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RICHARD D. D E ~ ~ AAND R S IRVINGSHAIN TABLE I1 X-RAYDIFFRACTION DATAON SULFUR SAMPLES
Monoclinic after 24 hr. (I/Zl)
n
4.11 3.90 3.48 3.36 3.24 3.14
14 100 37 38 i6 59
2.63
8
2.43 2.39
54 10
2.15 2.13 1.996
8 16 35
Pure orthorhombic2 d (I/II)
7.62 5.75 4. t i 4.06 3.85 3.56 3.45 3.34 3.22 3.11 3.09 2.85 2.68 2.62 2.57 2.50 2.43 2.38 2.29 2.14 2.11 1.988
Monoclinic after 24 hr. d (I/Ii)
6 10 1 16 100
.. I
19 31 39 21 15 15 3 11 5 9 10 9
1.906 1.829
25 6
1.760 1.i26
i 2
1.649
8
1.607
2
1.531
8
4 3 14 3
Pure orthorhombic2 d (I/Zl)
1.959 1.901 1.823 1.783 1.757 1.725 1.696 1.668 1.649 1.622 1.604 1.561 1.535 1.529 1.499 1.4i4 1.457 1,437 1.418 1.388 1.351 1.305 1,279
5 8
8 3 n
13 6 1 3 8 5 2 3 2 1 3 2
4 4 2 3 2 2
not be expected for the two samples for the same reason. In view of the recent experiments of Hartshorne and co-workers on the rate of transformation of monoclinic to rhombic sulfur,13 i t would appear certain that the conversion to orthorhombic sulfur should be complete in 24 hr. It would seem possible that the rate of this conversion may be increased by the X-ray irradiation. It has been shown that this is the case for the transformation (13) N. H. Hartshorne and hl. Thackray, J. Chrm. Soc., 2122 (1957), and earlier papers.
[CONTRIBUTION FROM
THE
Yol. SI
of plastic sulfur to the rhombic formI4 and more recently i t has been reported that rhombohedral sulfur (SB) is converted to a mixture of plastic aiid orthorhombic sulfur on exposure to X-rays.’j In some further experiments in which crpstallization of the melted sample was induced by various methods (touching with a spatula, tapping the side of the holder, etc.) various anomalous patterns were obtained which could not be indexed as completely, using the nionoclinic cell constants. l y e have not yet been able to assess completely the significance of these results. JThenever the sample was allowed to crystallize undisturbed, however, reproducible patterns were obtained. Acknowledgment.-The authors express their appreciation to the Texas Gulf Sulphur Co., Newgulf, Texas for a fellowship to J. s. I 0, T = 0 U+ masX--+ m,T>O dV _ bx - U[1 f Z exp( Y T ) ] x, = 0, T
A- SPHERICAL
a W
8 - LINEAR
a
0.31 I
1
2.o
I
I
I
I
I
2.5
I
2
3.0
of2 x
and
lo3
Fig. fL-Theoretical peak current as a function of p a n e and spherical electrodes.
TABLE I1 CURRENT AS A FUNCTION O F COXCENTRATIOS REDUCTION OF IODATE AT pH 4.7 Concn., moles/l.
x x x 1.00 x 5.00 x 1.00
10-4 10-6
Peak current," @amp.
3.83 1.86 0,750 ,381
. 4 ~ dev., . ~
5% 0.8 .6 .6
FOR TIlE iP/
concn. b
38.3 37.3 10-5 37.5 10-5 1.0 38.1 10-6 ,184 1.4 37.2 Average and average deviation of six determinations. Microamperes/millimole/liter.
5.00 2.00
=
* Do
exp
(14)
>0
[- an,F ( E ; - EO)] RT
(15) (16)
for
Effect of Diffusion Coefficient.-The current is also a complicated function of the diffusion coefficient, since U ( 0 , T )is a function of the diffusion coefficient. Since the data in Fig. 1 are all based on a diffusion coefficient of 1.00 X 10-5 cm.2/sec., i t is again necessary to convert all points to values for a real system by obtaining a ratio of peak currents corresponding to 1.00 X and the actual diffusion coefficient and then reducing all data to the same basis. This ratio can be obtained from Fig. 6, where, for comparison, the variation of current with D'/tOfor a plane electrode is included. For accurate work a large scale reproduction of curve A of Fig. 6 would be required.14 Current-Concentration Relation.-As expected (equation 11) the peak current is a linear function of the concentration, although the sensitivity is much lower than would have been expected had no kinetic effects been present. Table I1 demonstrates the linearity between peak current and concentration, and, a t the same time, indicates the general reproducibility of these irreversible waves. PEAK
where
1
(13)
Conclusion The use of the method of voltammetry with linearly varying potential for the investigation of slow electrode processes is rather cumbersome, due to the lack of an explicit relation between all kinetic and experimental parameters. On the other hand, i t is very easy to obtain the experimental data, since all kinetic information required can be taken from one current-voltage curve.
Replacing both differentials in equation 12 by central differences a t T l/zAT for the interval T AT,ll one obtains
+
+
+ AT) = 41 [ U ( X + A X , T f A T ) f U ( X + A X , I') f U ( X - A X , T + A T ) f
V ( X ,T
U((x
- A X , T)]
(18)
This equation holds under the restriction that AT/(AX)z= 1 and will be valid for all points in a rectangular X and T network except those points that lie on a boundary, provided AX and AT remain small enough. The boundary points in the X direction can be determined by extending the network far enough so that the value of U does not change during the electrolysis. Representing this distance by m u , where m is a sufficientlylarge integer, each boundary point in the A' direction will be of the form L'(mAX. 1')= 1
+ mAX
(19)
The boundary points in the T direction can be obtained by expressing equation 15 as a central difference in X and T and again in X and T AT. Combining these equations with equation 1s one obtains
+
+
V(0,T AT) = U ( A X , 7 + A T ) + U ( A X , T ) - A X U ( 0 , T)[1+ Z e x p ( YT)] 2 f AX ( 1 2 e x p [ Y ( T f A T ) ] ) (20)
+
Each boundary point a t X = 0 will be of this form. The problem now is to determine U as a function of X and T on a rectangular X and T network. For each increment in T there will be m simultaneous equations of which one will have the form of equation 19, one the form of equation 20 and m - 2 the form of equation 18. The procedure is started by calculating the points on the T = 0 boundary using equation 13, obtaining values of U for the range X = 0 t o X = mAx. Starting with these points, values of U on the T = T I increment are guessed, using a quadratic extrapolation as a guide. The guessed values are then iterated using the formulas outlined above until all the answers along the T increment are self consistent. The procedure is
CONDUCTANCE OF DILUTESOLUTIONS OF 1-1 ELECTROLYTES
June 5, 1959
repeated until all t h e variations of been satisfied.
X
and T have
TABLE TI1 PEAKCURRENTAS A FUNCTION O F AT A T X 104
ip(pamp.)
% ’ errorb
Aip“
0.7440 0.1412 26.0 20.00 ,0731 8.26 .8121 10.00 ,0373 4.21 .8479 5.000 ,0188 2.12 .8664 2.500 .0094 1.06 .8758 1.250 .0047 0.53 .8805 0.6250 ,0023 0.26 .8829 0.3125 .0011 0.13 .8841 0.1563 0 0 .8852’ 0 4 Difference between peak current a t indicated value of A T and peak current a t AT = 0. * (Ai,/0.@52) X 100%. Extrapolated.
This calculation is very time consuming and tedious and is not practical unless computer facili[CONTRIBUTION No. 1525 FROM
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2659
ties are available.’’ This is particularly so since the accuracy of the method depends on using small values of AT to reduce the error. The convergence of the calculations for decreasing values of AT was checked using the standard set of experimental and kinetic parameters (Table 111). As a compromise between accuracy and excessive machine time, a value of AT = 0.3125 X loe4 was used in this work. The results can be considered accurate within about 0.3%. Acknowledgments.-This work was supported in part by funds received from the United States Atomic Energy Commission under Contract No. AT(ll-1)-64 and the Research Committee of the Graduate School of the University of Wisconsin with funds received from the Wisconsin Alumni Research Foundation. (17) This problem was programmed for the I B M type 704 with the help of Mr. Lee F. Thompson of this Laboratory. It was Mr. Thompson’sability and patience which made this work possible.
MADISON,WISCONSIN
STERLING CHEMISTRY LABORATORY, YALEUNIVERSITY
Conductance of Dilute Solutions of 1-1 Electrolytes1 BY RAYMOND M. Fuoss RECEIVEDDECEMBER24, 1958 The conductance of solutions of 1-1 electrolytes decreases with increasing concentration because the mobility of the ions is decreased due to the action of long range interionic forces and because the concentration of free ions decreases due to association increasing with concentration. A conductance equation based on these effects is derived; it is applicable to 1-1 electrolytes in solvents of dielectric constant greater than about 15, at concentrations up to that corresponding to r(u 0.2 ( a = ionic diameter, K = Debye-Hiickel parameter). I n solvents of high dielectric constant, the equation reduces to Onsager’s limiting tangent while in solvents of low dielectric constant, it transforms into the classical Ostwald dilution law. In the intermediate range of dielectric constant, previously inaccessible to theoretical analysis, the equation describes conductance data in terms of three molecular parameters: the limiting conductance, the ion size and the association constant. Methods of applying the equation to experimental data are described.
The theory of the conductance of symmetrical electrolytes in dilute solutions now appears to be substantially complete; various parts of the derivation of the final equation are, however, scattered among a number of different publications.2-11 The purpose of this paper is to present a summary derivation for the case of 1-1 electrolytes, unobscured by mathematical details, and to describe the analysis of conductance data by means of the theory. Specific conductance, the experimentally observable quantity, is given by the ratio of current density t o field strength X and is therefore proportion to Zn,eiui, where ni is the number of ions of species i per unit volume and e; and ui are, respectively, their charge and mobility. The equivalent conductance A is defined as the ratio of 1000 times (1) Presented a t 135th meeting of the American Chemical Society, 5-10 April 1959, Boston, Massachusetts. (2) R. M. Fuoss and L. Onsager, Proc. N a t . Acad. Sci., 41, 274 (1955). (3) R. M. Fuoss. J . chim. phys., 63, 493 (1956). (4) R. M. Fuoss a n d L. Onsager, J . P h r s . Chem., 6 1 , 668 (1957). ( 5 ) R. M. FUQSS, THISJOURNAL, 7 9 , 3301 (1957). (6) R. M. Fuoss and C. A. Kraus, ibid., 79, 3304 (1957). (7) R. M.Fuoss, ibid., 80, 3163 (1958). (8) R. M.Fuoss, ibid., 80, 5059 (1958). (9) R.M.Fuoss and L. Onsager, J . Phys. Chem., 62, 1339 (1958). (10) R. M. Fuoss, ibid., 63, 633 (1959). (11) R.M .Fuoss and F. Accascina, “ L a Conducibiliti Elettrolitica,” Edizioni dell’Ateneo. Rome, 1959.
the specific conductance to the concentration equivalents per liter. Consequently A
-
zyiui
t
in
(1)
where yi is the fraction of the ions of species i which actually contribute to transport of charge a t a given concentration. The Debye-Hiickel-Onsager theoiy assdmes that yi equals unity and ascribes the decrease of A with increasing concentration to a decrease in mobility arising from the electrostatic forces between the ions. The mobility is found to be Ui
- A U i ) (1 f Ax/x)
= (u”
