Table I. Results in Amperometric Titration of Uranium(1V) with Iron(ll1)
U KO. of recovered, titraU taken, mg. c/o tions 172.8-209 2 16 97-67.88 5,231
99.98 99.99 99.10
17
39 6
Rel. std. dev., “;c 0.1 0.41 0.28
end point had been reached when iron was in 100% excess of the equivalence point. The first end point apparently corresponds to the point at which vanadium(I1) and uranium(II1) formed in the reductor have been converted to vanadiuni(II1) and uranium(1V). The vanadium(II1) however, is titrated along with the uranium(1V) causing high results for uranium. Molybdenum was added to the uranium solution in the forni of sodium molybdate solution. It interferes with the titration in a manner similar to vanadium. I n the case of molybdenum the reaction between iron(II1) and the loTTer oxidation states of molybdenum is very slow as evidenced by the very slow attainment of steady-state currents throughout the titration. The titration was discontinued when iron(II1) in 100% evccss of the equivalence point had been added without reaching the end point. The effects of excess iron and chromium on the titration were also investigated. Yo difficulty was en-
countered from iron at iron to uranium weight ratios as high as 8 to 1. Chromium at chromium-to-uranium weight ratios of one or less does not interfere providing the chromium is oxidized to chromium(V1) prior to the sulfuric acid fuming step and providing the fuming is discontinued a t the first appearance of heavy fumes. When dichromate samples were fumed for a prolonged period, crystals of anydrous Crz07 apparently formed and they could not be readily redissolved. With increasing concentrations of chromium the linearity of the anodic current prior to the first end point is affected and it is therefore necessary to accumulate not only more data, but data closer to the end point, if the accuracy of the titration is to be maintained. RESULTS
The results of a series of titrations of 5 to 209 mg. of uranium are given in
Table I. The low bias in the 5-mg. series is thought to be due to atmospheric oxygen despite the fact that stringent precautions were taken to exclude it, cf. Procedure. The titration can be carried out much more rapidly with little loss in precision by obtaining just two points prior to the first end point and two after the second end point and assuming that the residual current between the end points is zero. Using this approach on the data we gathered at the 200-ing. level, the relative standard deviation is 0.2% instead of 0.1% (Table I). ,Igain there is no apparent bias.
LITERATURE CITED
(1) Auger, V., Compt. Rend., 155, 647 (1912). (2) Belcher, R., Gibbons, D., West, T. S., ANAL. CHEM.26, 1025 (1954). (3) Booman, G. H., Rein, J. E., “Treatise on Anal. Chem.” Part 11, Vol. 9, p. 89, Interscience, New York, 1962. (4) Cellini, R. F., Lopez, J. A,, Anales Real SOC.Espan. Fis. Quina. (Madrid) Ser. B , 52, 163 (1956); Anal. Abstr. 4, 97 (1957). (5) Eskevich, V. F., Komarova, L. A., Zhur. Anal. Khim. 15, 84 (1960); C. A . 54, 13981i (1960). (6) Florence, T. ill., Anal. Chim. Acta 23, 282 (1960). (7) Gallai, Z. A., Kalenchuk, G. E., Zhur. Anal. Khim. 16, 63 (1961). (8) Halpern, J., Smith, J. G., Can. J . Chem. 34, 1419 (1956). (9) Issa, I. &I., Elsherif, I. &I., Anal. Chim. Acta 14, 466 (1956). (10) Kwiatkowski, S., Owens, J., Friess, S.,Grimes, W. R., Casto, C. C., U , S.
At. Energy Comm. Rept. C-4.100.23, p.
10, Dec. 1, 1945. (1:) Rodden, C. J., “Analytical Chemistry of the Manhattan Project,” p. 71, ilIcGraw-Hill, Ken, York, 1950. (12) Ibid., p. 70. (13) Ibid., p. 65. (14) Sagi, S., Rao, G. G., Talanta 5, 154 (1960). (15) Udal’tsova, T‘I. I., Zhur. Anal. Khim. 17. 476 11962’1. (16) ’Vortmann,’G., Binder, F., Z. Anal. Chem. 67, 2169 (1926). (17) Weiss, (i., Blum, P., Bull. SOC. Chim. France 1947. 735. (18) CHEIII. Zittel, 36. H. 45 (1964) E., ihller, F. J., ANAL.
RECEIVEDfor review July 6, 1964. Accepted November 6, 1964. Work performed under the auspices of the U. S. Atomic Energy Commission. Argonne National Laboratory is operated by the University of Chicago under Contract W-3 1-109 eng-38.
Comparison of Theoretical Limit of Separating Speed in Gas and Liquid Chromatography J. CALVIN GlDDlNGS Departmenf o f Chemistry, University o f Utah, Salt lake Cify, Utah
b An investigation is made of the principal factors affecting separating speed in gas and liquid chromatography. Following an earlier proposal by Knox, it i s assumed that the maximum inlet pressure is one of the basic limitations on analysis speed. This assumption is fully justified b y this work. It i s further shown that the comparative speed of separation depends, to a large extent, on the relative viscosity and diffusivity of liquids and gases. For moderately difficult separations gas chromatography is superior because of its small C term. For extremely difficult separations liquid chromatography is superior because of its low critical inlet pressure, the 60
ANALYTICAL CHEMISTRY
latter mainly resulting from the slow diffusivity of liquid systems. The approach to further increases in speed is discussed for both methods.
I
(4) a comparison was made between the ultimate separating power of gas chromatography (GC) and liquid chromatography (LC). In this work we shall compare the theoretical limit of separating speed between the two methods. If we are given a pair of solutes with a certain degree of molecular similarity and we wish to separate them at a fixed level of resolution, it is desirable to know how rapidly this resolution can be achieved using one method as compared to the N A PREVIOUS PAPER
other. We shall investigate some of the factors involved in this comparison. It is usually valid to assume that the speed of any given separation can be increased by imposing a larger pressure drop (and thus flow rate) across the column. The argument for this was first presented by Knox (6) in connection with GC. The reasons for this assumption will be made clear later. In any case we shall assume that one of the limitations imposed on any chromatographic system is the inlet pressure, p , (this concept does not apply directly to paper and thin layer chromatography, although the capillary pressure has an analogous role). The magnitude of the maximum p , may be comparable in GC
and LC although the latter has a certain advantage in its low compressibility and reduced danger from equipment fracture. The equations of this paper, as for the preceding one, will be based on uniform columns. Although the high compressibility of gases may be responsible for large nonuniformities in GC, it will be shown (repeating our experience with the earlier paper) that such nonuniformities can be closely approximated by taking the proper averages. If it is assumed that separation speed will, in general, increase with the pressure drop available, then one may increase the speed by reducing the outlet pressure, p,, to a minimum as well as through increasing p,. The use of vacuum outlet, or as near to this as possible in view of experimental limitations, has been shown (2) to be advantageous for GC. SEPARATION TIME
The time, t , required for completing a chromatographic run is simply the column length, L , divided by the velocity, Rv, of the zone or zones of interest. The quantity v is the mobile phase velocity and R is the usual retention factor indicating the degree to which the zone is retarded by the column. Thus t = L/Rv. The length, L , may be replaced by N H where iV is the number of plates and H is their height. This gives an expression of the general form first derived for GC by Purnell (8).
t = NI’I/Rv (1) This, as just indicated, is the time necessary to complete the run, but it may not be the proper time expression for separation because the latter is not guaranteed, and may well fail if the run is somehow inadequate. Thus Equation 1 is valid as a separation time only if we impose the condition that the number of plates, iV,must be sufficiently large for the separation in question. This condition, that N must equal some predetermined minimum value, will be a tacit assumption of the development below. [It has been pointed out (1) quite correctly thaut, strictly speaking, the completion time should include the time needed to elute the back half of the last zone. In this case Equation 1 is replaced by (H/Rv) ( N 2N1’2),with the R value being that applicable to the trailing zone.] The number of lilates needed for any given separation (‘particularly for the most difficult pair of components) will usually depend upon the similarity of molecular structure>and upon the selectivity which can be achieved through the choice of a stationary phase. As indicated in the previous paper ( 4 ) ) GC and LC have essentially the same potential for selectivity. Consequently the number of plates needed for a certain separation will bfb roughly the same in the two methods. Thus the question of maximum speed reduces to the
+
question of how a given number of plates can be most rapidly achievedi.e., with a fixed N , how t can be reduced to a minimum. If one is not concerned with pressure limitations, the t of Equation 1 is minimized by reducing the H / u ratio to its lowest value ( N and R fixed). Chromatographic theory shows that this ratio approaches a lower limit as v (along with column length) approaches infinity. This limit can be pursued but never reached in practice because the great lengths and velocities require equally unobtainable pressure drops. Thus, even in this case, one is forced into considering the imitation imposed by pressure. Since pressure drop ( A p ) , length, and velocity are all integral parts of separation speed, it is important to use a hydrodynamic expression relating them. Accordingly! the pressure gradient, A ~ / Lis, written as A p / L = 2417u/dP2 (2) where 4 is a constant of proportionality (-300/@ in packed columns in which @ is the fraction of mobile phase in interparticle space), q is the viscosity of the mobile phase, and d, is the mean particle diameter. If L is written as N H , this becomes an equation relating H and v. v = ApdP2/2+qSH (3) The plate height, H , is also a function of velocity, giving another relationship betveen H and v . For most chromatographic columns H can be adequately approximated by the form
+
H = B/v Cz: (4) where B allows for longitudinal molecular diffusion and C allows for all the possible nonequilibrium effects. It is necessary here, in contrast to the earlier paper on separation power, to introduce a specific term in the form of
cv .
While a slightly more rigorous and elaborate expression could be used here, this would complicate the subsequent development enormously. lye will consequently retain the equation above and think of C as an effective nonequilibrium term, departing from the real nonequilibrium terms in such a manner as to best implement the use of Equation 4. Khen the v from Equation 3 is substituted into Equation 4 and H solved for from the latter equation, we obtain
(5) This is an equation containing only the measurable parameters of the system. The variable, v , has been eliminated. A similar solution can be obtained for v itself. Upon substituting the above H expression into Equation 3 we have
The important ratio appearing in Equation l, H / v , can be obtained as the ratio of the last two equations.
The separation time, t thus be expressed as
=
NH/Rv, can
If we write p , = 29q.VB/dp2 the separation time becomes t =
E R
(9)
(5) Ap
(10)
The quantity p,, with the dimensions of pressure, is known as the critical inlet pressure (2, 4 ) . It is the minimum inlet pressure (or pressure drop) which will yield the N plates needed for separation. No manipulation of column length or flow velocity will give the N plates with A p < p,. Equation 10 shows, in fact, that the separation time approaches infinity as A p is reduced to Pc*
If the quantity B in Equation 9 is mitten as 2yD,, the critical inlet pressure becomes
pc = 4+yqD?nY/dp2 (11) This equation is identical to that given in the previous paper; a further discussion of p , can be found there. The quantity D , in these equations is the diffusion coefficient of solute in the mobile phase and y is an obstruction constant (-0.6) for the column packing. Equation 10 is the principal equation for separation time. This equation confirms, in general, the earlier assumption that t can be reduced by increasing the pressure drop A p . -in exception does exist, however, in GC. If gas phase nonequilibrium is dominant, the C term becomes pressure dependent and increases in magnitude as the inlet pressure increases. Thus t is often a minimum a t some finite value of pi and Ap. This is confirmed by a more rigorous approach ( 2 ) t o the gradients found in GC. COMPARISON OF METHODS
K e shall make a comparison of GC and LC based on the preceding equations, particularly Equation 10. R e shall first take the most common case in which the C term in GC is not dominated by gas phase nonequilibrium. In this way we avoid the anomalous situation in which a finite inlet pressure is optimal for the achievement of high speed. If we assume that S,R, and A p are comparable in GC and LC, Equation 10 shows that the relative merits of the two methods depends mainly on the value of C and p,. GC shows a distinct advantage by generally having a small C, and LC holds the advantage by VOL. 37, NO. 1, JANUARY 1965
61
virtue of its smaller critical inlet pressure. For simple separations (low N requirement), where p , is well below the obtainable pressure drop, t approaches NC/R in value. In these cases GC has a major advantage because of its low C value. For difficult separations (high w requirement) the correspondingly larger value of p , (the latter being proportional to A') may make GC separation extremely slow. ;It a certain level of difficulty, where p , exceeds the available A p , a separation cannot even be made using GC. LC, because of its smaller p , , will still be capable of making the separation a t this level and well beyond (4). The advantage held by LC with respect to its small p , value is easily calculated from Equation 11 to be roughly 100-fold with moderate pressures up to about 10 atmospheres. This is caused by the differences in diffusivity and viscosity, mainly the former. The advantage of GC in connection with the smallness of its C term is uncertain. While the overall C term of GC has been established in the approximate range from to 5 X 10-2 second, information about LC is very scanty [a few values can be obtained from Hamilton, Bogue, and .Anderson (5) and Mallik and Giddings (r)]. The C term caused by mobile phase effects would normally be 104-105 (the ratio of diffusivities) times larger than in GC, but this may be rendered ineffective by the coupling process which acts at very low velocities in LC. The stationary phase C term is probably much larger in most cases of LC because a greater diffusion path through the stationary phaqe is required. In ion exchange and paper chromatography, for instance, the entire bead and fiber must be traversed in a single diffusion step. The length of such steps may be 1 to 300 times longer than in the isolated liquid pools on a gas chromatographic support. The C term, proportional to the square of this, may consequently be from 1 to lo5 times larger in LC than in GC. The exact value depends largely on the particle size used, a point to be emphasized later. .Is a working hypothesis, assume that C generally ranges from 10-1 to 10 seconds. To reduce the above results to graphical form, it is necessary to consider the gradients of gas chromatography in slightly greater detail. The expression for critical inlet pressure, Equation 1 1 , has a mobile (gas) phase diffusion coefficient, D, = Do, which depends qtrongly on pressure. This dependence may be written as Do = Do'/P (12) where D,' is the gaseous diffusion coefficient at unit pressure. It was shown in the previous paper that by using the average column pressure, equal to 2 p , / 3 62
ANALYTICAL CHEMISTRY
LOG
LOG
N
Figure 1. Comparison of separation time in gas and liquid chromatography as a function of the number of required plates
when the outlet pressure is negligible, a critical inlet pressure could be obtained as a function of the constant parameter D,' rather than the variable D,. The approximate value of this is p,' = (6&y~D,'N/dp2)"2
(13)
The ratio A p / p c , pertinent to Equation 10, may be replaced by p i / p , when the outlet pressure is small comparted to p i . If the p in Equation 12 is replaced by its average value, 2 p i / 3 , and the resulting D, (= D,) is substituted back into the critical inlet pressure expression, Equation 11, the ratio p i / p , becomes pi2/pc'2. Thus for gas chromatography a more meaningful expression for separation time, replacing Equation 10, is
It is instructive to observe the change in t for the two methods of chromatography as the number, N , of plates required for separation becomes larger. This is shown in Figure 1. The curves for LC were computed from Equation 10 using the p , of Equation 11 and with 7 = poise and D, = 5 x 10-6 sq. cm./second. The GC plots were obtained from Equation 14 with p,' expressed by Equation 13 and with 7 = lo-* poise and D,' = 0.7 X lo6 gm.-cm./sq. second. For both GC and LC the following reasonable parameters have been used in comman: A p = lo' dynes/sq. cm. ( w 10 atm.), d,2 = 2 X sq. cm. (-100 to 120 mesh) +y = 5 X l o 2 and R = 0.1. The range of C values was stated earlier. The figure illustrates the previous qualitative discussion. Although the
parameters have been arbitrarily chosen, their precise value is not critical to the order of magnitude considerations used here (note logarithmic scale). The figure shows clearly that GC is superior by several orders of ten for easy separations. This superiority is lost as the required N approaches lo5 plates. -1bove this LC, only, is capable of separation. (Capillary GC has a somewhat greater range because of the reduced + value.) However, as noted by the time parameters at the right of the figure, LC becomes superior only for separations requiring 1 day or more. The most difficult of possible separations might require the order of 1 year. This is beyond the range of practical laboratory analysis. For many research purposes one or a few days, only, may be tolerated. For routine analyses, laboratory and industrial, it is highly desirable to reduce the separation time to a point between 1 hour and 1 second. For this purpose GC, where applicable, seems to be far superior. On the whole GC appears to show a real advantage over LC in terms of rapid analysis because its range of superiority is a more practical range. Figure 1 should not be construed as showing the ultimate limit of chromatographic performance. TF'ith judicious changes in the systems involved, the time could undoubtedly be reduced considerably, particularly in LC. The separation time, except as one approaches the theoretical limit of achievable plates, is directly proportional to the C term. -It sufficiently high flow velocities the C term caused by mobile phase effects largely disappears because of the coupling phenomenon. Such flow velocities can be obtained, if
necessary, with a multistage pumping system (3). The C term caused by stationary phase effects is probably also subject to big reductions. This may involve reducing the length of the diffusion path when using partition methods and reducing surface heterogeneity when using adsorption chromatography (S). In theory, using the above approach, the total C term can be reduced to approximately the same low level in LC as in GC. This would essentially eliminate the disadvantage of LC and make both methods more rapid. Since C generally increases with the particle diameter, d,, speed can often be This increased by reducing d,. approach works until the decrease in d, forces p , (or pc‘) up near the available pressure drop. In the case of liquid chromatography, for instance, C may often be assumed proportional to dP2. The critical inlet pressure, p,, is inversely proportional to dP2as shown by Equation 11. Thus in reducing particle size C is reduced and p , is correspondingly increased. These changes are very similar, as far as rapid separations go, to the change in performance from liquid to gas chromatography. Referring again to Figure 1 we see that this means that the analysis is more rapid for separations of moderate difficulty, but the ability to make difficult separations is lost. The choice of an optimum particle size can be based on the earlier equationq of this paper. The procedure used here is similar to that of Knox (6) who first explored the limiting pressureparticle size relationship in GC. If me restrict our treatment to LC and to the cases in which C is proportional to dP2,we can write Equation 10 as
which means that the optimum occurs for such a particle size that the maximum pressure drop available is just twice the critical inlet pressure. This is the optimum p, and not Ap; the latter is optimal only when it is as large as possible. If C increases less severely with d, then the optimum particle size is larger. This can be shown by assuming that C increases with some power of dp-i.e., as d,”. The foregoing procedure leads to d,
=
+ 2) ~ / n ] ’ / ~
[(n
This value is larger than that in Equation 16 by ( 3 / 2 ) 1/2 when n = l and when n = 0.1. The by (21/2) optimum critical inlet pressure, to be obtained through particle-size adjustment, is given by Ap = (n+2)pC/n in place of Equation 17. ROLE OF C, I N G C
To an approximation the effective C term may be represented as a sum of stationary phase and mobile phase contributions. In GC this becomes C = C L C,. The former is essentially constant with changes in pressure. The latter, because of its dependence on gaseous diffusion, is proportional to pressure and may be approximated as C,’p, where p is the average column pressure. The latter was shown earlier to equal 2p,/3. Thus
+
C
= Ci
+ (2/3)C,’pi
= c
dD2- a
(15)
where c is a constant of proportionality and a = dP2p,/Ap; the latter is independent of d, because of the inverse dependence of p , on dP2. When this is differentiated with respect to d, and set equal to zero we find for an optimum
a,
=
(2a)’/2 = ( ~ W ~ L N / A P ) (16 )
where the p , of Equation 11 has been used. If dP2p,/Ap is substituted directly for a we get the equdity
(19)
If the Cl term is dominant, as assumed earlier, the separation time from Equation 14 becomes t = (NC,/R)pz”(p22
t
(18)
- pc’?
= (2NC0’/3R) pL3/(pt’
-
pc”)
(21)
Upon differentiation this expression shows a minimum time for the inlet pressure p, =
1/3pc’
obtained earlier by more rigorous methods. It can also be shown that the optimum pi is finite (although usually quite large) for all combinations of C l and C,. If pi is fixed, an optimum can be obtained for d,. If C, equals a constant times d,” (where n is usually about 2 ) , the optimum particle size is given by
This equation can be used to show that a t optimum d, the critical inlet pressure is p,’ = [n/(n
(22)
+ 2)1”2Pi
(24)
A combination of this and Equation 22 can be used to demonstrate, assuming that p , and d, are both free parameters, that the best p , is its highest possible value when n > 1. This is the usual case. Some unusual cases in which the particle size is within a factor of 2 or 3 of the tube size may show n < 1 and thus a finite ultimate optimum inlet pressure. We may tentatively conclude, however, that in the great majority of cases where d, is subject to adjustment, the Knox hypothesis, that separation speed is increased with larger pressure drops, is correct. The foregoing theory is valid for capillaries as well as for packed columns. It is, in fact, more exact for capillary columns because the effective C term in packed columns is actually a variable as a result of coupling. The theory of minimum analysis time in the light of the coupling phenomenon has not yet been fully developed.
(20)
This decreases as p , increases over the entire range of operation. This can be verified by differentiation. This illustrates our earlier contention, first proposed by Knox (6), that maximum speed (minimum t ) can usually be obtained a t the highest possible inlet pressure. When the C, term is dominant the time becomes
t
Thus the optimum p , (for a fixed p e t ) is finite in this case, a conclusion
LITERATURE CITED
(1) Dal Xogare, S., E. I. du Pont de Semours & Co., Wilmington, Del.,
private communication, August 1964.
(2) Giddings, J. C., ANAL. Cmni. 34, 314 (1962). (3) Giddings, J. C., Ibid., 36, 1483 (1964). (4)Ibid., p. 1890. ( 5 ) Hamilton, P. B., Bogue, D. C., Snderson, R. A., Ibid., 32,1782 (1960). (6) Knox, J. H., J . Chem. Soc. 1961, p. 43.1.
( 7 ) - k d i k , K. L., Giddings, J. C., ANAL.
CHEM.34, 760 (1962). (8) Purnell. J. H.. Ann.
Y . Acad. Sci.
RECEIVEDfor review July 23, 1964. Accepted October 8, 1964. This work was supported by the C. S. Atomic Energy Commission under contract AT11-1)-748.
VOL. 37, NO. 1, JANUARY 1965
63
