Instrumental Peak Distortion 111.
The Analysis of Overlapping Curves
I. G . McWilliam' and H. C . Bolton Monash University, Clayton, Victoria, 3168, Australia An analysis is given of the distortion of two overlapping Gaussian input curves which arises because of the finite time constant of a measuring instrument. The condition for separability of two such curves i s taken to be the existence of a flex in the total output curve and earlier results of Westerberg (1969) on pure Gaussian curves are extended. The separability condition is used analytically, allowing a new function to be defined, and also numerically using digital search techniques. Results are given which relate height and area ratios, within 1% and 3% of the true values, with the time constant, the separation between the two curves, their relative heights, and the height at the minimum point between the curves.
INPARTI of this series ( I ) , we have discussed the distortion of a Gaussian curve due to the effect of a finite time constant of an instrument. Some examples were given of the separation which can be expected between two peaks at different concentration levels and for different time constants. It was also shown that the total area under a single curve was independent of the time constant, but in considering the overlap between two peaks the precise distribution of this area is of course important. The height of a peak is commonly used as an alternative measure of the area under a peak; this is justified if the peak shape is independent of the peak height as would be expected for a linear system. Height and area measurements are not equivalent when there are two partially resolved peaks, and it is therefore relevant to examine the effect of a finite time constant on both the peak height and apparent area for peaks which are not completely separated. We deal first with the simple case of two symmetrical peaks, and then go on to examine the asymmetrical case. SYMMETRICAL GAUSSIAN PEAKS
Consider the function &(T)= E(T)
+ HE(T - R )
(1)
which represents a Gaussian function E(T) followed after a time R by another, E(T - R), of equal width and of relative height H (Figure 1). With the usual convention, we call E(T) the leading function and E(T - R ) the trailing function, and we have
E(T) = exp{ - * / z T 2 )
(2) We define the necessary condition for the two functions to be just separated (2) as the solution of the two equations d&(T)/dT = 0
(3)
Present address, Department of Applied Chemistry, Swinburne College of Technology, John Street, Hawthorn, Victoria, 3122, Australia. (1) I. G. McWilliam and H. C. Bolton, ANAL. CHEM., 41, 1755
(1969). (2) A. W. Westerberg, ibid., p 1770.
and d2&(T)/dT2= 0
(4)
This defines the point Tp at which there is a flex in the E(T) curve. Using Equation 1 in Equations 3 and 4 we find
TF' - TFR
+1=0
or Tp'
=
( R i d R 2 - 4)/2
(5)
and also HF =
Expressions 5 and 6 have been given by Westerberg (2). We can eliminate R between Equations 5 and 6 to get
HF = TF' exp( - ' / z ( T F ~- 1/TF2)}
(7)
for both T F + and T F - . There are generally two values of TF; the positive sign corresponds to values of HF less than unity and the negative sign to values of HF greater than unity (Figure 2). The corresponding values of Hp are reciprocals of each other. The two values of TF satisfy TF+
+ TF- = R
This means that if TF+ is the distance of the point at which Equations 3 and 4 hold from the origin of E(T), which is the center of the leading function (T = 0), then TF- is the distance of this point from the centre of the trailing function. This is shown in Figure 2. It can also be seen from this figure that the two curves are of the same form but reversed in time. A critical value of TF,which we shall call Tc, is given for R = 2 at which value we have TC = TF- = TF+ = 1, Hc = 1, and the function E(T) is then symmetrical. We wish now to see how the true H values compare with the ratio of the two peak heights and the ratio of the apparent areas of the two peaks. The latter is obtained by dividing the peaks at the minimum TNin the combined curve, as shown in Figure 1, and ascribing the area on each side of this dividing line to each of the separate peaks (3). This of course can be done only up to the point where the peaks are just resolved, as defined by Equations 5 and 6 and shown in Figure 2. Referring again to Figure 1, the maxima of the two curves, which depend on the degree of overlap, and the minimum point can readily be obtained by a digital search technique similar to that shown in Figure l l b of part I1 of this series ( 4 ) . By inserting various values of peak separation R and height H , we can also determine the conditions at which the (3) E. Proksch, H. Bruneder, and V. Granzner, J . Chromatogr. Sci., 7, 473 (1969). (4) I. G. McWilliam and H. C. Bolton, ANAL.CHEM., 41, 1762 (1969). ANALYTICAL CHEMISTRY, VOL. 43, NO. 7, JUNE 1971
883
1-
Figure 1. Partially resolved Gaussian peaks, showing position of minimum point, TN
c (1)
Figure 3. Effect of peak separation on overall curve shape, for Gaussian peaks
I
n
16-
141210-
0.8060 L-
SEPARATION
--R:25-
- L - 2
2
0 T
L
Figure 2. Gaussian peaks which are just separated, showing the two solutions to Equations 5 and 6 for R = 2.5
peaks are just resolved, since for the unresolved peaks a minimum point cannot be obtained. This behavior is shown in Figure 3 which depicts the change in peak shape for H = 0.5 and R values of 2.5, 3.0, 3.5, and 4.0. It is evident from this figure that, for H = 0.5, the critical value of R which satisfies Equations 3 and 4 lies between 2.5 and 3.0. Having determined the maxima of the two curves for R values greater than the critical value, limiting values of R can be obtained (from the computer printout) corresponding to height ratios which are accurate to within a specified figure, for example within 1% and 3 %, respectively, of the inserted value of H. In Figure 4 the left hand curve defines the resolution limit according to Equations 5 and 6, and the separate points are the critical values of R given by the computer results for increments of 0.1 in R. The upper limit values have been used. The next two curves correspond to the limiting values of R for height ratio accuracies of 3 % and 1%, respectively. R values to the right of these lines correspond to peak separations which give an accuracy better than that specified; values to the left represent increased peak overlap and correspondingly greater errors in the observed heights. The positions of the two peak maxima were determined by the computer to 10.001. If it is assumed that both peaks are '
884
ANALYTICAL CHEMISTRY, VOL. 43, NO. 7, J U N E 1971
(R)
Figure 4. Peak separation as a function of H , for two Gaussian curves, corresponding to just-resolved, 3 % and 1 % peak height ratio errors, 3% and 1 % peak area ratio errors
sharply defined--i.e., free from overlap-which would maximize the error, the maximum error in the peak height ratio would be 1- exp( - (0.001)2] which amounts to less than 1 part in 106. We choose only H < 1, since, as has been discussed above, the system is symmetrical. Also shown in Figure 4 are the limiting values of the peak separation corresponding to errors in the area ratio of 1 % and 3z. Whereas the absolute value of the height ratio increases as the peak separation decreases, the area ratio is found to decrease. The area ratio is also more sensitive to the effect of peak overlap than is the height ratio, as is seen from Figure 4. The position of the minimum (TN, Figure 1) determines the area ratio which can be calculated from the equations A =
-sTN 1
v'G
exp( - l/zT2)dT
(9)
-m
The integrals required in the above equations were evaluated using an approximation (suitably modified) accurate to better
FIT)
+oAl
i
-1 0
\ +l 0
\
\
+20
T
’
-.-0 2
Figure 5. Effect of time constant on curve shape for two partially resolved peaks (cf. Figure3, R = 3.0)
-- -04 ---0 6
‘A ! \
i-08
Figure 6. The F function, Equation 20 for r / u and 1.0
T
than 1 part in lo7 (5). For small values of H, the maximum error in the area ratio from this source would therefore be in the order of 1 part in H X lo7. The position of the minimum was determined to within k0.0005 by the computer routine, and the corresponding maximum error in the area would be within 2 parts in l o 7 at the minimum point (TN = 3.88) for H = 0.01. These errors are obviously negligible.
Now let us examine the argument for the asymmetrical function D(T) which is the solution of
and d2D(T) __- -0 dT2
From Equation 16 we get, using Equation 12 u dD
U
(T - R) exp{ -I/?(T - R)?)- -r dT -(T adD
rJ
E(T)
dT
H = -Texp{-1/,T2)/(T-
d 2 exp(1/2S2 ~ - TS)
Q(T) = D(T)
=
0
and noting that Equation 15 is incorporated in 17, we at once get
The expression for D(T) is
where X = (T - S ) / d F a n d S consecutive peaks of equal width,
1
- R)
(17)
dD(q + ;D(T) = =
0.1,0.3, 0.6,
- -7 ~ e x p { - - l / ~ ~-2-r) -dT +
ASYMMETRICAL PEAKS-GRAPHICAL SOLUTION
D(T)
=
exp(-x2) dx
= u ] ~ .We
(13)
have, for two
+ HD(T - R)
(1 4)
where R is the distance between the centers of the two original input functions E(T) and E(T - R), and H i s again the ratio of the height of the second peak to that of the first peak. The time constant introduces another variable into the system which can lead to loss of resolution between two peaks. This is most pronounced when the trailing peak is comparatively small, Le., H 6.595 .lo85 ,279904 a1102 ,237751 .1115 ,19*JY60 ,1136 .la6325 ,1152 ,137521 ,1169 * llL12r) * ll8h u 9 0 h5 0 ,1233 ,0725fi.I ,1221 ,119754: .L239 .L1431111 I 1256 .u55111 1275 ,027024 ,1293 .u73594 I 1 3 1 2 U1524h 1331 .bliOlJ 1351 .UIJW59A ,1371 I, 3 0 3 0 1 ,1392 .004¶72 1413 UO328S ,1434 0 0 2J3 3 ,1457 eU4164R 1479 .U01150 .1502 0 (I 0 7 9 4 1526
.
'.
. .
. . I
..
.
I
,l e 4 8 .'1007?2 ,1470 ,1493 ,001944 ,001405 ,1517 I 1542 ,002170 ,002978 I1568 1595 .Q04140 ,1623 ,505701 ,1652 ,907774 ,n i n 4 o 7 ,1682 ,1114035 e 1714 ,0195R4 ,1747 ,024369 ,1781 ,11316 4 4 ,1818 ,1740172 ,1888 ,n 5 1 4 0 8 I 1918 ,365187 ,1946 ,081579 ,1986 ,101095 ,2028 ,2075: ,123977 ,150492 I 2127 ,180878 2184 ,215291 2246 ,253781 ,2313 ,296271 * 2387 ,2469 ,347550 ,398262 e 2561 ,2665 ,444894 ,499770 2705 ,556047 ,2928 ,612819 ,3101 ,4689.ll I3319 ,721305 e 3608 ,774678 ,4020 ,821870 4677 .R63776 ,5941 ,R 9 9 l R I a 9 6 3 1 ,927332 .947491 -0.4783
,0 0 0 4 9 4
0194 0870 a 1309 e 1630 ,1886 ,2104 ,2298 ,2477 ,2648 ,2814 ,2979 ,3145 a 3314 ,3489 ,3669 ,3359 ,4c58 ,4269 ,4493 ,4733 ,4991 I 5269 ,556Y ,5896 ,6252 ,6642 e 7071 * 7544 .BO68 8650 ,9300 ,'713630 1 , 0 0 2 8 ,nil1493 1.0840 .lJO$O36 1,1776 ~ 1 0 ~ 1 1 19. 2 8 2 9 ,004634 1.4031 .i)O3491 1.5410 e
,7552151 ,940236 ,916945 ,880040 ,848365 .PO4915 ,7557R6 ,705136 ,651133 ,595930 ,540568 ,486054 ,433279 I38'80 7 ,135353 ,791206 ,?511bR1 ,714293 ,18151 7 ,157513 ,12711 6 ,103112 ,o e h 2 4 1 ,070218 ,a 5 6 7 4 3 ,')455!U I OJh251 ,'I21668 ,n z 2 5 1 7 ,1117569
,000319 ,080469
,00068I ,000980 ,001395 ,001969 ,002751 ,003806 ,005315 ,087077 ,009513 ,012665 ,016701 * 031812 ,028218 ,036159 ,045896 ,057704 ,071067 ,068665. ,108363 .131199 .157365 ,186994 .270142 ,256770 .296736 ,339776 ,305504 .4S3408 ,462854 ,533093 .5n3285 ,632516 ,619828 ,724251 ,764636 ,800693 .03in24 ,855154 .872556 ,882879 ,885943 ,881765 ,870540 ,852636 ,838575 ,7991106 ,764680 ,726419 ,685081 ,641532 .596616 .551133 ,505~15 ,461312 ,4101 80 ,376880 ,437769 ,3n111? ,267nt)~ .235771 ,207200 ,181328 ,158n70 ,13729Q ,111863 102593 ,088308 * 075825 064965 ,05555s so47424 ,040425 ,034416 .0?9?68 I O24@6R
,1892 ,1929 ,1968 ,2008 ,2051 ,2095 .2142 ,2191 ,2243 ,1297 ,7355 ,2415 ,2480 .2548 2620 ,2697 ,2780 ,2868 .2963 ,3066 .3178 ,3301 ,3435 .3584 ,3750 ,3937 ,4151 4398 ,4689 ,5039 ,5473
,6030 ,6781 ,7870 ,9629 1.3056 2.3133 -1,6481 -0,6379 -0,1969 -0.1090 ,0078 , OY28 ,1603 ,2177 ,3692 ,3173 ,3639 ,4103 ,4576 ,5067 ,5585 ,61S9 ,6739 ,7395 ,8119 e924 ,9827 1.0847 1,?01)6 1.33?4 1,4863 1,6638 1,8?09 2.1143 2.4022 2,7450 3.q562 3,6526 4.2563 4.9950 5.9085 7.n433 8.4654 10.2616 12.5485
-F( 1 * 0 )
,000217 ;0 0 0 3 1 9 ,000465 .000670 a 000957 * 001353 .001895 ,002629 ,003612 ,004917 ,006629 ,008853 ,011712 .015349 .019927 ,025629 ,032655 ,041222 ,051554 ,063882 ,078431 ,095413 .115013 ,137303 .162623 ,190772 ,221794 ,255574 ,291901 ,330472 ,370887 ,412657 ,455206 ,497890 .540012 ,580841 ,619638 ,655680 688284 .716833 ,740797 .759750 ,773389 .781535 .784143 ,781296 ,773201 ,760173 ,742623 ,721036 .695954 ,667953 .637623 ,605551 ,572304 .538410 ,504354 ,470568 .437422 .405227 .374234 ,344656 ., 32 19 60 51 73 2
.2151 .2199 .2250 .2303 ,2359 ,2417 ,2479 ,2544 * 2613 .2686 .2764 .2846 .2934 .3027 .3128 ,3235 .3351 .3477 ,3613 .3762 .3925 .4105 .4305 ,4528 ,4780 ,5067 .5399 ,5786 ,6248 t 6 8 1 1 ,7516 ,8433 ,9684 1s 1 5 1 6 1 4504
2 0371 3.7726
-3.0827 -1 3 4 3 4 -0.7504 -0.4424 -0 a 2473 6
-0,1072 ,0026 .0949 ,1770
,2533 3272 .4011 ,4771 .5570 .6527 ,7362
,8398 ,9539 1.0U76
1,2385
1.4131 1.6169 1.8567 2.1414 2.4821 2.0928 .265365 3.3922 .242262 4 OC40 4,7595 .220866 .201076 5.7COl 6.8805 .le2850 .lh6117 6.3741 ,150794 10.2798 .I36792 12.7317 15.9132 ,124023 20.0713 ,1123'95 25.5751 .lo1821 32.8980 ,092216 083498 42.7390 Continued on next page I
0
ANALYTICAL CHEMISTRY, VOL. 43, NO. 7, JUNE 1971
889
VALUfS OF E ,
t
E
4.0 4.1 4.2 4.3 4.4 4.5 4 6 4.7 4.8 4.9
3.3546t-04 2.2375t-54 I.4775t-04 9.6533k-55 0.25226-0> 4,0065t-05 2 a 5419E-05 1.5967€-05 9,9295t-Ob 6 ,11366-06 3.7267t-Ob
5.0 A
SINGULAqITY
D AhID T FOR T/u
APPENDIX (Continued) 0 . 1 ~ 013, 0 . 6 A N D
-F( 0,3)
D(0.31
OF
.OUJb44 .UB0369
.U110247 .UOO164 .U00107 .000070 .OOOC45 .ilni102~
.unooi~ .UCOOl’! .JOOOO~
,1551 ,1576 ,1602 ,1629 ,1627 ,1666 ,1711 ,1748 ,1784 ,1819 ,1854
,002617 ,001952 , noi4so e001075 I t l o o 791 ,no05~i ,000475 0 OOd310 ,000226 no0164 ,000119
.
T H E F F c l ‘ l C T I O N O C C U R S A T 1.0
* 021113
1 I 7000 1,8842 2,0989 2,3505 2 ,6470 2,9987 3.4102 3 0 9221 4,5313 5.2727 60 1814
017914 ,015192 ,012878 ,010913 009245 ,007831 006632 ,005616 ,004755 ,004nzs I
15.4041 19.2836 24,2430 30,7720 39,4419 51.0562 66,7531 88.1595 117.6134 %58,5155 215.8376
,U 7 J S 9 1
56.0637 74.3444 .Ob8424 99.5617 ,061930 ,056048 134.7087 ,050722 184,1524 ,045900 254.3611 ,041535 354 e 9975 .037564 500.6179 ,034009 713.3400 ,030773 1027.0608 , 0 2 7 0 4 5 1494.1852
( S E E F I G U R E 6).
Potentiometric Determination of Cyanide with Ion Selective Electrode Application to Cyanogenic Glycosides in Sudan Grasses W. J. Blaedel, D. B. Easty,l Laurens Anderson, and T. R. Farrell Departments of Chemistry and Biochemistry, University of Wisconsin, Madison, Wis. 53706 After accelerated hydrolysis with emulsin, cyanide may be determined in the hydrolyzate by direct potentiometric measurement with an ion selective electrode. The method was evaluated for Sudan grass samples which were also analyzed by volatilizing the cyanide away from the hydrolyzate before measurement. The root mean squared error of the procedure is around 1 ppm of HCN, or 2% relative, whichever is greater.
A BRIEF SURVEY has been made of methods of determining cyanide resulting from hydrolysis of cyanogenic glycosides in plant tissue ( I ) . The sensitivity, speed, and simplicity of the potentiometric method give it an indisputable advantage over other methods. However, despite their commercial availability, ion selective electrodes have not yet been applied to the sensitive and precise determination of cyanide in plant tissues. Gillingham and coworkers ( 2 ) have used a cyanide electrode to screen forage samples approximately for the cyanide content. Gyorgy and coworkers (3) employed an electrode for the potentiometric determination of cyanide in steam distillates from plant hydrolyzates. These authors stated without experimental confirmation that direct measurements might be made on the hydrolyzate.\ In this paper, a method is developed for the determination of cyanide in plant leaf hydrolyzates by direct potentiometric measurement with an ion selective electrode. The method is tested on Sudan grass samples which were also analyzed by
another method ( I ) involving removal of the cyanide from the hydrolyzate before measurement. THEORY
Ion selective electrodes are not completely specific, but are subject to interferences. It has been shown that the potential of an ion selective electrode is dependent not only upon the activity ( a x ) of the sought-for substance, but also upon the activities of other substances that may be present. For some electrodes,
E
=
Constant
RT + 2.303 ~log (ax + $1 zxF
+
The explicit form of is not known except for a few simple systems which do not include the cyanide electrode [Ref. 4, p 42 (G. Eisenman), p 64 (J. W. Ross), and p 98 (A. K. Covington)]. In general, may depend not only upon the activities of other substances present, but also upon the selectivity of the electrode for them. If there is interaction between the sought-for substance and interferences, may also depend upon the chemical properties of these substances. In an analysis, it is usually the concentration (Cx) which is sought. Since C, is related to ax through the activity COefficient yx, Equation 1 may be rewritten:
+
+
E = constant
RT + 2.303 log Y x + ZxF -
1 Present address, Institute of Paper Chemistry, Appleton, Wis. 54911
(1) D. B. Easty, W. J. Blaedel, and L. Anderson, ANAL.CHEM., 43, 509 (1971). (2) J. T. Gillingham, M. M. Shirer, and N. R. Page, Agron. J . , 61, 717 (1969). (3) B. Gyorgy, L. AndrC, L. Stehl, and E. Pungor, Anal. Chim. Acta, 46, 318-21 (1969). 890
ANALYTICAL CHEMISTRY, VOL. 43, NO. 7, JUNE 1971
In the methods of this paper, Equation 2 is rendered analytically useful by making all determinations in a medium of (4) R. A. Durst, Ed., “Ion Selective Electrodes,” National Bureau of Standards Special Publication 314, November 1969.
