has several questionable aspects. This author agrees completely; the limitations were obvious from the outset, but perhaps not sufficiently emphasized. This can be made clear by explaining the origin of the equation. Because of the inherent difficulty in making use of least squares on the “summed” form of the coupling expression [equation 6 of that paper (@], equation 7 was used as a simple but effective criterion of coupling. Nowhere was it stated that equation 7 would yield physically meaningful parameters. In other words the parameter d was used as a criterion of “flattening,” an asymmetric property of coupled curves. It serves this role admirably well for d will be negative for any data obeying coupling and average to zero for any data obeying classical theory. It is of no consequence whatsoever that a negative d will lead h back through zero since there is no suggestion that the equation
does, or should, fit points beyond the immediate experimental range. We see then that d is only a qualitative criterion, but nonetheless a decisive one, in choosing between mechanisms. In the final analysis equation 7 is meaningless only because of the inherent shortcomings of classical theory-if classical theory were adequate one could indeed attach physical meaning to all parameters except d and this, as an average, would equal zero. Finally, to put the now dubious reputation of equation 7 in perspective, the clear failure of classical theory makes the universally used classical equations just as erroneous as equation 7 . If the use of equation 7 is to be condemned, one might argue with like conviction that the use of all classical equations is equally objectionable-or even more so because the parameters of classical theory are invariably given a physical interpretation of doubtful significance.
LITERATURE CITED
(1) Deemter, J. J. van, Zuiderweg, F. J., Klinkenberg, A., Chem. Eng. Sci. 5, 271 (1986). (2) Giddings, J. C., ANAL. CHEM. 35, 1338 (1963). (3) Gidd&s,’J. C., “Dynamics of Chro-
matography. Part 1. Principles and Theory,” Marcel Dekker, Inc., New York, 1965. (4) Harper, J. Si., Hammond, E. G., ANAL.CHEM.37, 486 (1965). ( 5 ) Klinkenberg, A., Ibid., 38, 489 (1966). (6) K,nox, J. H., 3rd International Symposium on Advances in Gas Chrornatography, Houston, October 1965. ( 7 ) .Littlewood, A. B., private communication. Newcastle-upon-Tyne, England (1965). (8) Myers, M.N., Giddings, J. C., ANAL. CHEM.37, 1453 (1965). (9) Saha, N. C., Giddings, J. C., Zbid., 37, 830 (1965). J. CALVIN GIDDINGS Department of Chemistry University of Utah Salt Lake City, Utah WORK supported by Public Health Service Research Grant GM 10851-08 from the National Institute of Health.
The Nature of Eddy Diffusion in Gas Chromatography SIR: In previous correspondence Klinkenberg (8) and Giddings (6) have exchanged views on the merits and limitations of the original van Deemter, Zuiderweg, and Klinkenberg (1)theory of band-widening in gas-liquid chromatography as compared to Giddings’ coupling theory. It is felt that for a better understanding of the problems a t stake a general statement on the present position, especially with regard to the concept of eddy diffusion, would be useful. Such a statement is accordingly presented. It is, in part, based on some observations made recently (9) in a general paper on residence time distributions in chemical engineering. The classical equation by van Deemter et al. ( I ) was based on the concept of a uniform forward gas velocity (“plug flow”) with some superimposed spreading mechanisms-viz., axial diffusion, axial eddy diffusion, spreading by transfer to/from stationary fluid, with resistance to transfer in both phases. This accounts for a 4-term equation of the general form of equations 1 and 2 (6) but with both linear terms vanishing for a nonsorbing solute. Plug flow, however, is only a first approximation for the real flow pattern. In each pore the velocity ranges from zero to a maximum value. In any real packing, not being a closest packing of equally sized spheres, there are moreover wider and narrower pores in parallel and there may even be packing
irregularities extending over larger regions. I n the van Deemter et al. concept the total convective spreading by all these mechanisms was called “eddy diffusion.” A consequence of the flow not being plug flow is that, upon the passage of a chromatographic band, there are radial concentration gradients. These give rise to lateral diffusion, which reduces the axial spreading and ultimately leads to an apparent axial diffusivity (“Taylor diffusion”). Accordingly “eddy diffusion” is reduced if there is appreciable lateral diffusion. A statement that this is to be expected in gases for heat transfer was made by Klinkenberg and Sjenitzer (IO), who have, however, not examined its consequences for GLC. Giddings dealt with this subject in many publications ( 2 , S , 6 , 7 ) . Such lateral diffusion occurs most easily in the case of the single pore. It is less important in the cases of the pores in parallel and of the packing irregularities, where the distances to be covered are greater and the particles are moreover obstructing the diffusion. This means that eddy diffusion should be split up into its component parts before the effect of lateral diffusion on it can be studied. Giddings accordingly considers the sum of “coupling” terms between “various eddy diffusion and mass transfer effects” ( 9 , s ) . Lateral diffusion becomes appreciable if the time of diffusion r 2 / D across a pore
radius r is comparable to the time of passage d,/v through a pore. Since r and d, are proportional, this leads to the conclusion that the concept of eddy diffusion as a purely convective process must be modified if D/vd, = 1 / u surpasses a \ alue in the neighborhood of unity. See Giddings ( 7 ) . The chromatographer is interested in working a t minimum h-Le., for h = 2y/u 2~ C Y , at h,,, = 2d2ic
+ +
I-
+
If the three parameters in this equation are all of the order of unity, as they seem t o be, Y 5: 1 is the preferred working parameter. Thus it is concluded that the eddy diffusion concept is going to fail when it is most neededLe., in a region where its classical contribution 2X is relatively most important in respect to the other terms, the sum of which is to be minimized. Giddings ( 7 ) arrived a t this same conclusion by comparing the velocity a t which the influence of radial diffusion becomes important to gas velocities used in practice. Comparing the classical prediction h = 2y/u 2X for a nonsorbing solute to the Kieselbach data, descending branch reaching h = 2 a t v = 1, horizontal branch h = 2 for Y > 1, (4) we see indeed that the limiting cases for low and high Y are well represented but for the region Y ‘v 1 this is not the case. The value of h approaches its limiting
+
VOL. 38, NO. 3, MARCH 1966
491
value of 2X = 2 much faster than if the plot of h against v were a hyperbola with an asymptote a t h = 2-i.e., in the region of v N 1 the value of h is reduced below that which the classical theory would require. Giddings’ coupling equation, equation 4 of reference ( 5 ) , in which he replaces the sum of contributions by two mechanisms-viz., 2X and wv-by the inverse of the sum of the inverses has precisely this effect. Moreover he writes this coupling equation in general as a sum of such terms, simplifying it to a single term only because of lack of complete information on these various terms. This is the consequence of the fact that “eddy diffusion,” as discussed earlier, is caused by a number of different mechanisms. Even if we do not know the exact
equation, we can in any case accept the empirical relationship between h and Y as it is provided by experiments. Unfortunately such an empirical relationship cannot be determined once and for all since it depends on several parameters, the tortuosity 7 , the various contributions X i to the eddy diffusion constant and mass transfer characteristics (wi or C in two theories). In various experimental investigations these parameters assume different values. In particular some packings are less regular than others and therefore produce more, less easily “coupled,” eddy diffusion than others. All this may well account for the fact that some authors attack the classical theory based on a uniform forward fluid velocity whereas others seem to be reasonably content with it.
LITERATURE CITED
(1) Deemter, J. J. van, Zuiderweg, F. J., Klinkenberg, A., Chem. Eng. Sci. 5, 271 (1956). (2) Giddings, J. C., ANAL. CHEM.34, 439 (1962). (3) Zbid., p. ’1186. (4) Ibid., 35, 1338 (1963). (5) Zbid., 38, 490 (1966). (6) Giddings, J. C., J . Chromatog. 5, 61 (1961). (7) Giddings, J. C., Nature 184, 357 (1959). (8) Klinkenberg, A,, ANAL.CHEM.38, 489 (1966). (9) Klinkenberg, A., Trans. Znst. Chem. Engrs. ( L o n d o n ) 43, T 141 (1965). (10) Klinkenberg, A., Sjenitzer, F., Chem. Eng. Sei. 5,258 (1956).
A. KLINKENBERG Bataafse Internationale.Petroleum ~ . Maatschappij N.V. (Royal Dutch/Shell Group) TIle Hague, The Nether1and.s
Solvent Enhancement of Emission Lines for Plasma Arc Determination of Vanadium in Petroleum Fractions SIR: hlany phases of petroleum research require the measurement of naturally occurring metals in asphaltenes, resins, and other petroleum fractions. The most abundant of these metals occurring in petroleum are vanadium and nickel. Within the past few years great strides have been made in the development and use of the gasstabilized arc, or plasma arc, as an emission source for solution analysis. Many investigators (1-6) have given detailed descriptions of the various parameters each has studied, and then they have recommended certain optimum sets of conditions to be used for each version of the plasma source. This work reports on an emission spectrographic technique that has been employed in conjunction with the determination of vanadium in chloroformsoluble petroleum fractions using a plasma arc. EXPERIMENTAL
A plasma arc assembly, constructed at our laboratory, was designed to fit a National Spectrographic Laboratory (NSL) excitation stand and gas manifold. A Bausch and Lomb dual grating spectrograph equipped with a 30,000 line per inch grating was used along with an NSL Model 110.91 power source delivering 300 volts open circuit. The technique used for the plasma arc analysis of vanadium is briefly as follows. A sample solution is prepared by placing from 2 to 200 mg. of sample into solution for each milliliter of chloroform. A solution of cobalt cyclohexanebutyrate (National Bureau of Standards) in pyridine may be used as an internal standard if desired. This sample is atomized with a Beckman medium bore atomizer into the plasma arc for 10 seconds with 9 liters per minute of helium atomizer gas and 60 492
ANALYTICAL CHEMISTRY
liters per minute of helium tangential gas flow. A current of 17.5 amperes is maintained during the entire exposure. Light 2 mm. above the upper orifice of the source is recorded onto 511-1 spectrographic plates through a 20micron slit. The plates were developed in D-19 developer a t 70” F. for 3 minutes with continuous agitation and fixed 3 minutes in a rapid fixer with a hardener. The intensity ratio of the vanadium line a t 2924.0 A. to an internal standard line, such as cobalt a t 2632.2 A. or the carbon-chlorine band peak a t 2789.8 A,, is then measured and the vanadium concentration determined from a working curve. A Control Data G-15 computer was programmed to handle emulsion calibrations and intensity ratio calculations and to develop a working curve. An extremely large gap of 21 mm. is maintained between the graphite anode and the graphite cathode which is the counter electrode. The other dimensions are: an analytical gap of upper orifice to cathode of 8.5 mm.; a lower anode orifice diameter of 4.0 mm.; an upper neutral graphite orifice diameter of 6.0 mm.; and a sample flow rate of 0.9 ml./minute. These conditions were found by trial and error to permit the most stable operation and were noted later to nearly coincide with those of Sirois (6). By using methods similar to those of Margoshes and Scribner (3), the average temperature of the plasma source was found to be 5800” K. with a standard error of determination of 400’ K. Recent studies by Sclirenk, Show-jy Ho, and Lehman (4) point out that the precision obtained without an internal standard is almost as good as with an internal standard as long as a constant sample flow rate is maintained. To permit reasonable control of sample flow rates it is necessary to put the crude petroleum or heavy petroleum
fractions into suitable solvents. Among those tried were benzene, pyridine, chloroform, and carbon tetrachloride. RESULTS AND DISCUSSION
When a choice is made for an internal standard, the major constituent is often used to supply a convenient reference line for the element being sought. For instance, Leistner (1) used the OH band as an internal standard in the analysis of aqueous solutions by the vacuum cup technique. I n our method, the chloroform solvent supplies a system of five prominent carbon-chlorine band peaks, any one of which may serve as an internal standard. One variable which influences line intensity is arc current. When the plasma arc current increases, the intensity of emission of the vanadium ion line also increases; however, the intensity of the carbon chlorine band stays nearly constant. Thus, as an internal standard, use of a carbon chlorine band would introduce an error proport,ional to any arc current fluctuation. Another variable which influences intensity is the composition of hydrocarbon solvent used. Two hydrocarbon solvent pairs, benzene-chloroform and pyridine-chloroform, were employed to determine their relative influences. The intensities of both the vanadium line and the carbon-chlorine band were inversely proportional to the weight % carbon present in the solvent system. Carbon tetrachloride behaved in the same manner as chloroform. Thus, among the solvents investigated, chloroform or carbon tetrachloride should permit the detection of the lowest concentrations of vanadium. These intensity relationships are shown in Figure 1. The intensity ratio of the vanadium line to the
