columns packed with 25 weight yo Apiezon L on 30- to 60-mesh Chroinosorb u'. Two sets of chromatograms were obtained on these columns to correspond to those on the bleeding columns. The same flow rates, temperature programs, and sample sizes were used. Under these conditions the naphthalene was eluted a t 175" C. and the p-terphenyl below 320" C. where a single-column base-line would not have drifted up-scale more than 10% a t an attenuation of 1. Six to eleven chromatograms of the binary mixture of naphthalene and p terphenyl were obtained at the two flow rates on both the bleeding and nonbleeding columns. The naphthalene and p-terphenyl used in making the mixture were each checked for purity on both columns to make certain that no minor impurity in either component was being resolved on one column and
counted with a major coinponent on the other column. The results of the study are given in Table I in terms of the averages of the rat'os of the two component peak areas on each of the two columns. Comparison of the average peak area ratios a t each individual flow rate shows very excellent agreement indicating that there is no appreciable change in the relative response for a component eluted when the column is bleeding to the extent studied here. This conclusion applies to thermal conductivity detectors; quite different results might be obtained if detectors with more limited linear response ranges were used. The 8% difference in the peak area ratios a t the two different flow rates cannot be explained entirely by the effect on relative response of the quite different flow rates and sample sizes. If the flow controller on the sample side
reproducil)ly failed to maintain the flow a t 120 cc. per minute during the entire temperature program, a peak area larger than expected nould be obtained for the p-terplienyl and would lend to the ~ i p parent difference in peak area ratios which was observed. ACKNOWLEDGMENT
The authors acknowledge the assistance of L. M. Olszewski in obtaining some of the chromatograms required for the quantitative study. LITERATURE CITED
(1) Emery, E. M., Koerner, 11'. E., AXAL. CHEM.33, 523 (1961). RECEIVEDfor review March 8, 1962. Accepted July 13, 1962. Presented in part at the Symposium on Gas Chromatography, Divisions of Analytical and
Petroleum Chemistry, 139th Meeting, ACS, St. Louis, Mo., March 1961.
Gas Chromatographic Response as a Function of Sample Input Profile CHARLES N. REILLEY, GARY P. HILDEBRAND, and J. W. ASHLEY, Jr. Department of Chemistry, The University o f North Carolina, Chapel Hill, N. C.
b A series of equations based on a simple principle is developed to describe the shape of chromatographic curves in terms of the mode of sample injection. This principle is used to develop useful parameters to describe step chromatography curves, and comparison is made to the corresponding parameters of impulse chromatography curves. The use of various input functions leads to many novel response shapes, thus considerably broadening the scope of gas chromatographic techniques. Applications of various sample input profiles to quantitative analysis, preparative chromatography, the study of column phenomena, and the monitoring of industrial process streams are discussed.
terpret. Also, only one component of a mixture could be isolated in pure form, so the technique, understandably, did not find wide use as a preparative tool. Therefore, its major use has been confined to the determination of adsorption isotherms (6, 7, 11, 12). By coating an inert solid support with a partitioning liquid and by using sufficiently dilute solutions of solute in the carrier gas, as suggested by Boeke (1, d ) , linear isotherms can be obtained,
a
and introduction of sample in the form of a step becomes a feasible technique, I n any chromatographic experiment the type of chromatogram obtained (the response) depends on the shape of sample input profile (the input function). Although the number of possible input functions is almost unlimited, a few interesting types are shown in Figure 1. I n Curve A , the sample is introduced as a step function-i.e., the concentration of sample
B
n h
.?
0
I-
a cc
F
introduced by Tiselius (13, 14) and developed by Claesson (6),has not been exploited in gas chromatography as fully as its counterpart, peak elution analysis. Undoubtedly, the reason t h a t little enthusiasm has been shown for the frontal technique lies in the fact t h a t early experimenters employed solid adsorbants, which exhibit interdependent As a result, nonlinear isotherms. frontal chromatograms of multicomponent mixtures were difficult to inRONTAL
1198
ANALYSIS,
ANALYTICAL CHEMISTRY
+ Z
E
F
G I
W
I
0
z
i
0 0 W
i
J
L
VOLUME _____j
Figure 1 .
Various input profiles
J-Il
._ STEP
INPUT PRCFILE
v is the transfer number (1, 2, 3,
I
RESPCiNSE PI;OFILE
. . .),
k is the distribution ratio of the solute , is (mobi1e:stationary phases), and nV the amount of the solute leaving tube p during transfer v. From Equation 1 it can be shown that t,he response, npv, to a step input, no*,occurring when v equals s is given by
7 1
T i
v-p-aS1 npv
2
C
= no*
x
i-1
+
( p i - 2)! ( p - l ) ! ( i- l)!(k
I AND
2 i
V
Figure 2. Input and response profiles illustrating additivity in the case of a single step input of two components
entering the first plate suddenly rises from zero to some finite value and remains a t that value thereafter. Chromatography employing a step input function is herein termed step chromatography. Curve B represents a double step. The first step suddenly places the sample into the column as in Curve 1,and the second, being negative and of the same magnitude, corresponds to the sudden removal of sample from the carrier gas entering the column. Curve C is a special case of Curve B where the second step occurs a t a time such that the sample as a whole may be considered to have entered only the first plate of the column. This type of double step represents the ideal case of impulse chromatography. Curves D through I , and L represent various other combinations of positive and negative step inputs of differing magnitudes. Several of these input functions lead to novel responses which may be of interest for analytical work and for studying the fundamental properties of chromatographic columns. In this paper a simple principle which correlates input an8 response functions is described. The basis for this development is founded on' two criteria: 1) The chromatographic action on each part of the input is independent of all other parts of the input, and 2) the amplitudes of the response and input are linearly related. Under these conditions, the response to an input can be expressed as the sum of the responses to the individual parts of the input. This principle is illustrated for Craig machine operation and for certain chromatographic processes.
CRAIG MACHINE ILLUSTRATION
The addivity or superposi+,ion principle stated above is aptly illustrated by a Craig machine operated in a singlewithdrawal process (9). The transfer of an ideal solute through a Craig machine with constant phase volumes per tube can be described by npQ
=
1 npe-l + k + lnp-lv-l k + 1
u
np* =
INPUT PROFILE
T
~
(2)
- p -SI+
1
1%"
x
i= 1
k,i-l + i - 2)! - l)! (i - l)!(kl + 1)i*-1 u - p - SI+ 1 x (p
(p
2no"
+
i=l
(1)
kzi-l + i - 2)! - l)!(i - l)!(ke + 1)'+*-'
-
(p
where p is the tube number (1,2, 3, . . .),
.OMPONENl
l)'+p-'
where i is a mathematical parameter having the values indicated. Thus, a plot of npu us. v gives the response that results from a single-step input. Further inspection of the model shows that the response a t any value of v to a combination of step inputs of a multicomponent mixture is the sum of the responses to each step input and each component a t the same value of v, as illustrated in Figures 2 and 3. Figure 2 shows the addition of the step inputs for two components to yield a single-step input and the corresponding addition of the responses for each component to yield a doublestep response curve.
i
1
ki-1
+
(p
1
(3)
RESPONSE PROFILE
L
7 1 T
I
2
I AND 2
\
V
Figure 3. Input and response profiles illustrating - additivity in the case of two successive step inputs of ti single. component VOL 34, NO. 10, SEPTEMBER 1962
1199
Figure 3 illustrates tliv atltlition of two step inputs of on^ component, occurring a t s1 and sz, to yield a doublestep input and the coi wspoiiding addition of the responses to each step input to yield a double-qtcp response profilr. npv
= no
0:
x
- 2)! 1)!(i - l ) !( k
ki-I
+
l)‘+P-’
+
u - p --a+1
x i=l
(i
+p
-
(p
kC-I
- 2)!
l ) !(i - l ) !
ic +
l)’+p-1
(4)
P
By a similar application of Equation 2, a double-step input of the form shown in Figure 1,B, gives rise to a response given by u - p -SI+
npz
=
noal
C
+p
1
x
hi-1 - 2)! - I ) ! ( k + l)’+’-’
( p - l ) ! (i
- SP+
u -p
(k =
+ l)p
(8)
({I)
kp
=
k(k
+ 1)p
(10)
VRVR‘
=
I(l1)
T
where u is conveniently experimentally approximated by one fourth the volume between the intersections of the v-asis with tangents to the inflection points as shown in Figure 5 , A . “Step” Analysis (Response to a Sirigle-Step Input). In “step” analysis the equation for the front step can be obtained from Equation 6 by setting s1 equal t o unity and letting the initial response, n,~,be zero.
i=l
(2
=
nheie 1 - H ic the “retention \ olunie,” VR‘ is the “adjusted retention volume,” and u is one half the volume b e t w e n the two inflection points (5). The number of tubes, p , can be obtained from the r e y ~ ~ lby iv
,=I
nos=
l‘n
and
j
(i + p
(p -
For the ”inipul~e“response, ~llustinted schematically for a single component 111 Figure L.4. it can be shown that
1.8’
jno
np“ =
-p--s,fl
IC I
“Impulse” Analysis. The equation for ideal “impulse” analysis can be obtained from Equation 5 by letting s2 - s1 equal unity and summing over j components.
1
noaa i=l
+
(z p - 2)! ( p - l)! ( 2 - I)! ( k
It,-l
+
l)’+P-’
(5)
where the two step inputs, n,s1 and n,az, have the same magnitude but opposite sign. The shape of the response profile obtained from Equation 5 depends on the value of the sample input period, sz - s1 = As, as showi in Figure 4. Curve d corresponds to a very large sample input period, As = 400, and can be regarded as consisting of a front step (v = 0 to 400) and a back step (v = 400 to 800). Curve D represents “impuke” analysis, As = 1, while curves B and C correspond to intermediate sample input periods. For a n input of the form given by Figure 1,G, where the sample input and hence the response, npo,is constant prior to the step occurring at v equals 8, the total response is obtained by simply adding the initial response, n,’, to Equation 2. From the above discussion the inputresponse relationship for a Craig machine can be written in its most general form as
Since the value of the second suniniation approaches unity as v becomes larger, t h e n n d = Z j n o l , and hence the
> a
c
i
magnitude of the response, nni, beconies virtually equal t o the magnitude of the step-input. This means that the concentrations of solute(s) in the niohile phase and in the stationary phase are constant throughout the machine and that the conrentration(s) in the mobile phase is equal to the concentration(s) of solute(s) entering the machine. The equation for the back steps in “step” analysis is also obtained from Equation 6 as follows. Prior to the occurrence of the second negative step a t v = s2 the machine is equilibrated as described above, and hence the initial response, rips, for the back step is Z,no1. I
Thus, from Equation 6, by setting sl equal to unity and replacing nP* n-ith Z,no1, the response for the back step j
becomes
0
+
(p i - 2)! ( p - l)! (i - l ) !( k j
+
1y+p-1
jJ
j components and I steps. 0
ANALYTICAL CHEMISTRY
400
600
500
V
(6)
where 2 indicates the summation over
1200
200
kji-l
Figure 4. Input and response profiles for various sample input periods
--SI
Input proflle Response profile = 0 in all curves
+
kji-l (i p - 2)! ( p - l)!(i - l)! ( k j 1)”P-l
+
(13)
Thus, t,he back step response for a given component is geometrically symmetrical to that of its front step.
Table 1.
Maximum, Inflection, and Third-Derivative-Equal-Zero Points for Step and Impulse Response Curves of a Single Component Issuing from a Craig Machine
Step Response
nP’
‘P”
+ i - 2)!
nlox U-P
=
i=l
(p
(p
- I ) ! (i -
Hence, for each of the principles in the following discussions of the front step, there is a n equivalent principle applicable to the back step. When Equation 12 is subjected to finite differentiation, the slope of the “step” response curve is obtained.
(p
(v - 2 ) ! - l ) !( V - p
k jv-p-l
- l)!( k j + 1)w-l
(14)
As can be seen from Equations 7 and 14, the derivative of the “step” response is identical to the ‘limpulse” response; this integro-differential relationship is illustrated by Figure 6. Therefore, the peak maximum and the inflection points of the “impulse” response curve correspond to the inflection point and the points where the third derivative equals zero, respectively, of the “step” curve. These relationships are summarized in Table I. For a “step” response curve, the value of u (one half the volume between the two points where the third derivative equals zero) can be determined by differentiating once and then treating the derivative as a n “impulse” response curve. Because this procedure for determining u is not experimentally convenient, an alternate method is proposed. The “step” response curve can be partly described by W, the volume between the intersections of the tangent at the inflection point with the u-axis and the line where np*/n.lequals unity, as illustrated in Figure 5,B. The slope of the curve a t its inflection point is given Equation 14 when v = k ( p - 1) p 1 (Table I). Since the final height of the curve is nDl, the width, W , is given by nol divided by the slope at the inflection point. Thus,
+
+
0
-
( p - l)! [ k ( p l ) ]! ( k [(k 1) ( p l)]!
+
+
l)(k+’l(p-’)+l
kk(p-’I (15)
By applying Stirling’s approximation, z! = (z/e)s 2/%, it follows that w =
[2ak(p - 1 ) ( k
+ 1)]1’2
(16)
I)! (k
Impulse Response
kj-1
+ l)~+i-I
npv = no1( p
When p is much greater than unity, Equation 14 can be combined with Equation 10 to yield w =
dSC7
-
(v - 2)! kv-p-1 l)! ( V - p - l ) ! ( k I)”-’
+
the input function, Zct); and the actio11 function, A($): &P,
(17)
Therefore, the number of plates, p , can be obtained from the response by
Thus, the use of w allows one to obtain as much information directly from a “step” response curve as from its derivative. At this point it is important to recognize that no implication is made that a chromatography column operates as a discrete-plate model. However, there are certain mathematical similarities, which are made apparent by the discussion below. APPLICATION T O GAS CHROMATOGRAPHY
I n linear, independent gas chromatography the response to a complex input can, in principle, be expressed as the sum of the responses to the simple elements of the complex input. For example, if the response to an impulse (Figure 1,C) is known, then the response to a step (Figure 1,A) can be obtained by adding together the responses to an infinite series of successive impulses of equal area. In this case the summation of the individual responses can be replaced by the integral of the response to the first impulse because the summation of the successive impulses can be replaced by the integral of the first. Unfortunately, for many compIex inputs-e.g., a sine wave, Figure 1,Jthis procedure becomes impractical. However, by the use of Laplace transformations the response to any input can be obtained from a knowledge of the individual responses to an impulse of each individual component. For a gas chromatographic process exhibiting linear response, independent action, and only a single component, the following mathematical relationship exists among the Laplace transforms of the response function, R(t);
=
f(P)&)
(19)
xcp)
where represents the Laplace transformation of X ( t ) . Because the Laplace transformation of a unit in]I(p) = 1-then pulse is unity-e.g. A ( L )is identical to the response, R ( l ) , to a unit impulse. In many rases in gas chromatography the response to a unit impulse of a single component (Figure 1,C) can be closely approsimated by a Gaussian distribution function :
where the values of t R and o are experimentally determined (see Figure 5 , A ) , and for a given solute depend only on the nature of the column and operating conditions-Le., the values of these parameters are unaffected by the type of input. Taking the Laplace transform of Equation 20
Equation 19 then becomes
which is in agreement to that stated by Boeke ( 2 ) . The power of the Laplace transform method embodied in Equation 19 (or 2 2 ) is that the response to any singlecomponent input can be obtained from Equation 19 (or 22) by replacing with the Laplace transformation of the particular input function and subsequently taking the inverse Laplace transformation of the resulting equation. For more than one component the response is the sum of the responses for each component: (23)
VOL. 34, NO. 10, SEPTEMBER 1962
1201
1 I
, /
a
3
+
f - - - - - - - - - - - - - - i
c
w
- - - - _ _ - - _ - _ _ -
0
1
where 2 indicates
the
1
,
,
summation
j
over j components. The term, RO(p), occurs when the input is of the form shown in Figure 1,E. In this case an impulse is superimposed on a constant base line of finite concentration whose response has reached a steady-state value of R,. On the other hand, in Figure l,C the value of R,, and hence R,,(P),is zero. Impulse Analysis. The response equation for ideal impulse analysis can be obtained from Equations 21 and 23 by letting Roc,) = 0 and I [ p , j ) = C'i,
where C', equals J Cj dt and Ci is the concentration of component j in the input. The impulse response, illustrated schematically for a single component in Figure 5 , A , is conventionally described by
where N is the column efficiency for a given solute. Step Analysis. I n step analysis the response t o the step input (Figure I, A) can be obtained from Equations 21 and 23 by letting I(p,i) = C j / p and
Adp)
=
1202
0. ANALYTICAL CHEMISTRY
(26)
Equation 26 has the form of the integral of Equation 24; hence the step response profile is identical in shape to t h a t of the integral of the impulse response, as pointed out by Boeke (I,@. The step response curve for a single component is illustrated schematically in Figure 5,B. At large values of t the response approaches Cj, an inflection point occurs a t t = t R , and the slope a t that point is equal to C j / d % ~ . Hence, W , the ratio of C j to the slope a t the inflection point, is given by 6J
= v4ro
(27)
and the column efficiency for a given solute, in terms of step response parameters, becomes
The response for a double step input (Figure 1,B) with the second step occurring at t = t2 is given by
where the two step inputs are ZC,and j
-Xi.
As the v d u e of tz approaches
j
zero the response to the double step input approaches the response to an impulse containing an ideritical number of moles. =It values of t such that t > t 2 >> t~ t,he first summation in Equation 29 apC. proaches a value of 2-'; hence the j2 back step response (Figure 1,D) can be expressed by
where t2 is arbitrarily set equal to zero. Thus the back step for a given component is geometrically symmetrical to the front step and for each principle applicable to the front step response there is an equivalent principle applicable to the back step response. In the above discussions a Gaussian action function is used to illustrate the principle expressed by Equation 23. I n cases where the response to an impulse is not best approximated by a Gaussian distribution function, then the appropriate function itself must
be used; the principle expressed by Equation 23 is still applicable provided that linear, independent column behavior exists. Resolution. Although it is quite evident t h a t different mechanisms are involved in gas chromatography and in the operation of a Craig machine, the mathematical form for "impulse" response obtained with a Craig machine is very similar to a Gaussian distribution when the values of the distribution ratio, k , and the number of tubes, p , are large. I n this sense the quantity p , for a gas chromatographic peak is operationally defined by P
tRtR'
=
F
where tR, tR', and u are experimentally determined quantities. The relationship between p and N can be expressed by
For large values of t ~ / ( t ~t i ) , p and N become virtually equivalent. I n impulse chromatography the resolution, Ri, of two components can be defined as the ratio of the separation of the peaks to the average of the peak widths a t the base line. h' - t R ' , ' - 4 U l + 401
E.=
(33)
2
Qualitatively, the resolution of two components might be described (1) for
a step response, in terms of the slope of the plateau between the two step responses, and (2) for an impulse response, in terms of the height of the minimum between the two peaks. Since the derivative of the step response is identical in shape to the impulse response as shown in Figure 6, the slope of the plateau in the former is proportional to the height of the minimum in the latter, and therefore the value of the resolution in either case may be considered the same. For this reason, it seems appropriate to define the resolution of two components in such a way that its value is the same for ideal step chromatography as for ideal impulse chromatography. Thus, the combination of Equations 27 and 33 suggests that the resolution, R,, of two components in step chromatography be d e fined as 4 7 8 times the ratio of the separation of the two inflection points to the average of the curve widths,
The above considerations also indicate that, when separation is not complete, quantitative data obtained from a step response has the same degree of uncertainty as data obtained from an impulse response. EXPERIMENTAL
Apparatus. A Perkin-Elmer 154C Vapor Fractometer, equipped with a thermistor detector and a one-meter
y;p Figure 7.
i;
Apparatus for step input of sample
'/d-inch 0.d. borosilicate glass coluniii, packed with 8% Carbowax 400 on 30to 60-mesh Chromosorb P, was used throughout this study. For the quantitative work the output from the detector was measured by a 10-mv. Brown recorder. For the comparison of the impulse and step chromatograms a Sargent recorder, Model-SR, with a chart speed of 4 inches per minute, was used. The apparatus used for obtaining the step input is shown in Figure 7 . This apparatus consists of a flow restrictor, A , (Adjustable Restrictor, Foxboro Co.), a 1-cm. sample injection cavity sealed off by a rubber septum, B, a %way, 2-mm. capillary stopcock, C, connected to the injection block by an O-ring assembly, a mercury manometer, a flow restrictor, D, (Hoke needle valve), and a stopcock, E. The constant delivery apparatus was constructed using a synchronous 2 r.p.m. motor to drive a linear screw which, through a ballbearing, drives the plunger of a 50pl. gas-tight syringe (Hamilton Co. No. 1705) equipped with a 27-gauge needle. This apparatus gave a sample delivery rate of 0.843 pl. per minute. A considerable amount of care was given to the alignment of each part of this apparatus. A ball-bearing, slightly peened into the center of the end of the rotating screw made the best connection with the centrally tapped plunger top. The integral of the impulse response was obtained by feeding the output from the detector into an operational amplifier (Philbrick Model UPA-2 with a RB-100 power supply) using a 10-pf. feedback condenser and a 100-kilohm input resistor. The derivative of the step response was obtained by replacing the feedback condenser with a 4megohm resistor in parallel with a 0.01pf. condenser and the input resistor with a 1-pf. condenser in series with a 20-kilohm resistor. The output from the operational amplifier was attenuated and fed to a recorder. Procedure for Step Analysis. The Vapor Fractometer was operated in the conventional manner with the column inlet pressure set to give the desired outlet flow which was measured by a 100-ml. soap-film meter. To obtain the step input, a carrier gas stream a t one p.s.i. greater than the column inlet pressure was connected to restrictor A , shown in Figure 7. The restrictors, A ana D, were adjusted so that the pressure between them equalled the column inlet pressure, and the sample flow rate was approximately one fourth the column outlet flow rate. The constant delivery appartitus was positioned so that the syringe needle was in the center of the sample injection cavity. The syringe drive was started and the sample stream allowed to reach a steady-state, after which it was rapidly switched onto the column. The combining of the sample stream with the carrier gas stream in the injection block effected a decrease in carrier gas flow by the reference side of the detector, producing a switching transient which lowered the base line. Although this combination produced an unwanted VOL 34, NO. 10, SEPTEMBER 1962
1203
transient, it had the advantage that the column outlet flow remained constant. After a set time, determined by the breakthrough of the last component, bhe sample stream was shut off, which allowed pure carrier gas to purge the column, and the base line returned to ita initial value. This method of obtaining a step input function is applicable for volatile solutes and can be carried out with a minimum of alterations t o a commercial gas chromatograph. Although further alterations could be made to eliminate the switching transient, such aa disconnecting the reference side of the detector from the carrier gas stream, the use of this system has revealed several interesting applications of step chromatography. RESULTS
Comparison of Impulse and Step Chromatograms. The chromatograms shown in Figure 8 were obtained to test experimentally the relationship between w and u expressed in Equation 27, w = For each peak in chromatograms A and D, u was determined as described in Figure 5 . A and corresponding w valie calculated
4%~.
Table II. Comparison of Impulse and Step Chromatograms
Impulse response Integral of impulse rwponse S t e reswnse ~ F;ontr - ~-
sec.
MEK ~ec.
I20
180
205
18.8
7.5"
Back Derivative of step response Front
108 112
10.8 10.0
4.3"
206 208
22.0 17.9
8.8"
4.P
Back
112 lo8
10.1" 10.0"
4.0 4.0
206 208
21.1" 17.0"
8.5 6.8
a
Values calculated for comparison
(W
=
by Equation 27. For each step response in chromatograms B and C, w was determined as described in Figure 5,B and the corresponding u value calculated by Equation 27. To facilitate the determination of the retention time, tR, for curves B and C, the inflection point was approximated by the point a t one half the step response height. The values for U, W , and t~ are shown in Table 11.
240
300
360
1 420
1
480
Impulse response curve, 0.1 pl. sample B. Integral of A (obtained b y electronic integration of A, see text) C. Step response curve, sample feed rate 0 . 8 4 3 pl./min. D. Derivative of C (obtained by electronic differentiation of C, see text) Conditions far curves A and B: one-meter column of 8% Carbowax on 3 0 - to 60-mesh Chromosorb P; outlet flow rate, 169 ml. helium/min.; column and detector temperature, 24' C.; column inlet pressure 2 6 6 mm. Hg above atmospheric pressure; air peak, 5.0 sec. Conditions for curves C and D: same a s for A and B with the sample inlet pressure equal to the column nlet pressure. Sample flow into injection black equivalent to 4 2 ml./min. at atmospheric pressure
ANALYTICAL CHEMISTRY
sec.
4.0a
TIME, sec
1204
7.3
10.0
Figure 8. Comparison of experimental impulse and step chromatograms for equivolume mixtures of methyl ethyl ketone (MEK) and diethyl ketone (EEK) A.
18.2"
110
u,
.- ..
L 60
u,
206
10.0"
i) s
EEK W , sec.
111
W,
C
.
t ~ sec. ,
see. 4.0
tB,
7.2"
4%
0).
As one would expect, good agreement is found among the t R values for each component. For either w or u there is, in general, good agreement among calculated and measured values. A notable difference is observed in the W ' S for the front and back step responses for diethyl ketone. This difference is concentration dependent as shown in Figure 9. Interestingly, the average of the two w values for the front and back responses is fairly close to that calculated from u of the impulse response peak. This deviation from ideal behavior points out an advantage of using step inputs-namely, that they may be a useful tool for studying column properties. This aspect of the use of input functions is described below. Quantitative Analysis. Quantitative results can be very easily obtained from the height of the step response for a given solute because this height is proportional to its concentration ( I ) . However, certain conditions must be met. The prerequisites for quantitative analysis using a step input function are: (1) the isotherms of the solutes must be independent, and preferably should be linear; (2) the detector response should be linear over the concentration range used; and (3) no solute-solute interactions adversely affecting the linear detector response should exist. By using a very low sample feed rate and a high carrier gas flow rate, the concentration of the solutes in the gas stream may be kept a t a very low value, hence tending to satisfy these prerequisites. The achievement of the prerequisite conditions for quantitative analysis may be experimentally verified by examination of step chromatograms obtained from standard solutions. By way of illustration, the chromatograms of methyl ethyl ketone, diethyl ketone, and three mixtures of the two are shown in Figure 10. Examination of the data obtained in Table I11 shows that the retention time t R , for any one component remained constant over the concentration range used. Because of this constancy and the fact that the
20
0 Q)
v)
3"
l2
10
-I
I
C
0
'
0
i 0.06
0.03
0.12
0.09
0.15
CO NC ENTR AT I ON ("A) Figure 9. Variation of w with concentration volume in carrier gas stream)
(% by z
0
x ,
0. MEK. EEK.
a cn w a
value of w determined from front step value of w determined from back step methyl ethyl ketone diethyl ketone
I
shape of each step response was symmetrical, it may be concluded that the ratio of the amount of solute in the gas phase to that in the liquid phase is constant for each solute, and hence the isotherms are linear. For any one component, the height of its front step response was found t o be equal to the height of its back step response, thus indicating the absence of solute-solute interactions adversely affecting the linearity of the detector response. Also the height of the step response for any one component is proportional to its concentration, thus indicating that the detector response for each component is linear and independent. The linearity of the detector response may also be checked by replacing the column with a short piece of tubing and varying the concentration of the solute in the carrier gas stream. The quantitative results shown in Table I11 were obtained by taking the average response (to the front and back step inputs) ,of a component in the mixture and dividing by the average response of the pure component. However, either the front or the back step response alone could be used for quantitative calculations
Table 111. t ~ ,
tR,
Front, Back, A. 25.0% MEK and 75,070 EEK B. 50% MEK and 5070 EEK C. 75.2% MEK and 24.8% EEK D. lOO%MEK E. 1 0 0 ~ o E E K
/ I 0
I 100
1 200
I
I
300
400
TIME. sec Figure 10. Experimental chromatograms for samples containing different concentrations of methyl ethyl ketone and diethyl ketone A.
25.0% methyl ethyl ketone and 7 5 .O%
diethyl ketone by
volume
B.
50.0% methyl ethyl ketone and 50.0% diethyl ketone b y
ketone volume 75.2% methyl ethyl and 24.8% diethyl ketone b y volume D. 100% methyl ethyl ketone E. 100% diethyl ketone Conditions same as in Figure 8 with the sample inlet pressure a p proximately 10 mm. of H g greater than the column inlet pressure. C.
Quantitative Data by Step Analysis of MEK and EEK Mixtures
Ht. Front
MEK Ht.
h,
Found, %
Front, see.
sec. ResponseResponse
15.7
15.8
25.4
178
180
45.3
30.8
30.5
30.6
49.3
179
180
48.0 62.1
47.2 62.1
47.6 62.1
76.2
180 178
sec.
Response Response
96
97
15.9
96
98
97 96
98 97
Ave.
Back,
EEK Ht. Ht. Front Back
Ht.
see.
Back
tR,
Ht.
Found, %
44.9
45.1
74.9
30.0
30.2
30.1
50.0
180
14.8
15.3
15.0
24.9
179
60.8
59.5
60.2
Ave.
VOL. 34, NO. 10, SEPTEMBER 1 9 6 2
1205
since they each yield virtually the same answer. DISCUSSION
I n comparing step analysis with impulse analysis, one can see that the same amount of information can be obtained by either method. For qualitative identification of a component, the retention times in step and in impulse analysis are equal and are independent of concentration. Needless to say, in both techniques the sample concentration must be sufficiently small so that one remains on the linear part of the isotherm. Just as u is used in impulse analysis, o can be used in step analysis to calculate efficiency and resolution. I n quantitative impulse analysis, although the peak height may be used, usually i t is necessary to determine the peak area. The method of integration used to evaluate the peak area will, of course, introduce another source of error (10). However, in step analysis only the step response height is used, thus eliminating the need for additional computing apparatus. I t may be pointed out that the recorder pen is moving rapidly when recording an early peak. Therefore, any overshoot or sluggishness in the recorder response may cause errors when computing the area under the peak. In contrast, the critical measurement in step analysis is made at the step response height, where the pen is moving slowly or not a t all. In step analysis, sensitivity does not decrease for later components whereas in impulse analysis the relative peak height, hi, decreases fdr later components according to (35)
Also, the response height in step analysis is independent of column efficiency, Ni.
Column Properties. I n impulse chromatography the effects of partitioning into and out of the liquid solvent are difficult to study separately. Also, because the solute concentration decreases considerably while traveling through the column, the effects caused by varying solute concentrations are difficult to describe accurately. B u t by using step anslysis, in which solute concentrations can be controlled and the effects of partitioning kinetics are somewhat separated, these phenomena can be studied more effectively. Bosanquet (3, 4 ) compared the slopes of the front and back responses resulting from step analysis and reported that the front should be sharper than the back, the difference becoming greater with increasing concentration. He attributed this effect to a change in flow rate when the solute condenses or vaporizes in the column. I n his work the 1206 *
AN’ALYTICAL CHEMISTRY
Concentration range studied was from 3 to 30%; hence, the solute was a significant fraction of the gas stream and undoubtedly the flow rate did change during partitioning. Boeke (1) observed this effect in one system and also attributed i t to a change in flow rate. However, in another system the published chromatograms appear to indicate an opposite effect, the back step being sharper than the front step. The data in Table I1 shows that for methyl ethyl ketone the front and back step responses are equally sharp. This would be expected because the concentration of solute in the gas stream was only 0.1% and the flow rate change caused by partitioning would be insignificant. However, for diethyl ketone a t the same concentration, the sharpness of the front and back step responses are not equal. This difference, as shown in Figure 9, is concentration dependent and the back step response is sharper than that of the front step. This latter phenomenon is opposite in direction to that predicted by Bosanquet’s study. The value of w obtained by extrapolating to zero concentration is in agreement with the value of w calculated from impulse response data for a very small sample. Therefore, this phenomenon does not result from a change in flow rate caused by partitioning of the solute, and its cause must be sought elsewhere. The use of various input functions promises to be helpful in studying this type of effect. Some work of this nature is now in progress. PREPARATIVE CHROMATOGRAPHY
Impulse-type chromatography has found only limited application in the field of bulk separation. The greatest barrier retarding widespread use of this versatile and powerful separation tool is the small amount of sample that most columns are capable of handling a t one time. The magnitude of this problem may be partially minimized by employing larger amounts of liquid substrate to increase the column capacity and then counter-balancing the increased retention volume with higher temperatures. Although the distribution ratio, k , remains the same and the retention volume is the same, the concentration of solute in the stationary liquid phase is smaller (being inversely proportional to the amount of stationary liquid phase). The use of elevated temperatures also allows larger inlet concentrations of high-boiling components and aids the efficiency of condensation in collection tubes a t the column outlet. However, the extent of these improvements is limited in that large amounts of liquid substrate cause a decrease in column efficiency. Also, large concentrations of sample entering the column may cause the sub-
strate to be washed through the column as the sample dissolves in it. For these reasons it is of interest to consider the use of single- and doublestep modes of sample input for separation methods and to compare their merits with those of the impulse method. In Figure 4, Curve D represents the separation of two components by the impulse method. The first component, collected between v equals 125 and v equals 200, is of very high purity and its amount is represented by one unit. Curve C, which results from a sample input period of 50 plate volumes, shows the separation of the same two components. The first component, collected between v equals 125 and v equals 335, is 50 units of very high purity material. Curve B, which represents a sample input period of 100 plate volumes, yields nearly 100 units of the first component (collected between v equals 125 and v equals 270) with only a slight contamination by the second component. The front steps of Curve A show that collection of the first component between v equals 125 and v equals 250 yields about 100 units of high purity material, while collection between v equals 125 and v equals 270 yields about 120 units having the same amount of contamination as the 100 units collected in Curve B. ‘rhus, for these easily separated components, the step method yields a much greater quantity of separated material even though all four curves represent the same sample inlet concentration. Comparison of Ideal Impulse and Step Methods. Now let us compare the bulk separation of three difficultly separable pairs of components by step and by impulse methods. T h e three pairs chosen (A,B1; A,B2; and A,B,) represent separations of u, 20, and 3u, respectively. For simplicity it is assumed that the mixtures are one to one, that the k’s and N are sufficiently large so that the responses of all four components will have the same size and shape, and that component collection begins a t v = 0. I n Figure 11, Curves A and C show the totaI accumulated quantity of each component collected by the impulse and step methods, respectively, as a function of v . The v-axis is graduated in a-units for generality, and very importantly, the ordinate scale of Curve C is fifty times that of Curve A . While the maximum amount of component obtained by the ideal impulse method is limited to one unit, the amount obtained by the step method has n o theoretical limit. I n Curve B, the impurity ratio (the amount of component B1. Bz, or Bs collected to the amount of component A collected) is plotted as a function of v, which now represents the end of sample collection. The solid and dashed lines cor-
50 0
10 0
0 01
IMPURITY, (B/A) Figure 12. Comparison of the theoretical amount of component obtained as a function of the impurity ratio for impulse (--) and step (- - -) techniques
,
,
v - VR. Figure 1 1. Theoretical comparison of impulse (-) and step (- - -) techniques when employed for preparative chromatography A and C, amount of each component collected up to V - VRA by impulse and step methods respectively. 6, impurity ratio, B/A, when sample is collected up to V - VRA.
respond to the impulse and step methods, respectively. In all cases except the A,& pair the impurity ratio at any value of v is always less by the step
method. Even for the A,B1 pair, the additional amount of component A collected by the step method permits termination of collection a t an earlier v
and this feature completely offsets the increase in impurity. This is illustrated in Figure 12 where the total amount of component A collected is plotted as a function of the impurity ratio-i.e., sample is collected by both methods up to the same impurity level. The solid and dashed lines again represent the impulse and step methods, respectively. This plot not only shows that, when collection is stopped a t a given impurity level, a much larger quantity of the component is obtained by the step method, but also that this additional quantity becomes greater as the pair becomes farther separated. In obtaining component B1free from component A , the back step response in the step technique may be employed. The same arguments presented above and Figures 11 and 12 still hold. One simply interchanges in the discussion and figures the components A and B and reverses the sign of u values. Hence the application of back step chromatography is superior to ideal impulse chromatography for separating the second component of a two-cornponent mixture. Similar arguments also hold in general for the separation of the last component in a poly-component mixture. Poly-Component Step Separation. Although the preceding discussion considered the isolation of only the first (or last) component of a mixture, the increased amount of solute obVOL. 34, NO. 10, SEPTEMBER 1962
* 1207
A
w
z n W
IL
' Figure 13. mixture
i
' L m d
RECYCLED TO B
0
, RECYCLED
ic c
Step separation scheme for a polycomponent
tained per run indicates t h a t it might be advantageous t o make additional runs to isolate a single component from a complex mixture. For this purpose a schematic representation of a three-component separation is given in Figure 13. Curve d illustrates a normal step chromatogram (front and back step responses) of a mixture of components 1, 2, and 3, and the sections labeled a through e are collected separately. These sections contain pure 1, a mixture of 1 and 2, the original mixture, a mixture of 2 and 3, and pure 3, respectively. Subsequently, the sections b and d are re-subjected to step chromatography as shown in Curves B and C, respectively, to isolate component 2. Section c and the unseparated section of Curves B and C are not wasted because they can be recycled. The amount of section c can be minimized by changing from sample to carrier gas input a t a volume ~ V R 1from the start of close to V R sample input. Similarly, the volume between Section e and the repetitive curve can be minimized by changing from carrier gas to sample input a t a ~ V R ]from the volume close to V R start of carrier gas input. Similar principles of volume programming will minimize the quantities to be recycled in Curves B and C. By appropriate selection of column parameters. teniperature, programming of sample and carrier gas inputs, and sampk collection, three columns may be used in an integrated fashion to permit automatic operation. The general principle illustrated here is that any one component with an intermediate k value may be isolated from among any number of others by eliminating all slower components with a 1208
~
I
ANALYTICAL CHEMISTRY
I
I
r
0
50
100
I TIME.
I50
I 200
2:c
5 e c
Figure 14. Experimental chromatograms illustrating the use of programmed steps for preparative separations Column conditions and sample composition are identical to those given in Figure 10. Sample injected continuously for 50 seconds as shown
front-step separation (Curve A ) and then eliminating all faster components with a back-step separation (Curve B ) , or vice versa (Curyes A then C), or combining these two ' schemes for a larger yield. . Also: the maximum amount of collected material per unit time can be obtained by properly programming sample volumes. Double-Step Separations. As illustrated in Figure 4, Curve B , the volume programming of the doublestep curves may be adjusted so t h a t the back-step response of one component immediately precedes the front-step response of the next. If the components are easily separated, the amount of component collected t o a given impurity level is in this case only slightly less than t h a t achieved by the single-step method. This technique is particularly advantageous for mixtures of very widely separated components because it eliminates the additional procedures outlined in Figure 13. Three experimental curves (using an analytical-scale column) illustrating this method are shown in Figure 14. Application to Other Forms of Chromatography. T h e discussion on preparative methods (and input functions in general) is also applicable t o other forms of chromatography provided that the response characteristics are sufficiently similar. T h e similarity is readily checked experimentally by noting the shape of the front and back steps of the doublestep method and checking the additivity of the responses. INITIAL BASE LINE TECHNIQUES
The preceding discussion illustrates that observable effects ingas chromatog-
raphy result, not from the rate of sample input into the column, but from a change in that rate. Normally, chromatography involves a change in rate from zero to some finite value and a return back to zero. If, however, an input pattern is superimposed on a constant base-line-Le., the column IS pre-equilibrated as a result of :t previous step from zero to eonie finite value-then the total response is the algebraic sum of the constant base-line response, R,, and the response t o the input pattern RI. This technique is advantageous because adsorption effects may, in some cases, be eliminated. For example, if both partitioning and adsorption mechanisms of distribution are operative in the same column, and if the adsorption is of the BET Type I (Langniuir monolayer type), operation with an initial base line concentration above that required for saturation of the adsorption sites allows partitioning effects to be observed without interference from adsorption. Furthermore, even for adsorption columns, one may operate by a differential approach. That is, by applying an initial base line and subsequently causing only small changes, one may operate on a linear part of the isotherm; thus favorable distribution ratios among the sample components may be achieved and the response shape becomes more Gaussian. The rates of adsorption and desorption also depend upon the degree of surface coverage; hence, favorable kinetic parameters may be achieved by the use of the initial base line technique. Several input functions having interesting responses are discussed below. Detailed experimental studies on these
and otliw functions will appear separately. Single-Step Input Techniques. The total response to inputs of the type illustrated in Figure 1, Curves A , L), nnd G. is given by
SAMPLE
RESPONSE
*
sample step combinations (single step methods)
where R, = 0 in Curve -4,and the sign of the last two terms is negative in Curve D. Consider the simple case presented in Figure 1 5 , A . Here the column is equilibrated by continuous flow of a sample and at some later time the sample concentration level is suddenly altered by a known fraction. As a result, the response to each component is altered by the same fraction, as shown. In this technique, the total response to sample 1 is used as the reference for subsequent measurements of A l l A2, and A3. In cases where adsorption effects need not be minimized, the input to the column may simply be changed from that of the sample to that of the carrier gas whenever an analysis is desired. Such techniques may have value in monitoring industrial process streams in a semicontinuous manner. A second method is illustrated in Figure l5,B. In this technique, the diference in sample composition between two streams (one of which may be a reference of known composition) may be monitored. First one sample (or reference) containing components 1, 2, and 3 in a certain ratio is equilibrated with the column. At the appropriate time the input to the column is suddenly changed from this sample to the second, with which i t is to be compared. The changes observed in the response, denoted by A l , A2, and A3, are attributed to differences in the composition of the two samples. The negative value of A 1 corresponds to a smaller concentration of component 1 in the second sample relative to that present in the first. Analogously, the positive values of A 2 and A 3 indicate that the second sample is richer in these two constituents. Had the two samples been of identical composition, no response would have been observed. This difference technique may be employed advantageously to eliminate, or minimize, the effect of one or more components in the sample. See Figures 15,C and D. In Figure E , C , the sample contains three components of interest (1, 2, and 3) and a fourth component, 2, whose relative concentration is high. The response obtained by a single step input (Figure 15,C) does not permit sensitive detection of the
1
Z
0 c 0 ir
+ z W
0
z
0
c
$3-1
0
sample constituents 1, 2, and 3 . This difficulty may be circumvented by operation according to Figure 15,D. Here the column is first equilibrated with a concentration of Z equal to that present in the sample. Subsequently, the input to the column is altered to that of the sample. The resulting changes in response are attributable only to the components 1, 2, and 3 . Had the concentration of 2 been allowed to flow through the reference part of the thermal conductivity cell, the initial base line of the response-Le., on the recorder-would have been at zero and by suitable amplification, the components 1, 2, and 3, although present in traces, could be detected. Even if matching of the concentration of Z in the reference and sample streams were not perfect, the response obtained would be more sensitive for detection of the other constituents than the response obtained by operation according to Figure 15,C would be. Kote that 2 could also be a mixture of several components and hence each of their effects could be eliminated. This difference technique may also be of value in cases where the sample has to be dissolved or diluted with a solvent prior to introduction into the sampling system of the chromatograph. A modification of the techniqiie
I
illustrated in Figure l5,B should be useful in studying certain column effects. Imagine that the first sample in Figure 15,B consists of substance X and the second sample of radiolabeled substance X ; by using an appropriate detecting device, the response to the radioactive substance can be obtained. I n this manner the entire chromatographic process occurs without a change in the concentration of substance X in various parts of the column. This technique should prove useful for studying the chromatographic response of systems exhibiting nonlinear characteristics. Consider the following cases. In ordinary step analysis the temperature of the stationary phase preceding and following the solute front as it travels through the column may be slightly different as a result of the heats of solution and vaporization involved. This temperature change may, in some cases, be sufficiently great to affect the distribution ratio, k; which would result in different w and t g values for the front and back step responses. I n the technique described above, the concentration and temperature remain constant, and partitioning of the radio-labeled species occurs only because of entropy differences. Hence, if different w values are obtained for the front and back step responses, the reason for this VOL 34, NO. 10, SEPTEMBER 1962
1209
difference cannot lie iii s tenilxrsture effect. Similarly, this technique cannot give rise to a Bosanquet effect (3, +$), This procedure may also he useful in the study of solid absorbants and stationary phases that eshibit nonlinear isotherms with certain solutes; the advantage being that the shape of the fronts may be observed a t various concentration levels without a concentration gradient being present. Single step input functions such as shown in Figure 1,G are also useful for determining a n incremental step in an isotherm. Single Impulse Technique. The total response t o impulses of the type illustrated in Figure 1. Curve, C, E , and F , is given by
(37)
where R, = 0 in Curve C and the sign of the last term is negative for Curve F'. The response for the impulse illustrated in Figure l , C is the normal response obtained for ideal gas chromatography. By superimposing the impulse on a n initial base line, R,, of a more strongly absorbed component, undesirable adsorption effects and tailing may be reduced. For eurnple, Knight ( 8 )
used water-saturated carrier gas to minimize tailing in gas liquid chromatography. In this case the constant response corresponding to water is represented by R,. Another interesting application of the single impulse technique (Figure 1,F) is illustrated in Figure 16,D. This type impulse function is obtained by injecting a plug of pure carrier gas onto a column previously equilibrated with a multicomponent sample. The response appears as an inverse chromatogram with negative peaks occurring a t retention volumes corresponding to each of the components present in the sample! Furthermore, the "shadow" area for each negative peak is related to the concentration of that component in the sample as shown in Equation 37. This response may be compared with that of Figure 16,C, Thich represents the usual impulse chromatogram. This novel type of chromatogram (Figure 16,D) is readily explained qualitatively from consideration of Figures 16,A and 16,B. In Diagram A , which represents the case of normal impulse chromatography, the components 1, 2, and 3 are eluted through the column at different rates; hence the three components emerge from the column a t different retention
r
volumes (Figure 16,C). Analogously, the behavior illustrated in Figure 16,B corresponds to the case of the initial base line procedure. Here the sample flow is interrupted momentarily by introduction of carrier gas. Again component 1 moves through the column more rapidly than component 2, etc., as shown in the right of Figure 16,B. The result of this movement is given in Figure 16,D. Because of the simplicity of introducing plugs of carrier gas, this technique may find practical application in appropriate situations. Appreciation of the behavior illustrated in Figure 16,D is important. Any pulse-Le., a switching transient or pressure peak-will result not only in response occurring early in the chromatogram-Le., where component 1 is indicated-but also at the retention time of each of the subsequent components. It is all too easy to forget that reaction to a single change in column operation will occur a t several points on the observed chromatogram. The reason for this ease of oversight is that the effect tends to be washed out a t longer times in the same n a y that u increases. Thus, for example, an erratic pulse occurring periodically in the syringe drive of Figure 7 exhibits a simple and periodic response on the initial base line only if a single component sample is being issued into the column. If several components w e present, each
RESPONSE
IMPULSE
I
I
c
1
1 I
z
0 ta cr: t-
z
W
0
z
c
0 0
VOLUME
Figure 16.
1210
Graphic illustration of inverse chromatography ANALYTICAL CHEMISTRY
Figure 17. Responses combinations
to
(or TIME)
various
> sample
impulse
uill exhibit a response of its own character (degree of “washing out”) and phase relationship (relative to the injected pulse) with the result that a complrx, though repetitive, pattern is observed in the response. h i impulse technique, where the column is first equilibrated with a referelice mixture and a narrow plug of a sample is subsequently introduced onto the column through swapping sample streams momentarily, permits ready differential comparison of the two, the response to common constituents of equal concentration being eliminated. ilpplication of this technique to control analysis may be appreciated by inspection of Figure 17>.4, B, and C. The column is equilibrated with a reference miuture containing the component concentrations desired. The response observed can be considered as an error signal indicating differences in concentration betmen the reference and the sample. In Figure 17,*4, the concentration of component 1 in the sample is less than in the reference and a negative m o r signal proportional to tlic differenre is observed; as the concentrations of componmt 2 are equal, no error signal is observed, whereas the concmtration of component 3 is greater in the sample than in the reference and a positive error signal is observed. After adjustment of the concentration of component 3 in the process until it is equal to the reference, the response will be as shown in Figure 17,B. When the sample and reference compositions have been made equal, no error signal will be observed as shown in Figure 17,C. ,4n advantage of this type of technique is that gradual changes in the column flow rate, solvent bleeding, instrument drift, and temperature will not affect the results because the technique is essentially a true nulling method. The technique should also be of use in trace analysis as illustrated in Figure 17,0, where Z may be one or more major components. In this case, the concentration of Z entering the column is constant; hence the response peaks (solid curve) are attributed only to the presence of compcnents 1, 2, and 3. The dotted curve a corresponds to the normal technique in which 2 is absent in the carrier and gives a response peak which masks the peak due to component 3. Curves b and c represent cases in which the concentration of Z in the carrier and sample are only approximately matched and peak 3 is unveiled. Because this initial 2 base line technique may suffer a IOSS in detector sensitivity, its use may be ..ndesirable for detection of components ;Ath which 2 does not interfere. T i e ixpulse illustrated in Figure l,E may be used for the determination of isotherms. The initial response, R,, for ,.ven solute is proportional t o the
we more e n s i i ~ secured experimentally than are ramp functions. Repetitive input functions, such as those illustrated in Figure 1,1, J, and L , may be employed advantageously in gas chromatography. Consider first the application of the repetitive input of Figure 1,J which represents a sine wave of constant amplitude H I , superimposed on an initial base line, R,. The steady-state response to this input function is given by R(r)sa= I?,
+ e - 2 ( d d 3 sin 277f(t
-t ~ ) (40)
\
\-.
wheref is the frequency of the sine wave. For a single component the response to a sine wave input will be a sine wave of the same frequency (phase shifted by the amount - 2 n f t ~ )but the amplitude. H B , will depend on the values off and U . Appreciation of this response can be obtained by inspection of Figure 18, where the relative amplitude, H R I H I , is plotted against 2nfu. Consider a situation where the sample component of interest has a u value of 1 and the other sample components habe u values greater than 3. By employing an input frequency such that 2nf = 1, the amplitude of the response, H R , will be 0.6 times the input amplitude, H I , for the component of interest and less than 0.01 H I for the other components. Thus a continuous read-out will give a sinusoidal response amplitude that is proportional to concentration for the component of interest with relatively little interference from the other components. The response to a multicomponent sinusoidal input is given by
L-LA--d’4 L 4
38
2
6
20
2.
i 6
11
Figure 18. Dependence of relative amplitude on f and u for the response to sine wave input amount of solute in the vapor phase, n,. By superimposing a small impulse of the solute on a constant n,, a differential distribution ratio, Ak. may be determined from the response. Thus a plot of Ak us. n, may be made by determining Ak at various levels of no. If Ak remains constant over various values of n,, the isotherm is linear. The shape of nonlinear isotherms may be obtained graphically by integrating the area under the Ak us. n, curve. Complex Input Techniques. The response to the input function illustrated in Figure l , H for a single component is given by
(38;
where At is the time between evenl!. spaced steps and m is the step number. If the time between the steps is sufficiently long so that equilibration of the column can be achieved prior to the next step, the technique can be applied for determining isotherms. Obviously, an automatic device for obtaining isotherms can be constructed on this principle. The response to the input function illustrated in Figure 1,R for a single component is given by
This input function could also be used for determining isotherms; however, the rise rate would have to be sufficiently small to permit nearly complete equilibration of the column. The staircase input pattern of Figure l , H has the advantage in that faster equilibration is possible-as well as a measure of the degree of attainment of equilibration. In addition, programmed step changes
where C, corresponds to the amplitude of the sinusoidal variation in component j in the input function, hence the amplitude of the sinusoidal response is the vector sum of the amplitudes of the individual component responses. Analysis of a multicomponent mixture can be achieved, in principle, by solving j simultaneous equations of the form given in Equation 41. Suitable coefficients for the simultaneous equations may be obtained by appropriate experimental variations in any one or more of the parameters upon which the coefficients depend, f, u , , t, and t ~ ~ . Of course, components which have identical u and t~ values a t each of the chosen experimental conditions cannot be distinguished because f and t are the same for all components-i.e., no two components may be allowed to behave exactly alike under all conditions. The values of u , and t R i can be altered only by varying the action functions of the chromatographic processVOL. 34,
NO. 10, SEPTEMBER 1962
* 1211
c
e.g., flow rate, temperature, amount and nature of the stationary phase, column length, etc. Thus, for a given chromatographic network, simple or complex, the action functions for each component will generally be different a t each point of the system. The number of components which can be distinguished depends on the number of action points monitored. Because response is a vector quantity, not more than two independent equations can be obtained from measurements a t each action point. Hence, for a single frequency, the maximum number of components which can be simultaneously determined, n,, is given by nc = 2n.
=
2n,nf
(43)
A number of techniques can be based on the use of more than one frequency. 1212
ANALYTICAL CHEMISTRY
z 0 ~
a
(42)
where no is the number of action points monitored. Although monitoring the column input can yield only a single independent equation because here all h i ' s are zero-Le., all components are in phase-this input serves as a useful time reference for defining phase shifts a t other monitored points. However, when the input is employed as an action point, the maximum number of components is one less than the value expressed by Equation 42. Although response maximum amplitudes and phase angles can be used to set up simultaneous vectorial equations, a more expedient approach is to use instantaneous response heights a t judiciously selected times to set up linear simultaneous equations. This linear relationship is expressed by Equation 41 when frequency and time are constant a t each action point for all components. For a single component the sinusoidal responses measured with different initial base lines should be identical if the isotherm is linear and the kinetics of condensation and vaporization are sufficiently rapid. If not, and the amplitude of the sine wave is sufficiently great, harmonic distortion of the sine wave response occurs. Studies of this type may eventually prove useful in the study of nonlinear column parameters. Also, if two components are normally eluted together, but have somewhat different nonGaussian shapes, arising from different degrees of nonlinearity of their column response, the measurement of harmonics by the sine wave input technique could permit measurement of one in the presence of the other. Linear simultaneous equations can also be achieved when the input function is a linear combination of sine waves of various frequencies. In such cases, the maximum number of components determinable a t each action point is twice the number of sine wave frequencies contained in the input, nf. Hence, nc
z
c
Figure 19. Response characteristics to square wave input functions (schematic instantaneous sample profiles) Both distribution ratio and u values increase in the order 1
The repetitive input of Figure 1,1 represents a square wave superimposed on an initial base line, R,. This periodic square wave input function, by Fourier analysis, may be said to consist of a fundamental sine wave of frequency identical to that of the periodic frequency of the square wave plus sine waves of higher, odd harmonics and may be expressed as 1(f)= R ,
+ -4 H I sin 2 ~ f +t
4 - H I sin Saft 3a
4 +HI sin 1 0 ~ f + t ... 5lr (44)
Thus the response is given by R(t)=
sin K2n
+ 1 ) ( 2 a f )( t -
tR)1
(45)
As this input function travels down the column, the sharp steps become rounded to an extent dependent upon the u value of the component under consideration. This rounding is equivalent to a more rapid decrease in the amplitude of the higher harmonics relative to that of the fundamental. Hence, a frequency analysis of the resulting response for a mixture should permit analysis of the mixture in a continuous manner. In this connection, consider Figure 19. Here sample components are issued in a square wave manner (superimposed on the initial base line) into the column. I n the column, the wavelengths of components 2 and 3, because of their respectively larger k values, and hence lower veloc-
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