T H E APPLICATION OF T H E RAYLEIGH INTERFEROMETER TO THE MEASUREMENT O F REACTION VELOCITY DANIEL B. LUTEN, JR. Department of Chemistry, University of CaliJornia, Berkeley, California Received M a y $9, i934 INTRODUCTION
Duane (6), in 1901, demonstrated that the progress of a reaction in the liquid phase could be determined by measuring the changes in refractive index of the reacting solution. A t the same time he showed that the same result could be obtained by measuring the changes in volume of the solution during the course of the reaction. Koelichen (11) had first employed this “dilatometric” method the preceding year. The latter method has since been employed frequently in rate investigations; the former has been almost abandoned. Recently, however, with the increased use of the Rayleigh interferometer, a few investigators (9, 16, 17, 19) have employed that instrument in the determination of rates of reactions. None of these, however, has devoted much space to a description or analysis of his method, which deserves more consideration. There is a close relationship between the dilatometric and refractometric methods; this is a necessary consequence of the exactness of all the formulas for specific refraction, in which the refractive index is closely inversely proportional to the density. Wagner, Fajans, and Kohner (12), among others, have shown that for completely ionized salts and acids, as well as for unionized substances in aqueous solutions, the molecular refractions calculated from the Lorenz-Lorentz equation and the Biot-Arago law of mixtures are very nearly constants, rarely deviating by as much as 0.5 per cent from their mean values except over very wide ranges of concentration. Since the Lorenz-Lorentz equation is also the soundest, theoretically, of the relations between density and refractive index, we shall employ it in an investigation of the relations between the two aforementioned quantities and the molecular refraction. For a general reaction
A1+ . . . A ~ + . . . A ~ ~ B ~ + . . . B ~ + . . (1) . B z the Lorenz-Lorentz equation before any interval of the reaction is
199
200
DANIEL B. LUTEN, JR.
and after the interval is
(n2- 1) for the various components are designated by d(n2 2) M with sub- and super-scripts as indicated below: M for the solution before the interval of reaction, M’ for the solution after the interval of reaction, Mi for the reactant Ai, M j for the product Bj and M , for the solvent. pi and q i are the weight fractions, respectively, of Ai and B j before the reaction, and pi’ and yj’ their weight fractions after the reaction. Subtracting equation 3 from equation 2 we obtain
where the terms
+
The weight fractions of the terms remaining on the right-hand side of equation 4 correspond to the quantities consumed and produced according to the stoichiometric equation. That being so,
Mol. wt. Mol. wt. and P - P’ 4 - Y‘ are the same for all terms. Multiplying and dividing the right-hand side of equation 4 by this quantity and defining the molecular refraction,
R=
Mol. wt.
(n2
d
(n2
- 1)
+ 2)
we have
The right-hand side of equation 5 we will call K . K is a function only of molecular refractions, which may, in general, be calculated with a fair degree of accuracy, and of concentrations; hence it is readily determinable. Equation 5 may be solved as a quadratic in (n‘ -n), the change in refractive index, in.terms of the parameters n, d, d’, and K . The result is
(n’ - n) = -n f
(n2
+2) d’
(n2
(n2
- 1) (d’ - d ) - Kdd’
+ 2 ) (6)
(a2
- 1) - d (1 + Kd’) (n2+ 2)
201
MEASUREMENT O F REACTION VELOCITY
From equation 6 it is seen that the changes in density, concentration, and molecular refraction must be known in order to calculate the change of refractive index. It is also to be observed that there may be reaction systems wherein there will be a change in density but not in refractive index, and vice versa. Secondly, (n‘ - n) is very nearly proportional to the change in density, (d‘ - d ) , and to K , when these quantities do not exceed the magnitudes encountered in rate work, viz., when [ d’ - d [ < 0.01, K < 0.001.’ This is illustrated by table 1.
I
I
TABLE 1 Change of refractive index as a function of change of density and molar refraction (n’ - n ) = f [ K , n, (d’ - d ) , d ] K
= 0,d =
1 , n = 1.333
-n U’-d n’
n’ - n
d‘-d
-
I
K
d‘
I
-0.01 -0.00001
-0.01773 0.001783 0.00001784 0 +0.00001783 0.001785 0.01794
,001
.00001 0 -0.00001 -0.001 -0.01
-1.773 1.783 1.784 -1.783 1.785 1.794
d = 1, n = 1.333 (nc
1
- n)
I
0.01 0.01 0.00001 0.00001
-0.01 -0.00001
$0.01
+o +o
0,3670 0.3669 0.366
1
-d
-
0,3885
(a) K
(n’ - n)
Ii
0.3844
0.03844 0,003685 0.0003670 0.00003669 0.00000366 0
0.1 0.01 0.001 0.0001 0.00001 0.0
- d ) = 0,d = 1, n = 1.333
(d‘
- n)from obtained as sum of limiting values columns 3 and 6 above
(n’
I
0.02184 0.003691 0.01786 0.00002150
0.01784 0.0000178 0.01784 0.00001784
+ 0.00367 = 0.02151 + 0.003669 = 0.003687 + 0.00000367 = 0.01784 + 0.0000367 = 0.00002151
1 To show that the limiting values for K are reasonable we have calculated the change in molecular refraction for the following reactions:
RCN RCOOR
+ HIO + HIO
+ ROH
ZR ZR
- ZR’ - ZR‘
+ (CH3)RCO
ZR
- ZR’
+ RCONHz + RCOOH
=
+ 0.39
+ 0.17
R
/
-+ RICO
RnC(OH)CH&
\\
=i
- 0.69
0
I n dilute aqueous solutions, ( p - q)/mol. wt. = AN.lO-r, A N being the change in normality, and K = N ( 2 R - ZR’).10-3
If we take N
= 0.1,
then K falls between 10-4 and 10-6.
202
DANIEL B. LUTEN, JR.
In dilatometric investigations it has generally been assumed that the small changes occurring in the density are strictly proportional to the changes in concentration. We see from the above considerations that, if the assumption is valid for changes in density, it is also valid for changes in refractive index. However, with the interferometer it is not necessary, although it is convenient and generally satisfactory, to make this assumption, for usually a direct determination may be made of the relation between concentration and refractive index. EXPERIMENTAL METHOD
The general methods for using the Rayleigh interferometer have been adequately described (1, 2, 4, 7, 10, 20). A very brief description of the instrument will be sufficient here. A beam of light is split into two portions, one of which passes through a length of a “reference” substance, then through a fixed glass compensator plate, the other through an equal length of an “unknown” substance and through an identical compensator plate; finally the beams are reunited through slits to form interference fringes. If the optical paths through the reference and unknown substances are not the same, i.e., if the refractive indices of the solutions are made different, there will result a shift of the fringe system. By adjusting one of the compensator plates by means of a micrometer screw this shift is overcome, and the fringes are brought back to the zero position. Differences of refractive index may be determined as a simple, almost linear, function, of micrometer readings. Two corrections must be applied to the readings to make them a true linear function of the differences in refractive index. These have been treated fully by Adams (l),Barth (2), and others. Crist, Murphy, and Urey ( 5 ) have noted recently that in some instruments errors may be introduced, owing to imperfections in the compensator plate mechanism. The substances (liquids or gases) to be compared are placed in the two troughs of a chamber fitted with glass endplates, and the chamber is then inserted in the instrument so that the two beams of light pass each through one of its troughs. The instrument used in this investigation was a Zeiss portable liquid interferometer. Ordinarily it is advisable that the reaction be allowed to proceed in the chamber rather than that the instrument be used for analysis of samples of the reaction solution at suitable times. This is largely a matter of convenience, but uncertainties, which are better avoided, are introduced by the large amount of manipulation required by the latter method. PREPARATION OF SOLUTIONS
The following discussion is restricted to reactions in the liquid phase. In the simplest case the reaction mixture is obtained by mixing equal amounts of two solutions, A and B, each containing one of the reactants,
MEASUREMENT OF REACTION VELOCITY
203
their concentrations being, of course, double those desired in the experiment, I n addition a third solution, C, is prepared of some inert substance such as potassium chloride, which will not react with anything present in solution B. The concentration of C is such that when diluted by half it will give a zero reading in the interferometer when compared to solution A when the latter is diluted by half.2 Then equal amounts of C and B, previously brought to thermostat temperature, are placed in one chamber; equal amounts of A and B are mixed, the time of mixing being taken as the zero time of the reaction, and are placed in the other chamber. As a consequence of the manner of preparation of the solutions, the readings a t zero time should approximate the zero of the instrument. If the kinetic picture is simple, this may be shown to be true by extrapolating the readings to zero time. If, however, there is a rapid preceding reaction, the extrapolation to zero time will give a reading which differs from the zero of the instrument by the change in refractive index of that reaction. Using this method it is possible then to establish with a fairly high degree of certainty3 the existence or non-existence of preceding reactions. In B similar manner, by using in the reference solution a substance thought to be a product of the reaction, the high probability that it is or is not a product may be established. Szego (19), in investigating the oxidation of nitric oxide to nitrogen dioxide, ingeniously took advantage of the fact that there is a change of refractive index in the oxidation, but none in the reaction of nitrogen dioxide to give nitrogen tetroxide. Consequently, with the interferometer he could follow the oxidation without regard for the existence of the following equilibrium. A few other applications will suggest themselves. After such information on stoichiometry has been obtained, much less care need be taken with the reference solution; it is better, however, to have the readings during a run on the lower part of the scale, so that the corrections will be small and the high differences in dispersion accompanying high differences in refractive index will be minimized. REDUCTION O F OBSERVATIONS
The problem of the reduction of observations for first-order reactions has been treated satisfactorily by Guggenheim (8). I n the method described by him a series of readings V I . . . v i . . . vn are made a t 2 The comparison of C and A must be made after dilution because of the shifting of the achromatic fringe, owing to the almost inevitable difference in dispersion of the two Solutions. If the differences in dispersion are too large, the fringes will disappear (cf. Hirsch (9)). a This degree of certainty is dependent upon the extent to which reactions are accompanied by changes of refractive index. We have experience only with nitrile, amide, ester, and diacetone alcohol hydrolysis, and with acid ionization (shift of equilibrium in buffers). The change of refractive index is of the same order of magnitude in each of these.
204 times times
DANIEL B. LUTEN, JR. tl tl
. +
. . t i . . . t n and another series vi . . . vi. . . . . . t i + . . . t n + Then from the equation
v: a t
7.
lcti =
2.303 log
IO
(v: -
vi)
+ constant
(7)
the velocity constant may be obtained. Guggenheim obtained the constant by plotting log (v’ - v) against t , but that is entirely incidental to the method. The constant may be obtained by any of four standard methods, viz., by plotting, by a method of averaging, by an unweighted least squares solution, or by a weighted least squares solution. The advantage of Guggenheim’s method is that each reading is used once and once only; every point on a graph and every value in a calculation is independent of all other points or values. It is reasonable then to apply sophisticated methods of analysis to the data. Reed and Theriault (13) have criticized Guggenheim’s treatment on the grounds that, since the weights theoretically to be assigned to the points diminish as the reaction progresses, no treatment which neglects this factor can yield better than very approximate results. The statement implies that the graphical reduction of data is essential to the method, and that is not true. Reed and Theriault (14) conclude a paper on the least squares treatment of first-order reactions with the rather extreme statement that, “. . . . the least squares treatment should be used whenever the velocity constant is to be determined within an allowable error of, say 5 per cent.” It is a fact not very generally recognized that whenever a very considerable body of data is a t hand, almost any reasonable method of reducing that data to a final result will yield a result very close to the best value obtainable from a theoretically perfect least squares solution. A preliminary demonstration of the truth of this statement may be given by calculating velocity constants from the sample data given by Guggenheim. There are two sets of data taken during a single run, with twenty pairs of points in each. The constants are calculated by the four formulas given below, and the constants obtained are given in the right-hand columns: k ( X 0.4342 X (1) Graphically (by Guggenheim) : 0.4342kti
+ log (v: - v,) = Const.
0.4342k =
-
n
log (vi
- Vi) - n / 2 + 1 log ( V l ti 1
-
2
ni2S-1
t;
6.54
6.56
6.558
6.566
(8)
(2) Method of averages 7Ll2
loa)
- Vi) (9)
MEASUREMENT OF REACTION VELOCITY
205
(3) Unweighted least squares:
6.565 6.568
(4) Weighted least squares :
6.556 6.565
0.43421c = -
It is fairly obvious that the graphical method is not sensitive enough to do full justice to the data, but the uncertainty introduced by using any of the first three methods as alternatives for the weighted least squares is of an order of magnitude far less than 5 per cent. While with accurate data it is thus apparent that the uncertainty introduced by using a less exact method of calculation may be reduced to a small quantity, there is yet no indication of the relationship between that uncertainty and the probable error which arises from accidental errors in the data. To shed light on this point four identical runs were made at low concentrations (0.005 N ) to insure large probable errors in the results, which are given in table 2. The data are admittedly not sufficiently ample to be entirely convincing, but the indications are clear. The weighted least squares result is to be taken as the standard with which the other three are to be compared. The graphical method appears to be the least reliable, since it gives the largest deviations from the weighted least squares solutions. The solutions by averages and the unweighted least squares solutions give results which are of about the same reliability; since the latter method is somewhat more laborious, we shall discard it in favor of the method of averages. The differences between the constants obtained by averaging and those obtained by weighted least squares are all less than the probable error in the latter constants, averaging about two-t,hirds of the probable error. The principle that the uncertainties introduced by calculations must be negligibly small in comparison with the experimental error, is indisputably sound in the common situation where the calculations form the lesser part of the investigation. There is, on the other hand, practical justification for using a less precise method of calculation in those situations where the time required for a least squares solution would be a major fraction of the
206
DANIEL B. LUTEN, JR.
total time required for the work. In rate investigations there is a place for each of the methods with the exception of the third. The graphical method is the best when there are trends toward the end of the experiment due to complications such as often appear in organic reactions. These are usually of such an involved nature, or are so incompletely understood, that they may not be treated exactly, but the observer is, none the less, perfectly justified in giving less weight to points obtained after the appearance of the trend. On the other hand, if there are no systematic trends, if a fairly large amount of data can be obtained in each run, and if there are a large number of runs to be handled, then the method of averages is the most satisfactory solution. If the quantity of data is comparatively small and is characterized by a relatively high probable error, then the fourth method will give best results. Furthermore, if the data are known TABLE 2 Rate constants f o r runs calculated b y f o u r methods
I
R U N NO.
B A T E CONSTANTS
i '
1 . .. . . . . . . . . . . . . . . .
4 Weighted least squares
3
2
Graphical
Unweighted least squares
Averages
0.03853
0,03823
0.03809
0.03805
f O ,00057
f O ,00059
f 0.00066
=to,00056
4..................
0,03959 0.04003 0.03894
0.03973 0.03924 0.03779
0.04007 0.03853 0.03735
0.03982 0.03878 0.03802
Mean .............
0,03927
2 . .. . . . . . . . . . . . . . . . 3.. ................
..I
I
0.03868
1
0.03841
1
0.03885
Probable errors determined for each method by the external consistency of the four constants.
to be characterized by certain simple constant errors of known type, then the method described by Reed and Theriault finds its application. In the interests of greater accuracy two cautions are to be observed. The time between the two readings of each pair should be equal to two or three half-times of the reaction. Roseveare (15) has shown that the rate constant is most accurately determined from two points, if the second occurs a t two-thirds completion of the reaction. Accordingly, it is well to plan a concentration of points a t the beginning of the reaction and another concentration near two-thirds completion. Higher order reactions must still be treated by the methods developed for ordinary chemical analysis, which are apt to be, with this instrument and the dilatometer, incapable of yielding very reliable results. Roseveare has given an equation which eliminates the necessity of obtaining the infinite time reading directly for the bimolecular reaction, where dx/dt = k(a - xy.
207
MEASUREMENT O F REACTION VELOCITY
RELIABILITY OF THE INTERFEROMETER
Returning to an examination of the data given in table 2, and adding the data from which the’ constants in run No. 1 (see table 3) were obtained, we find evidence that the systematic errors introduced by the interferometer are very small. If a number of readings be made with the interferometer chambers empty, the probable error of a single reading, determined from the degree of consistency of the readings, is on our instrument about 0.2 division. In TABLE 3 Results obtained in run No. 1 k = 0.03809 1’
-v
’ - v (cslcd.)
26.4 25.1 24.4 23.4 22.3 21.4 20.3 20.1 18.9 18.5 17.9 17.1 16.4 15.4 15.2 14.4 14.2 13.9 13.4 12.8
26.0 25.0 24.2 23.2 22.4 21.5 20.7 19.9 19.2 18.4 17.8 17.1 16.5 15.9 15.2 14.7 14.1 13.6 13.1 12.6
v‘
V‘
RESIDUALS
minutes
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
106.8 108.3 109.3 110.5 111.7 112.9 114.0 114.5 115.6 116.1 117.1 117.8 118.6 119.6 120.0 120.8 121.0 121.2 122.0 122.5
63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82
133.2 133.4 133.7 133.9 134.0 134.3 134.3 134.6 134.5 134.6 135.0 134.9 135.0 135.0 135.2 135.2 135.2 135.1 135.4 135.3
0.4 0.1 0.2 0.2 0.1 0.1 0.4 0.2 0.3 0.1 0.1 0 0.1 0.5 0 0.3 0.1
0.3 0.3 0.2
run No. 1 the probable error in (v’ - v) is 0.18 division as determined by the internal consistency of all the readings, that is, by the formula T
= 0.6745
,/= n-2
where Wi is the residual (vi’ - vi) obed. - (vi’ - v i ) oalod. The readings vi’ were each taken three times; assigning them a proper weight, the law of propagation of errors gives the probable error in v i equal to0.16. The agreement between this value and the one obtained from direct readings
208
DANIEL B. LUTEN, JR.
indicates that the first-order rate equation is satisfactory. The probable error in the velocity constant, arising from this probable error in TJ, is found to be 0.00046. The probable error in the constant obtained from the external consistency of the four runs is 0.00059. Birge (3), in his treatment of the problem of internal and external consistencies, has shown for the case where there are four functions being compared, that owing t o statistical fluctuations, the ratio of probable error from external consistency to that from internal consistency should be 1.24. It is in the case given above, fortuitously, 1.28. A much less close agreement would still be a strong indication that there are no systematic errors, which vary only from run to run! TEMPERATURE CONTROL
i
The commercial instrument is very inconveniently thermostated. Fairly satisfactory results were obtained within five or ten degrees of room temperature by pumping distilled water through 30 ft. of copper tubing in a thermostat, and then through the reservoir which surrounds the chamber containing the solutions. By making heavy brass cover plates for the chamber, of the type described by Cohen and Bruins, but hollow so that water could be drawn through them, the temperature regulation was made very satisfactory within ten degrees of room temperature and reasonably so within fifteen. The difficulties arising are due not to temperature variations which affect the rates of the reactions, but rather to much smaller inhoniogeneities of temperatures, both in the reacting solution and in the surrounding bath, which affect the refractive indices enough to make the fringes very unsteady. A thermostat controlling the temperature to 0.01OC. should be satisfactory provided there are no sudden changes of Since the above was written it has been found that the instrumental defect described by Crist, Murphy, and Urey (5) is more common than might have been suspected. Results as consistent as those given above cannot be obtained from an instrument having an appreciable error of the type described. This error appears when the t i p of the micrometer screw is not in the axis of the screw. A deviation of 0.05 mm. is sufficient to produce the results described by Crist, Murphy, and Urey. The error is periodic, going through a cycle in one complete turn of the micrometer screw. It disappears when the lever arm on which the screw acts is perpendicular to the screw. It was under these special conditions t h a t the runs described above were made. Crist, Murphy, and Urey calibrated their instrument by measuring each fringe throughout the whole scale. Adopting different tactics, we removed the micrometer screw from our instrument and reground the tip so as to center it. This can be done only on a very good lathe. After this treatment no regular periodicity could be found. If this correction is not made, systematic errors as large as 2 per cent may appear in the rate constants when the concentrations are as low as those described above. At higher concentrations the errors become much'less important.
MEASUREMENT OF REACTION VELOCITY
209
temperature within it amounting to more than 0.002"C. It is not possible with the present design of the instrument to obtain temperature coefficients over a wide range of temperature. If the heat of the reaction is large, the temperature of the reacting solution may rise slightly above that of the thermostat and the reference solution; this results in a very easily perceptible change of refractive index. From the nature of the relations involved it appears that the temperature difference reaches a maximum in a very few minutes, but is dissipated slowly, and exists as a potential source of error throughout three-quarters of the life of the reaction. For first-order reactions this potentiality disappears when we examine the situation more closely. The temperature difference is given by the equation
( T - To) =
AHA& 1000(K - k)
[e-kl
-
e-KI]
which is the solution of the differential equation, dT AH dA . _ - - R ( T - To) - dt
1000 dt
(14)
k is the velocity constant, K the Newton's law of cooling constant, AH the heat of reaction in calories per mole, and A. the initial concentration of the reactant. We may set the observed reading at any time equal to the correct reading a t that time plus the difference due to the temperature difference, so that vi =
vi + xi
(15)
xi is directly proportional to the change in refractive index due to the temperature disturbance and so is directly proportional to the temperature difference. Therefore we may say in any given case that
xi =
c1(e--k'
- e-K'i)
(16)
Now the exponential forin of the first-order rate equation is
(vi - v,)
=
(v0- v,) e-"i
(17)
Eliminating xi and vi from equations 15, 16, and 17, we have, z being zero a t both zero and infinite time, (vi
- v,)
=
(vo
- v,
- C,)
e--kti
+
CI
(18)
As soon as the second exponential term becomes negligible with respect to the first, the temperature difference will have no influence on the rate constant. In a plot of log (vi - v,) against time the observed result would be that the y-intercept would be log (vo - v, - Cl) rather than log
I
-
210
DANIEL B. LUTEN, JR.
(DO - D-), while the slope would be unchanged. I n a particular case a t hand AH was about 18,000 cal., k was 0.0201 min.-', K for the interferometer chamber (all the metal chambers should give about the same value) was roughly 0.85 mim-l, and AOwas 0.05 N . The maximum temperature difference should be O.O19"C., occurring at 4.51 minutes. At that time the second exponential term is nearly negligible, and by eight minutes it is completely so. The temperature difference does not fall to the point where its effect on the refractive index is negligible for over two hours. In two runs where the first point was at fifteen minutes, there were no trends; in a third where the first point was a t three minutes, there was a very definite trend, and the rate constant was two or three per cent Iower. For higher order reactions the same situation may be found; the matter has not been investigated, and there is no certainty that such is the case. After several of the chambers had been used for some time, baffling irregularities began to appear in the results. It was finally found that the cement holding the end-plates in place had been partially dissolved away and the chambers had developed small leaks which resulted in a slow change of the refractive index of the solution. This is undoubtedly the source of the drift which was attributed by Stewart and Kung (18), in their use of the method, to unequal evaporation from the two chambers. A COMPARISON O F THE MERITS OF T H E DILATOMETER AND T H E INTERFEROMETER
In its present design the interferometer is less flexible than the dilatometer; it can not be used over a wide range of temperatures; certain waxed joints must be protected if non-aqueous solvents are to be used; it is probably not applicable to react,ions which are highly photosensitive throughout the range of visible light. Moreover the instrument is expensive. On the other hand the interferometer will normally handle faster reactions; it requires less care in thermostating; and when calibrated interferometers are available (in use for ordinary analytical work) results may be obtained with a minimum of effort. If only small amounts of reagents or solvents are available, experiments may be made in favorable cases with as little as 0.05 cc. of solution. It is difficult to compare the sensitivities of the two methods, since the practical consideration of temperature control constitutes the main restriction to the extension of the sensitivity of the dilatometer. Finally, as noted above, with the interferometer the presence or absence of rapid preceding reactions or of equilibria may be demonstrated with high probability. SUMMARY
1. The quantitative relabions between change of density and change of refractive index during reactions have been derived. It has been shown
* MEASUREMENT O F REACTION VELOCITY
211
that the change of refractive index is as reliable a measure of the progress of a reaction as is the change in density. 2. Practical methods for the application of the Rayleigh interferometer to rate measurements have been described. 3. It has been shown that the interferometer can be of value in the investigation of the stoichiometry of a reaction. 4. The reduction of observations has been discussed in some detail. Guggenheim’s method is recommended for first-order reactions, and the method of averages is found to be the most convenient and, in general, a sound and thoroughly reasonable means of obtaining velocity constants. REFERENCES
(1) ADAMS,L. H.: J. Am. Chem. SOC. 37, 1181 (1915); J. Wash. Acad. Sci. 6, 267 (1915). (2) BARTH,W.: Z. wiss. Phot. 24, 146-66 (1926). (3) BIRGE,R. T.: Phys. Rev. 40, 207 (1932). (4) COHEN,E., AND BRUINS,H. R.: Proc. Acad. Sci. Amsterdam 24, 114 (1924). G. M., AND UREY,H. C.: J. Chem. Physics 2, 112 (1934). (5) CRIST,R. H., MURPHY, (6) DUANE,W.: Am. J. Sci. 161, 349 (1901). (7) EDWARDS, J. D.: J. Am. Chem. SOC.39, 2382 (1917); Bull. Bur. Standards 14, 474 (1917). (8) GUGGENHEIM, E. A.: Phil. Mag. [7] 2, 538 (1926). A. E.: Fermentforschung 6, 302 (1923). (9) HIRSCH,P., AND KOESUTH, (10) HOUBEN, J.: Die Methoden der organischen Chemie, pp. 977-98 (1925). (11) KOELICHEN, K . : Z. physik. Chem. 33, 129 (1900). Tabellen. Springer, Berlin (1923, 1927, 1931). (12) LANDOLT-B~RNSTEIN: (13) REED,L. J., AND THERIAULT, E. J.: J. Phys. Chem. 36, 673 (1931). (14) REED,L. J., AND THERIAULT, E. J.: J. Phys. Chem. 36, 950 (1931). (15) ROSEVEARE, W. E.: J. Am. Chem. Soc. 63, 1651 (1931). K., AND BARTH,W.: 8. wiss. Phot. 24, 166 (1926). (16) SCHAUM, ~ , Z. physik. Chem. 122, 405 (1926). (17) S Z A BR.: (18) STEWART, T. D., AND KUNG,H. P.: J. Am. Chem. SOC.66, 4813 (1933). (19) S Z E GL.: ~ , Gazz. chim. ital. 60, 212 (1930). (20) WILLIAMB, W. E. : Applications of Interferometry, pp. 8-16. Methuen and Co., London (1930).
