gallol chelate, but this difference may also be partially due to a kinetic complication, such as a preceding chemical reaction. The slight hump in Figure 2, curve -1, is caused by reduction of the tin(1V)-tartrate complex which shows a pronounced dip in the limiting current.
LITERATURE CITED
(1) Alberts,
1895 (1963). (2) Bard A J. Zbid 34 266 (1962). ( 3 ) Koltioff; M,: ~ i J . J.,~ “Polarography,” p. 526, Interscience, CHEM.
(6) Phillips, S. L., Morgan, E., ANAL.
G. S., Shain, I., ANAL.
‘lg2(lg61).
33y
359
New Yorkj 1952. (4) Ibid., p. 782. ( 5 ) Phillips, S. L., Ph.D. Thesis, Univ. of Wisconsin, Madison, Wis., 1964.
~
~
S.L.PHILLIPS R.~ A. TOOMEY ,
~
Data Systems Laboratory Dept. C27, Bldg. 702 International Business Machines Corp. Poughkeepsie, N. Y.
Evidence on the Effect of Column-to-Particle Diameter Ratio in Gas Chromatography SIR: Recently Sternberg and Poulson (3) pointed out that columns with a small column-to-particle diameter ratio, d,/d,, perform unusually well because of reduced plate, H , and reduced pressure drop. Earlier Giddings (1) speculated, follouing the then surprising observation ( 2 ) of H < d, for dJd, -4, that the small plate height might be due to the effective shortening of the diffusion distance between points of unequal flow. We wish to test and discuss this hypothesis, insofar as possible, in the light of Sternberg and Poulson’s experimental results. THEORY
Each velocity inequality between two neighboring points (or regions) of a column will create a degree of nonequilibrium and thus contribute to plate height. Other things being equal the contribution will increase with the square of the diffusion distance (the distance between the two points). This is assuming that the velocity is not too high so that the relevant term, C,, has not yet been rendered negligible by coupling. Although in a random packing the extremes of the flow inequalities will be varying distances apart, it is likely that we can approximate the main effect by a single characteristic length, 1. Assuming that the principal effect arises from particle bridging (the so-called short range interchannel effect) we expect 1 to be 1.25 d, or thereabouts ( 1 ) . In Figure 1 we show in schematic fashion points of unusually high velocity (+ sign) and unusually low velocity
WALL
Figure 1. Regions with strong nonequilibrium polarization and their intersection by column wall
(- sign). In effect these neighboring points form nonequilibrium “dipoles” covering roughly the elliptical regions shown between the points. Some of these nonequilibrium “dipoles” would be cut in two if a small diameter column were somehow inserted into the chromatographic medium without disturbing it. The nonequilibrium effect would thus be reduced since some of these “dipoles” would exist only in part, and contribute less to plate height. For the sake of simplicity it will be assumed that the “dipoles” are randomly placed and that each one is weighted in effect by the length remaining within the circle (representing the column cross section). Thus the “average” length, assuming as above that one length characterizes the nonequilibrium, is i = 2 lL2/21t.The C, or w term (where C, = wdP2/D,) is thus reduced in small columns in proportion to i2/12. (If one did not use a single length to characterize nonequilibrium, a slightly better result might be gotten from i2 = Z , 3 / E l , . ) The results of this paper are calculated, therefore, on the assumption that W / W , = i2/12,where w and W, are the coefficients corresponding to the effect under discussion, with w, applying to a column approaching infinity in diameter. CALCULATIONS
The average length, i, remaining within the column cross section can be obtained in several ways. The problem can be set up in a straight-forward way by geometric arguments, but the result is a double integration involving an elliptical integral. A nonanalytical approach consists of drawing out a circle and letting sticks of a given length fall in its neighborhood; the average length of stick within the circle can then be identified with 2. Our approach was somewhere beta een the two in which the calculational problems were easily handled by computer. I n effect the computer averages the “stick” length by placing the center of the “stick” successively in 100 radial positions (properly weighted), and then rotating it through 200 angular positions. This
program could undoubtedly be streamlined, but there was no need for this since it was no particular burden to the I B M 7040. Each of the 20,000 such numbers, which are then averaged (after weighting), is obtained by simple trigonometric equations. There is no restriction on column diameter in these calculations other than the obvious one, d,/d, > 1. The value of d, can be larger or smaller than 1. RESULTS
Figure 2 shows a plot of W/W, (= i2/ P) us. the particle-to-column diameter These results were calratio, d,/d,. culated by assuming l / d , values of 1.25 and 2.5, respectively. For comparison the results of Sternberg and Poulson are shown. The latter plots are rather arbitrary because W , cannot really be
-
CALC. RESULTS ---EXP, STERNBERG 8 POULSON
.9-
.0-
.7.6-
Y w- . 5 .4
-
.3-
04
0
.2
.4
.6
I
.8
1.0
dJd, Figure 2. Comparison of experimental and theoretical values of relative w term, w / w , VOL. 37, NO. 4, APRIL 1965
e
609
established. Nonetheless these results tend to show that the proposed mechanism is capable of reducing w roughly in the observed way. I t is interesting to note that the nonequilibrium distance, 1, must be assumed to equal at least 2.5 d, before reasonable agreement is obtained. This is perhaps due to the domination of short range nonequilibrium by more heavily weighted, long range effects. I t would also tend to explain why the theoretical W , based on l/d, = 1.25, is apparently too low in normal columns ( 1 ) . The comparison of Figure 2 is greatly oversimplified since a "single" theoretical effect is being compared to a number of experimental contributions. h slightly more realistic approach involves the calculation of absolute w values instead of the relative values w/w,. For this purpose we use the equation w = 0.11 0.5 w/w, f 10-3(d,/d,J2, where the first term accounts for very short range effects and the last term accounts for transcolumn effects. These terms are very approximate, having been obtained from a simple random walk picture, but should nonetheless account for order-ofmagnitude effects. I n figure 3 we have plotted w us. d,/dc and compared the values once again with Sternberg and Poulson's experimental plots. The agreement is still moderately satisfactory in the case of Chromosorb P with l / d , = 2.5. The foregoing results do not account for all the factors which might change
+
--- EXP,
.3
It is not fully understood why the glass bead columns have a smaller w than Chromosorb P. While the former pack somewhat more uniformly than the latter, there are enough observable inhomogeneities in a packing of beads to make this explanation seem incomplete. Since w drops so rapidly in glass bead columns it would appear that long range effects (ltdp-lOj are dominant. However the velocity difference cannot be severe or the absolute value of w would be larger. It is interesting to note that the glass bead results conform fairly well to the w equation if only the last term, accounting for transcolumn effects, is used. The problems discussed here are of enough practical interest in chromatography to justify a good deal more experimental and theoretical effort toward their solution.
.2
LITERATURE CITED
CALC. RESULTS STERNBERG 8 POULSON
1.0
.9
.a .7 CHROMOSORB ' P" .6
.5 .4
.I 0
'------GLASS BEADS .2
I
.4 .6
,8
I
1.0
44 Figure 3. Comparison of experimental and theoretical values of w
the plate height as column diameter is varied. One can observe, for instance, severe changes in packing structure as the column becomes smaller.
(1) Giddings, J. C., ANAL.CHEM.34, 1186 (1962). (2) Giddings, J. C., Robison, R . A., Ibid., 34, 885 (1962). ( 3 ) Sternberg, J. C., Poulson, R. E., Ibid., 36, 1492 (1964). P. D. SCHETTLER C. P. RUSSELL J. C. GIDDINGS Department of Chemistry University of Utah Salt Lake City, Utah 84112 WORKsupported by the Atomic Energy Commission under contract AT-( 11 1b748.
Potentiometric Determination of Single Halides and Mixtures of Halides by Coulometric Generation of Silver(1) in Fused Sodium Potassium-Nitrate Eutectic at 250" C. SIR: Few analytical techniques applicable to high temperature fused salt systems exist. It has been demonstrated that it is possible to generate silver(1) a t 1 0 0 ~ ocurrent efficiency in sodium-nitrate, potassium-nitrate eutectic melt a t 250" C. (3). Further, silver halide salts are relatively insoluble in the same eutectic melt (1, 6). Halides and mixtures of halides have been previously potentiometrically determined using a glass reference electrode in lithium, potassium-nitrate eutectic a t 165" C. by adding weighed amounts of silver nitrate t o melts containing the halides ( 5 ) . This method of addition of silver(1) is less accurate and more timeconsuming. This work demonstrates that halides and mixtures of halides can be conveniently determined in microequivalent amounts by the in situ coulometric generation of silver(1) in sodium, potassium-nitrate eutectic. 610
ANALYTICAL CHEMISTRY
EXPERIMENTAL
Apparatus and Reagents. The electrolytic cell used in this study consisted of a 200-ml. 3-necked roundbottom flask. Because of the danger involved in using ordinary rubber in conjunction with a highly oxidizing medium, silicone rubber stoppers (No. 5) bored with a variety of holes to admit necessary apparatus were used a t the entry ports. Equipment, preparation of the reference electrode [hg/dg(I) 1, and compartmentalization of the melt have been described elsewhere (3, 4). All chemicals were reagent grade and silver wires served for both generating and indicating electrodes. h Sargent coulometric current source, Model IV, supplied the constant current and a Leeds and Northrup Model K-3 potentiometer was used to measure cell potentials. Procedure. The technique used for preparation and purification of the eutectic melt has been described (4). The method for determination of the
individual halides as well as the mixtures consisted of weighing the samples directly into fritted glass sealing tubes which were then placed in the melt and filled with the fused eutectic mixture by gentle suction. The silver indicating and generating electrodes were then placed in the compartment and the cell potential was measured after generation of various increments of silver(1). Stirring of the solution was accomplished throughout generation by vibrating the generating electrode which had a loop wound a t the lower end. The volume of solution in the fritted compartments containing the dissolved halides was in all cases approximately 21/2 to 3 ml. RESULTS A N D DISCUSSION
Titration curves for the pure individual halides show the expected solubility trend. The solubility products for silver iodide, silver bromide, and silver chloride calculated from these
