215 K for Na2S203.5H20)there is little evidence of an approach to the limiting low-frequency plateau in either the permittivity or the conductivity. The conductivity data display a low-frequency decline, especially apparent in the case of Na2S2O3-5Hz0,which does not appear to give rise to an additional low-frequency dispersion in the imaginary part of the modulus spectrum. This drop off in conductivity is thus associated with a circuit element which also contains a high capacitance and is thus probably associated with the electrode impedance. As discussed above, ideally blocking electrodes act as pure capacitances; however, should any electrode reaction take place then the overall conductance will decrease at low frequencies as the resistance associated with the charge-transfer reaction (which is normally considered as being in parallel with the double layer capacitance) becomes important in the overall impedance of the cell. This effect is thus insignificant in terms of the modulus plots but frustrates any attempt to obtain limiting dc conductivities from the present data. This low-frequency falloff has been observed in other similar but has not been found to complicate (21) J. R. Macdonald, J. Schoonman, and A. P. Lehnen, J . Electroanal. Chem., 131, 77 (1982).
, I000
000
,
1
I 000
2 000
1
I
3 000
4
coo
Log(f/Hz) Figure 4. Imaginary part of the dielectric modulus as a function of frequency over a range of temperatures for Na2S2O3.SH20.
attempts to fit data in the modulus plane to the various distribution functions. In Figures 3 and 4 are shown the imaginary modulus spectra for Ca(N03).4H20 and Na2S203-5H20,respectively, calculated from u and e as described above. The imaginary modulus spectra are the more informative for the purpose of detecting the number of different relaxation phenomena which are occurring in the systems, each ideally giving rise to a separate peak in M”. In the M”spectra at each temperature the peak is characterized by a rather sharp onset on the low-frequency side relative to the more gradual decline at higher frequencies and is considerably broader at half-height than the expected Lorentzian (single relaxation time) half-width. In the case of Na2S2O3.5Hz0there is a clear secondary relaxation apparent in the lower temperature spectra. Similarly the fact that the spectra for Ca(N03)2.4H20show little sign of tending toward zero at high frequencies, even at the lowest temperatures, suggests that the broadness of the main relaxation may be due to the presence of a secondary relaxation in this case also. Previous ~ o r k suggests * ~ ~ that there are three main varieties of relaxation which we might expect to detect in these liquids. The first is likely to be, at least in part, the main conductivity relaxation; however, it is somewhat surprising that dispersion peaks should be observable in the frequency range at such low temperaures. It would appear that there is a significant degree of decoupling of the structural and conductance relaxations at these temperatures, a point which will be discussed further below. The remaining varieties are dielectric relaxations and, if observable, must be characterized by more rapid relaxation times than that of the conductivity relaxation. In liquids such as these dielectric relaxation might take place via reorientation of either the water molecules or, in the case of Na2S,03-5H20,the thiosulfate anion. In Ca(N03).4H20 water molecules associated with the Ca2+ions
The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 5853
Ca(NO3),*4H20and Na2S203.5H20Admittance Data
TABLE I: Best Fit Cole-Davidson Parameters for Ca(NO3),-4H20and Na2S2O34H20
T/K
TO/S
Y
M,
2 12.6 213.2 213.9 215.2 215.9 217.2 218.7 219.4 221.4 224.1 225.3 228.9
7.08 f 0.4 4.2, f 0.2 2.30 f 0.1 0.95 f 0.05 0.59 f 0.03 0.22, f 0.01 (7.4 f 0.5) X IO-, (5.3 f 0.4) X ( l . 6 2 f 0.2) X IO-, (3.7 f 0.5) x 10-3 (2.3 f 0.4) X IO-’ (3.6 f 0.3) X
0.274 f 0.003 0.273 f 0.004 0.278 f 0.004 0.257 f 0.009 0.25, f 0.01 0.25, f 0.01 0.271 f 0.02 0.259 f 0.01 0.240 f 0.03 0 . 2 1 ~i 0.01 0.20, f 0.02 0.177 f 0.02
0.254 f 0.004 0.246 f 0.004 0.249 f 0.004 0.259 f 0.005 0.261 f 0.006 0.263 f 0.008 0.256 f 0.009 0.26, f 0.01 0.272 f 0.02 0.294 f 0.03 0.29, f 0.03 0.32, f 0.02
213.0 215.2 217.4 219.9 222.4
2.1 f 0.2 0.124 f 0.008 (1.07 f 0.1) X (1.63 f 0.3) X l r 3 (2.4 f 0.3) X
0.21, f 0.01 0.20, f 0.01 0.24 f 0.02 0.150 f 0.05 0.17 f 0.05
0.248 f 0.007 0.267 f 0.009 0 . ~ 2 6f~ 0.01 0.35 f 0.1 0.33 f 0.05
---------
freq range, Hz
0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.5 0.5 2.0
5.0 200
std dev in M”
100 1592 1592 1592 1592 1592 1592 1592 1592 1592 104 2 x 104
In
0.0003 0.0005 0.0008 0.001 0.002 0.002 0.002 0.003 0.003 0.003 0.003 0.002
0.1 5.0 0.1 20 0.5 200 1 1592 I O -r 1592
((T,,)/s)
0.663 f 0.08 0.137 f 0.06 -0.45 f 0.05 -1.41 f 0.07 -1.91 f 0.09 -2.86 f 0.08 -3.9, f 0.1 -4.29 f 0.1 -5.55 f 0.2 -7.13 f 0.2 -7.64 f 0.3 -9.7 f 0.2 -0.82 f 0.1 -3.70 f 0.1 -5.96 f 0.2 -8.3 f 0.5 -10.1 f 0.2
0.002 0.002 0.002 0.003 0.005
log ( c T / ~ - cm-l) ’ -12.74 f 0.03 -12.51 f 0.03 -12.25 f 0.00 -1 1.98 f 0.04 -1 1.64 f 0.04 -1 1.23 f 0.05 -10.76 f 0.06 -10.61 f 0.07 -10.07 f 0.1 -9.4 f 0.1 -9.2 f 0.1 -8.4 f 0.1 -12.10 f 0.05 -10.89 f 0.06 -9.88 f 0.06 -9.0 f 0.3 -8.2 f 0.2
TABLE II: Best Fit VTF Eouation Parameters ~~
~
property Ca(N02),.4H,0 Ca(NO;j;.4H;O Ca(N03),.6.15H20 Ca(N0,)2.8H20 NazS2O3.5H2O Na2S2O3-5H2O
(Til)
1/6,
‘2-l
cm-l
( T ~ )
l/o,
0-l
cm-l
(70)
1/c,
cm-’
To/K 160 f 4 204 161 132 187 f 7 204
In ( A / s ) or In (Ao/R cm-I) -43 f 3 -0.71 -30.9 -12.3 -37 f 4
are thought to be bound through the oxygen atom along a line bisecting the H-O-H bond angle. Under such a restraint the only remaining degree of rotational freedom left to the water molecule is about the axis of the Ca-0 bond and hence involves no dipole reorientation. This rotation cannot, therefore, serve to relax an applied electric field and would not be observable as a dielectric relaxation. Water molecules bound to either of the anions along the direction of the 0 - H bond would, however, still retain a rotational degree of freedom which would allow dipole reorientation and hence interaction with the electric field. At the water contents present in the hydrates studied in this work we would expect to find most of the water molecules bound in the hydration of the cation in the Ca(NO3),-4H2O case rather than associated with the anions and thus make little or no contribution to the relaxation of the applied field. In Na,S2O3.5H20, on the other hand, we would expect to find water molecules bound via strong hydrogen bonds to thiosulfate oxygens, these water molecules retaining a rotational degree of freedom which might serve to relax the electric field as described above. Also in this case reorientation of the whole anion could serve to relax the applied field. Thus the dielectric relaxation observed in this liquid can probably be attributed to either anion bound water or the anion itself. From the foregoing it appears that the broadness observed in the main conductivity relaxation peaks in Ca(NO3),.4H,O and also to some extent in NaS2O3.5H2Omust be due to a distribution of conductivity relaxation times, rather than to dielectric relaxation, and we proceed to analyze the data in terms of such a distribution.
Conductivity Response Functions As discussed above the Davidson-Cole response function appears to be one of the most useful currently available in providing an accurate fit to the experimental data, especially in the region of the modulus peak maximum. Unless it is suspected that a single peak arises from two or more distinct relaxation processes, perhaps involving different species, then an accurate description in the region of this most probable relaxation frequency must be seen as one of the most important criteria for the acceptance of the applicability of a response function. An indication of the fit to a typical experimental M”spectrum of the Davidson-Cole function is shown in Figure 5 , this providing a good description of the data, especially around the peak maximum. In the case of the concentrated aqueous acidsg the fit was indistinguishable at the lower frequencies from that of the Williams-Watts function; however,
B/K
std dev in In ( T )
temp range/K
T,/K
ref
0.058 0.003 0.05 0.07 0.091
213-229 278-343 195-211 190-203 213-220 278-323
218 218 194 183 216.5 216.5
this work 24 7 6 this work 2
2300 f 300 567 933 1810 930 f 240 484
.
Experimental -+Cole-Davidson
fit
,061
M “ .04
I
0’
, 0
I
1 Log ( f / H z 1
,
I
2
3
1
Figure 5. Example of Davidson-Cole function fit to experimental data for Ca(N03)2.4H20 a t 218.7 K.
around the maximum the latter function clearly overestimated the most probable relaxation frequency and fell off more rapidly than the data at the higher frequencies. In contrast, the Davidson-Cole function appeared to provide a good fit over the whole range of frequencies as is also found in the present case.*, Nonlinear least-squares fitting parameters from fits of the Davidson-Cole function to the spectra as a function of temperature for both Ca(N03),.4H20 and Na2S203.5H20are shown in Table I. Error limits indicated are one standard deviation, these being obtained from the variance-covariance matrix of the fit. In the (22) Attempts to obtain a complex nonlinear least-squares2’ fit of the complete data set (Le., M’ and M “ simultaneously) to the Davidson-Cole function uniformly failed to reach convergence. Part of the cause of this seemed to be the fact that the M’data does not tend to zero as w 0, there being always a small residual probably due to electrode polarization. The inclusion of extra, additive parameters in the fit failed to completely take account of this offset. However, such effects tend to be quenched in the M” data and it was for this reason that we have accepted the relatively good fits to these data sets. Fits reported in ref 9 to this function were also only to the imaginary part of the data.
-
5854
T h e Journal of Physical Chemistry, Vol. 89, No. 26, 1985
MacFarlane and Wong TABLE 111
l
/
l
C a (N0,124H,0
I
-30;
/
I
-401I
?
/
I
I
1
I
I
44
45
L6
47
IO~/T(K) Figure 6. Arrhenius plots of the conductivity relaxation time for Ca(N03),.4H@ and Na2S203.5H,0. Points are experimental data and curves are fitted VTF functions. Inset. VTF parameter To as a function of composition. Tovalues are those obtained (see text for sources) from data relevant to circa T, temperatures, T,' are those obtained from higher temperature data. T g data from ref 24.
case of Na2S2O3e5H20 the presence of a secondary relaxation close to the main relaxation caused the fitting parameters to be dependent on the high-frequency extent of the frequency range chosen. To overcome this, data points were progressively excluded until the fitting parameters became constant. The frequency range to which the parameters are applicable is indicated in the table. The average relaxation time for charge migration, ( T,,),is given , by the product of the Davidson-Cole parameters y and T ~ i.e. ( T o ) = Y70
and is also shown in Table I. It was found from the correlation matrix produced by the fitting routine that there was a high degree of correlation between the parameters y and T ~ .This indicates that almost equivalent fits could be produced by a compensating variation of these two parameters. On examining the value of ( T ~ during ) convergence toward the best fit it was found that the value of this factor remained relatively constant despite oscillations in the values of y and T ~ .This finding allows us to have more confidence in the value of ( T ~ than ) might be indicated by the error bounds shown for the parameters in the table, these being directly obtained from the individual parameter standard deviations. Also shown in Table I are the dc conductivities calculated via no = to/( s , ) M , . These connect smoothly as a function of temperature with the much higher temperature data of ref 18. Temperature Dependence of the Average Conductivity Relaxation Times Arrhenius plots of ( 7 , ) for both liquids are shown in Figure 6. There is a distinct curvature present, this being characteristic of all transport properties in the temperature region above Tg. Fitting to eq 1 again using the nonlinear least-squares technique yields the parameters listed in Table I1 where the meaning of the error limits is as for the Davidson-Cole fitting described above. Also shown are the Tovalues obtained for the same liquids at much higher temperatures, the diminution in Towith the temperature
liauid
( ( 7s) / ( T o ) IT&
Ca(NO3),+4H20 Ca(N03),.6.15 H 2 0 Ca(N03),.8H20 LiCI.4.83H20 LiCb5.77H20 LiCI.9H20 Na2S20,.5H20 0.4Ca(N0,),/0.6KN03
3 x 103 2 x 103 104 4 8 500 2 x 104 I 04
ref this work 7 6 8 8 25 this work 26
range of the data from which it is obtained showing the extent of return to Arrhenius behavior. Comparison of the value of To for Ca(N03),.4H20 with that obtained by Ambrus, Moynihan, and Macedo6 for Ca(N03)2. 6.1 5 H 2 0 of 16 1 K indicates an unexpected coincidence of values, although it should be noted that the distribution function fitted in the present work is different from that used by the latter authors. Fitting parameters obtained from dc conductivity data measured over a slightly higher temperature range relative to Tg for Ca(NOJ2.8H20 indicate (see inset to Figure 6) that the parallelism observed between To and T gin ref 22 is maintained, although a partial return to Arrhenius behavior has increased the gap in temperature between Tg and To. If the maintenance of this parallelism is accepted as the most probable behavior then the deviation from this trend observed for the Tovalue for the 6.15 H 2 0 solution suggests that the log Gaussian distribution actually alters the temperature dependence of the relaxation time, perhaps because of the changing half-width with temperature. This modification of the temperature dependence would only be manifest in the value of To. On the other hand, the concordance of results obtained from the use of the Davidson-Cole distribution function in the present work with that of a true dc measurement result lends support to the validity of this function. In contrast to the calcium nitrate solutions the data in Table I1 indicate that there is only a weak tendency to return to Arrhenius behavior in the case of Na2SzO3-5H20.This conclusion is also supported by the Tovalue obtained from similar experiments using an emulsified sample over a slightly wider temperatu 3 range.25 Conductivity relaxation time data obtained at temperatures in the vicinity of Tg allow us to estimate directly the difference between the conductivity and structural relaxation times at the where the liquid is no longer able to relax point in temperature, Tg, toward equilibrium on the time scale of the observation. Tgcan be taken as an isostructural-relaxation time for samples of identical thermal history and, if determined at the usual 10 K min-l, is thought to correspond to a structural relaxation time in the range 10-100 s . ~Interpolating the relaxation time data of Figure 6 to yield values of ( 7 , ) at Tgfor the two liquids, we obtain values of 0.03 and 0.005 s for Ca(N03),-4Hz0 and Na2S203.5H20, respectively. Thus the ratio of structural to conductivity relaxation times can be estimated for these liquids. These ratios are shown in Table I11 along with values obtained by various, in most cases more precise, means by other workers for a variety of liquids. From this it can be seen that, despite the lesser degree of return to Arrhenius behavior in Na2S203.5H20,when compared with Ca(NO3),-4H20,the value of this ratio is, if anything, higher in the former case. Thus the fact that the conductivity relaxation process has become decoupled from structural relaxation, as evidenced by the disparity in the relaxation times, seems to be unrelated to the degree of return to Arrhenius behavior at Tg. Presuming that the two relaxation times are of comparable magnitude in these liquids at much higher temperatures, this low-temperature disparity must arise from a divergence in the To's for the two processes, Le., from a differing degree of return to Arrhenius behavior between the two processes. This is not necessarily inconsistent with the fact that Na2S2O3-5H20 shows only (23) C. A . Angell and R. D. Bressel, J . Phys. Chem., 76,3244 (1972). (24) C. A. Angell and E. J. Sare, J . Chem. Phys., 52, 1058 (1970). ( 2 5 ) D. R. MacFarlane, Ph.D. Thesis, Purdue University. 1982.
J . Phys. Chem. 1985, 89, 5855-5862 a minor decrease in To(a),even though ( T J T ~ )is~ large, ~ since the latter quantity is dependent on the degree of divergence between To(a)and To(s),rather than the degree of return to Arrhenius behavior. Thus the suggestion from the data is that the two liquids retain a similar degree of decoupling of the relaxation processes at Tgeven though there is a marked difference in their measured To(a)’s. It is also apparent from the comparison in Table 111 that values of this ratio of the order of 103-104 are the norm for nitrate hydrates and molten nitrates as well as Na2S20,.5H20. On the other hand, the LiCl hydrates are somewhat more unusual in that the ratio of relaxation times is much smaller. Recent on the conductivity of LiCl solutions as a function of (26) R. Bose, R. Weiler, and P. B. Macedo, Phys. Chem. Glasses, 11, 117 (1970). (27) S. I. Smedley and D. R. MacFarlane, J. Electround. Chem., 118,445 (1981).
5855
pressure has shown that the conductivity rises rapidly above the 1-atm level over the first few kilobars of applied pressure. This effect becomes even more pronounced at lower temperatures, yet the glass transition temperature has been found to increase slightly with pressure over this pressure range. Thus we might expect the liquids in Table I11 to behave more similarly if sufficient data were available for the comparison to be made at say 2 kbar. The reason for this unusual behavior is the LiCl solutions is not clear at this point in time.
Acknowledgment. The authors are indebted to Professor C. A. Angel1 for his encouragement of this work and many useful discussions. Registry No. Ca(N0J2, 13477-34-4; Na2S20,, 7772-98-7 (28) S. I. Smedley, D. R. MacFarlane, and J. Scheirer, to be submitted to J . Phys. Chem.
Optical Selection in Double-Resonant Two-Photon Photodissociation: Near-Threshold State-to-State Fragmentation Dynamics of NO2 2hv NO(% 2111,2, Y = 0, J , A) -t
+
-+
O(lW Laurence Bigio and Edward R. Grant* Department of Chemistry, Baker Laboratory, Cornell University, Ithaca. New York I4853 (Received: August 19, 1985)
Optical-optical double resonance is employed to achieve state-to-state resolution in two-photon photodissociation. NOz seeded in a free-jet expansion is dissociated by sequential two-photon absorption in the region of 488 nm, just above the threshold for production of O(’D). The channel to produce excited oxygen is observed to dominate the formation of vibrationally cold NO(% 211, u = 0), which is detected by resonant two-photon ionization. For two-photon photolysis energies within 100 cm-’ of threshold, photodissociation dynamics, manifested in NO rotational and A-doublet state distributions, are observed to be a sensitive function of parent intermediate state. NO populations are an oscillatory function of J , with varying patterns of A-doublet preference that change with selected NO2 intermediate state. Certain features in the photodissociation spectrum of NOz in this region are seen to correlate with product NO state total parity. At energies higher above threshold (>200 cm-’), the oscillatory behavior in J is less pronounced and, accordingly, less sensitive to total photolysis energy. In this region a uniform A-doublet preference also emerges.
Introduction Photofragment spectroscopy has developed rapidly over the past decade to offer an increasingly refined view of the scattering dynamics associated with a growing number of molecular halfcollision processes.’-3 Theory has kept pace and in many respects led progress in the field.&’ Starting most often with a separation that factors the photodissociation process into a Franck-Condon quantum selection of initial conditions followed by scattering on (1) J. P. Simons, J . Phys. Chem., 88, 1287 (1984). (2) R. Bersohn, J. Phys. Chem., 88, 5195 (1984). (3) S. R. Leone, Adv. Chem. Phys., 50, 255 (1982). (4 M. Shapiro and R. Bersohn, Annu. Rev. Phys. Chem., 33,409 (1982); .G. G. Baht-Kurti and M. Shapiro, Chem. Phys. 61, 137 (1981). (5) K. F. Freed and Y. B. Band in “Excited States”, Vol. 3, E. C. Lim, Ed., Academic Press, New York, 1977, p 109; Y. B. Band and K. F. Freed, J. Chem. Phys., 63, 3382 (1975); 67, 1462 (1977); 68, 1292 (1978); M. D. Morse, K. F. Freed, and Y. B. Band, Ibid.,70, 3604, 3620 (1979); M. D. Morse and K. F. Freed, Ibid.,74, 4395 (1981); 78, 6045 (1983). ( 6 ) R.W. Heather and J. C. Light, J. Chem. Phys., 79, 147 (1983); 78, 5513 (1983). (7) W. M. Gelbart, Annu. Rev. Phys. Chem., 28, 323 (1977).
the upper state dissociative surface, theory operates with a state-testate resolution that for the most part has been unavailable to experiment. Indeed, quantum interference effects, manifested in oscillating rotational population distributions, which are a feature of most theoretical calculations of state-to-state photodissociation cross sections, have yet to appear in conventional photodissociation results.8 While exit channel interactions are no doubt a factor, in some cases at least such smoothing can best be explained by the coarse graining introduced in the laboratory by averaging over initial states. Even for narrow initial state distributions, such as those typically found in free-jet expansions, bound-continuum photodissociation transitions overlap the contributions from a significant number of parent initial states and orientations. Thus an averaged view is presented, more readily interpreted in terms of Ehrenfest dynamics or phase space t h e ~ r y . ~ Clearly, for confirmation of quantum theories of photodisso(8) See, for example, W. G. Hawkins and P. L. Houston, J . Phys. Chem., 76, 729 (1982). (9) C . Wittig, J . Chem. Phys., in press.
0022-3654/85/2089-5855$01.50/00 1985 American Chemical Society
