Maximum Nlumber of Components Resolvable by Gel Filtration and Other Ellution Chromatographic Methods J. Calvin Giddings Department of Chemistry, University of Utah, Salt Lake City, Utah 84112 CONSIDERABLE THEORETICAL WORK has been done on the factors controlling peak width and retention in chromatographic systems. The translation of this knowledge into information on the total number of resolvable components has not been made. While a precise determination of resolvable peaks will depend on the nature of solutes existing in a particular mixture, one can define and estimate a "peak capacity" which approximates the maximum number of peaks to be separated on a given column. Gel filtration and related methods are rather unique in that, under normal conditions, there is a well defined limit to the peak capacity. The reason is that these peaks are confined, in the absence of adsorption, to a definite retention volume range, constrained by the column interstitial volume, V", on one end and by the maximum solvent volume (both in and out of grains), Vs, on the other (1) (see Figure 1). This range lies roughly betwzen 0.4 and 0.9 times the total column volume. The largest molecules, with no penetration, will elute at the first limit anti the small molecules at the second. In most forms of chromatography, by contrast, there is a similar lower limit to the retention volume, this being the column free volume, again equal to a fraction of the total column volume. However, the upper limit is indefinite; retention volumes may in practice, extend to values ten or a hundred times larger than the minimum volume. Because the peaks from all chromatographic columns have a finite width as dictated by plate height, only a limited number of such peaks can crowd into the range accessible to them. We assume in the Following treatment that the column possesses a fixed number of plates, N , equal for each solute. Thus, if the retention hrolume of the ith peak is Vi, its width (defined as mu where usually m = 4)is 4 Vi/N112= aVt, where a is (for present purposes) a column constant, m/N112. The next peak, i 1, with a corresponding volume V,+l and width aV,+l, can therefore be no closer to the ith peak than the mean of the two width!;, (a/2)(Vi Vi+1), if a minimum ma separation is required. Therefore at closest spacing
+
+
- V , = a-2 (V*+ Vi+1)
vi+1
(1)
From this we obtain the ratio of retention volumes for adjacent peaks vt+1 - 1 4 2 v, 1 - a/2
+
Because this ratio is a function only of a , it too is a column constant. If the first peak elutcs at volume VI, the minimum elution volume of the nth and final peak, by comparison to VI, will be given by an (n - 1)-fold compounding of the ratio in Equation 2 Vn 1 a/2 '-' (3) VI == (-2)
+
(1) J. Porath, Biochim. Biophys. Acta, 39,193 (1960).
.8 I
i
GEL FILTRATION
RANGE
Figure 1. Generation of closely spaced peaks as a function of available elution volume range When logarithms are employed, the following expression for n results
(4) which gives the maximum number of peaks (the peak capacity) which can be resolved between volumes Vl and Vn. In most practical situations, N is sufficiently large (lO"lO4) and a sufficiently small that the term (1 a/2)/(1 - 4 2 ) s 1 a. Furthermore, In (1 a ) E a , when a is small. Thus to a good approximation Equation 4 becomes
+
+
+
This is the most general and convenient form of ,the desired relationship. In the gel filtration case, barring complications caused by adsorption, etc., peaks are confined to the elution volume range Vo to Vs,as explained earlier. Thus the peak capacity is
-
n
=
1
+ a-1 In(VS/Vo)
(6)
-
Because Vs/Vo 2.3, the peak capacity is approximately n 1 0.8/a. Quantity a can, in turn, be repaced by m/N112. Because a 4u separation (m = 4) is generally considered adequate, we have the following highly simplified expression for the peak capacity of a gel filtration or permeation column.
+
n
1
+ 0.2
"12
(7)
Table I shows some peak capacities based on this equation along with those for other systems (see subsequent discussion). Although little information has been acquired on the effi-
Table I. Comparative Peak Capacity of Gel Filtration and Other Columns for Given Numbers of Theoretical Plates Peak capacities, n Theoretical Gel Gas Liquid plates, N methods chromatog. chromatog. 100 3 11 7 400 5 21 13 lo00 7 33 20 2500 11 51 31 loo00 21 101 61
VOL 39, NO. 8, JULY 1967
1027
ciency of gel filtration columns, most evidence indicates a typical N of about 400 (2). A typical column can therefore be expected to separate a maximum of only about five peaks. As mentioned before, other forms of chromatography have no clear maximum retention volume; additional peaks can be eluted until time becomes excessive or until their dilution hinders proper detection. In gas chromatography, with its considerable speed and sensitivity, it is practical to work with peaks whose retention volumes are 50 or so times that of the air or initial peak (in many such cases, programmed temperature would actually be preferred). Thus V,/VI in Equation 5 can be replaced by a number the order of 50. This shows, very roughly, that the peak capacity of a gas chromatographic column is 5-fold greater than that of a gel filtration column of 1 "12. Peak capacities based on equal plates- i.e., n this assumption are also shown in Table I. If we limit Vn/Vlto 10, as would be reasonable in many gas and most liquid chromatographic columns, the peak capacity would be only 3 times better-i.e., n 1 0.6 N1I2(Table I). The above indicates that for a separation of comparable overall complexity, gel filtration and permeation columns must have many more plates than conventional columns. Again using Equation 5, with a = m/N1/2,we see that to fully offset the 3- to 5-fold increase in peak capacity of conventional columns, plate numbers of gel filtration and related columns
-
+
-+
(2) J. C . Giddings and K . L. Mallik, ANAL.CHEM., 38, 997 (1966).
+
would need increasing by 3 and 52, respectively-roughly one order of magnitude. While such increases may not be easy to achieve, the enormous importance of the gel techniques in complex biological and macromolecular separations makes an effort worthwhile. A preliminary discussion of factors affecting plate height in such systems may provide a basis for improvement (2). While the foregoing conclusions are based on highly simplified assumptions, they are probably valid as a general rule. For instance, while N is not constant for all components in a given column, as assumed, values generally lie within a twofold range. An average value for several peaks should be adequate. Also the two approximations leading from Equation 4 to 5 give exceptionallygood results. It so happens that the errors of the two steps cancel one another to a first order; even for N = 100 and thus a = 0.4, the approximations together are valid to nearly 1%. The question of separation time has not been considered above. Clearly the limited elution range of gel chromatography is to its advantage timewise. This would compensate to some extent for increases in time which might accompany the search for more theoretical plates. RECEIVED for review February 6, 1967. Accepted May 1, 1967. Investigation supported by Public Health Service Research Grant G M 10851-10 from the National Institutes of Health.
Determination of Trace Amounts of Phosphorus in a Composite Propellant by Fast Neutron Activation Analysis M. H. Rison, W. H. Barber, and Peter Wilkniss Research and Development Department, U.S . Naval Propellant Plant, Indian Head, Md. A RAPID, SENSITIVE METHOD to determine small amounts of phosphorus in a composite propellant was needed at this laboratory. The propellant consisted of approximately 60% ammonium perchlorate, 20 % aluminum, and 20 % binder. The small amounts of phosphorus to be determined were known to be in the binder phase of the propellant. Fast neutron activation analysis seemed to be a promising approach to the problem, and the results obtained with the method are described. To obtain the best sensitivity the 31P(n,a)BAl reaction was used. This reaction had already been used by Lbov and Naumova (1) and Maen To-on et al. (2) to perform fast neutron activation analysis of phosphorus in different materials. Interfering reactions are 28Si(n,p) %A1and nAl(n,y)"Al. In this investigation small amounts of silicon were detected in aluminum, but no silicon could be detected in the other propellant ingredients and materials used. The interference of the reaction 27Al(n,y)28Al which has a cross section of 0.53 mb for 14.5-MeV neutrons is serious. This compares with 150 mb for 31P(n,~~)~~Al for neutrons of the same (1) A. A. Lbov and I. I. Naumova, Atomnaya Energ., 6,468 (1959), (In Russian), NSA 13, 13335 (1959). (2) Maen To-on,F. Sicilio, and R. E. Wainerdi, Trans. Am. Nucl. SOC.,7 , 328 (1964). 1028
ANALYTICAL CHEMISTRY
energy [Gillespie and Hill (3)]. Using these cross sections one can calculate that the same amount of %A1will be produced during the irradiation in the same flux of 14.5-Mev .neutrons when the ratio of aluminurn to phosphorus is the order of 300. This was the case with samples which contained 200 mg of aluminum and approximately 0.4 mg of phosphorus per gram of propellant. Two methods were used to overcome this interference: chemical separation of the phosphorus and aluminum, and spectrum stripping. EXPERIMENTAL
Neutron Generator. The generator used was a Kaman Nuclear 1001 200-KV unit; 14.5-MeV neutrons are produced by 3H(d,n)4He. With a new target (4 curies 3H/in~h2) output is 1011 n/sec, usable flux several times lOgn/cm2-sec at 1 cm from the target. The flux decreases, of course, with use of the target, and, therefore, all results are adjusted to lo9 n/cm2 sec. Flux Monitoring. A piece of copper wire was included with each sample irradiated. The reaction 63Cu(n,2n)6 2Cu was employed. The annihilation radiation ofe2Cu was counted using a 1.75 X 2-inch NaI(T1) well crystal and a single channel (3) A. S. Gillespie and W. W. Hill, Nucleonics, 19, (No. ll), 170
(1961).
