4165
NOTES Table I: Ratios of the Integrated Intensity of the trans to the cis Bands Temp,
Ratio
O C
2.1333 1.8975 1* 4959 1.3368 1.2565
25 10 - 15 -25 - 30
the OH stretching mode of o-trifluoromethylphenol is assigned to the cis isomer where intramolecular hydrogen bonding could stabilize this conformation. If the ratio of the extinction coefficients of the cis (3606 cm-l) and trans (3624 cm-I) bands is approximately constant over the temperature range studied, the ratios given in Table I are related to the equilibrium constants for interconversion merely by a constant. A plot of the log of the ratios given in Table I ws. the reciprocal of the absolute temperature is shown in Figure 2. From the slope of the straight line obtained, the enthalpy difference between the cis and trans conformation was determined by standard thermodynamic relationship to be 1.4 kcal/mol.
.4t .3 ’
LOG RATIO
.2
’
I ’
comparisons it appears that an increase in ring size from five t o six members does not influence appreciably the magnitude of the hydrogen bond interaction. These results cannot be adequately compared with the o-fluorophenol system since the OH stretching mode in this system appears to have no detectable asymmetry. They do suggest, however, that the interactions in the cis isomer of o-fluorophenol are too weak to result in an observable frequency shift.
Diaphragm Cell Diffusion Studies with Short Prediffusion Times
by Michael J. Piltal Chemistry Department, University of Tennessee, Knoxville, Tennessee 57916 (Receined May 87, 1970)
The diaphragm cell has proven to be a very useful tool in the study of liquid The basic feature of a diaphragm cell is the porous membrane, which is usually a glass frit separating an upper and a lower compartment containing liquids of different composition. If the concentration of diffusing species is initially zero in the upper compartment, the classical analysise shows that from the duration of the diffusion experiment, t, and the concentration of the diffusing species in the upper and lower compartments at the end of the run, denoted by W and V, respectively, the diffusion coefficients may be calculated from the equation
Here, P is a cell constant determined by calibration, and f is defined by
~
3 00
4.00
5 00
LXIO’
Figure 2. A plot of the log of the ratios tabulated in Table I us. l/T°K.
A separation of 18 cm-I between the cis and trans conformation is considerably less than that found for intermolecular association of phenol with a covalently bound fluorineg (39.7 cm-I). However, the frequency change is almost identical with that reported for the intramolecular association postulated for monofluoroethano15 (15.5 cm-l) and trifluoroethanole (19.0 cm-l). The enthalpy difference derived from these data is significantly smaller than found in the above examples (2.13, 2.07, and 3.32 kcal/mol, respectively). From these
where Vu and VLare the volumes of the upper and lower compartments, respectively, and VD is the volume of the frit. The derivation of eq 1 assumes the concentration distribution in the frit is linear throughout the experiment, although it has been shown’ that as long as the (1) J. N. Northrop and M. L. Anson, J . Gen. Physiol., 12, 543 (1928). (2) A. R. Gordon, A n n . N . Y . Acad. Sci., 46, 285 (1945). (3) R. H. Stokes, J . Amer. Chem. Sac., 72, 763 (1950). (4) J. G. Albright and R. Mills, J . Phys. Chem., 69, 3120 (1965). (5) R. Mills and L. A. Woolf, “The Diaphragm Cell,” Australian National University Press, Canberra, Australia, 1968. (6) See R. A . Robinson and R. H. Stokes, “Electrolyte Solutions,” Academic Press, New York, N. Y., 1959, Chapter 10. (7) C. Barnes, Physics, 5, 4 (1934). The Journal of Physical Chemistry, Vol. 74, No. 83,1970
NOTES
4166
initial concentration distribution is linear, eq 1 is still valid in all practical cases. For the purpose of developing an initial linear concentration distribution, a prediffusion step is introduced into the experimental procedure. I n a tracer experiment (ie.,where the diffusing species is present in small concentration and moves in an essentially homogenous environment), the upper compartment and the diaphragm are initially filled with solution devoid of tracer species. Solution containing the tracer (labeled solution) is placed in the lower compartment, and diffusion is allowed to proceed for prediffusion time, r . Next, the solution in the upper compartment is removed, and the compartment is rinsed with unlabeled solution. Fresh unlabeled solution is then placed in the upper compartment, and the actual diffusion run is started. The assumption is that after a sufficiently long prediffusion time, the concentration distribution in the frit is linear and eq 1is valid. The question is, of course, how long must the prediffusion time be to preclude serious error due to the failure ~ ~determine ~~ a of eq 1. The usual p r o ~ e d u r eis~to “rough but safe” lower limit for from the Gordon inequality, Dr/P > 1.2, where D is the diffusion coefficient in cm2/sec and 1 is the “apparent thickness” of the frit (2 x 1.6 X actual thickness). For the usual type of frit, 1 = 0.4 cm, and if D = 2 X cm2/sec, r is of cm2/sec, the order of 3 hr. However, if D = 2 X the estimated preliminary diffusion time is 30 hr. Thus, the diaphragm cell technique does not appeal. to be practical when the diffusion coefficients are small, particularly if short-lived isotopes are to be studied or the solution undergoes slow chemical decomposition. However, there is evidence which suggests that the Gordon inequality overestimates the time required to establish a linear concentration distribution.g The purpose of this note is to present the results of a combined theoretical and experimental investigation of the effect of variation in prediffusion time on the “apparent” diffusion coefficient calculated from eq 1. A theoretical “correction” to be applied to eq 1 for short prediffusion times will be derived. Experimental data will be presented which indicate the theoretical correction is accurate within *30% or better. The practical result of this study is that a prediffusion time smaller by a factor of five than those given by the Gordon inequality may be used without introducing serious error. Theory The following discussion refers to a tracer diffusion experiment where the tracer is originally in the lower compartment and is allowed to diffuse into the upper compartment. The diaphragm is devoid of tracer at the start of prediffusion. Both compartments are taken to have the same volume. The theory developed here is basically an extension of the earlier work of The Journal of Phvsical Chemistry, Vol. 74, N o . 2.3, 1970
Barnes.’ The general procedure is to first evaluate the concentration distribution in the frit at the start of the diffusion run. The distribution function derived contains a nonlinear term depending on r . Next, this distribution function is used to relate the concentrations of diffusing species in the upper and lower compartments to the product Dt. Algebraic manipulation of the resulting equation then yields the diffusion coefficient as a sum of two terms. The first term is the right-hand side of eq 1, while the second term is the theoretical “correction” for a short prediffusion time. The symbols used are defined as follows.
x = V,/VU
=
V,/VL
V O= concentration of tracer in the lower compartment at the start of the diffusion run (ie,,at t = 0 ) ; Vo’ = concentration of tracer in the lower compartment at the start of prediffusion; W , = concentration of tracer in the upper compartment at the conclusion of prediffusion; Wo = 0 = concentration of tracer in the upper compartment at the start of the diffusion run; a = effective volume of a frit of unit area; Uo(x) = concentration of a tracer in the frit at position x (0 5 II: 5 a) at the conclusion of prediffusion (and at the start of the diffusion run). Barnes’ has given several solutions of the problem of diffusion through a membrane based upon a solution of Fick’s second law subject to the appropriate boundary conditions. In the case where the concentration distribution in the frit is initially linear (gradient filled case), Barnes’ equations reduce to eq 1 for all practicalpurposes, with the cell constant being given by @ = 2X(1 - X/6 X2/45)/a2. Another situation considered by Barnes was the case in which the frit is initially free of tracer, and no prediffusion period is used. I n this case (solvent-filled case), omitting the small quadratic terms in X and the vanishingly small series term, Barnes’ results may be written in the form
+
D
=
+
D 1 (l/@r)In [(l - X/6)/(1 - X/2)]
(3)
where D1 represents the value of the diffusion coefficient calculated from eq 1. We now consider the solution of the diffusion problem in the more general case where the initial concentration distribution in the frit is determined by the prediffusion conditions. Our solution will correspond to the SYStematic retention of all significant terms linear in A ; the quadratic terms in X will be omitted. From the equations given by Barnes, we find that the concentration distribution in the frit at the start of the diffusion run is given by (8) A. L. Geddes and R. E.Pontius in “Physical Methods of Organic Chemistry,” Part 2, A. Weissberger, Ed., Interscience Publishers, Inc., New York, N. Y., 1960,p 895. (9) R. Mills, L. A. Woolf, and R. 0 . Waats, A.1.Ch.E. J., 14, 671 (1968).
4167
NOTES
needed to evaluate S1 and Sz. The approximation, p g 2X/a2,may be used for this purpose. It is a simple matter to verify that our solution reduces to the solvent filled case as T -+ 0, as it should. When r = 0, S1 = l/nz = ~2/8, and Sz = 2 l/n2 =
where
n
(n = 1, 2, . . .)
=*Dn2r2/a2
n even
r2/24. Therefore, as r + 0, a1-+ X/2, a z -t X/6, and eq 11 reduces to the solvent-filled case.
(6)
Experimental Section
and
W , = (v0’/2)(1 - e-pDT)
odd
(5)
To demonstrate the essential validity of eq 11, one Following the procedure outlined by Barnes with UO(S) tracer diffusion run was made with each of two diagiven by eq 4, and using the approximation BO’= phragm cells. In each cell, a small volume of solution Bo in eq 4, we find containing trace amounts of 24Naf,45Ca2+,and 88Y3f V = (v0/2)[1 - 4h&/T2 W,h/2v0 was added to the solution in the lower compartment. (1 - 4X8z/T2 - W , X / ~ V O ) ~ - (7) ~ ~ ~ IFor each cell, the concentration ratios W / V were determined for the three isotopes by taking advantage of the = (v0/2)[1 - 4h8i/T2 w,h/2v0 short half-life of 241\’a+and the counting characteristics (1 - 4h8z/a2 - W,h/6V0)e-~’~] (8) of the isotopes. Diffusion coefficients were then calculated using eq 11. The diffusion conditions were with chosen so that the function D - D’,for each isotope, was significant for cell 1 but insignificant for cell 2’. (9) Magnetically stirred diaphragm cells of the Stokes odd d e ~ i g n with , ~ Teflon bottom plugs14were used in the experiments. The stirring rate was 60 rpm. The calibration constants, @,were evaluated in the usual even way by allowing 0.5 N potassium chloride to diffuse into pure water. Rearrangement of eq 7 and 8 to eliminate V Oand using The potassium chloride and sodium chloride were the good approximation, W, = VoPDr/2, yields the reagent grade chemicals. The potassium chloride was desired result recrystallized once from water. All solutions were D = D1 (l/Pt) In [(l - a ~ ) / ( 1- 4 1 (11) filtered through a fine glass frit before use and were then degassed. Triple-distilled water was used in this rewhere D’ is, again, the value of D calculated from search. The 1.00 m sodium chloride solution mas preeq 1. The symbols a1 and a2 represent the functions pared from weighed quantities of the dried salt and disa1 (4S1/n2 - @D7/4)X (12) tilled water. The radioactive isotopes, yttrium-88 and calcium-45, were obtained from the Isotopes DiviCY^ E (4Sz/d PD7/12)X (13) sion of Oak Ridge National Laboratory, Oak Ridge, When a1 and az are small compared to unity, the Tenn. The sodium-24 used in this research was prefunction D - Dl may be simplified to pared by irradiation of reagent grade sodium carbonate using the Oak Ridge research reactor at Oak Ridge D - D1 = (SI - SZ) - PDr/3] (14) National Laboratory. The radioactive purities of the Pt yttrium-88 and of the sodium-24 samples were verified The presence of SI and Sz reflect the lack of “filling” by checking their respective y-ray spectra. the frit to a lineqr concentration profile, while the presThe following procedure was employed. Trace ence of ~ D reflects T the partial “emptying” of the frit at amounts of the radioactive species, z4Na+, 88Y3+,and the start of diffusion when the upper solution containing 45Ca2+were added to 1.00 m sodium chloride in the tracer is replaced with fresh unlabeled solution. The lower compartment. Diffusion was allowed to proceed terms involving X1 and Sz may be significant if the prefor a prediffusion time, T . The upper compartments diffusion time is very short, while the term in ~ D isT were then emptied and rinsed four times with 20-ml significant only when the prediffusion time is of the portions of 1.00 m sodium chloride. The diffusion run 8ame order of magnitude as the length of the diffusion was begun when the final filling of the upper compartrun. ment was started. Within a few hours after the conIt should be noted that the calculation of D - D1 by clusion of the diffusion run, “initial” counting rates eq 11 requires a preliminary estimate of D to evaluate of weighed samples of the upper and lower compartand a2. As a first approximation, D Z D Lmay be ments were determined on a well type sodium iodide used to evaluate a1 and a2. Also, the parameter a is scintillation counter. After sufficient time had elapsed
+
+
+
w
+
+
>[$
The Journal of Physical Chemistry, Vol. 74, No. 2.3, 1970
4168
NOTES
for the sodium-24 to decay away, the same samples were again counted on the sodium iodide counter to give final counting rates. To detect a possible change of counting efficiency between the initial counting period and the final counting period, a standard solution of yttrium-88 was counted each time. The initial and final counting rate data, plus the experimental observation that the p decay of calcium-45 did not register on the sodium iodide counter, allowed calculation of XN*+and X y s + , where X denotes the ratio W / V . Appropriate decay corrections were made for decay of sodium-24 during the initial counting period and for the decay of yttrium88 between the time of the initial counting period and the time of the final counting period. To evaluate X c a a + , samples from the upper and lower compartments were counted on a Model 314 EX Packard Tri-Carb liquid scintillation counter using the scintillation cocktail described by Bray. lo The experimentally determined counting efficiency of yttrium-88 on the Tri-Carb (measured relative to the efficiency on the sodium iodide counter) allowed the contribution of the yttrium-88 to the observed counting rates to be eliminated, since the yttrium-88 counting rate on the sodium iodide counter was known. Corrections for background were, of course, applied in all of the counting experiments. The counting rates were low enough so that coincidence corrections were not needed. The experimental values of X are given in Table I along with other relevant ex-
Table I : Experimental Parameters Parameter
P
x f 7,
sec
t , sec
102 X X ( N a + ) 102 x X(Caa+) 102 x X(Y3+)
Cell 1
0.13985 0,0148 1.0124 2.28 x 103 3.750 X lo4 3.172zt0.014 1.887zt0.012 1.454 iz 0.008
Cell 2’
0.12000 0.013 0.9117 3.765 x 104 4.956 X lo4 3.8782~0.027 2.420iO.Olg 1.975=!=0.005
perimental data. The error limits on the X values correspond to a 90% confidence limit, considering counting errors only. To provide a test of the theory under conditions where D - D’is large, both the prediffusion time and the diffusion time for cell 1 were kept very short, while the prediffusion time for cell 2’ was long enough for a linear concentration profile to be established. Unfortunately, this procedure also has the effect of making the results very sensitive to any small disturbance of the frit during handling. However, past experience in this laboratory indicates that the “error” in the D 1values reported here introduced by “frit disturbance” should be less than the corresponding “error” introduced by the uncertainty in the counting data. The Journal of Physical Chemistry, Vol. 74, No. $3,1070
Results and Discussion From the data in Table I, a value of D 1was calculated using eq 1, and the “correction term”, D - D1,was calculated using eq 11. To evaluate 011 and aZ,the value of D 1from cell 2’ was used as an approximation for D. The approximation, u2 = 2h/P was used to evaluate u2. The results of these calculations were then combined to give a value of the diffusion coefficient, D. Since, for cell 1, the “correction term” is significantly larger than the experimental uncertainty in the value of D ,a test of the theory may be made by comparing a diffusion coefficient obtained from cell 1 with the corresponding quantity obtained from cell 2’. The results of this comparison are given in Table 11, Table 11: Comparison of Theory with Experiment: Tracer Diffusion Coefficients in 1.00 N Aqueous KaC1 a t 25” -1O6(0
- D1)theory-
Ion
Cell 1
Cell 2‘
a4Na+ 45Ca2 +
0.030 0.049 0.056
-0.004 -0.002 -0.002
m y 3
+
,-.----io5D------Cell 1
1,247“ 0.773 0.614
Cell 2’
1.244“ 0.777 0.633
a For sodium-22, 1 0 6 D ~ , + = 1.236 [R. Mills, Rev. Pure A p p l . Chem., 11, 78 (196l)l.
where theoretical values of D - D 1 and the corresponding diffusion coefficients are listed. Each entry represents the result of a single experiment. We note that good agreement between the cell 1 and cell 2’ diffusion coefficients is found for both 24Na+and 45Ca2+. For **Y3+,however, the cell 1 value is about 3.1% lower than the cell 2’ value. This difference may be due to experimental error. However, the possibility of a small defect in the theory cannot be ignored. More extensive experimentation would be needed to settle this point. Since we expect no significant isotope effect between 24Na+diffusion and 22Na+diffusion,ll it is significant to note that the diffusion coefficient for 24Na+ determined in this research is in good agreement with the literature value for z2Na+. With due consideration of the experimental uncertainties, me conclude, from the data, that the theoretical expression for D D 1is in error by no more than -30% for the range of Dr values spanned by the data. Thus, the data, although sparse, do demonstrate that the theoretical expression for D - D‘ is, at worst, semiquantitative. Indeed, the expression for D - D 1may be quantitative. However, in the absence of additional data, it would seem advisable to restrict the use of eq 11 to systems (10) G. A. Bray, Anal. Biochem., 1, 279 (1960). (11) Preliminary measurements in this laboratory have indicated ) that there is no significant isotope effect (i.e., above ~ 0 . 5 % between 24Na+ and zzNa+ diffusion in 4.0 M NaC1. Therefore, we also expect the absence of an isotope effect in 1.00 M NaCl.
COMMUNICATIONS TO THE EDITOR
4169
where an error of -30% in D - D’ has a negligible effect on D. In practice, this is not a serious restriction. To carry out more stringent tests of the D - D’ correction, one should use diaphragm cells having values of X considerably larger than the cells used here. The results of this research demonstrate that one may use prediffusion times significantly less than those calculated from the Gordon inequality. For example, in the case of Na+ diffusion in 1.00 M NaC1, the Gordon prediffusion time is about 4 hr. Yet, as the data in Table I1 indicate, accurate diffusion data may be obtained when the prediffusion time is only about 0.5 hr, provided the correction term, D - D’,is used. The practical significance of this research is more obvious when diffusion coefficients are very small. For exam-
cm2/sec, the Gordon prediffusion ple, if D = 2 X time is about 30 hr for our diaphragm cells, which are typical of those in common use. However, by using the correction term, D-D’ (assumed to be accurate within 30%) diffusion data accurate to within 1% may be obtained even when the prediffusion time is only 5 hr and the diffusion time is 24 hr.
Acknowledgments. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, and to the University of Tennessee Faculty Research Fellowship Fund for partial support of this research. The author wishes to thank Dr. G. E. Boyd and Oak Ridge National Laboratory for the use of laboratory facilities.
C O M M U N I C A T I O N S T O THE E D I T O R Photodissociation of an ea,- Complex in Hydrogen-Saturated Alkaline Solutions
Sir: We have discovered an unstable transient species in continuously photolyzed H2-saturated NaOH solutions at pH 11. This species may be ionized by light of wavelengths as long as 340 nm and produces eaq-, the hydrated electron. Since the concentration of ea,- produced on photoionization increases with added Na+ concentration, we tentatively attribute this species to a dissociable complex involving Na+ and ea,-. Assuming its structure to be (Na+.e-)&, we write (Na+.e-),,
+ hv + Naas+ + ea,-
We interpose Corning filters 0-52 and 0-53 in holder i in order to absorb the shorter wavelengths. See the inset of Figure 2 for the optical transmission properties of these filters. I n juxtaposition with (i) about 300 nM eaq- is produced by the xenon flash without the cutoff filters. The reliability of this unit is greatly improved over the earlier modell by accumulating the signals from several carefully controlled flashes for each experiment. The eaq- signals are stored in the 400-channel analyzer and read out on an XY recorder.
(1) complex,
However, a t this stage, an impurity-e,,H-, or excited water cannot be excluded. The apparatus used for creating and detecting this species is diagrammed in Figure 1. It consists of a sensitive ea,- detector, an Hg vapor lamp for generating a steady-state concentration of e,,-, and an analyzing xenon flash lamp. The tungsten lamp (a), Suprasil irradiation cell (b), 700-nm filter (c), photomultiplier (d), 400-channel analyzer (e), and X Y recorder (f) constitute the ea,- detecting system. With it we may study transient ea,- signals with a sensitivity of 0.1 nM. Hydrated electrons are continuously generated in the irradiation cell (b) a t a rate of about 10 nM/ sec by a low-pressure Hg vapor lamp (g). A xenon lamp (h) provides a 50-joule light flash of 30-~sec half-width and generates about 50 nM ea,- when placed at a distance of 20 cm from cell b. Lamps g and h produce and transmit light down to
