emission) line profile in the case of low optical thickness (i.e. low concentration) irrespective of the detuning and intensity of the exciting laser beam. This means that there is a full spectral redistribution of the fluorescence radiation. This redistribution is expected because at 1-atm flame pressure, collisions of the excited metal atoms with flame particles restore Maxwellian velocity distribution and bring about full collision broadening. However, it is not the fluorescence line profile that is found by measuring the excitation profile at low concentration; the latter is related to the absorption line profile (in the absence of saturation). Besides, doublet-mixing collisions will make both Na-D components appear in the fluorescence spectrum at an intensity ratio that is determined by the ratio of the statistical weights of their upper levels, irrespective of which component is excited by the laser beam ( 4 ) . This has been confirmed experimentally for the type of flames used in our experiments.
LITERATURE CITED (1) R. 6. Geen, J. C. Travis, and R.A. Keller, Anal. Chem., 48, 1954 (1976). (2) R. Herrmann and C. Th. J. Alkemade, “Chemical Analysis by Flame Photometry”, 2nd revised ed., translated by P. 1.Gilbert, Interscience,
001
002
l o s e r detuning ( n m ) Flgure 1. Na fluorescence intensity (in relative units) as a function of laser detuning In an H,-02-Ar flame. Curve a is the excitation profile measured when the whole hser-illuminatedNa cloud is detected; curves b and c are found when the laser-facing edge of the flame is screened
off
length-scanning was done by hand, not too precise a meaning should be attached to the shape of the curves. On theoretical grounds, one may expect the true normalized fluorescence line profile to conform to the absorption (=
New York, N.Y., 1963. (3) C. Th. J. Akemade, Pus Appl. Chem.. 23,73 (1970). (Lecture presented at International Atomic Absorption Spectroscopy Conference, Sheffield, U.K., 14-18 July 1969.) (4) C. Th. J. Alkemade and P. J. Th. Zeegers, Chapter 1 in “Spectrochemical Methods of Analysis”, J. D. Winefordner, Ed., Wiley-Interscience. New York, N.Y., 1971, p 3.
C. Th. J. Alkemade* T. Wijchers Fysisch Laboratorium, Rijks-Universiteit Princetonplein 5 Utrecht, The Netherlands RECEIVED for review June 17,1977. Accepted August 15,1977.
Calculation of the Velocity of a Desolvating Aerosol Droplet in an Analytical Flame Sir: A great deal of interest has recently developed among atomic spectroscopists in calculating the movement of aerosol droplets in high temperature flames and plasmas. L’vov and co-workers have attributed some of the lateral spread of atoms in a slot burner to a horizontal acceleration of the droplets as they enter the flame ( I , 2). Li (3)has used a formula similar to that employed by L’vov to express the vertical velocity of a droplet as it moves in a flame. Unfortunately, these calculations of the acceleration of aerosol droplets do not match data collected in our laboratory ( 4 , 5 ) . In previously reported experiments ( 4 , 5 ) ,we injected individual aerosol droplets into a flame in such a manner that their initial horizontal velocity was zero and found that the droplets quickly reached a velocity indistinguishable from the velocity of the flame itself. In contrast, the formula used by Li (3) predicts a relatively slow approach of the aerosol to the flame velocity. We offer here a modification to that formula which gives strikingly faster acceleration of the aerosol. The acceleration of a droplet under a driving force, F , can be expressed as dt
m
where u, is the droplet velocity, t is time, and m is the mass of the droplet. In an analytical flame, two such driving forces 2112
ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
act upon a droplet. One force is the gravitational attraction of the earth, mg,where g is the gravitational constant. The other force, the viscous drag of the droplet in the rapidly rising flame gases, can be expressed as 3~17d(u- u,) where d is the diameter of the droplet, 17 is the viscosity of the flame gases, and u is the velocity of the flame. From this relation, Li derives the following formula for the droplet velocity (3):
where p is the droplet density, uo is the droplet velocity at t = 0, and do is the initial droplet diameter. The foregoing treatment unfortunately neglects the effects of desolvation. In reality, a droplet evaporates as it accelerates toward the flame velocity, and the consequent reduction in the droplet size and mass affect its acceleration. The degree to which this acceleration is increased can be calculated with knowledge of the droplet’s evaporation (desolvation) rate. During desolvation, the square of the droplet diameter, d2, decreases as a linear function of the time the droplet spends in the flame (4, 5 ) and follows the equation
This relation has been experimentally confirmed only for the portion of the desolvation period where any dissolved solids remain a small fraction of the droplet's mass. Because Equation 3 predicts the complete disappearance of the droplet a t the end of the desolvation period, this relation does not therefore accurately model the system during the latter stages of vaporization. However, in the absence of more complete experimental data and because Equation 3 is valid over most of a droplet's evaporation history, it will be used to describe the change in droplet diameter during desolvation. The change in the droplet mass during desolvation can also be deduced from Equation 3 with the assumption that the density of a droplet remains constant as it evaporates. Of course, this assumption will also be valid only for droplets of a pure solvent. For droplets containing a dissolved solute, density will change gradually from a value nearly equal to that of the solvent to one essentially that of the dried solute material. However, a sample calculation shows that such a change in density is of minor importance throughout most of the droplet's history. Assume an aqueous droplet which contains a total of 1 % dissolved solids and that the overall density of the dry solids is 2.2 g/cm3 (approximately that of CaClZ (6)). To a good approximation, the droplet density changes by only 1% when desolvation is 50% complete. Also, after 90% of the time required for desolvation has passed, the density has changed by only 13%, although the radius of the droplet will have decreased to 32% of its original value. Thus, the relative magnitude of the density change is much less than the reduction in droplet radius. In addition, the greatest part of the change occurs near the end of the desolvation period where the desolvation model employed in the foregoing derivation has not been experimentally investigated and will probably not hold (5). Consequently, inclusion of a density factor in the calculation of droplet velocities in the flame is an unnecessary complicating factor and can be justifiably omitted. The effects of desolvation on droplet acceleration in a flame may now be evaluated. Because the viscous force serving to accelerate the droplet is directly proportional to the droplet diameter, there is less viscous force on the droplet as desolvation proceeds. However, this reduction in viscous drag is more than offset by the concurrent reduction in droplet mass which is in turn proportional to the droplet diameter cubed. Inserting the time dependence of the diameter into the viscous drag and mass equations given before, we obtain:
1
L ,
u, = u
+ (1- h t / d o ' ) A ( u , - u + (5)
where A = 18 v / p k . Equation 5 now represents the velocity of an evaporating aerosol droplet in a moving gas stream. The foregoing two models, represented by Equations 2 and 5 , can best be compared by plotting the calculated droplet velocity during the droplet desolvation time. Values for parameters required in this calculation were chosen to ap-
'
3
10'05
R a i d e n C e Tima
4
8"
'
"
0
'
01
5
6
7
8
9
6 0 ~ m dioplat lmilI!secmdsl
"
"
'
0 15
'
"
" 02
'
"
'
' 0 25
Flame for a IO pm d r o p l i t I I F I I I I ~ ~ S O ~ ~ S I
Figure 1. Comparison of the vertical velocities predicted by Equatlon 2 and Equation 5 for a 60-pm diameter droplet during its desolvation period in a laminar air-acetylene flame. The conditions chosen for this calculatlon approxlmate the condltlons of previously reported experiments ( 4 , 5). The flame velocky Is 1000 cm/s, denoted by the broken line, and the droplet's initial vertical velocity is zero. The curve labeled A represents the velocity profile predicted by Equation 2, which does not account for the droplet's changing mass or diameter during desolvation. The c w e labeled B , which represents the velocity profile predicted by the model Including the desolvatlon effects, Equation 5, shows a significantly faster approach to the flame's veloclty. Curves Aand Bcanalsobeusedtodeducethepredictedvelocitlesofa 10-pm aqueous droplet In the same flame. The only change required Is the use of the atternatbe abscissa (labeled "for a 10-pm droplet") which includes the entlre desolvatlon time preducted by Equation 3
(4) Equation 4 recognizes the droplet's existence only over the time interval 0 5 t 5 d o 2 / k . In addition, one must recognize that flame velocity ( u ) can change, even in a laminar flame, with position in the flame. When these changes in v are significant, Equation 4 must be evaluated directly. However, for the usual case when the flame rise velocity may be considered constant, a solution to differential Equation 4 is:
2
Residence T h e rn Flame for
proximate the conditions found in the droplet desolvation experiments of Clampitt and Hieftje (5). In these experiments, an air-acetylene flame was used; thus, the viscosity was approximately that of Nza t 2350 K (v = 8.28 X g cm-' s-*) and the flame velocity measured to be about 1000 cm s-l. A water droplet ( p = 1.0 g ~ m - with ~ ) an initial diameter of 60 pm (do = 6.0 x cm) and an initial velocity of zero ( u g = 0.0 cm 8-l) was used in this calculation. The results are shown in Figure 1. The curve representing the velocity predicted by Equation 2 shows a much slower approach to the flame velocity than that of Equation 5 , which accounts for changing diameter and mass of the aerosol. After the period during which desolvation occurs, the droplet velocity predicted by Equation 2 reaches only 97% of the flame gas velocity. The changing diameter and mass model, Equation 5 , predicts that the aerosol will reach this same velocity when desolvation is only 63% complete, or when 5.6 ms have passed. When desolvation is complete, Equation 5 predicts that the resulting particle velocity will be less than 0.005% slower than the flame gases, while Equation 2 predicta a terminal velocity 0.5% slower than the flame velocity. Thus, Equation 5 predicts that even before desolvation is complete, the aerosol has reached a velocity very close to that of the flame gases. Figure 1can also be used to express the velocity of a droplet whose initial diameter is more typical of those found in commercial flame spectrometric instruments. Let a 10-pm diameter droplet enter an air-acetylene flame with conditions identical to those expressed above for the 60-pm droplet. The velocity of the 10-pm droplet predicted by the two models can then be deduced from Figure 1 by the use of the alternative (lower) time scale. Each time axis represents the entire desolvation period for the initial droplet size considered. Thus, despite a droplet's initial diameter, Equation 5 predicts that the droplet will essentially reach the flame velocity before it completely evaporates. Further improvements in this model are possible. Droplets used in flame spectrometry are not pure water and, upon
ANALYTICAL CHEMISTRY, VOL. 49, NO. 13. NOVEMBER 1977
2113
desolvation, produce particles of finite diameter and mass. This expectation is not consistent with the model employed herein, in which complete evaporation is predicted. One would also expect that as desolvation occurs, the density of the droplet would change because of the increasing analyte concentration. Finally, vapor-phase water and its decomposition products, in the region of the droplet, might change the viscosity of the flame gases. However, these changes are minor refinements and would not significantly alter the result predicted here, that before desolvation is complete, the aerosol will have attained a velocity which is experimentally indistinguishable from that of the flame gases.
LITERATURE CITED (1) B. V. L’vov, L. P. Krugiikova, L. K. Polzik, and D. A. Katskov, J . Anal. Chem. USSR, 3 0 , 545 (1975). (2) B. V. L’vov, L. P. Krugiikova, L. K. Poizik, and D. A . Katskov, J . Anal. Chem. USSR,3 0 , 551 (1975).
( 3 ) Kuang-pang Li, Anal. Chem., 48, 2050 (1976). (4) G. M. Hieftje and H. V. Malmstadt, Anal. Chem., 40, 1860 (1968). (5) N. C. Clampiti and 0 . M. Hieftje, Anal. Chem., 44, 1211 (1972). (6) “Handbook of Chemistry and Physics”, 54th ed.,R . C. Weest, Ed., CRC Press, Cleveland, Ohio, 1973.
C. B. Boss’ G.M. Hieftje*
Department of Chemistry Indiana University Bloomington, Indiana 47401 Present address, Department of Chemistry, North Carolina State University, Raleigh, N.C. 27607.
RECEIVED for review May 9,1977. Accepted August 17, 1977. This work was supported in part by the National Science Foundation through grant NSF MPS 75-21695 and by the National Institutes of Health through grant P H S GM 17904-05.
Deceptive “Correct” Separation by the Linear Learning Machine Sir: The linear learning machine (LLM) has the promise of being an extremely useful and simple tool for distinguishing among different groupings of items, including chemical compounds, into a meaningful order or into defined categories based on measurable quantities (1-3). The equations upon which the linear separations are made have then been used to predict the category of unclassified items. This is done by using the same physical measurements determined on the unknown as in the categorized groups and observing the categorized group association of the unclassified item. Kanal ( 4 ) reported on problems concerning the dimensionality and sample size in pattern recognition techniques. This warning has not been explicitly stated in the chemical literature except for a recent theoretical discussion and demonstration on artificial data by Gray ( 5 ) . We attempted to categorize chemical compounds into appropriate electrical fire-hazard classes for bulk water-transport using the LLM. Our use of the LLM procedures consisted of first trying to separate chemicals into the categories developed experimentally by the National Academy of Sciences (NAS) which ranked the electrical fire-hazard of the compounds (6). Our separators were based on variables which were a composite of physical measurements and structural information. The compounds with their NAS classification are listed in Table I and the variables are listed in Table 11. Both binary- and multi-group linear learning machine procedures, found in the computer package ARTHUR (8, were applied to this data set. With either routine, we were able to completely separate the compounds listed into their appropriate NAS assigned category using either the complete variable set listed in Table I1 or a number of smaller subsets of the variables. It was essential in our work with the LLM that we obtain correct separation of the experimentally classified compounds a t greater than 95% confidence. Any applications of the LLM that have substantial human or material risk associated with them, such as this fire-hazard problem, cannot have any error in the correctness of the predictability and, since perfect prediction of the training set does not guarantee that the test data will be predicted with any great accuracy, other tests must be performed. Applications where only a trend is needed to indicate the direction for future work can benefit from the less accurate predictions of the LLM. The high percentage of correct separation of the training set into their assigned categories has been used by others to indicate that unknowns could be categorized with a high degree of accuracy (8). Since 2114
ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
we achieved complete separation of our training set and were able to measure the variables in Table I1 easily, we had hoped to use this premise to categorize a large number of compounds. To test this prediction, we used a leave one out iterative procedure (JACKKNIFE) which involved treating one compound in the training set as an unknown and then classifying it based on the rest of the data. Comparing the predicted classification of each chemical with its correct classification would provide a better estimate of the prediction capability of the LLM than the percentage separation of the training set. The LLM had already separated the training set with 100% separation, whereas the JACKKNIFE procedure only classified 68% and 66% of the compounds correctly for the multi- and binary-linear learning machines respectively. These precentages were obtained with the first 13 variables listed in Table 11, which were the best results we obtained for any subset of variables tested, including the complete data set. The variables used in the subsets tested were chosen based on a series of stepwise discriminate analysis. This routine gives an ordering of the variables and their approximate importance according to the percentage of variance they contribute to the data set while retaining the compounds in their respective categories. We found that both fewer and greater numbers of variables than 13 decreased the reliability of our predicting the NAS assigned classification. Decreasing the number of variables below 13 most likely leads to insufficient information being present in the data set to make a correct decision concerning a compound’s classification. This idea is supported by the fact that we often did not get the 100% separability of the training set with fewer than 13 variables. An explanation for why adding more variables decreased the percentage of correctly predicted compounds is more difficult to find, especially since the 100% separation of the training set was retained. One possible explanation is similar to what Gray ( 5 ) proposed. He did a series of calculations which suggest that a “noise” feature can be found in a data set which can be manipulated by linear learning machines to shift the equations describing the hyperplane used to make the category decision for a compound. This allows the training set to be separated more rapidly but the optimal choice hyperplane or even a correct one for separation is not always found. This inclusion of a random component would lead to incorrect classification of unknowns, especially those near the true boundary of different categories because the hyperplane is shifted slightly in its orientation. Two other possible explanations for why we were able to
