cases, argon resulted in a slight decrease in limit of detection as would be eq)ected from the discussion presented by Alkemade ( I ) . S o analytical curves will be presented in this communication because they are all similar t o the ones previously given ( 5 ) . Essentially the same accuracy of analybis and signalto-noise ratio as preLiously measured ( 5 ) were again found in t,hcse studies. S o detectable fluo1,escence was observed for the 2874, 4033, and 4072 .I. lines of gallium, for the 2560, 3040, 3256, 4102, and 451 1 A. lines of indium, and for the 2040 and 2591 A. lines of selenium when esciteti by their respectilre electrodeless discharge lamps. This was primarily a result of the estremely low intensity of the spe-tral lines emitted
by the respective electrodeless lamps. These results are certainly not indicative of the method. At the present time additional studies, particularly with respect to sources of excitation, methods of sample atomization, flame type, and amplifier type, are being conducted. These studies should result in increased sensitivities of analysis of the metals already studied and the e\tension of atomic fluorescence flame spectrometry to other metals. LITERATURE CITED
(1) Alkemade, C. T. J., International Conference on Spectroscopy, College Park, Md., June 1962. (2) Gilbert, P. T., Jr., Pittsburgh Conference on Analytical Chemistry and Ap-
plied Spectroscopy, Pittsburgh, Pa., February 1961. ( 3 ) Mavrodineanu, It., Boiteux, H., “L’Analyse Spectrale Quantitative Par La Flamrne,” Masson et Cie, Paris, 1954. (4) Winefordner, J. D., >Tansfield, C. T., Vickers, T. J., ANAL.CHEM.35, 1607 (1963i. -, ( 5 ) Winefordner, J. I)., Staab, It. A , , Ibid., 3 6 , 165 (1964). (6) Winefordner, J. I ) . , 1-ickers, T. J., Ibid., 36, 161 (1964). J. 11. WINEFORI)NER Department of Chemistry University of Florida Gainesville, Fla. R. A. STAAR Ivorydale Technical Center Proiter & Gamble Co. Cincinnati, Ohio \ - -
RECEIVEDfor review March 20, 1964. Accepted April 14, 1964. This communication taken in part from the Ph.1). thesis of R. A. Staab.
Gas Chromatographic Response to Plug-Shaped Sample Inputs: Calculation of Plate Height from Response Curve SIR: Recently, the desire has been expressed for a met’hod whereby plate heights can be computed from gas chromatographic response curves when the sample input profile is in the form of a plug ( 3 ) . In previous work ( I , 4 ) , we discussed the calculat,ion of response profiles for various sample input profiles in linear chromatographic systenxi.e., no solute-solute interactions. If it is assumed that the response, S ( t ) , to an ideal impulse can be adequately expressed as a Gaussim,
S(t)
plate height can be computed as for a Gaussian response. When the plug width becomes sufficient’ly large, the peak is flat on top-approaching the response to a step- and the plate height can be computed as for a step response ( I , 4 ) . However, for intermediate plug widths, neither method yields the correct value for the plate height of the column. Figure 2 illustrates four of the more convenient parameters that can be used to calculate plate height in the case of a
=
(2~)-~”% ex],[-~
(t -
tR)’/2U2]
(1)
where t R is the retention time and u is the standard deviation of the peak. Then the response, R(t),to a plug input of unit, height is
R(t) = (2x)-“2
S
(t -
(t -
tR),’U
tE
- s)/u
e--’*%y
(2)
where s is the width of the sample plug. Time, t , is taken as zero at, the beginning of sample input, and t equals s a t the end of the plug input. Equation 2 is equivalent to Equation 29 in our earlier paper (4) and is similar in form to Equation 9 of Peniston, Agar, and McCarthy (2). Thli shape of this Gaussianed-plug response is illustrated in Figure 1 for various relative plug widths. When the plug width becomes sufficiently small, the peak approaches the response to an impulse and the
( t - tR)/U
Figure 1 . Chromatographic response to plug-shaped sample inputs ( I )
Gaussianed-plug
re:l)onse.
Methods
a, b, and c all give the correct value for
the true plate height, H , when the input is an ideal impulse. Method d gives the correct value for true plate height when the input is an ideal step. However, when applied to the response to plug inputs of intermediate widths, these four methods yield four different numerical values for the apparent Idate H,*, height,, H * . Values of /lo*, and H,* have been comlnited for various plug widths by standard mathi3matical analysis of Equation 2. I3ecause the equations do not appear in rlosed form and iterative numerical computations are necessary, no details will be presented here. The results of these calculations are givcn in Figures 3 and 4 in a form which permits their direc,t use in converting an apparent plate height, H*, into the true plate height, H, nhen the plug width, s, and one of the measures of peak width- At,,, Atli2, or Ati-or the tramition time, 7 , are experimentally known. In Figure 3, the i,rxlative value of the apparent plate height is plotted as a function of the ratio of sample plug width to peak width for each of the three different ways of measuring the peak width of the response curve. Thus, the true plate height, H ,can be calculated from an experimental apparent plate height-- H0*, or H,*-in the following n a y . First, the value of s / A t is determined from the experimental values of the input plug VOL. 36, N O . 7, JUNE 1964
1369
-7-Pi-Rmax 1.8 r
I -
'*
= 1.4 1.0
0
0.2
0.4
0.6
1.0
0.8
s/At Figure 3. Effect of sample plug width on Apparent Plate Heights, calculated by methods a, b, and c
TIMEFigure 2. Rough sketch indicating some convenient measurements of response peaks H H*
sum of the variances, respectively, of the input function and the system function. Therefore,
LU2/tRZ
= LU*~/~R' Method a. UO* = A6/4 Method b. u1/**= At1/*/2(1 n 41'12 Method C. mi* = AIi/2 Method d . uT* = s / ( 2 a ) ' / ~
width, s, and one of the measures of the response peak width, Ab A t 1 p , or At,. For this value of s/At, the numerical value of H * / H is then read directly from the appropriate curve (a, b, or c) of Figure 3 . Finally, the true plate height, H , is obtained by dividing the experimental value of the apparent plate height, H*, by the value of H * / H read from the graph. In Figure 4 the relative value of apparent plate height is plotted as a function of the ratio of the input plug width to the transition time, 7 . This figure is particularly useful in calculating true plate heights from apparent plate heights when the input plug width is so large that the value of H * / H is too sensitive to the exact value of s / A t in Figure 3. The calculation of true plate height via Figure 4 and method d is analogous to that described above for Figure 3 and methods a, b, or c. I t might a t first appear from Figure 3 that an apparent plate height calculated by method a requires a much larger correction than would a value calculated by methods b or e. However, in comparing these three methods, it is necessary to consider the values of Hoc, Hllz*, and H,* that would be obtained from the same chromatographic response curve-Le., obtained from the same values of s/2u rather than equal numerical values of s//Ab, s/Atllz, and s/At,. For this reason, Figure 3 includes tie lines which connect curves a, b, and c a t points corresponding to the same input plug width. Sotice that, for a given value of s/2u, the values of H * / H are roughly the same for all three curves. Hence, from mathematical considerations only, the three methods are equally ad\-antageousexcept when the input plug width is so 1370
ANALYTICAL CHEMISTRY
= L(fi2.e -
large that method d is superior to all three. The preceding discussion is only applicable in the case of a plugshaped sample input and a Gaussian column. More generally, but still requiring that the system be linear, the true plate height of the column is given by H = Lpzs/~is~ (3) where u l s and p25 are the delay and variance, respectively, of the system function-i. e., the response to a unit impulse-S(t). [The delay and variance of a function, say F ( t ) , are defined by Y1.r
p2.r
J(t
z JtF(t)dtjJF(t)dt
- viF)'F(t)dt/JF(t)dt
(4)
PZI)/(YIR
UIZ)'
(6)
where the subscripts I and R refer to input and response, respectively. Equations 4,5 , and 6 provide a general method for computing the value of true plate height from experimental input and response curves, regardless of their shapes or the shape of the system function. The integrals in Equations 4 and 5 could be evaluated, for example, by analog or digital computers. I n the case of a plug-shaped sample and Gaussian column, the delay and variance of the input plug are, from Equations 4 and 5 ,
PZZ =
(t
- s/2)zdt/[
dt =
s2/12 (8)
(5)
where the integrals are evaluated over all values of time for which F ( t ) exists.] If the system is linear-i. e., if the superposition principle holds- the delay and variance of the response function will equal the sum of the delays and the
-
However, the computation of the variance of the response peak is not as simple, so it is advantageous to make an approximation that will simplify the calculation. As ' c a n be seen from Figure 1, the response shape appears to be approximately Gaussian when the input plug width is not too large-say, when s / 2 u is less than about 2.5. Hence, the response is approximately represented by
R(t) = k exp[-
(t
- t,,,)2/2u*21
(9)
where k is a proportionality constant, tmaxisthe time at the response maximum, and U * is the approximate standard deviation of the response peak. The value of u* can be determined by methods a, b, or c of Figure 2. The value of k does not need to be determined because it cancels in the expressions for delay and response which are Figure 4. Effect of sample plug width on Apparent Plate Heights, calculated by method d
It can be readily seen that Equation 10 is quite correct, but Equation 11 is only an approximation because the response shape is not actually Gaussian. Substitution of the above expressions and t,, = tR 4 2 into Equation 6 yields
I
I
I
I
+
H H
L(U*' z
- s2//12)tR2
H * - LS2/ L2tR2
(12)
Thus, the true plate height, H , can be approximately calculated from the apparent plate height, H *, by first determining the value of H * and then subtracting from it the value of h2/12tR2, which might conveniently be termed the sample plate height, H " , for a plug. It is important to realize that Equation 12 is an approximation only because of the approximate measurement of the variance of the response and not because of a non-Gaussian sample input shape nor because of any inapplicability of Equation 6. The error resulting from this approximation is considerably larger than might be anticipated after a qualitative inspection of Figure 1. Figure 5 shows that the relative error in the approximate plate height (calculated by Equation 12, using method a, b, or c evaluate u * ) as a function of H P / H * , which equals S ~ / ~ ~ C,\gain, T*~. tie lines are included in the figure to
I
0
I
I
0.2
I 0.4
I
LITERATURE CITED
0.6
HP/ H" Figure 5. Relative error in approximate plate heights, calculated by Equation 12 and methods a, b, and c Hp
the sample plate height, H P ,is less than about one fourth of the apparent plate height, H*. On the other hand, the usefulness of Equation 12 can be demonstrated by comparing Figures 3 and 5. For example, a t s/2a = 1.0, the initial error of about 40y0 in the apparent plate height is reduced to about 5y0 by subtracting the sample plate height.
S2/12tR2
facilitate comparison a t equal values of input plug width. Figure 1 shows that the response appears roughly Gaussian 2.5, a t plug widths , u p to 4 2 0 whereas Figure 5 shows that the Gaussian approximation (and method a) gives a 4% error in plate height when s / 2 u is only 1.0-with the error increasing rapidly for larger plug widths (at 4 2 0 = 2.5, the error is 4301,). Hence, Equation 12 is useful only when
-
(1) Hildebrand, G. P., Ph.D. dissertation, University of North Carolina, Chapel Hill, N. C., 1963. ( 2 ) Peniston, Q. P., Agar, H. D., McCarthv. J. L.. ANAL. CHEW 23. 994
(1951j.' (3) Purnell, H., Sawyer, D. T., Ibid., 36, 668 i 1964). (4)Re'illey,'C. N., Hildebrand, G. P., Ashley, J. W., Jr., Ibid., 34, 1198 (1962).
J. W. ASHLEY,JR. G. P. HILDEBRAND~ CHARLES N. REILLEY DeDartrnent of Chernistrv Unheraity of North Carolina Chapel Hill, N. C. Present address, Plastic Department, E.I. du Pont de Sernours & Co., Experimental Station, Wilmington 98, Del. RECEIVEDfor review March 19, 1964. Accepted April 6, 1964. Information developed during work supported by the Advanced Research Projects Agency, Contract No. SD-100.
Coulometric Titration of Organic Acids in Acetone SIR: Although acidic species have been generated electrolytically in nonaqueous solvents for the coulometric titration of organic bases (3-6), the only recent reference in the literature to a similar electrolytic generation of base for the titration of acids is a paper by Crisler and Conlon ( I ) . These men used a 1 : l mixture of benzene and methanol containing lithium chloride as the electrolyte. d n antimonyglass electrode. pair was used as the detection system. KO clear inflection points were obtained in the titration curve. Coulometry as a technique offers the advantage of precision in the detection of small quantities of materials; nonaqueous solvents are useful for the determination of acids and bases not readily determined in water either because of insolubility, low ionization in that solvent, or other reasons. The analytical interest in nonaqueous solvents over the past decade attests t o their utility. During the past yc'ar we have attempted to generate a basic species in a
nonaqueous medium containing a minimum quantity of water. The initial choice of solvent was acetone. It has a reasonable dielectric constant ( e = 21.3), dissolves a variety of organic acids, and is readily available commercially in a high degree of purity. Fritz and Yamamura (2) have demonstrated the analytical possibilities of this solvent for the titration of carboxylic acids and phenols as well as other weak organic acids. In the current work two methods were used. EXPERIMENTAL
Apparatus. A Sargent Model IV coulometric current source was employed for both methods. Using this, a current of 9.65 ma. was applied to two platinum generating electrodes immersed in separated half cells. I n one method the cells were electrically connected by a salt bridge consisting of two sintered glass filter sticks ( E , Ace Glass Co., Vineland, N. J.) joined together to form a U-tube containing 10% aqueous potassium nitrate. One end of the bridge was immersed in a
half cell containing 10% aqueous potassium nitrate and a generating electrode. The other end was protected by Whatman KO.44 filter paper, glass wool, and a second sintered glass frit and was placed in the sample half cell. The latter compartment contained a generating electrode, a calomel reference electrode, and a glass indicating electrode all surrounded by a n acetone solution which was 0.05M to tetrabutylammonium perchlorate (Southwestern dnalytical Chemical Co., Austin, Texas) and 0.lM to water. h magnetic stirrer was placed under the sample compartment. The calomel reference electrode was filled with a saturated solution of tetramethylammonium chloride in acetone. The reference and indicating electrodes were connected to a Leeds & Northrup Model 7401 pH meter. .I Model 152 voltmeter (Accurate Instrument Co.) was connected across the terminals of the coulometric current source. The second method made use of the same apparatus described above except that tetrabutylammonium bromide was employed as supporting electrolyte, and the water content was 0.5%. The salt bridge in this case consisted of a glass VOL. 36, NO. 7, JUNE 1964
1371
