CORRESPONDENCE Determination of Chromatographic Resolution for Peaks of Vast Concentration Differences SIR: One of the most important relationships for determining operational parameters in chromatography is resolution. Therefore, it is often necessary to determine resolution for adjacent peaks of vast concentration differences. This is particularly true when deciding optimum sample loading in trace analysis. Chromatographic resolution is normally determined by the definition ( I )
where the resolution, R , for two adjacent peaks is a function of the width of each peak, Y & , d f , at the base and the difference in retention distances, dRj,i,for the peaks of interest. Other descriptions of resolution or resolving power have been proposed (2-7), but they are not as generally applicable in the laboratory. The definition proposed by Kaiser (8) is given in Equation 2 .
e
=
f/g
pi24 Figure 1. Relationship between unresolved peaks
(2)
As shown in Figure 1,e is a function of peak overlap. Resolution, 8, has no restrictions on peak quality and is particularly suited to adjacent peaks whose heights differ greatly. The term f represents the distance between the valley separating two peaks and a line joining the apexes of the peaks. The term g represents the distance from the base line to the line joining the peak maxima. It should be noted that f3 varies only from 0 to 1.0 and therefore makes a very practical value for laboratory use. Even though construction lines for the determination of 0 can generally be drawn more accurately than those required for the determination of Ri,i, 6 becomes very difficult to ascertain accurately when the sizes of adjacent peaks are grossly different. As the peaks become more disproportionate, the line joining to the peak apexes and the perpendicular through the valley approach parallelism. This makes the intersection point of these lines difficult to locate. When attenuation is required to display both peaks, neither procedure can be used directly. The method of determining resolution proposed here eliminates the need for construction lines other than those required to establish a base line, peak maximums, and the location of the valley between adjacent peaks. The equation (1) D. Ambrose, 0. T. James, A. I. M. Keulemans, E. Kovats, H. Rock, %. Rouit, and F. H. Stross in “Gas Chromatography 1960,” R. P. W. Scott, Ed., Butterworths, London, 1960, p 429. (2) W. E. Harris and H. W. Habgood, “Programmed Temperature Gas Chromatography,” John Wiley & Sons, New York, N.Y., 1966, pp 121-7. (3) J. H. Purnell, J . Chern. SOC.,1960, 1268. (4) P. Chovin and C. Guiochon, Bull. SOC.Chim. Fr., 1965, 3391. (5) P. R. Rony, Separ. Sci., 3,239 (1968). (6) Ibid., p 351. (7) A. B. Christophe, Chrornatographia, 4, 455 (1971). (8) R. Kaiser, “Gas Phase Chromatography,” Vol. I, Butterworths, London, 1963, p 39.
may be derived using the definition for 6 as given in Equation 2. The equivalents for f and g , as taken from Figure 1, may be substituted into Equation 2 giving
e=
BC
+ H, - H, + H,
FC
-1--_
BC
H”
+ H,
(3)
Since the two triangles ABC and AEG are similar, their respective sides are proportional. BC -
EG
- Hi
AB
KE
dR1
- Hs - dRs
(4)
-
AB, the distance from dRsto the valley between adjacent peaks, may be designated dRu - dRsand Equation 4 may now be solved for FC.
BC=
(dR0
- dRs) (Hi- Hs) dRi - ~ R S
(5)
Substituting in Equation 3 the terms from Equation 5, 0 becomes ffddR1 - dRs) (6) (dR0 - dRs) ( H I - Hs) $- H S ( ~-R~ZR S ) If the antipodal configuration of peaks is used for the derivative of 6, all the terms remain constant save those representing AB. AB becomes (dRs - dRa). Therefore, to describe both configurations with one equation and to retain absolute identity with the 0 of Kaiser, this term is replaced with the absolute value as shown in Equation 7.
e = i -
The equation can be further generalized by inserting the factors X,,X l , X , for those respective values that are affected
ANALYTICAL CHEMISTRY, VOL. 44, NO. 11, SEPTEMBER 1972
1905
by attenuation. The general form of the equation would then become
DEFINITION OF SYMBOLS
Resolution ( I ) . Resolution ( 2 ) . = solute designation: solute i, relative to solute j eluted later. Subscript 1,s = solute designation: large peak, small peak. Subscript D = designates valley between adjacent peaks. = height of respective peak. Hi,, = distance from base line to valley between H, adjacent peaks. = retention distance for respective peaks. d R i . j , dRl.8 = width of respective peaks. Ydj.z = retention distance to valley. dR B = attenuation factor for respective peaks and Xl,s,v valley.
R
8 Subscript i,j
The linearity of the detection system including the attenuator will directly affect the accuracy of 0 and should be considered when making calculations. Using Equation 8, resolution may be calculated for adjacent peaks that may vary in height as much as the dynamic range of the instrument. All that is required to make an accurate determination of 0 is the peak and valley heights and their respective retention distances. More information is required to calculate 0 in this derived form, but each measurement is relatively easy to make and awkward construction lines are entirely eliminated. This method could easily be applied to systems using computers for data reduction. Those computer systems using continuous sampling techniques would, in fact, acquire all necessary data and need only sufficient programming to make the calculation.
= =
GLENNC. CARLE Ames Research Center, NASA Moffett Field. Calif. 94035 RECEIVED for review February 2, 1972. Accepted May 12, 1972,
Comments on Smoothing and Differentiation of Data by Simplified Least Square Procedure SIR: Some years ago, Savitzky and Golay (1) proposed a series of numerical tables for the smoothing of experimental data and for the computation of their derivatives. The main advantage of the method is that they obtain universal numerical functions, the convolution of which with the original vector data gives a smoothed vector as well as its successive derivatives. Unfortunately the published tables seem to include some numerical errors. Moreover, Equation IIIc in (1) fails in the case i = k = r = 0 since it does not give the identification b,,o = y o as it follows from Equation IIIa. The correct equation is
izm [(y
i=-m
i;=o bn,k .
ik) - y * ] ir
+ b,,o - yo = 0
I I I1
I1 IV IV VI VI a
25 23 23 13 21 19 15 5
zt5
343 805
norm all
divisible by 3
f 4 $3 +3 f 2 $2
-135 +79564 E ! 7372 -44 +2
Refers to tables given in (I).
Thus, we may write
i#O
The tables were recalculated by introducing this correction in Savitzky and Golay’s reasoning and a check was performed by using a more general matrician formalism (2, 3). Let Y be the (n X 1) vector of observations and a the (n x 1) vector of “error” random variables with E(€)= 0
(1)
V ( € )= E ( € € ’ )= u2I
(2)
and dispersion matrix
where I is the (n X n) identity matrix. The relations 1 and 2 imply that the ei are uncorrelated, they all have zero mean value and the same variance u2. (1) A. Savitzky and M. J. E. Golay, ANAL.CHEM., 36,1627 (1964) (2) J. Steinier, Bull. Rech. Agron. Gembloux, in press. (3) M. Kendall and A. Stuart, “The Advanced Theory of Statistics,” Vol. 2, Charles Griffin et G. Ltd., London, 1958-1966, pp 75-87. 1906
Table I. Corrections of Values given by Savitzky and Golay Tablea Column Point Value
y = D + e
(3)
where D is the (n X 1) vector of exact values that are searched. Let us suppose now that D
=
X8
(4)
where X is a (n X k ) matrix of known coefficients, with n > k , and 8 a ( k X 1) vector of parameters. This means that the n true Dt values can be expressed by means of a polynomial of degree (k - 1). Thus from Relations 3 and 4, we have
y=xe+€
(5)
which allows the determination of 8 and thus of D by Equation 4. Then the least square method of estimation of this vector of parameters requires that we minimize the scalar sum of squares.
ANALYTICAL CHEMISTRY, VOL. 44, NO. 11, SEPTEMBER 1972
