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Absorbance vs. time curves at high heating rates in electrothermal atomic absorption spectroscopy
Spectrochimica Acta Part B 55 Ž2000. 65᎐73
Absorbance vs. time curves at high heating rates in electrothermal atomic absorption spectroscopy G. TorsiU , F.N. Rossi, D. Melucci, P. Reschiglian, C. Locatelli, D. Di Cintio Department of Chemistry ‘G. Ciamician’, Uni¨ ersity of Bologna, Via F. Selmi 2, I-40126 Bologna, Italy Received 15 April 1999; accepted 4 November 1999
Abstract The absorbance vs. time curves that can be obtained in electrothermal atomic absorption spectroscopy with special atomizers and high heating rates can show a very steep increase of the signal from the baseline at the beginning of the atomization step, followed by a more or less extended flat region and by an exponential decay. This type of curve can be mathematically described by simple equations if rather drastic assumptions are made, i.e. if the atom supply is a delta function, if the absorbing species are homogeneously distributed in a plane at the center of a container of constant cross-section, and if the atom removal occurs only by diffusion. In this paper, some experimental curves are presented which are in satisfactory agreement with the theoretical model, thus supporting the claim that, with our atomization system and power supply, the existence of the flat region is indeed justified. With a new atomizer, specifically designed, it is possible to measure the diffusion coefficient of atoms in a gas at high temperature. However, for accurate measurements, the model must be refined and better measurements of the temperature of the atomizer must be obtained. 䊚 2000 Elsevier Science B.V. All rights reserved. Keywords: Electrothermal atomic absorption spectroscopy; Spectroscopic constant
1. Introduction The accepted model for describing absorbance Ž A. vs. time curves in electrothermal atomic absorption spectroscopy ŽETAAS. is a convoluU
Corresponding author. Fax: q39-051-209-9456.
tion integral of a source and removal functions w1,2x given by: NŽ t . s
t⬘
H0 S Ž t⬘. R Ž t y t⬘. dt⬘
Ž1.
where N Ž t . Žatom. is the number of atoms present in the atomizer at time t Ž s ., SŽ t . is the rate
0584-8547r00r$ - see front matter 䊚 2000 Elsevier Science B.V. All rights reserved. PII: S 0 5 8 4 - 8 5 4 7 Ž 9 9 . 0 0 1 6 8 - 8
G. Torsi et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 65᎐73
66
of atoms production Žsupply function. and RŽ t . is the normalized removal function. When SŽ t . is fast compared to RŽ t ., becoming at the limit a delta function, Eq. Ž1. can be simplified w2x as follows: N Ž t . s N Ž0.
t
H0 R Ž t . dt
Ž2.
where N Ž0. is the total number of atoms present in the injected sample if there are no losses in the pre-atomization steps. As claimed in our previous papers w6᎐8,10x, under the assumption that: 1. all atoms injected, at the beginning of the atomization step, are in the vapor state and homogeneously distributed in a plane perpendicular to the optical source beam; 2. all atoms are found at the center of the atomizer Žinitial conditions. and the atom concentration at its ends is always zero Žboundary conditions. which is equivalent to say that the atoms are stable only inside it; and
3. only one-dimensional diffusion is considered as the removal mechanism w3,4x, then the integral of the removal function can be easily calculated by integrating the atoms density along the tube length as follows: RŽ t . s
l
H0 nŽ x,t . d x
Ž3.
where n is the density of the analyte at position x and time t. The solution of the above equation leads to two mathematical expressions, both giving equivalent results. These expressions are series which provide the desired accuracy by simply adding terms in the calculations w3,5x. N Ž t . s N Ž0.
4 eyŽ9 D t .r l y 3 2
yŽ 2 D t .r l 2
= e
ž
eyŽ2 5 D t .r l ⭈⭈⭈ 5 2
q
2
2
/
Ž4.
Fig. 1. Theoretical curves calculated according to Eqs. Ž4. and Ž5. by retaining only five terms of the series. The insert shows the different results obtained at low t values. is a dimensionless parameter.
G. Torsi et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 65᎐73
N Ž t . s N Ž 0 . 1 y 2 erfc
ž
l 4'Dt
3l q2 erfc ⭈⭈⭈ ' 4 Dt
67
takes the form:
/
Ž5. 2
y1 .
In the above equations, D Žcm s is the diffusion coefficient of the atoms and l Žcm. the length of the graphite tube. Fig. 1 shows that there is no difference in an N Ž t .rN Ž0. vs. time curve at reasonably high values of t using five terms in both equations. On the contrary, at low t, Eq. Ž4. results in higher values than physically possible. Such deviation, which is due to the low number of terms used in Eq. Ž4., does not influence our results because, at low t, only Eq. Ž5. is used. The experimental data to be compared with the theoretical curve, are the absorbances measured every 9 ms with the Perkin-Elmer Model 1100B Atomic Spectrometer. This sampling period is sufficiently low to prevent any distortion of the signal at the heating rate used. The experimental N Ž t . values are obtained through the Lambert᎐Beer law, which in our case
AŽ t . s k
NŽ t . S
Ž6.
where k Žcm2 atomy1 . has been defined in our previous works as a spectroscopic constant and represents the optical cross-section of the atom. S Žcm2 . is the geometrical cross-section Žsupposed constant. of the atomizer. In the same way we can define AŽ0. together with N Ž0. which is the asymptotic value of N Ž t . in Eq. Ž5.. The simplest form of Eq. Ž4., i.e. when only the first term of the series is retained, has been extensively used in studying the final portion of the A vs. time curves obtained with commercial atomizers. In fact at sufficiently high t, the other terms of Eq. Ž4. approach zero, making the deviation of the real situation at t s 0 from the ideal conditions Žall atoms are homogeneously distributed in a small volume at the center of the graphite tube. less and less important. On the other hand, Eq. Ž5. has never been used because sufficiently fast sources and appropriate atomizers were not available on the market and
Fig. 2. Experimental data obtained with 60 pg of Pb Žq200 ng of Pd.. The continuous curve is the best fit of the points from 100 to 50% of AŽ0.Žexp.. AŽ0. s 0.29 and D s 4.9. T ; 2500 K.
G. Torsi et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 65᎐73
68
therefore, no experimental data could be used to test the model. Indeed the experimental conditions are rather demanding when Eq. Ž5. is used. These conditions can never be met but only more or less closely approached, because the transformation of the analyte from the condensed phase on the atomizer surface to the vapor state followed by its homogeneous distribution in a small volume at the center of the atomizer, can never be as instantaneous as required by the model. In this interval of time, as can be seen from Fig. 2, A is steadily increasing up to a point where, more or less clearly, it becomes constant. This interval of time is a function of the heating rate, the physical state of the analyte Žas a solid more or less dispersed or embedded in a matrix on the surface of the atomizer., the final temperature of the atomization step and especially the physical characteristics of the analyte, in particular its volatility. In fact, the limits of our current atomization system are such that the data analyzed in this paper are relevant only to volatile elements like Hg, Cd, Pb and Ag. There are other elements of rather high volatility like Tl and Bi that can certainly be studied with our instrumentation. However, our current interest lies in extending the method to those elements of medium and relatively low volatility through atomizers of new design and through different modes of heating. The aim of this paper is to present the comparison between the experimental curves and those calculated from Eq. Ž5. obtained with the new ŽPb. and old ŽHg and Ag. atomizers in order to ascertain how well the experimental data are described by the rather simple model adopted. Moreover, the asymptotic value AŽ0. thus obtained is used to evaluate the feasibility of standardless analysis, already proposed when curves of this type are obtained. Finally, it will be shown how the experimental data can be considered valid for the measurement of the diffusion coefficient of atoms at high temperature.
2. Experimental The experimental data relevant to Hg and Ag
have already been published w8x and are here analyzed with the new model. The data analyzed with the new atomizer were obtained in the same way as previously described w6,7x. They should give better curves for testing the model, due to a more homogeneous temperature distribution compared to the previous ones and a more defined value of its length l. The new atomizer is more similar to commercial atomizers, because it is fixed to a base and there is no change of atomizer during one run. The geometrical dimensions are the same because the most important part, i.e. the graphite tube, is the same Ž l s 36 mm; internal and external diameter 3.2 and 4.6 mm, respectively.. The difference is given by the electrical contacts which are arranged as shown in Fig. 3. In this way, the extremities are at a temperature determined by the form and geometry of the contacts. In order to obtain temperatures as much as possible close to those of the center of the graphite tube, the contacts were made with graphite rods of 4.6 mm in diameter Žthe same as the graphite tube., modeled at their ends in order to give the highest contact area and thinned near the graphite tube by drilling a hole of 2.5 mm at approximately 5 mm from its center. With this geometry, it is believed that the graphite tube has a homogeneous temperature over its whole length and that l is much better defined when compared with the previous atomizers, in which the contacts were made with circular pieces of graphite with a hole that extended the length of the atomizer by a quantity difficult to define w7x. On the contrary, with the new atomizer, the inert gas surrounding the tube ends is strongly heated and therefore, should have a high upward velocity when, during the atomization step, the temperature of the extremities is high. In these conditions, the analyte concentration at these points should be practically zero, as required by the model. The maximum heating rate obtained with this new atomizer should not be much different Ž10 000 K sy1 . than that of the previous atomizer since the increased resistance of the contacts has been compensated by the use of three batteries in series instead of two. With this atomizer, only
G. Torsi et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 65᎐73
69
1. The points that were judged as belonging to the part of the curve, after the beginning of the atomization step, where A is supposed practically constant, were summed and averaged. This value was called AŽ0.Žexp.. The reason why A is, in this interval, considered only practically constant follows from the fact that mathematically wsee Eq. Ž5.x A is never constant and AŽ0. is only an asymptotic value. 2. The starting time Ž t s 0. is taken as the time of the first point at which A G AŽ0.Žexp.. 3. The starting value of Drl 2 was determined by the slope of the plot ln A vs. t in the interval of A from 80 to 20% of AŽ0.Žexp. by using Eq. Ž4. while retaining only the first term.
Fig. 3. Sketch of the atomizer showing its side contacts.
graphite tubes of 36 mm length were used while, with the old atomizers, measurements were made with graphite tubes of 36 and 50 mm in length. Sample introduction, also in the case of the new atomizer, was made with a bridge, which could slide on the rims of the Plexiglas base, to which a 5-l Gilson pipette was fastened. The pipette tip was substituted by a Teflon tubing of 1 mm external diameter. This tip could be inserted inside the graphite tube in order to deliver the sample solution at its center. All other conditions were those already described w6,7x.
The points used for the best fit were from t s 0 to the value of t at which A is a given, chosen fraction of AŽ0.Žexp.. The fitting program was then run to obtain the best values of AŽ0. and Drl 2 . A difference no higher than 1% was found between AŽ0.Žexp. and AŽ0. if the plateau in the atomization curve is recognizable and the final point used is 20% higher than AŽ0.Žexp.. The values of Drl 2 were generally decreasing with an increase of the number of points considered Žsee Table 2.. Practically, for a fixed starting point t s 0, no difference was found between the initial and final values when the starting values of AŽ0.Žexp. and Drl 2 were changed by "20%. It is obvious that, with this procedure, the most critical choice is that of t s 0. 3.2. Validation of the model
3. Results and discussion 3.1. Best fit procedure After having obtained an experimental curve, which is a series of A values as a function of time, one must find the best values of AŽ0. and D which fit Eq. Ž5. Žthe data of Fig. 2 were obtained with the new atomizer and therefore l was considered constant and equal to 3.6 cm.. The starting values were chosen in the following way.
The validation of the model was accomplished by comparing the experimental data with the theoretical curves. Figs. 4 and 5 show the experimental data obtained in our laboratory with old atomizers w7,8x together with the best-fit curves obtained as described above. These data are shown here because only with these atomizers data, at different tube lengths and at different pressures, are available. It is clear that, apart from the expected steady increase of A at the beginning of the atomization step, which cannot be considered in the model adopted and is, there-
70
G. Torsi et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 65᎐73
Fig. 4. Best-fit curves wthrough Eq. Ž5.x and experimental points for Hg Ža. and Ag Žb. with graphite tubes of different lengths. Ž ⌬ . 36 and Žx. 50 mm.. T ŽAg. ; 2600 K. T ŽHg. ; 2000 K.
fore, not shown, the rest of the experimental curves follows the model quite closely. Of great importance is, however, not only the form of the curve, but also the constancy of AŽ0. which holds for graphite tubes of 36 mm as well as of 50 mm. Another important point is that, over more than 4 years of investigations and measurements on ele-
ments like Cd, Pb and Hg, k has always been found to remain constant within 10% R.S.D. During this period, many instrumental settings have been changed, except the spectrometer and the hollow cathode lamps type. A comparison of k values with those calculated from theoretical consideration w9x shows that there
G. Torsi et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 65᎐73
71
Fig. 5. Best-fit curves wthrough Eq. Ž5.x and experimental points for Hg Ža. and Ag Žb. at 1 Ž ⌬ . and 3 Žx. atm. Temperature values are the same as in Fig. 4.
is a difference with the theoretical values always higher than those found by us Žsee Table 1.. No values of k obtained through measurements of
the type described here has been found in the literature. The good agreement between experimental and theoretical values reported by L’vov
G. Torsi et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 65᎐73
72
Table 1 k values = 10 13 Žcm2 atomy1 . Element
Ag Cd Hg Pb a
Theoretical values
Experimental values
1 atm
3 atm
7.5 20 0.17 1.5
3.6 7.6 0.09
1 atm 36 mma
1 atm 50 mma
3 atm 50 mma
13 0.06 1.0
6.7 13 0.07 1.1
3.1 6.8 0.07
Length of the graphite tube. Temperature ( 2500 K.
w9x, where peak areas are considered, could be attributed to a combination of systematic errors w10x. The discrepancies between the theoretical and the experimental data must, therefore, await a more detailed knowledge of the line profiles of the source and the absorber. Another problem observed with Hg and related to the line profile, is the practical constancy of k with pressure which is not explained by the accepted model even when the hyperfine structure of the line is taken into account w8x. 3.3. Measurements of D As reported before, the value of D can be calculated from the experimental data obtained with the new atomizer. Table 2 collects a series of D values, calculated using different portions of the same experimental curve. It is clear that the value of D is not constant. It is almost always found that, using Eq. Ž5., D decreases by increasing the number of points considered. The interpretation of this trend points to the inadequacy of the model, which assumes that all the atoms at t s 0 are homogeneously distributed in a plane at the center of the atomizer and that diffusion is the only removal mechanism. However, this cannot be true because, as already observed, there is an interval of time during which the atoms are vaporized and uniformly distributed. Moreover, as long as the temperature of the atomizer walls increases, there will be a corresponding increase in the volume, with a not yet defined time lag, of the inert gas present, to which must be added a further increase due to the gases formed during the atomization step. Therefore, the apparent
decrease of D can be explained as a decrease of the convection removal mechanism proportional to wŽdTrdt .rT x w11,12x. As already calculated w3x and found w13,14x with other atomizers, the incidence of convection, as seen by the change of D in the first three lines of Table 2, seems rather small also with our present system. At large times, convection should be practically negligible. With the current atomization system, any increase in the temperature should stop after about 0.1 s Žplus the above cited time lag., assuming that the final temperature is 1000⬚ higher than the ashing temperature and the heating rate is 10 000 K sy1 . The best points for measuring a true value of D are, therefore, the middle points Ž80r20%. of the curve where the influence of convection and a potentially wrong choice of the baseline are minimized. The values shown in Table 2 are in relatively satisfactory agreement with those that can be Table 2 D values Žcm2 sy1 . for Pb and Cd in Ar a Points used
b
100᎐70% 100᎐50%b 100᎐20%b 80᎐20%c 50᎐15%c 30᎐10%c
Pb
Cd
( 2700 K
( 2200 K
( 2100 K
5.6 5.1 5.1 5.3 5.1 5.4
4.3 4.1 3.9 4.1 4.2 4.3
3.6 3.6 3.6 3.6 3.7 3.9
a The values calculated from L’vov w9x are: Pb Ž2700 K.: 5.4; Pb Ž2200 K.: 3.7; and Cd Ž2100 K.: 4.6. b D from the best fit obtained using Eq. Ž5.. c D from the slope obtained using Eq. Ž4. Žretaining only the first term..
G. Torsi et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 65᎐73
calculated by extrapolating data obtained at low temperature w9x. It is, however, necessary to point out that also the temperatures of the atomization step, given in Table 2, have a low accuracy because of the rather long extrapolation used in obtaining those values w15x. Moreover, it is difficult to maintain a pyrometric system well calibrated. A better and more meaningful comparison must therefore await for a more refined model in which convection is taken into account and the accuracy of the temperature measurement is improved. On the other hand, it is worth stressing that these deficiencies are important only if a very precise value of D is sought, but are of little relevance if one measures AŽ0., since AŽ0. is not related to D. The importance, for us, of measuring AŽ0. stems from the fact that, if there are no losses in the pre-atomization steps, then N Ž0. is the number of atoms injected: as a consequence, provided that k is known, standardless analysis is possible. As already mentioned, AŽ0. remains practically constant in the fitting procedure, its variation being well within the noise level of our measurements.
4. Conclusions It has been shown that a simple model describing the A vs. time curves, in which only diffusion is considered, gives good agreement with the experimental curves obtained with our atomization system in ETAAS. The agreement is remarkably good for the values of AŽ0. while is not as good for the diffusion coefficient D, due to the simultaneous presence of convection as an additional removal mechanism. The data presented clearly support our claim that AŽ0., found as the asymptotic value resulting from a best-fit procedure, is given by the simultaneous presence of all atoms injected in the optical beam. As a consequence, the statement can
73
be made that the method proposed has the highest sensitivity that can be obtained with this type of measurement. Finally, once the spectroscopic constant Žoptical atomic cross-section. of the line under study is known, standardless analysis is possible.
Acknowledgements We thank Prof. S. Paveri-Fontana and Prof. N. Omenetto for their helpful suggestions. This work has been partially supported by the University of Bologna ŽFund for Selected Research Topics.. References w1x S.L. Paveri-Fontana, G. Tessari, G. Torsi, Anal. Chem. 46 Ž1974. 1032. w2x W.M.G.T. van den Broek, L. de Galan, Anal. Chem. 49 Ž1977. 2176. w3x A.K. Gilmutdinov, I.S. Fishman, Spectrochim. Acta Part B 39 Ž1984. 171. w4x H. Falck, C. Schnurrer, Spectrochim. Acta Part B 44 Ž1989. 759. w5x H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford Press, 1959, p. 53. w6x G. Torsi, P. Reschiglian, F. Fagioli, C. Locatelli, Spectrochim. Acta Part B 48 Ž1993. 681. w7x G. Torsi, S. Valcher, P. Reschiglian, L. Cludi, L. Patauner, Spectrochim. Acta Part B 50 Ž1995. 1679. w8x F.N. Rossi, D. Melucci, C. Locatelli, P. Reschiglian, G. Torsi, A. Millemaggi, Fresenius J. Anal. Chem. 361 Ž1998. 504. w9x B.V. L’vov, Spectrochim. Acta Part B 45 Ž1990. 633. w10x G. Torsi, F. Fagioli, S. Landi, P. Reschiglian, C. Locatelli, F.N. Rossi, D. Melucci, T. Bernardi, Spectrochim. Acta Part B 53 Ž1998. 1847. w11x J.A. Holcombe, Spectrochim. Acta Part B 48 Ž1983. 609. w12x S.L. Paveri-Fontana, G. Tessari, Prog. Anal. Atom. Spectrosc. 7 Ž1984. 243. w13x W. Frech, B.V. L’vov, Spectrochim. Acta Part B 48 Ž1993. 1371. w14x C.M.M. Smith, J.M. Harnly, J. Anal. Atom. Spec. 10 Ž1995. 187. w15x G. Torsi, G. Bergamini, Ann. Chim. ŽRome. 79 Ž1989. 45.
Absorbance vs. time curves at high heating rates in electrothermal atomic absorption spectroscopy G. TorsiU , F.N. Rossi, D. Melucci, P. Reschiglian, C. Locatelli, D. Di Cintio Department of Chemistry ‘G. Ciamician’, Uni¨ ersity of Bologna, Via F. Selmi 2, I-40126 Bologna, Italy Received 15 April 1999; accepted 4 November 1999
Abstract The absorbance vs. time curves that can be obtained in electrothermal atomic absorption spectroscopy with special atomizers and high heating rates can show a very steep increase of the signal from the baseline at the beginning of the atomization step, followed by a more or less extended flat region and by an exponential decay. This type of curve can be mathematically described by simple equations if rather drastic assumptions are made, i.e. if the atom supply is a delta function, if the absorbing species are homogeneously distributed in a plane at the center of a container of constant cross-section, and if the atom removal occurs only by diffusion. In this paper, some experimental curves are presented which are in satisfactory agreement with the theoretical model, thus supporting the claim that, with our atomization system and power supply, the existence of the flat region is indeed justified. With a new atomizer, specifically designed, it is possible to measure the diffusion coefficient of atoms in a gas at high temperature. However, for accurate measurements, the model must be refined and better measurements of the temperature of the atomizer must be obtained. 䊚 2000 Elsevier Science B.V. All rights reserved. Keywords: Electrothermal atomic absorption spectroscopy; Spectroscopic constant
1. Introduction The accepted model for describing absorbance Ž A. vs. time curves in electrothermal atomic absorption spectroscopy ŽETAAS. is a convoluU
Corresponding author. Fax: q39-051-209-9456.
tion integral of a source and removal functions w1,2x given by: NŽ t . s
t⬘
H0 S Ž t⬘. R Ž t y t⬘. dt⬘
Ž1.
where N Ž t . Žatom. is the number of atoms present in the atomizer at time t Ž s ., SŽ t . is the rate
0584-8547r00r$ - see front matter 䊚 2000 Elsevier Science B.V. All rights reserved. PII: S 0 5 8 4 - 8 5 4 7 Ž 9 9 . 0 0 1 6 8 - 8
G. Torsi et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 65᎐73
66
of atoms production Žsupply function. and RŽ t . is the normalized removal function. When SŽ t . is fast compared to RŽ t ., becoming at the limit a delta function, Eq. Ž1. can be simplified w2x as follows: N Ž t . s N Ž0.
t
H0 R Ž t . dt
Ž2.
where N Ž0. is the total number of atoms present in the injected sample if there are no losses in the pre-atomization steps. As claimed in our previous papers w6᎐8,10x, under the assumption that: 1. all atoms injected, at the beginning of the atomization step, are in the vapor state and homogeneously distributed in a plane perpendicular to the optical source beam; 2. all atoms are found at the center of the atomizer Žinitial conditions. and the atom concentration at its ends is always zero Žboundary conditions. which is equivalent to say that the atoms are stable only inside it; and
3. only one-dimensional diffusion is considered as the removal mechanism w3,4x, then the integral of the removal function can be easily calculated by integrating the atoms density along the tube length as follows: RŽ t . s
l
H0 nŽ x,t . d x
Ž3.
where n is the density of the analyte at position x and time t. The solution of the above equation leads to two mathematical expressions, both giving equivalent results. These expressions are series which provide the desired accuracy by simply adding terms in the calculations w3,5x. N Ž t . s N Ž0.
4 eyŽ9 D t .r l y 3 2
yŽ 2 D t .r l 2
= e
ž
eyŽ2 5 D t .r l ⭈⭈⭈ 5 2
q
2
2
/
Ž4.
Fig. 1. Theoretical curves calculated according to Eqs. Ž4. and Ž5. by retaining only five terms of the series. The insert shows the different results obtained at low t values. is a dimensionless parameter.
G. Torsi et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 65᎐73
N Ž t . s N Ž 0 . 1 y 2 erfc
ž
l 4'Dt
3l q2 erfc ⭈⭈⭈ ' 4 Dt
67
takes the form:
/
Ž5. 2
y1 .
In the above equations, D Žcm s is the diffusion coefficient of the atoms and l Žcm. the length of the graphite tube. Fig. 1 shows that there is no difference in an N Ž t .rN Ž0. vs. time curve at reasonably high values of t using five terms in both equations. On the contrary, at low t, Eq. Ž4. results in higher values than physically possible. Such deviation, which is due to the low number of terms used in Eq. Ž4., does not influence our results because, at low t, only Eq. Ž5. is used. The experimental data to be compared with the theoretical curve, are the absorbances measured every 9 ms with the Perkin-Elmer Model 1100B Atomic Spectrometer. This sampling period is sufficiently low to prevent any distortion of the signal at the heating rate used. The experimental N Ž t . values are obtained through the Lambert᎐Beer law, which in our case
AŽ t . s k
NŽ t . S
Ž6.
where k Žcm2 atomy1 . has been defined in our previous works as a spectroscopic constant and represents the optical cross-section of the atom. S Žcm2 . is the geometrical cross-section Žsupposed constant. of the atomizer. In the same way we can define AŽ0. together with N Ž0. which is the asymptotic value of N Ž t . in Eq. Ž5.. The simplest form of Eq. Ž4., i.e. when only the first term of the series is retained, has been extensively used in studying the final portion of the A vs. time curves obtained with commercial atomizers. In fact at sufficiently high t, the other terms of Eq. Ž4. approach zero, making the deviation of the real situation at t s 0 from the ideal conditions Žall atoms are homogeneously distributed in a small volume at the center of the graphite tube. less and less important. On the other hand, Eq. Ž5. has never been used because sufficiently fast sources and appropriate atomizers were not available on the market and
Fig. 2. Experimental data obtained with 60 pg of Pb Žq200 ng of Pd.. The continuous curve is the best fit of the points from 100 to 50% of AŽ0.Žexp.. AŽ0. s 0.29 and D s 4.9. T ; 2500 K.
G. Torsi et al. r Spectrochimica Acta Part B: Atomic Spectroscopy 55 (2000) 65᎐73
68
therefore, no experimental data could be used to test the model. Indeed the experimental conditions are rather demanding when Eq. Ž5. is used. These conditions can never be met but only more or less closely approached, because the transformation of the analyte from the condensed phase on the atomizer surface to the vapor state followed by its homogeneous distribution in a small volume at the center of the atomizer, can never be as instantaneous as required by the model. In this interval of time, as can be seen from Fig. 2, A is steadily increasing up to a point where, more or less clearly, it becomes constant. This interval of time is a function of the heating rate, the physical state of the analyte Žas a solid more or less dispersed or embedded in a matrix on the surface of the atomizer., the final temperature of the atomization step and especially the physical characteristics of the analyte, in particular its volatility. In fact, the limits of our current atomization system are such that the data analyzed in this paper are relevant only to volatile elements like Hg, Cd, Pb and Ag. There are other elements of rather high volatility like Tl and Bi that can certainly be studied with our instrumentation. However, our current interest lies in extending the method to those elements of medium and relatively low volatility through atomizers of new design and through different modes of heating. The aim of this paper is to present the comparison between the experimental curves and those calculated from Eq. Ž5. obtained with the new ŽPb. and old ŽHg and Ag. atomizers in order to ascertain how well the experimental data are described by the rather simple model adopted. Moreover, the asymptotic value AŽ0. thus obtained is used to evaluate the feasibility of standardless analysis, already proposed when curves of this type are obtained. Finally, it will be shown how the experimental data can be considered valid for the measurement of the diffusion coefficient of atoms at high temperature.
2. Experimental The experimental data relevant to Hg and Ag
have already been published w8x and are here analyzed with the new model. The data analyzed with the new atomizer were obtained in the same way as previously described w6,7x. They should give better curves for testing the model, due to a more homogeneous temperature distribution compared to the previous ones and a more defined value of its length l. The new atomizer is more similar to commercial atomizers, because it is fixed to a base and there is no change of atomizer during one run. The geometrical dimensions are the same because the most important part, i.e. the graphite tube, is the same Ž l s 36 mm; internal and external diameter 3.2 and 4.6 mm, respectively.. The difference is given by the electrical contacts which are arranged as shown in Fig. 3. In this way, the extremities are at a temperature determined by the form and geometry of the contacts. In order to obtain temperatures as much as possible close to those of the center of the graphite tube, the contacts were made with graphite rods of 4.6 mm in diameter Žthe same as the graphite tube., modeled at their ends in order to give the highest contact area and thinned near the graphite tube by drilling a hole of 2.5 mm at approximately 5 mm from its center. With this geometry, it is believed that the graphite tube has a homogeneous temperature over its whole length and that l is much better defined when compared with the previous atomizers, in which the contacts were made with circular pieces of graphite with a hole that extended the length of the atomizer by a quantity difficult to define w7x. On the contrary, with the new atomizer, the inert gas surrounding the tube ends is strongly heated and therefore, should have a high upward velocity when, during the atomization step, the temperature of the extremities is high. In these conditions, the analyte concentration at these points should be practically zero, as required by the model. The maximum heating rate obtained with this new atomizer should not be much different Ž10 000 K sy1 . than that of the previous atomizer since the increased resistance of the contacts has been compensated by the use of three batteries in series instead of two. With this atomizer, only
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1. The points that were judged as belonging to the part of the curve, after the beginning of the atomization step, where A is supposed practically constant, were summed and averaged. This value was called AŽ0.Žexp.. The reason why A is, in this interval, considered only practically constant follows from the fact that mathematically wsee Eq. Ž5.x A is never constant and AŽ0. is only an asymptotic value. 2. The starting time Ž t s 0. is taken as the time of the first point at which A G AŽ0.Žexp.. 3. The starting value of Drl 2 was determined by the slope of the plot ln A vs. t in the interval of A from 80 to 20% of AŽ0.Žexp. by using Eq. Ž4. while retaining only the first term.
Fig. 3. Sketch of the atomizer showing its side contacts.
graphite tubes of 36 mm length were used while, with the old atomizers, measurements were made with graphite tubes of 36 and 50 mm in length. Sample introduction, also in the case of the new atomizer, was made with a bridge, which could slide on the rims of the Plexiglas base, to which a 5-l Gilson pipette was fastened. The pipette tip was substituted by a Teflon tubing of 1 mm external diameter. This tip could be inserted inside the graphite tube in order to deliver the sample solution at its center. All other conditions were those already described w6,7x.
The points used for the best fit were from t s 0 to the value of t at which A is a given, chosen fraction of AŽ0.Žexp.. The fitting program was then run to obtain the best values of AŽ0. and Drl 2 . A difference no higher than 1% was found between AŽ0.Žexp. and AŽ0. if the plateau in the atomization curve is recognizable and the final point used is 20% higher than AŽ0.Žexp.. The values of Drl 2 were generally decreasing with an increase of the number of points considered Žsee Table 2.. Practically, for a fixed starting point t s 0, no difference was found between the initial and final values when the starting values of AŽ0.Žexp. and Drl 2 were changed by "20%. It is obvious that, with this procedure, the most critical choice is that of t s 0. 3.2. Validation of the model
3. Results and discussion 3.1. Best fit procedure After having obtained an experimental curve, which is a series of A values as a function of time, one must find the best values of AŽ0. and D which fit Eq. Ž5. Žthe data of Fig. 2 were obtained with the new atomizer and therefore l was considered constant and equal to 3.6 cm.. The starting values were chosen in the following way.
The validation of the model was accomplished by comparing the experimental data with the theoretical curves. Figs. 4 and 5 show the experimental data obtained in our laboratory with old atomizers w7,8x together with the best-fit curves obtained as described above. These data are shown here because only with these atomizers data, at different tube lengths and at different pressures, are available. It is clear that, apart from the expected steady increase of A at the beginning of the atomization step, which cannot be considered in the model adopted and is, there-
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Fig. 4. Best-fit curves wthrough Eq. Ž5.x and experimental points for Hg Ža. and Ag Žb. with graphite tubes of different lengths. Ž ⌬ . 36 and Žx. 50 mm.. T ŽAg. ; 2600 K. T ŽHg. ; 2000 K.
fore, not shown, the rest of the experimental curves follows the model quite closely. Of great importance is, however, not only the form of the curve, but also the constancy of AŽ0. which holds for graphite tubes of 36 mm as well as of 50 mm. Another important point is that, over more than 4 years of investigations and measurements on ele-
ments like Cd, Pb and Hg, k has always been found to remain constant within 10% R.S.D. During this period, many instrumental settings have been changed, except the spectrometer and the hollow cathode lamps type. A comparison of k values with those calculated from theoretical consideration w9x shows that there
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Fig. 5. Best-fit curves wthrough Eq. Ž5.x and experimental points for Hg Ža. and Ag Žb. at 1 Ž ⌬ . and 3 Žx. atm. Temperature values are the same as in Fig. 4.
is a difference with the theoretical values always higher than those found by us Žsee Table 1.. No values of k obtained through measurements of
the type described here has been found in the literature. The good agreement between experimental and theoretical values reported by L’vov
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Table 1 k values = 10 13 Žcm2 atomy1 . Element
Ag Cd Hg Pb a
Theoretical values
Experimental values
1 atm
3 atm
7.5 20 0.17 1.5
3.6 7.6 0.09
1 atm 36 mma
1 atm 50 mma
3 atm 50 mma
13 0.06 1.0
6.7 13 0.07 1.1
3.1 6.8 0.07
Length of the graphite tube. Temperature ( 2500 K.
w9x, where peak areas are considered, could be attributed to a combination of systematic errors w10x. The discrepancies between the theoretical and the experimental data must, therefore, await a more detailed knowledge of the line profiles of the source and the absorber. Another problem observed with Hg and related to the line profile, is the practical constancy of k with pressure which is not explained by the accepted model even when the hyperfine structure of the line is taken into account w8x. 3.3. Measurements of D As reported before, the value of D can be calculated from the experimental data obtained with the new atomizer. Table 2 collects a series of D values, calculated using different portions of the same experimental curve. It is clear that the value of D is not constant. It is almost always found that, using Eq. Ž5., D decreases by increasing the number of points considered. The interpretation of this trend points to the inadequacy of the model, which assumes that all the atoms at t s 0 are homogeneously distributed in a plane at the center of the atomizer and that diffusion is the only removal mechanism. However, this cannot be true because, as already observed, there is an interval of time during which the atoms are vaporized and uniformly distributed. Moreover, as long as the temperature of the atomizer walls increases, there will be a corresponding increase in the volume, with a not yet defined time lag, of the inert gas present, to which must be added a further increase due to the gases formed during the atomization step. Therefore, the apparent
decrease of D can be explained as a decrease of the convection removal mechanism proportional to wŽdTrdt .rT x w11,12x. As already calculated w3x and found w13,14x with other atomizers, the incidence of convection, as seen by the change of D in the first three lines of Table 2, seems rather small also with our present system. At large times, convection should be practically negligible. With the current atomization system, any increase in the temperature should stop after about 0.1 s Žplus the above cited time lag., assuming that the final temperature is 1000⬚ higher than the ashing temperature and the heating rate is 10 000 K sy1 . The best points for measuring a true value of D are, therefore, the middle points Ž80r20%. of the curve where the influence of convection and a potentially wrong choice of the baseline are minimized. The values shown in Table 2 are in relatively satisfactory agreement with those that can be Table 2 D values Žcm2 sy1 . for Pb and Cd in Ar a Points used
b
100᎐70% 100᎐50%b 100᎐20%b 80᎐20%c 50᎐15%c 30᎐10%c
Pb
Cd
( 2700 K
( 2200 K
( 2100 K
5.6 5.1 5.1 5.3 5.1 5.4
4.3 4.1 3.9 4.1 4.2 4.3
3.6 3.6 3.6 3.6 3.7 3.9
a The values calculated from L’vov w9x are: Pb Ž2700 K.: 5.4; Pb Ž2200 K.: 3.7; and Cd Ž2100 K.: 4.6. b D from the best fit obtained using Eq. Ž5.. c D from the slope obtained using Eq. Ž4. Žretaining only the first term..
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calculated by extrapolating data obtained at low temperature w9x. It is, however, necessary to point out that also the temperatures of the atomization step, given in Table 2, have a low accuracy because of the rather long extrapolation used in obtaining those values w15x. Moreover, it is difficult to maintain a pyrometric system well calibrated. A better and more meaningful comparison must therefore await for a more refined model in which convection is taken into account and the accuracy of the temperature measurement is improved. On the other hand, it is worth stressing that these deficiencies are important only if a very precise value of D is sought, but are of little relevance if one measures AŽ0., since AŽ0. is not related to D. The importance, for us, of measuring AŽ0. stems from the fact that, if there are no losses in the pre-atomization steps, then N Ž0. is the number of atoms injected: as a consequence, provided that k is known, standardless analysis is possible. As already mentioned, AŽ0. remains practically constant in the fitting procedure, its variation being well within the noise level of our measurements.
4. Conclusions It has been shown that a simple model describing the A vs. time curves, in which only diffusion is considered, gives good agreement with the experimental curves obtained with our atomization system in ETAAS. The agreement is remarkably good for the values of AŽ0. while is not as good for the diffusion coefficient D, due to the simultaneous presence of convection as an additional removal mechanism. The data presented clearly support our claim that AŽ0., found as the asymptotic value resulting from a best-fit procedure, is given by the simultaneous presence of all atoms injected in the optical beam. As a consequence, the statement can
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be made that the method proposed has the highest sensitivity that can be obtained with this type of measurement. Finally, once the spectroscopic constant Žoptical atomic cross-section. of the line under study is known, standardless analysis is possible.
Acknowledgements We thank Prof. S. Paveri-Fontana and Prof. N. Omenetto for their helpful suggestions. This work has been partially supported by the University of Bologna ŽFund for Selected Research Topics.. References w1x S.L. Paveri-Fontana, G. Tessari, G. Torsi, Anal. Chem. 46 Ž1974. 1032. w2x W.M.G.T. van den Broek, L. de Galan, Anal. Chem. 49 Ž1977. 2176. w3x A.K. Gilmutdinov, I.S. Fishman, Spectrochim. Acta Part B 39 Ž1984. 171. w4x H. Falck, C. Schnurrer, Spectrochim. Acta Part B 44 Ž1989. 759. w5x H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford Press, 1959, p. 53. w6x G. Torsi, P. Reschiglian, F. Fagioli, C. Locatelli, Spectrochim. Acta Part B 48 Ž1993. 681. w7x G. Torsi, S. Valcher, P. Reschiglian, L. Cludi, L. Patauner, Spectrochim. Acta Part B 50 Ž1995. 1679. w8x F.N. Rossi, D. Melucci, C. Locatelli, P. Reschiglian, G. Torsi, A. Millemaggi, Fresenius J. Anal. Chem. 361 Ž1998. 504. w9x B.V. L’vov, Spectrochim. Acta Part B 45 Ž1990. 633. w10x G. Torsi, F. Fagioli, S. Landi, P. Reschiglian, C. Locatelli, F.N. Rossi, D. Melucci, T. Bernardi, Spectrochim. Acta Part B 53 Ž1998. 1847. w11x J.A. Holcombe, Spectrochim. Acta Part B 48 Ž1983. 609. w12x S.L. Paveri-Fontana, G. Tessari, Prog. Anal. Atom. Spectrosc. 7 Ž1984. 243. w13x W. Frech, B.V. L’vov, Spectrochim. Acta Part B 48 Ž1993. 1371. w14x C.M.M. Smith, J.M. Harnly, J. Anal. Atom. Spec. 10 Ž1995. 187. w15x G. Torsi, G. Bergamini, Ann. Chim. ŽRome. 79 Ž1989. 45.