Computer modelling of atomization processes in graphite furnace atomic absorption spectrometry

Specmchimiw

Acra.

Vol. 388,

No.

IO. pp. 1287~1300.1983.

0584-8%7/83 t3.00+ .w @

Printed in Gnal Britain.

1983.Perymon PressLtd.

Computer rn~e~li~g of atomi~tion processes in graphite furnace atomic absorption spectrometry C. Ottawa-Carleton

L. CHAKRABARTI,* S. B. CHANG,S. R.

LAWSON

and S. M. WONG

Institute for Research and Graduate Studies in Chemistry, The Department of Chemistry, Carleton University, Ottawa, Ontario KiS 5B6, Canada (Reeeiued 15 March 1983)

Abstract-A

theoreticai’modei, based on a first-order or a pseudo-first-order consecutive rate process to describe transient charaeteristies of an atomic absorption pulse generated in a pulse-heated graphite furnace. has been established. The effectsofchange in the parameters on the shape and height of the atom population-timecurves have been studied using a numerical integration method. The model has been tested experimentally, and the experimental results are in reasonably fair agreement with those predicted by the model.

1. I NTR~IXJCUON THE TRANSIENT atomic

absorption signal generated by rapid atomization of an analyte in a pulse-heated el~trothermal atomizer provides a temporal history of the atom population. Several dynamic models for interpretation of this temporal signal generated by various types of electrothermal atomizers have been proposed [l-21). L’vov [l-3] formulated the change in the atom population within the analysis volume with time in terms of atomization time (TJ and residence time (z2). Minimization of the r1/r2 ratio was proposed as a means of improving the sensitivity of eiectrothermal atomizers. However, many limitations on achieving this minimi~tion are not apparent from the model. A comprehensive theoretical model for atomic absorption signals of analytes generated by an electrothermal rod atomizer has been developed by TORSI, TESSARI ef al. [4-l 1J in which the overall atomization process is described as the result of an interphase release process, followed by gaseous transport to the optical observation region, where the analytical signal is obtained. In the model, the authors expressed the supply of the analyte in terms of an Arrhenius-ty~ rate constant and related the removal of the analyte to diffusion and convection.

*The author to whom ail correspondence should he addressed.

[I] B. V. L’vov, Atomic Absorption Spec~rochemi~al An&y&, Engiish Edn. Adam Hiiger, London (1970). [t] [33 [4] [5] [6] [7] [S]

B. V. L’VOV, Pure Appt. Chem. 23, 11 (1970). D. A. KATSKOV and B. V, L’vov, Zh. Prikl. Spektrosk. 15, 783 (1971). G. TORSI and G. TESSARI, Anal. Chem. 45, 1812 (1973). S. L. PAVERI-FONTANA, G. TESSARIand G. TORSI, Anal. Chem. 46, 1032 (1974). G. TORSKand G. TESSARI, Anal. Chem. 47, 839 (1975). G. TORSI and G. TESSARI, And. Chem. 47,842 (1975). G. TESSARI, Sym~s~um on ~~eerrot~errnul Atomizntion in Atomic Absorption S~crromerry. p. 81. Chlum u Trehone, Czechoslovakia (1977). [9] S. L. PAVERI-FONTANA, G. TESSARIand G. TORSI, Ann. Chimica 68, 943 (1978). [lo] G. TESSAR~and G. TORSI, Ann. Chimica 68,967 (1978). [ 1l] G. TORSI and G. TESSARI, Ann. Chimiea 68,991 (1978). [12] W. M. G. T. VAN DEN BROEK and L. DE GALAN, And. Chem. 49,2176 (1977). [i3] C. W. FULLER, Anafysf, 99, 739 (1974). [14] D. J. JOHNSON, B. L. SHARP, T. S. W~s~and R. M. DAGNALL, Anal. Chem. 47, 1234 (197.5). [I 51 J. ZSKAO, Anal. Chem. 50, 1105 (1978). [ 163 H. FALK, Speclrochim. Acta 338, 695 (1978). [17] B. SMETS, Spectrochim. Aclo 3JB, 33 (1980). [18] B. V. L’vov, P. A. BAYUNOV and G. N. RYABCHUK, Spectrochim. Acta 368,397 (1981). [i9] J. A. HOLCOMBE, Spectrochim. Acta, 38B, 609 (1983). [20] G. TE~~ARI and G. TORSI, Anal. Chem. 5X,2039 (1979). 1211 J. ZSAKO, Anal. Chem. §1,2040 (1979). 1287

1288

c. L. CIIAKHAUAHII VIul.

Later, VAN DEN BROEK et uI. [ 121 expressed the transient number of analyte atoms in a graphite furnace as the convolution integral of two functions-a supply function and a removal function. They concluded that, (1) the supply function is determined by the heating of the furnace and by the frequency factor and activation energy of the dominant analyte release process; (2) the removal function is determined either by convection or by diffusion, which is related to the convective flow rate and the furnace volume or to the diffusion coefficient and the furnace length, respectively. Rcccnt cxpcrimcntal studies by HOLCOMW ZI ul. [22 241011 the atomization process and the spatial distribution of gaseous analyte atoms within a CRAM graphite furnace have revealed the complexity of the processes involved in atom formation and loss. The authors [22-241 have concluded as follows. A uniform, free atom distribution within a furnace atomizer does not exist at all times during atomization of a number of metals, and the degree of nonuniformity appears to be (1) element-dependent, (2) affected by the chemical nature of the surface, and (3) dependent on mass transport processes. Diffusional processes alone cannot account for the concentration gradients observed in all cases and, whereas gas phase reactions may influence the spatial distribution, these reactions cannot justify many of the observed signals. For many elements studied, the desorption kinetics appear to govern the absorbance-time profiles, including the falling edge of the signal. This paper presents an alternative, simplified model of the atomization based on simplifying assumptions specified in the following section. The proposed model describes the characteristics of a transient atomic absorption pulse under linear thermal perturbation in an electrothermal graphite tube furnace. The effect of the major parameters on the shape., position, and height of the atomic absorption pulse under the postulated conditions has been theoretically investigated using a numerical integration method. The validity of the theoretical prediction based on this model of the atomization process has been tested experimentally and the results are presented in this paper. 2. GENERAL EXPRESSION 2.1. The model

In the simplified model considered here, the following assumptions have been made. (a) The atoms are dispersed on an extremely small area at the middle part of the graphite tube surface with a monolayer distribution. (b) The atomic release from the graphite surface-gas interface is irreversible, i.e. no re-deposition of the released atomic vapour is allowed. (c) Atom distribution in the gas phase over each transverse cross-sectional plane is uniform. (d) The loss of atoms from the analysis volume is diffusion-controlled (Fick’s first law) with the density gradient falling linearly from a maximum at the middle part of the graphite tube to zero at the ends, i.e. the loss of atoms by convection and expulsion is not considered in the present study. (e) There is no temperature gradient along the length of the graphite tube and no temperature difference between the tube-surface and the gas-phase. (f) The effective path length is equal to the graphite tube length. (g) Temperature-dependence of the atomic partition function is ignored. (h) Absorbance is proportional to the number ofanalyte atoms in the analysis volume (the latter is equal to the cross-sectional area of the incident beam of radiation times the effective path length). The major species inside the graphite tube furnace during the atomization cycle include carbon (graphite), inert purge gas (usually argon), and analyte species (usually in the form of metal oxides or metal halides). The large quantities of carbon (graphite) and argon present relative to the analyte atoms are essentially unchanged by the atomization process. Therefore, it is reasonable to assume that the atomization reaction is a first-order or a pseudo-first-order rate process for atom formation. Furthermore, if it is assumed that the rate of atom loss is diffusion-controlled (assumption (d) above), the atom loss is a first-order rate process provided that the distribution of the gaseous atoms along the length of the graphite tube [22] J. A. HOLCOMBE, G. D. RAYSON and N. AKERLIND, JR, Spectrochim. Acta 37B. 319 (1982). [23] G. D. RAYSON and J. A. HOLCOMBE, Anal. Chim. Acra 136,249 (1982). [24] G. D. RAYSON and J. A. HOLCOMBE,Sprcrrochim. Acra, Part B, in press.

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ETA-AAS: computer modelling of atorqization

furnace results in a linearly-decreased concentration gradient from the middle part of the graphite tube furnace to the ends. The atom loss process will be treated as a first-order rate process. With the above simplifying ~sum~tions, the time-de~ndence of the atom popuIation in a graphite tube furnace can be described by a first-order or a pseudo-first-order consecutive rate process as follows: k,(T)

X-Y-Z

k,(T)

(1)

where X is the pre-atomi~tion analyte species, Y is the absorbing analytc atomic vapour formed; 2 is the analyte species that is lost from the analysis volume, k, (7”)and k,(T) are respectively the first-order rate constants for the formation and the loss of the observable atomic species, Y as a function of temperature, T. Since the temperature of a pulse-heated graphite furnace is a function of time, t, the expressions k, (T) and k,(T), may be described as kl (T(~))and k~(T(~)), respectively. One uses the k, (t)and k,(t) instead of k~(~(~))and k~(T(~)) in the following text to simplify notation for the rate constants. Since the process involved in Eqn (1) takes place in a constant-volume graphite tube furnace, the amount of change for each species can be described by a set of differential equations as follows:

dX,_ dt

-k,(t)mX,

s= &

k i’t-2’t (t) X

k (t) Y

dZ = k,(t)* y, dt wherek,(t)andk,(t)haveunitsofs-*, and X,, r;, and Z, have units of number of atoms. Since the volume of the graphite tube furnace is constant, the unit of x in terms of number of atoms allows its number density at any time to be readily determined. When the absorbing analyte species, Y,is the only observable species, the first and the second terms on the right-hand side of Eqn (3) are the rate of atom formation and the rate of atom loss, respectively. Rate of the gaseous atom formation. For a first-order (or a pseudo-first-order) atom formation reaction, the rate of formation of the gaseous atomic analyte species, Y,is equal to the product of the rate constant, k,(t), and the amount of the preatomization species, X,, at time t. The Arrhenius equation, when applied to this case, results in the equality given by Eqn (9. Rate of atom formation = k,(t) =X, = A’+e -E~IRrw.X,

(5)

where E,, A’, R, and Tare the activation energy of the rate-determining step, frequency factor, gas constant, and absolute temperature, respectively. Rate of the gaseous atorn loss. For a first-order loss process, the rate of atom loss can be expressed as the product of the rate constant, k2(t), and the amount of the gaseous analyte atomic species, x, in the analysis volume at time t. Rate of atom loss = - k2(t). x.

(6)

Although the loss of atoms from the analysis volume can be attributed to five different processes [ZS], only the diffusion process will be considered for the sake of simplicity. The rate of atom loss by diffusion is given by the folIowing equation [ 1, 261: Rate of atom loss = - D.2.S where D is the diffusion coefficient; dp,/dx [25]

(7)

is the analyte atom density gradient in the

R. E. STURGEONand C. L. CHAKRABARTI,Prog. Anal. Atom. Spectrosc. 1,

1 (1978).

1290

C. L. CHAKRABARTIet al.

direction of diffusion at time c along the length of the graphite tube which is aligned parallel to the x-axis; S is the transverse cross-sectional area of an imaginary plane perpendicular to the x-axis in the graphite tube furnace across which diffusion occurs. Assuming that the atomic vapour is lost by diffusion symmetri~lly from the middle part of the graphite tube towards its ends, and that the density of atomic vapour falls linearly from P,X= ‘at the middle part of the graphite tube (where the sample is deposited and atomized) to zero at a distance x = l/2 away (where 1is the length of graphite tube), one can derive Eqn (8). Ptx=0 -&=“2

dp, -= dx

p;=” (8)

=r’

i/2

Since the density falls linearly, the average density, &, inside the graphite tube furnace at time t is

r, p,“=o p,=-_,=,-*

(9)

Thus,

dp, W, dx=1/2

2(H,/v) =-=m== 112

41:

4X

(10)

where V = S*l is the volume of the graphite tube furnace. Since the diffusion of atomic vapour is from the middle part of the graphite tube to both ends, substitution of Z(dp,/dx) into Eqn (7) gives Rate of atom loss = -E 12 * YI’

(11)

The relationship between the diffusion coe~~ent, D, of the diffusing gas, and the temperature has been empirically established [l, p. 287; 261 as follows. D==D,

;

n (12)

0 0

where Do is the diffusion coefficient at S.T.P. (To = 273 K), and the value of n (to be called gas combination factor) varies from 1.5 to 2.0 for various combinations of gases [I, p. 2041. Substituting Eqn 12 into 11, one gets ‘, Y,.

(13)

From Eqns (6) and (13), one gets the rate constant of the atom loss, k&): k

(14)

2

Finally, the kinetic expressions of Eqns (2~(4) can be written respectively as: dX, - _A’.e dt

-E/RW.X . t

dYt=pe-“,R”t’ . . x -800 dt

(15)

(16) (17)

As mentioned earlier, the temperature of a pulse-heated graphite furnace is a function of time. When the temperature of the furnace is increased at a certain heating rate, a, (from the charring temperature, T,, to a preset final atomization temperature, T,) and then held [26] G. I. NKOLAEV and V. B. ALEKSOVSKH,Zh. Tekh. Fiz. 34,753 (1964).

ETA-AAS: computer modeifing of atomization

1291

constant at Tf, the relationship between the temperature and the time can be expressed as follows: t


t>T’-T’ a

T(t)

= T, + ut

T(r)

= Tf.

Since the rate constants are temperature-dependent and the temperature is a function of time, the above differentiai Eqns (15~(17) are too imprinted to be solved aM]yti~Ily. A numerical integration method has therefore been used to study the change in the number of atoms inside the graphite tube furnace as a function of time. 3.

EXPERIMENTAL

methods are available for numerical iterative integration of a first-order differential equation such as the consecutive process of atom formation in and loss from graphite tube furnace, e.g. see Eqns (1S)-( 17).A multi-step integration, and Adams-Bashforth method, has been applied in the present studies to the numerical solution of differential equations such as Various

[27,28-J,

dX, dt

=

-

A’ *e - URW). X, =f(t,

x,).

The algorithm of the Adams-Bashforth method is commonly expressed as

X r+41

=

X,+~[55/(t,X,)-59/(r-Al,X,_,d -I- 37fO - 21% Xt-tbt)

-9fft

-

3A4 X8_,,)].

(21)

If the solution X has been computed for the four points, t, t - At, t - Zht, and t - 3At,then Eqn (2l)can be used to compute X,+dl for different temperatures, since the temperature is a function of time. However, the initial amount of X is known at t = 0, i.e. XIsO = X0. In order to obtain enough values to initiate the multi-step iterative integration, a single-step, fourth-order, Runge4Cutta method has been employed to evaluate the first three values according to the following equation: X ,+A(=

X,+~(So+2S,+2S,+S,)

where SO=f(t* X,) At.&,

s, =f t+q, x,+--ys* =f

(

t+$

At*&

x,+-j--

S, = f(t -I- At, X, + Ar*S,).

After that, the solution of X is done systematically by the A&ms-Bashforth method. In practice, Bqn (21)is not used by itself. Instead, it is used as a predictor and then another equation is used as a corrector. The corrector usually employed with Eqn (21)uses the Adams-Moulton method as follows: X ~+nt=X,+~[9f(f+At,X:+,,l+l9f(r,X,) -Sf(t-At,

X,_,,)+f(t-2Ar,

X,-J].

(23)

Here X:+br is the tentative value of X ,+&,computed from Eqn (21).Thus, Eqn (21)predicts a tentative

[27] A. C. NORRIS,

Com~u~aIia~~ Chemistry. John Wiley, New York (1981). [2g‘J C. F. GERALD, Applied Numetlcaf Anafysis, 2nd &In. Addison-Wesley, Don Miffs, Ontario, Canada (1978).

1292

C. L.

CHAKRABARTIet

at.

value of X, +*, , and Eqn (23) computes this X value more accurately. The combination of the two equations results in a predictor-corrector scheme which has been used for the present studies. 3.2. Apparatus and reagents A Honeywell CP-Vcomputer which empioys two Sigma 9central processing units was used to do the numerical integration. The capacitive-discharge-heated graphite furnace atomic absorption spectrometer used in this study is essentially the same as described in earlier publications [29-321. Only a larger capacitor bank of 2.OF total capacitance was used in place of the 0.15 F bank of the previous papers [29-321. Tubes of different length were bored from an anisotropic pyrolytic graphite block in the same way as mentioned in the previous paper [29], i.e. the longitudinal axis of the graphite tube is ~r~ndicu~r to the c-axis, and the dimensions of the tubes were: 3.0 mm i.d. and 6.0 mm o.d. The length of the tube was varied from 10 to 25 mm. For studies of the peak-height absorbance as function of the heating rate, 15 mm-long tubes were used. A 30 s charring cycle was employed prior to atomization at preset final temperatures. The internal gas flow was interrupted during the entire duration of the atomization cycle. Heating rates were calculated from oscilloscope traces of an automatic optical pyrometer (Ircon series 1100, Ircon Inc., Niles, Ill., U.S.A.)from its output voltage by m~suring the time interval between the triggering of the capacitor bank and the attainment of the final temperature. The temperature change during this time is the difference between the final and the charring temperature. Low charring temperatures (below WC) could not be measured by the above pyrometer. However, another pyrometer (Ircon series 300, Ircon Inc., Niles, III., U.S.A.) was used to measure low charring temperatures. Chemicals used were of ACS reagent grade purity. A stock solution confining 1000 rg/mL of each metal studied (except MO) was prepared from pure metal or its oxide by dissolving it in IJLTREX nitric acid (J. T. Baker Chemical Co.). The stock solution of MO was prepared by dissolving pure MOO, in 407, (v/v) ammonia aqueous solution. Each standard solution was diluted to the desired concentration with ultrapure water obtained direct from a Milli-Q2 water system (Millipore Corporation) immediately prior to its use. Argon gas of 99.995 “/, purity was used as the internal purge gas and the external sheath gas.

4. RESULTS AND DISCUSSION Theoretical studies on the relationship between the atom population and time The shape of atom population-time curves can be characterized by the atom-popuiation

4.1.

maximum (peak height), the position of the maximum (time at the peak height), and the haifwidth (the pulse width at half of the peak height). However, it should be noted that the simulated results of the present studies serve only as a model, the validity of which must be tested by experimental verification. 4.1.1. E&t of the temperature on the atom population-time curves. In pulse-heated graphite furnace atomic absorption spectrometry, the temperature of the furnace at a given instant during the atomization cycle is described by Eqns (18)and (19). It is well known (also, Eqns (4) and (15) show) that the rate constants for atom formation and loss are temperaturedependent. Therefore, the maximum rate constants for atom formation and loss are determined by the final (highest) atomization temperature. However, the role of the heating rate to the release of maximum atom population is clearly seen if the degree of change in the rate constant for atom formation and loss with temperature is compared. Figure 1 shows the effect of the heating rate on the shape of atom population-time curves. The atom population-time curves A, B, C, D and Eare simulated under the temperature-time profiles a, 6, c, d and e, respectively. As can be seen from Fig. 1, an increase in the heating rate increases the maximum atom population and the area of the atom population-time curve which lies in the constant temperature region of the final atomization temperature. Figure 2 shows the effect of the final atomi~tion tem~rature on the atom popu[29] C. L. CHAKRABARTI,H. A. HAMED,C. C. WAN, W. C. LI, P. C. BERTELS, D. C. GREGOIRE~~~ S. LEE, Anal. Chem. 52, 167 (1980). [30] C. L. CHAKRABARTI, C. C. WAN, H. A. HAMED~~~ P. C. BERTELS, Anal. Chem. 53,444 (1981). [31] C. L. CHAKRABARTI, C. C. WAN, R. J. TESKEY, S. El.CHANG, H. A. HAMEDand P. C. BERTELS,Specrrochim. Acra 368,427 (1981). [32] C. L. CHAKRABARTI, C. C. WAN, H. A. HAMEDand P. C. BERTELS, Nature 288,246 (1980).

ETA-AA% computer modelling of at&%Mion

TIME /

1293

ms

Fig. 1. Effect of the heating rate, a, on the shape of the atom population-time curves. General parameters: E, = 480 kJ mol- *, A’ = l.O~lO’~s-‘, r,=7OOK. r,=ZSOOK, I=2.0cm, D, =O.lOcm*s-‘,n= i.S.Heatingrate:(a) IOOKms- r,(b) IOKms-‘,(c)5Kms-‘,(d)2.5Kms-‘, (e) 1.5 Kms-t.

TIME I ms Fig. 2. Effect of the final atomization temperature, ‘f,, on the shape of atom population-time curves. General parameters: E,=480kJmol-L. A’=lO~10’~s-‘, 7’,=7OOK, a=LOKms-‘, 1= 2.0 cm ,o.D = 0 10 cm’s_ ‘, n = 1.8. Final atom&&ion temperature: (a) 2700 K, (b) 2500 K, (c) 2400 K, (d) 2300 K, (e) 2200 K.

lation-time curves at a constant heating rate (5 K ms- ‘)_The atom population curves A, B, C, D and E are simulated under the temperature-time profiles a, b, c, d and e, respectively. No significant increase in the maximum atom population is observed at a final temperature exceeding 2600 K if one uses the present numericai values for the parameters in Eqns (15)--( 17). However, the more rapid fall in the decay part of the atom ~pu~ation-time curve at higher final atomi~tion temperatures reflects the effect of the greater rate of loss of the atoms at higher temperatures on the shape and the area of the atom population-time curves. 4.1.2. Effect of rhe tube length on the atom population-time curves. Figure 3 shows the effect of the tube length on the atom population-time curves when the other parameters are the same as in Fig. 2 but T, = 2500 K. As can be seen from Eqn (14), the rate constant for the loss of the atoms depends on the tube length, Increase in the tube length slows down the rate of atom loss because an increase in the tube length increases the residence time of the atoms, i.e.

1294

C. L. CHAKRABAKTI et al.

TIME /ms

Fig. 3. Effect of the lube length, I, on the shape of the atom ~pulation-time curves. General parameters: ES = 480 kJ mol- ‘, A’ = ~.OX~O’~S-‘,T,=~~~K,T~=~SDOK,~=~.OK~~-’,D, = 0.10cm2s-‘, n = 1.8. Tube length: (A) S.Ocm, (B) 4.0cm, (C) 3.0cm, (D) 2.0cm, (E) LOcm.

decreases the rate constant for atom loss, in the analysis volume, which results in a higher maximum atom population as well as a longer tail in the decay part of the atom population-time curves. 4.1.3. E$ect of dijiision coejicient, D,, gas combination factor, n, frequency factor, A‘, and activation energy, E,, on atom population-time curves. Figures 4-6 present the effect of the diffusion coefficient, the gas combination factor, and the frequency factor, respectively, on atom population-time curves at the same heating conditions. An increase in the diffusion coeficient and the gas combination factor reduces the maximum atom population because of the faster rate of loss of the atoms. On the other hand, a high value for the frequency factor increases the rate of atom formation by increasing the rate constant for atom formation and thereby increases the maximum atom population. Figure 7 shows the effect of the activation energy on the atom population-time curves. Equation (5) shows that for a given temperature, the rate constant for atom formation, k,(t), is governed by both the frequency factor and the activation energy. A very high activation

TIME / ms Fig. 4. EGct of the diffusion coefficient, D,, on the shape of the atom population-time curves. General parameters: E,, = 480 kJmol_‘, A’ = 1.0 x 10” s-‘, r, = 700 K, T, = 2500 K, a I = 2.Ocm, n = 1.8. Diffusion coefficient: (A) 0.05 cm’s_‘, (Bf O.lOcm’s- *+ = S.OKms-‘, (C) 0.20cm2s-‘, (D) 0.4Ocm*s-*.

1295

ETA-AA% computer modelling df atarilization

a60-

TIME / ms

Fig. 5. Effectof the gas combinationfactor, n, on the shapeof the atom population-timecurws. General parameters: E,=48OkJmol-‘, A’=10~lO’*s-‘, T,=7OOK, T,=25OOK, a = S.OKms-I,

I = 2.Ocm. D, = 0.10 cmzs- *. Gas’combination factor: (A) 1.5, (B) 1.8, (C) 2.0.

1200

600

I600

TIME / ms Fig. 6. Effect of the frequency factor, A’, on the shape of the atom population-time curves. General T,=25OOK. a=XOKms-‘, I=2.0cm, D, parameters: E, = 480 kJ mol- ‘, T,=7OOK, = O.lOcm* s-l, n = 1.8. Frequency factor: (A) 1.0 x 10’” s-l, (B) 1.0 x 10” s-‘,(C) 1.0 x 10” s-‘, (D) 5.0 x 10”‘s-I.

energy means a difficult-to-overcome energy barrier to a chemical reaction. The decrease in the height of the atom population-time curve in Fig. 7 indicates that for the given temperature, the higher the activation energy barrier the slower the rate of the atom formation, and hence the lower the maximum atom population. Also, a higher activation energy results in atom population-time curves which are shallower and broader. 4.1.4. The e&t ofthe heating rate and the tube length on the maximum atom population, Y fix, with the activation energy as a parameter. In Eqns (15~( 17), the parameters: diffusion coefficient, gas combination factor, frequency factor, and activation energy, are all natural properties of an analyte species under given experimental conditions. As can be seen from Eqns (15~(17), only two parameters, the heating rate and the tube length, can be varied by change in the instrumental setting. The effect of the heating rate and the tube length on the

12%

C. L. CHAKRABART~ et al.

TIME/ ms Fig. 7. Effect of the activation energy, E,. on the shape of the atom population-time curves. General T,=2SOOK, a=5.OKms-‘, 1=2.Ocm, D, parameters: A’ = 1.0 x 10” s- ‘, T,=?tMK, = O.fOcmzs-s, n = l.8. Activation energy: (A) 300 bfmol-‘, (B) 400 kJmol-‘, (C) 480kJmol-‘, (D) SOOkJmol-‘, (E) 530kJmol’~‘.

maximum atom population can be simulated by assuming that al the other natural properties except the activation energy remain the same, Figure 8 shows the effect af the heating rate on the maximum atom population. Figure 8 indicates that with the increasing heating rate, the smaller the activation energy, the greater the increase in the maximum atom population. Because of the relatively low atomization temperatures used in Fig. 8, no significant increase is observed for the high activation energy systems. Figure 9 presents the effect of heating rates on a system having a high activation energy (630 kJ moi- ’ ), with the final atomization temperature as a parameter. A comparison of the curves in groups A and B in Fig. 9 indicates the following, As expected, a higher heating rate shifts the atom population.-time curve to a lower time. Figure 9 shows that the maximum in

1.00’

0.60

0.40

0th I.

3

3

7

HEATING

10

30

50

70

100

RATE / K ins-’

Fig. 8. Effect of the heating rate on the maximum atom population, W,,, with activation energy as a parameter. General parameters: A’ = I.0 x 10” S-I, T, = 700 K, T, = 2500 K, I = 2.0cm, D, = O.i0cm*s-‘, tl = f.8, Activation energy: +400 kJmol_‘; A-450 kJmol_‘; e-480 kJmof_‘: I5OOkJmol-‘; 8-53OkJmol-‘.

ETA-AA%

computermodellingof atetiitation

1297

0.60‘

0.60

B-I

1 X0

0.40.

0.20

-

u 0

TIME / mr Fig. 9. Effect of the heatingrate and the finalatomizationtemperatureon a highactivationenergy system. General parameters: E, = 630 kJ mol- I, A’= l.O~lO’~s-~, r,=7OOK, 1=2.0cm, D, = O.lOd s-l, n = 1.8. Curves A: 01s 50 K ms-‘; r, = 3300 K (for A-l), 3000 K (for A-2), 2800 K (for A-3). Curves B: 01= 5.0 K ms- ‘; T, = 3300 K (for B-l), 3000 K (for B-2), 2800 K (for B-3).

atom population is sensitive to the heating rate only for the highest final atomization temperature (3300 K); it is not sensitive to the heating rate for lower final atomization temperatures. Since the rate constant for atom loss (by diffusion alone) is not as sensitive to the temperature change as the rate constant for atom formation, the effect of the final atomization temperature (and the heating rate) on a high activation energy system can be predicted from the Arrhenius equation (Eqn (5)), which states that the rate constant for atom formation is proportional to exp( - &/T(t)). For a given (high) activation energy, only a high enough temperature (especially when such a high temperature is attained at a high heating rate) can maximize the atom formation and minimize the atom loss (by diffusion alone), and hence can give the highest ratio of x/X,. A faster heating rate yields a shorter time required to reach a high atomization temperature and also a shorter time during which atom loss can occur by diffusion alone. Therefore, a system having a high value of E, will be sensitive to both a high heating rate and a high final atomization temperature only when these two parameters are coupled, i.e. only when a high heating rate is used to yield a high atomization temperature. Also, the highest value for the atom-population maximum for such a system is attainable only when a high heating rate is used to yield a high atomization temperature. In Fig. 9 curves A, the coupling of the atomization temperature of 3300 K and the heating rate of 50 K ms- ’ yields a value of x/X, * 70 7; against a maximum possible value of 100% that is theoretically possible. Absence of coupling of high heating rates and high atomization temperatures will make such a system not very sensitive to change in these two parameters. Because of the upper temperature constraints of the pyrolytic graphite used for the construction of the furnace, an atomization temperature higher than 3300 K was not practical and was therefore not used. As has been mentioned earlier, an alternative method of increasing the maximum atom population is to reduce the rate of atom loss by increasing the tube length. Figure 10 shows the effect of variation in the tube length on the maximum atom population with the activation energy as a parameter. A considerable enhancement in the maximum atom population for the high activation energy systems is observed. 4.2. Experimental study on the effects of the heating rate and the tube length 4.2.1. E&t ofthe heating rate. Figure 11 presents experimental results of the effect of the heating rate. Comparison of the simulated and the experimental plot (Figs 8 and 11,

C. L. CHAKRABARTI et al.

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Fig. 10. Effect of the tube length on the maximum atom population, Y_, with activation energy as a parameter. General parameters: A’ = 1.0 x 10” s- I, T, = 700 K, i”, = 2500 K, a = 5.0 K ms-I, Do = 0.10 cm2 s- I, n = 1.8. Activation energy: o-400 Id mol- ‘; A-450 kJ mol- ‘; +W kJ mol- ‘; V500 kJmol_‘; M-530 kJmol_‘.

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Peak-height absorbance as a function of the heating rate. Tube length = 15 mm. id. = 3 mm. DMn, 1. = 279.5 nm, T, = 880 K, T, = 2800 K. A-Ni, I = 232.0 nm, T,= 1020IL T, = 3200 K. O-Cd, 1 = 228.8 nm, T, = 470 K, TI = 1800 K. A-Co, I = 240.7 nm, T,= WOK T, = 3000 K. l-Pb, 1= 283.3 nm, T, = 770 K, r, = 2470 K. O-MO, 1. = 313.3 nm, r, = 1020 K, T, = 3300 K.

Fig. 11.

ETA-AA%computer modelling of atontization

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respectively) of the peak-height absorbance as a function of the heating rate shows a reasonable agreement between the effect predicted by the model and what is observed experimentally. The heating rate values represent the average heating rate measured over the temperature range from the final charring temperature to the preset final atorn~tion temperature. For the elements reported in this paper (see Fig. 11) the above maximum occurs at a heating rate between 15 and 35 K ms- *. There is also a general trend that high-volatility elements are more sensitive to changes in the heating rate than low- or medium-volatility elements, However, the decrease in peak-height absorbance observed for Cd and Pb at high heating rates is not predictable from the model. Both elements exhibit a very small range of heating rates over which the ok-height absorbance remains constant. This is especially true of cadmium, the low appearance temperature [25] of which occurs at a time when the expansion of the internal purge gas inside the furnace is very rapid [ 12,291 with consequent partial expulsion of gaseous cadmium species. Expulsion apparently plays an important role in the atom removal process for this element. At heating rate c 15 Kms- ‘, the increase in atom formation exceeds the increase in expulsion as more and more of the analyte vaporizes at higher tem~ratures. Eventually, expulsion begins to dominate as the heating rate continues to increase. For cadmium this occurs at approximately 17 K ms- ‘, but seems to reach a steady state beyond 35 K ms- ‘, At this point most of the cadmium absorption pulse lies in the isothermal region and therefore an insignificant fraction of cadmium exists in the vapour phase during the immediately preceding period of rapid expansion of the internal purge gas. Expulsion is less important for lead, but the lead absorption pulse also moves into the isothermal region, although more slowly than the cadmium absorption pulse, as the heating rate is increased. Therefore, decrease in the lead absorbance does not level off until the heating rate exceeds 55 K ms- I. Expulsion does not seem to be important for removal of medium- and low-volatility elements; this may be due to their higher appearance temperatures, the latter being much closer to the constant temperature attained during the atomi~tion cycle. 4.2.2. E$ect o~r~e~u~~uc~tube length, Figure 12 shows the effect of the furnace tube length on the peak-height absorbance. Experimental results for cadmium and lead agree well with

0.80’

0.60.

TUBE

LENGTH /cm

Fig. 12. Peak-height absorbance as a function of the tube length. Heating rate = 17 Kms-‘, i.d. = 3 mm. O-MIX, d = 279.5nm, 7’,= 880 K, T, = 2800 K. DCd, 1= 228.8nm, T, = 470 K, T, = 18OOK.o-Pb,I = 283.3 nm, T, = 770K,T,= 2470K.A-Co,1= 240.7 nm, T, = 1020K.T, =3OOOK.

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CHAKRABARTIet al.

the model. However, cobalt is relatively insensitive to change in the tube length. The peakheight absorbance for manganese shows a sharp increase with increasing tube length from 10 to 15 mm, but shows less change with longer tube lengths. The model predicts that, in general, the tube length will have a greater effect than the heating rate on the peak-height absorbance for elements which are relatively more difficult to atomize. The experimental results suggest that other processes, e.g. condensation, formation of thermally stable compounds, etc., in which an increase in the tube length will increase the amount of atom loss have not been taken into account in the mathematical formulation of the model. 5. CONCLUSIONS High heating rates enhance the peak-height absorbance by shifting the atomic absorption pulse towards the constant temperature region. Once the peak of the absorption pulse moves towards the constant temperature region with high heating rates, the peak-height absorbance reaches a maximum value, and no significant increase is observed at higher heating rates. For a given element, the heating rate required to achieve this maximum value is determined by the final temperature and the kinetics of atom formation and loss. For a relatively involatile element, i.e. an element having a relatively high E, value, because the appearance temperature is close to the final temperature, the entire analyte absorption pulse appears close to the isothermal region, and higher heating rates will have much less effect (than with relatively volatile elements) in shifting the analyte absorption pulse towards the isothermal region. Since loss by expulsion is less important for relatively involatile elements, high heating rates will not produce large enhancement in their peak-height absorbance. Both the experimental data and the results from the simple model based on first-order consecutive rate processes agree qualitatively. The importance of furnace length to the maximum atom population has been demonstrated. The results indicate that atom loss occurs by the time the maximum atom population is achieved. Long tubes as well as high heating rates which maximize the k, (C)/&~(C)ratio result in the greatest peak-height absorbance. More quantitative comparisons between theory and experiment are difficult to make because of uncertainties in the values of the frequency factors and activation energies, and also in the gas phase temperature during rapid change in the furnace wall temperature. Admittedly, this simple theoretical model has some limitations which stem largely from its simplifying assumptions and oversimplified treatment of the atom loss processes. However, such limitations do not seriously impair its usefulness, which lies not so much in its predictive power (which is considerable), as in its ability to offer a theoretical framework for understanding the effect of various experimental parameters on the analytical sensitivity. The model presented in this work and the above conclusions are not valid for the case of analyte atoms vaporized from an excess of salt matrix since in this case the loss by expulsion can become very high depending on the nature’and the amount of matrix. Increasing the analysis volume through use of longer tubes is a possible method of reducing the expulsion loss. Acknowledgements-The authors are grateful to Dr M. RAHAMAN and Dr J. S. WRIGHT for helpful discussionand to the Natural Sciencesand Engineering Research Council of Canada for financial support of this research project.