- Email: [email protected]
Condensed phases in flame spectrometry: an equilibrium model of the acetylene flame
0%4-8547/82/0403wA39so3.00/0 Petgamon Press Ltd.
Condensed
phases in flame spectrometry: An equilibrium of the acetylene flame ho
LANG and
model
VIKTOR DOBAI_
Institute of Geology and Geotechnique, Czechoslovak academy of Sciences, V HoleSoviCkzich 41, 182 09 Prague 8. Czechoslovakia and
GUSTAV
SEBOR,
JR.
Department of Petroleum Technology, Institute of Chemical Technology, 116 28 Prague 6. Czechoslovakia (Received 22 May 1981; in revised form 29 September 1981) Abstract-An equilibrium model for a wet acetylene flame has been used for the calculation of the distribution of carbon, aluminium and silicon between gaseous and condensed phases. Calculated values of free atom vapour fractions are in agreement with published experimental data. It is thermodynamically probable that the formation of condensed phases limits atom vapour formation. There is a relationship between the position of an element in the Periodic Table and the nature of the condensed phase.
1. INTR0Duflt0~ RECENTLY.RUBE&A has emphasized the importance of condensed phases as an interference source of some elements [l, 21. We have shown [3] using the example of the equilibrium distribution of titanium and zirconium among the different phases in the acetylene flame that in the atomizing medium some transition metals can be present in a condensed phase and the qualitative and quantitative description of the distribution depends on the !Iame conditions. The presence of condensed phases can also be significant in the atomization of metals from other groups of the Periodic Table. Also carbon itself is potentially a significant component of the condensed phases. The purpose of this work is the determination by calculation of the condensed phases of carbon, aluminium, and silicon in the atomizing medium of an acetylene flame. Our works start from the results of CHESTER et al. [4-6], RASMUSONet al. [73 and recently published works of WITIENBERG et al. [8] and ASAHINA et al. [9]. A more complete description of the condensed flame system is presented in this paper giving results which correspond satisfactorily with published experimental data. 2. THEORETICAL Equilibrium model assumptions From the point of view of analyte and sample matrix quality the objective choice of optimum operating conditions with a computer would mean definite progress in flame spectrometry. For this it is necessary to get as perfect an atomization model as 2.1
[l] I. RIJBESKA,Anal. Chem. 48, 1640 (1976). [21 I. RUBE~KA,Chem. Anal. ( Wurszuw) 22,403 (1977). [31 V. DOBAL, I. LANGand G. SEBOR,Chem. listy 74, 1233 (1980). 141J. E. CHIZSTER, R. M. DAGNALLand M. R. G. TAYLOR,Anolysl 95.702 (1970). 151J. E. CHESTER,R. M. DAGNALLand M. R. G. TAYLOR,Anal. Ckim. Actn 51,95 (1970). [61 R. M. DAGNALLand M. R. G. TAYLOR,Spectrosc. Lett. 4, 147 (1971). 171 J. 0. RASMUSON,V. A. FASSELand R. N. KNISELEY,Spectrochim. Acra ZSB,365 (1973); 31B. 229 (1976). @I G. K. WITTENBERG, D. V. HAUN and M. L. PARSONS,Appl. Spectrosc. 33,626 (1979). [91 T. ASAHINA,M. KOSAKAand I. SHIROYANAGI, Koon Gakkaishi 5, 35 (1979). 309
310
I.
possible. constants. the
A kinetic Although
et 121.
model requires of necessity the knowledge of relevant ratr it is known that the flame is generally in a non-equilibrium state
use of classical thermodynamics
at present.
LANG
Both the quantity
is the only approach
and the accuracy
which
of tabulated
also important factors in an equilibrium approach. The existence of a local steady state in the measured assumption of the equilibrium model. Following existence of a local steady state in the interzonal
is directly
thermochemical
applicable data are
zone of the flame
is an
ALKEMADE[IO] we assume region of an analytical
the acetylene
We believe that the analytical flame is realized in terms of the stability and reproducibility of the flame spectrometric measurements. These flame characteristics imply the existence of a local steady state. An additional reason which justifies a study of the possibilities of classical thermodynamics is the fact that a general non-equilibrium approach requires the knowledge of the thermodynamical brunch of solution [ 1I]. The true dynamic system would be characterized by rising and continually transforming superatomic and supermolecular structures which cannot be described without introducing time in the sense of a system history [I I]. Such a major dependence of signal fluctuations with time has never been observed in the practice of flame spectrometry. We support the opinion of JENKINSand SUGDEN[12] that the equilibrium state is a very valuable concept; the effect of non-equilibrium phenomena on the over-all composition of flame gas mixtures is generally small because the fraction of the total energy of the gas that resides in excited states or in free radicals is very small. Of course, effects such as heat losses to the burner, air entrainment, or the addition of significant amounts of liquid droplets may have a large effect on the composition, either directly or through temperature effects. These effects should either be avoided or otherwise taken into account in accurate calculation of the equilibrium compositions. This problem should be gradually solved. In the present paper we shall be concerned with the effect of the solvent first of all. Considering high temperature and normal pressures as further assumptions of the equilibrium model, an ideal behaviour of the gaseous phase may be assumed. The gas phase can be considered as mixed. So as not to introduce activity coefficients it is better to treat the condensed phases as being immiscible. flame.
2.2 Calculation of local equilibrium in the acetylene flame A concept of free energy minimization has been used for the calculation of the local equilibrium of flame gases and metal species. The advantages of this concept have been discussed by PARSONSet al. 181. We have utilized the method of HOLUB and VO~JKA[133 for the calculation of the equilibrium composition in homogeneous systems, and for heterogeneous systems the method of STOREY and VAN ZE~~EREN [14]. Data came from the JANAF Thermochemical Tables [15]. Equilibria were calculated using the programmable calculator WANG 2200. The results of the calculations of homogeneous equilibria were used as the initial data for the calculation of the heterogeneous systems. 3. RESULTS AND DWUSSION 3.1 Calculation of C(s) in wet acetylene flame The flame components considered in the calculations were taken from the JANAF [IO] C. Th. J. ALKEMADE and R. HERMANN, Fundamentals of Analytical Hame Spectroscopy, p. 30.Hilger, Bristol (1979). [I 11 G. NICOLIS and I. PRIGOGINE, Self-Organization in Nonequilibrium Systems. Wiley, New York (1977). [I21 D. R. JENKINS and T. M. SUGDEN, Radicals and Molecules in Flame Gases, in: Flame Emission and Atomic Absorption Spectrometry (Edited by J. A. DEAN and Th. C. RAINS), Vol. I. p. 170. Dekker, New York (1969). 1131 R. HOLULIand P. VOP~KA. Chemical Equilibrium of Gaseous Systems. Academia, Prague (1975). [I41 S. H. STOREY and F. VAN ZECCEREN, Can. I Chem. Eng. 42.54 (1964). [IS] D. R. STULL and H. PROPHET, JANAF Thermochemical Tables, 2nd Ed. NSRDS-NBS Publ. 37. Washington (197 I).
Condensed phases in flame spectrometry
311
the same as those considered in Refs. 3-9. To characterize the initial we make use of the C/O ratio. (This atomic ratio is usually used to characterize an initial mixture of industrial flames [16] and also in the petrochemical production of soot by thermal decomposition of the acetylene). it may be confirmed that it is only this elemental atomic ratio which determines the initial composition of the acetylene flame. In the cafcufations [7,8], the authors have taken both the amount of solvent and the aspiration efficiency to be constant (water; 2 and 4 ml min-‘, resp.; 10%). Thus the composition of the initial mixture is characterized by the ratio p = oxidant/CzH2. It is however not possible with these calculations to easily take into account a change of solvent [7,8]. One of the main purposes of our work was a general study of the enrichment of the flame by carbon with the use of organic solvents. By making use of the p ratio one can omit the influence of the solvent upon the composition of the initial mixture. In this case, the theoretical and the practical models cannot be consistent when considering the absolute difference in the initial states. This is clear from the values in Table 1 and also from the results of ASAHINA et ni. [9], who have shown that there are substantial differences between the composition of wet and dry flames. The number of moles of elements entering the wet flame per minute is listed in Table 2. Taking into account the major ~mponents in the condensed phase of the acetylene flame, calculation shows that C(s), i.e. carbon in the solid phase, is a single but very important component, Fie 1 demonstrates the dependence of the equiliTables
and
are
composition
Table 1. Composition of the mixture entering into the burner Atomizing
Flow
(Lmin-1)
solution
efficiency
oxidant
C2H2 2.3
11.3 =
water b
2.3
11.3
Water
2.3
11.3
Xylem
5.2
10.4 e
5.2
a.2
C/O
P
(0) 20
0.77
1.0 =
10
0.86
1.0
10
1.07
1.0
water
10
0.94
2.0
water
20
1.10
1.6
d
‘Air. ‘5 ml min-‘. ‘Assuming OI as oxidant. d2 ml min-‘. ‘Nitrous oxide.
Table 2. Initial composition of a wet acetylene flame mole
C/O
d
per
minute
C
H
N
0
0.20536
0.31656
0.79700
0.26748
1.11356
0.86
0.20536
0.26396
0.79700
0.23968
1.05796
1.07
0.22656
0.21066
0.79700
0.21188
1.00766
0.46428
0.57548
0.92857
3.49209
1.50405
3.46428
0.57548
0.73214
'3.42167
1.30762
0.11
0.94
b
1.19
“C&-air
flame.
*C2HrN20 ffame. [I61
K. H. HOMANN
e-
and H. G. WAGNER, Proc. Roy. Sot. 307A.
141 (1%8).
I.
312
LANGYf
a!.
/..A?---
/-& f/A .
d
Fig. 1. The dependence of the equilibrium concentration C(s) on the Cl0 ratio for chosen temperatures of acetylene fiame: 0, 2400 K; 0.2800 K; A, 3000 K.
brium concentration C(s) (number of mole C(s) per mole of the equilibrium mixture) upon different C/O ratios at temperatures 2400, 2800 and 3000 K, resp. The “limit of carbon formation*’ [ 161 at temperatures in a wet analytical acetylene flame (2400 K for air-acetylene, 28OO-3000K for nitrous oxide-acetylene) can be defined by the inflexion point of the curves. When the critical ratio C/O = 1 is reached, the concentration of C(s) in the equilibrium mixture increases by several orders. Thus the calculations confirm that it is quite reasonable to look for a relationship between the concentration C(s) and the high atomizing efficiency of the fuel-rich frame, as proposed by DE GALAN and SAMAEY [17]. 3.2 Condensed phases of aluminium in wet acetylene flames The following species of aluminium were included in the flame components: Al(g), Al(I), Al+(g), AU&, AlHW, AIN( AlNb), fob), AlO+( AIOI-W, AIOH+(g), AlOH-( AlO&), AlHOzW Al,O(d, Al2O+W, Al&(g), AlzOz+W, ~203th 0, HAlO( Calculation co&med that the ionization of aluminium is suppressed by the presence of 2000 pgK per milliliter of solution. As in Refs. 3-9, the flow rate of analyte and matrix elements into the ffame was calculated from the concentration of the element in the aspirated solution, assuming a given flow rate and an aspiration efficiency of 10%. The number of moles of aIuminium (3.706 x lo*) corresponds to the nebuli~tion of an aqueous solution of 100 pg Al ml-’ at a rate of 5 ml min-’ or xylene solution of 250 pg Al ml-’ at 2 ml min-‘. Table 3 presents the calculated results of the distribution of aluminium (%wt) in the important components of a wet acetylene flame at a temperature of 28OOK with various C/O ratios. The results prove the importance of the condensed phases in the acetylene flame atomization of aluminium. CHESTER et al. [5] have already calculated that the main aluminium species is A1203(s, I) in acetylene flames with Cl0 ratios less than unity. Their calculations, however, did not consider the condensed phases of aluminium nitride and aluminium metal, which are significant at C/O ratios greater than unity. RASMUSON et al. [7] have assumed total vaporization of the metal species as have WI~ENBERG et al. f8] who included the Al ions and halides in their calculations. In this work the calcuiated degree of atomization 13.7% (j3 = 0.14) at C/O = 0.94 (p = 2.0) agrees very well with the experimental results of DE GALANand SAMAEY[I73 who found a p value of 0.13 when p = 2.2 and of WILLIS [18] who found a B value of 0.15 with p = 1.9. [ 171L. DE [I81
GALAN and G. F. SAMAEY. Spectrochim.
J. B. WIIMS,Specfrochim.
Ada
Acta
258,487 (1970).
25B, 245 (1970).
313
Condensed phases in flame spectromctr~ Table 3. Calculated distribution of aluminium in a heterogeneous S!‘S~CIII (%wl) PI 2800 K ~.___ __.._- ~-~ ___~_ -___-____
C/O !;!>
0.57
0.36
0.94
1 .>>.I
1.1,
:..‘.!
I. i.’
1.41
?n . R
20.4 36.1
.__._________.
nl,I)3:l’
/
I_
85.8
54.1
0.1
1.7
7.5
22.1
.) 1 .7
2,.
Al(l)
0.1
2.2
11.6
35.4
I ‘, 6
3i.H
Ib.
n i iq 1
n. 2
2.G
11.7
.I I
II.3
Ai
41,s
32.1
Alo,,
1.0
3.4
7.‘)
L
t
t
t
3.0
t
’
91.7
AlN(.i’
1
‘I. I
L
2.6
11
t
3
il
L
J
All102(9)
6.3
AlO(9)
0.5
1.1
2.2
t
L
t
L
t
AlH(9)
t
0.1
3.4
1.3
1.3
1.3
1.4
1.4
For the C/O ratios greater than 1.0, the calculated value of /3 for Al(g) is practically constant at 0.41-0.42. Considering a temperature of 3000 K and the given initial compositions of the wet acetylene flame, the calculated /3 values increase to 0.67 with a corresponding decrease in the condensed phases of AlN(s) and Al(I). A temperature of 3000 K can be taken as a probable value for the rich flames. At 2400 K, condensed phases of AlZ03(s), AlN(s) and Al(I) play the same role, but to a greater extent than at 3000 K. A&O,(s) is the most sign&ant since it is not possible to reach a C/O ratio equal to unity when nebuliing aqueous solutions. In accordance with experimental data [171, the calculated fi values are very low (at C/O = 0.86 lo-’ max.). 3.3 Condensed phases of silicon in wet acetylene jlame The number of silicon moles (4.985 x 10”) entering the flame corresponds to the nebulization of an aqueous solution of 140 fig ml-’ Si or of a xylene solution of 350 pg ml-’ Si. We have included fluorine (0.002 mol) in the calculations, since HF is a component in the closed vessel silicate decomposition procedure. The following species have been included in flame components: Si(g), Si(f), Sir(g), S&(g), Sic(g), Sic(s), Sic(g), SiF(g), SiFH&), Si&(g), Si&H&), SiFD(g), SiF&), SiM(g), SiF,(g), SiH(g), SiH&), SiNtg), SiO(g), Si@(g), SiO&). SirC(g), SirNtg), Si&(g), and the fluorine components with the major flame elements (C, H, N, 0). The presence of fluorine, corresponding to a concentration of HF currently used for decomposing silicates has practically no effect on the atomization of silicon. Under the conditions in the acetylene flame fluorine is present almost exclusively as HF(g) in the equilibrium mixture. Similarly, we confirmed the conclusion [19] that most of the chlorine is bound in the form of HCl(g). Table 4 shows the silicon distribution (%wt) of the important components at temperatures of 2800 and 3000K. The atomization of silicon is negligible at the subcritical composition of wet flame (C/O = 0.94). The main component is SiO(g) (see also Ref. 5). The condensed phase of Si(f) is important at the supercritical composition of wet flame. DE GALAN and SAMAEY (171 have utilized the incandescent flame and found the experimental value of p = 0.055, which is in excellent agreement with our predicted fl value of 0.054. Under the given conditions in a wet acetylene flame, the assumption of complete vaporization of silicon into the gas phase is never fulfilled. 4. CONCLUSIONS The calculations have confirmed the relation between the concentration of C(s) and the atomizing efficiency of the acetylene flame. The analytical qualities of wet [I91 Ref. IO, p. 72.
314
+t = < O.OS%wt.
acetylene flames are defined by the C/O ratio and temperature. However, all major elements of the flame must be included in an accurate description of the atomizing medium. Our results from heterogeneous equilibrium calculations for titanium, zirconium, silicon and aluminium have shown that, depending on the operating conditions used, the formation of a condensed system (generally metal-oxide-carbidenitride) in the flame is thermodynamically probable. This fact acts as a limiting factor in the atom vapour formation of elements at the temperatures in an acetylene flame. It is possible to find a relationship between the position of the element in the Periodic Table and the nature of the condensed phase. In the present study calculated /I values are in excellent agreement with the experimental data of DE GALANand SAMAEYand those of WILLIS.Thus, the equilibrium model would be useful for numerical optimization of flame spectrometric measurements. Acknowfedgunenr-The authors thank Prof. Dr. ROBERT HOLUB of the Prague Institute of Chemical Technology for suggestions on the methods of chemical equilibria calculation. The help of Dr. R. J. DECKER.University of Zimbabwe, with the improvement of the English style of the manuscript is gratefully acknowledged. APPENDIX Computation of chemical equilibrium compositions There are in principle two approaches to the solution of the chemical equilibrium of a system [l3]. The first approach sets out from a description of the overall chemical conversion of given system by means of a set of chemical reactions, and the corresponding standard changes of free energy or equilibrium constants. Assuming R to be number of linearly independent reactions, this procedure converts to the solution of a non-linear set of R equations for R unknown variables (degrees of conversion). In the second approach, the system is considered as a whole, its thermodynamic properties being expressed by the overall free energy of the system. Calculation of the chemical equilibrium is equivalent to the problem of finding the minimum of the free energy of the system dependent on the numbers of moles of the constituents ni subject to the constraint of maintaining the validity of the balanced relations. We shall now expound some methods which use the second approach. Let us consider a closed system containing N constituents formed from M different elements, and distributed among (1 phases at a given temperature, T. and pressure, P. Not only a neutral molecule but also a radical can be regarded as a constituent. Provided an electron is additionally defined as one of the set of elements, ions may also be considered to be independent constituents. Only ideal systems will be specifically considered, although the method may also be applied to those non-ideal systems where the activity coefficients are known functions of the composition. The total free energy of the system divided by the term RT, is defined [20,2l] by the expression:
1201 F. P. BOYNTON, 1. Chem. Phys. 32, 1880 (1960). (211 F. J. ZELEZNIK and S. GORDON. fnd. Eng. Chrm. 60, 27 (1968).
Condensed
phases in flame spectromelry
31.5
where
The ~qujIibrium ~nsti~ueu~ values ~7 are those which minimize the total free energy of eqn (I), subject to the constraint of the conservation of all of the etemeotr. These limitations are expressed as
Xn a ~mogcoous system D = 1. ~en~e~n~tb we shall omit the superscript o. The summations in eqn (1) and (2) can also be omitted. By applying the welt known method of Lagrangian muhipiiers to eqn (f) with canstraints (2) we obtain a set of M + t equations for unknown Lagrange multipliers fIi and the total number of moles n in the system: N
I;a&-$4 j=f,2,...,M &=1-
a-
where yr are molar fractions given by the relations: (4)
The set (3) of not&tear equations cart be solved by, e.g. Newton’s method with mducti~~ parameter.We chose the set of Lagmnge multipliers R$@,i = 1,2,. . . , M @rst ~pF~~ation~ $0 that for the term in parentheses in eqn (4) the fortowing r&&ion hoids:
We dcvetop the feft-hand side of the eqn (3) into a Taylor series to the point we have t&en as the first approximation. i.e. neglecting second- and hiir-order terms of the series. This leads to a set of linear equations:
where Antl’“=rr,
ITi@second approximation
-Rf’
k==I , 2 t.“t
M
of lT* values is obtain& from the relation
where e is the reduction parameter. ‘L”bereduction parameter rE (0,1) is &en chosen so as to make the second approximation also satisfy the inequalities (5). In other words, e guatautees the validity of the condition y, E @,I) i = 1, 2, . . . , N thmghaut the numerical process. The whote procedure is repeated until the values of max fA&f is smaller than some predetermined number (usually IO* is taken).
Here, we shalt restrict ourselves to a description of the method of cafcuiation published by STOREYand VANZECJCEREN fI4& which we used. The method is called a steepest descent search for the minimum of the free energy of the system.
ii6
1. I.mr;
We shall assume pure condemed eyn (I) the following
Cl
ph:lses Only. Assming
ni. that condensed
phases do not mix. we obtain from
relations: L1
i=1.2
,:=cC+&E,,ln>
.___. N.
introducing a search parameter A which characterizes the rate of progress in the selected direction and also a set of variables .$ defined by n: =exp(fT).
i=
I,2 ,___. N
(7)
we may write eqn (1) as follows
(8) Similarly, eqn (2) assumes the form:
(9)
where n4 will always be non-negative in virtue of the definition (7). At a given suitable composition (specified by a set n;‘), the values of de/d,4 are taken to be those which minimize dr/dr\, while satisfying eqn (9) and an additional subsidiary condition
W) is chosen for convenience of scaling. The use of Lagrange’s method of multipliers leads to the set of equations:
where ll is the Lagrange multiplier from constraint (10). Multiplying eqn (11) by o&‘,
summing over i and
a and using eqn (Y), leads to the set of linear equations for the TI,:
(12) Once the f’f, have been found, the required dt:ldh specified by the & given by
are given by eqn (11). The next set of values n4 is
where the sign of dA and its magnitude are chosen so as to lead to a smaller value r. The computation is halted when no further reduction in f can be made.
List of Symbois constitution coetlicient, denotes the number of moles of element j in constituent totat number of gram-atoms of element j = (G’jIRT) + In P dimensionless partial molar free energy of constituent i index denoting gaseous state free energy in standard state number of elements total number of moles in a system number of moles of constituent i number of constituents in a system total pressure in a system gas constant absolute temperature molar fraction of constituent i
i
Condensed
phases in flame spectrometry
Greek Letters a
superscript
denotes
dimensionless Kronecker’s
a phase a. or number
function
delta (unit diagonal
matrix
of dimension
final difference reduction
parameter
search parameter =Inn, Lagrangian
multipliers
of phases
of free energy
in Newton’s
method
g x a)
317
Condensed
phases in flame spectrometry: An equilibrium of the acetylene flame ho
LANG and
model
VIKTOR DOBAI_
Institute of Geology and Geotechnique, Czechoslovak academy of Sciences, V HoleSoviCkzich 41, 182 09 Prague 8. Czechoslovakia and
GUSTAV
SEBOR,
JR.
Department of Petroleum Technology, Institute of Chemical Technology, 116 28 Prague 6. Czechoslovakia (Received 22 May 1981; in revised form 29 September 1981) Abstract-An equilibrium model for a wet acetylene flame has been used for the calculation of the distribution of carbon, aluminium and silicon between gaseous and condensed phases. Calculated values of free atom vapour fractions are in agreement with published experimental data. It is thermodynamically probable that the formation of condensed phases limits atom vapour formation. There is a relationship between the position of an element in the Periodic Table and the nature of the condensed phase.
1. INTR0Duflt0~ RECENTLY.RUBE&A has emphasized the importance of condensed phases as an interference source of some elements [l, 21. We have shown [3] using the example of the equilibrium distribution of titanium and zirconium among the different phases in the acetylene flame that in the atomizing medium some transition metals can be present in a condensed phase and the qualitative and quantitative description of the distribution depends on the !Iame conditions. The presence of condensed phases can also be significant in the atomization of metals from other groups of the Periodic Table. Also carbon itself is potentially a significant component of the condensed phases. The purpose of this work is the determination by calculation of the condensed phases of carbon, aluminium, and silicon in the atomizing medium of an acetylene flame. Our works start from the results of CHESTER et al. [4-6], RASMUSONet al. [73 and recently published works of WITIENBERG et al. [8] and ASAHINA et al. [9]. A more complete description of the condensed flame system is presented in this paper giving results which correspond satisfactorily with published experimental data. 2. THEORETICAL Equilibrium model assumptions From the point of view of analyte and sample matrix quality the objective choice of optimum operating conditions with a computer would mean definite progress in flame spectrometry. For this it is necessary to get as perfect an atomization model as 2.1
[l] I. RIJBESKA,Anal. Chem. 48, 1640 (1976). [21 I. RUBE~KA,Chem. Anal. ( Wurszuw) 22,403 (1977). [31 V. DOBAL, I. LANGand G. SEBOR,Chem. listy 74, 1233 (1980). 141J. E. CHIZSTER, R. M. DAGNALLand M. R. G. TAYLOR,Anolysl 95.702 (1970). 151J. E. CHESTER,R. M. DAGNALLand M. R. G. TAYLOR,Anal. Ckim. Actn 51,95 (1970). [61 R. M. DAGNALLand M. R. G. TAYLOR,Spectrosc. Lett. 4, 147 (1971). 171 J. 0. RASMUSON,V. A. FASSELand R. N. KNISELEY,Spectrochim. Acra ZSB,365 (1973); 31B. 229 (1976). @I G. K. WITTENBERG, D. V. HAUN and M. L. PARSONS,Appl. Spectrosc. 33,626 (1979). [91 T. ASAHINA,M. KOSAKAand I. SHIROYANAGI, Koon Gakkaishi 5, 35 (1979). 309
310
I.
possible. constants. the
A kinetic Although
et 121.
model requires of necessity the knowledge of relevant ratr it is known that the flame is generally in a non-equilibrium state
use of classical thermodynamics
at present.
LANG
Both the quantity
is the only approach
and the accuracy
which
of tabulated
also important factors in an equilibrium approach. The existence of a local steady state in the measured assumption of the equilibrium model. Following existence of a local steady state in the interzonal
is directly
thermochemical
applicable data are
zone of the flame
is an
ALKEMADE[IO] we assume region of an analytical
the acetylene
We believe that the analytical flame is realized in terms of the stability and reproducibility of the flame spectrometric measurements. These flame characteristics imply the existence of a local steady state. An additional reason which justifies a study of the possibilities of classical thermodynamics is the fact that a general non-equilibrium approach requires the knowledge of the thermodynamical brunch of solution [ 1I]. The true dynamic system would be characterized by rising and continually transforming superatomic and supermolecular structures which cannot be described without introducing time in the sense of a system history [I I]. Such a major dependence of signal fluctuations with time has never been observed in the practice of flame spectrometry. We support the opinion of JENKINSand SUGDEN[12] that the equilibrium state is a very valuable concept; the effect of non-equilibrium phenomena on the over-all composition of flame gas mixtures is generally small because the fraction of the total energy of the gas that resides in excited states or in free radicals is very small. Of course, effects such as heat losses to the burner, air entrainment, or the addition of significant amounts of liquid droplets may have a large effect on the composition, either directly or through temperature effects. These effects should either be avoided or otherwise taken into account in accurate calculation of the equilibrium compositions. This problem should be gradually solved. In the present paper we shall be concerned with the effect of the solvent first of all. Considering high temperature and normal pressures as further assumptions of the equilibrium model, an ideal behaviour of the gaseous phase may be assumed. The gas phase can be considered as mixed. So as not to introduce activity coefficients it is better to treat the condensed phases as being immiscible. flame.
2.2 Calculation of local equilibrium in the acetylene flame A concept of free energy minimization has been used for the calculation of the local equilibrium of flame gases and metal species. The advantages of this concept have been discussed by PARSONSet al. 181. We have utilized the method of HOLUB and VO~JKA[133 for the calculation of the equilibrium composition in homogeneous systems, and for heterogeneous systems the method of STOREY and VAN ZE~~EREN [14]. Data came from the JANAF Thermochemical Tables [15]. Equilibria were calculated using the programmable calculator WANG 2200. The results of the calculations of homogeneous equilibria were used as the initial data for the calculation of the heterogeneous systems. 3. RESULTS AND DWUSSION 3.1 Calculation of C(s) in wet acetylene flame The flame components considered in the calculations were taken from the JANAF [IO] C. Th. J. ALKEMADE and R. HERMANN, Fundamentals of Analytical Hame Spectroscopy, p. 30.Hilger, Bristol (1979). [I 11 G. NICOLIS and I. PRIGOGINE, Self-Organization in Nonequilibrium Systems. Wiley, New York (1977). [I21 D. R. JENKINS and T. M. SUGDEN, Radicals and Molecules in Flame Gases, in: Flame Emission and Atomic Absorption Spectrometry (Edited by J. A. DEAN and Th. C. RAINS), Vol. I. p. 170. Dekker, New York (1969). 1131 R. HOLULIand P. VOP~KA. Chemical Equilibrium of Gaseous Systems. Academia, Prague (1975). [I41 S. H. STOREY and F. VAN ZECCEREN, Can. I Chem. Eng. 42.54 (1964). [IS] D. R. STULL and H. PROPHET, JANAF Thermochemical Tables, 2nd Ed. NSRDS-NBS Publ. 37. Washington (197 I).
Condensed phases in flame spectrometry
311
the same as those considered in Refs. 3-9. To characterize the initial we make use of the C/O ratio. (This atomic ratio is usually used to characterize an initial mixture of industrial flames [16] and also in the petrochemical production of soot by thermal decomposition of the acetylene). it may be confirmed that it is only this elemental atomic ratio which determines the initial composition of the acetylene flame. In the cafcufations [7,8], the authors have taken both the amount of solvent and the aspiration efficiency to be constant (water; 2 and 4 ml min-‘, resp.; 10%). Thus the composition of the initial mixture is characterized by the ratio p = oxidant/CzH2. It is however not possible with these calculations to easily take into account a change of solvent [7,8]. One of the main purposes of our work was a general study of the enrichment of the flame by carbon with the use of organic solvents. By making use of the p ratio one can omit the influence of the solvent upon the composition of the initial mixture. In this case, the theoretical and the practical models cannot be consistent when considering the absolute difference in the initial states. This is clear from the values in Table 1 and also from the results of ASAHINA et ni. [9], who have shown that there are substantial differences between the composition of wet and dry flames. The number of moles of elements entering the wet flame per minute is listed in Table 2. Taking into account the major ~mponents in the condensed phase of the acetylene flame, calculation shows that C(s), i.e. carbon in the solid phase, is a single but very important component, Fie 1 demonstrates the dependence of the equiliTables
and
are
composition
Table 1. Composition of the mixture entering into the burner Atomizing
Flow
(Lmin-1)
solution
efficiency
oxidant
C2H2 2.3
11.3 =
water b
2.3
11.3
Water
2.3
11.3
Xylem
5.2
10.4 e
5.2
a.2
C/O
P
(0) 20
0.77
1.0 =
10
0.86
1.0
10
1.07
1.0
water
10
0.94
2.0
water
20
1.10
1.6
d
‘Air. ‘5 ml min-‘. ‘Assuming OI as oxidant. d2 ml min-‘. ‘Nitrous oxide.
Table 2. Initial composition of a wet acetylene flame mole
C/O
d
per
minute
C
H
N
0
0.20536
0.31656
0.79700
0.26748
1.11356
0.86
0.20536
0.26396
0.79700
0.23968
1.05796
1.07
0.22656
0.21066
0.79700
0.21188
1.00766
0.46428
0.57548
0.92857
3.49209
1.50405
3.46428
0.57548
0.73214
'3.42167
1.30762
0.11
0.94
b
1.19
“C&-air
flame.
*C2HrN20 ffame. [I61
K. H. HOMANN
e-
and H. G. WAGNER, Proc. Roy. Sot. 307A.
141 (1%8).
I.
312
LANGYf
a!.
/..A?---
/-& f/A .
d
Fig. 1. The dependence of the equilibrium concentration C(s) on the Cl0 ratio for chosen temperatures of acetylene fiame: 0, 2400 K; 0.2800 K; A, 3000 K.
brium concentration C(s) (number of mole C(s) per mole of the equilibrium mixture) upon different C/O ratios at temperatures 2400, 2800 and 3000 K, resp. The “limit of carbon formation*’ [ 161 at temperatures in a wet analytical acetylene flame (2400 K for air-acetylene, 28OO-3000K for nitrous oxide-acetylene) can be defined by the inflexion point of the curves. When the critical ratio C/O = 1 is reached, the concentration of C(s) in the equilibrium mixture increases by several orders. Thus the calculations confirm that it is quite reasonable to look for a relationship between the concentration C(s) and the high atomizing efficiency of the fuel-rich frame, as proposed by DE GALAN and SAMAEY [17]. 3.2 Condensed phases of aluminium in wet acetylene flames The following species of aluminium were included in the flame components: Al(g), Al(I), Al+(g), AU&, AlHW, AIN( AlNb), fob), AlO+( AIOI-W, AIOH+(g), AlOH-( AlO&), AlHOzW Al,O(d, Al2O+W, Al&(g), AlzOz+W, ~203th 0, HAlO( Calculation co&med that the ionization of aluminium is suppressed by the presence of 2000 pgK per milliliter of solution. As in Refs. 3-9, the flow rate of analyte and matrix elements into the ffame was calculated from the concentration of the element in the aspirated solution, assuming a given flow rate and an aspiration efficiency of 10%. The number of moles of aIuminium (3.706 x lo*) corresponds to the nebuli~tion of an aqueous solution of 100 pg Al ml-’ at a rate of 5 ml min-’ or xylene solution of 250 pg Al ml-’ at 2 ml min-‘. Table 3 presents the calculated results of the distribution of aluminium (%wt) in the important components of a wet acetylene flame at a temperature of 28OOK with various C/O ratios. The results prove the importance of the condensed phases in the acetylene flame atomization of aluminium. CHESTER et al. [5] have already calculated that the main aluminium species is A1203(s, I) in acetylene flames with Cl0 ratios less than unity. Their calculations, however, did not consider the condensed phases of aluminium nitride and aluminium metal, which are significant at C/O ratios greater than unity. RASMUSON et al. [7] have assumed total vaporization of the metal species as have WI~ENBERG et al. f8] who included the Al ions and halides in their calculations. In this work the calcuiated degree of atomization 13.7% (j3 = 0.14) at C/O = 0.94 (p = 2.0) agrees very well with the experimental results of DE GALANand SAMAEY[I73 who found a p value of 0.13 when p = 2.2 and of WILLIS [18] who found a B value of 0.15 with p = 1.9. [ 171L. DE [I81
GALAN and G. F. SAMAEY. Spectrochim.
J. B. WIIMS,Specfrochim.
Ada
Acta
258,487 (1970).
25B, 245 (1970).
313
Condensed phases in flame spectromctr~ Table 3. Calculated distribution of aluminium in a heterogeneous S!‘S~CIII (%wl) PI 2800 K ~.___ __.._- ~-~ ___~_ -___-____
C/O !;!>
0.57
0.36
0.94
1 .>>.I
1.1,
:..‘.!
I. i.’
1.41
?n . R
20.4 36.1
.__._________.
nl,I)3:l’
/
I_
85.8
54.1
0.1
1.7
7.5
22.1
.) 1 .7
2,.
Al(l)
0.1
2.2
11.6
35.4
I ‘, 6
3i.H
Ib.
n i iq 1
n. 2
2.G
11.7
.I I
II.3
Ai
41,s
32.1
Alo,,
1.0
3.4
7.‘)
L
t
t
t
3.0
t
’
91.7
AlN(.i’
1
‘I. I
L
2.6
11
t
3
il
L
J
All102(9)
6.3
AlO(9)
0.5
1.1
2.2
t
L
t
L
t
AlH(9)
t
0.1
3.4
1.3
1.3
1.3
1.4
1.4
For the C/O ratios greater than 1.0, the calculated value of /3 for Al(g) is practically constant at 0.41-0.42. Considering a temperature of 3000 K and the given initial compositions of the wet acetylene flame, the calculated /3 values increase to 0.67 with a corresponding decrease in the condensed phases of AlN(s) and Al(I). A temperature of 3000 K can be taken as a probable value for the rich flames. At 2400 K, condensed phases of AlZ03(s), AlN(s) and Al(I) play the same role, but to a greater extent than at 3000 K. A&O,(s) is the most sign&ant since it is not possible to reach a C/O ratio equal to unity when nebuliing aqueous solutions. In accordance with experimental data [171, the calculated fi values are very low (at C/O = 0.86 lo-’ max.). 3.3 Condensed phases of silicon in wet acetylene jlame The number of silicon moles (4.985 x 10”) entering the flame corresponds to the nebulization of an aqueous solution of 140 fig ml-’ Si or of a xylene solution of 350 pg ml-’ Si. We have included fluorine (0.002 mol) in the calculations, since HF is a component in the closed vessel silicate decomposition procedure. The following species have been included in flame components: Si(g), Si(f), Sir(g), S&(g), Sic(g), Sic(s), Sic(g), SiF(g), SiFH&), Si&(g), Si&H&), SiFD(g), SiF&), SiM(g), SiF,(g), SiH(g), SiH&), SiNtg), SiO(g), Si@(g), SiO&). SirC(g), SirNtg), Si&(g), and the fluorine components with the major flame elements (C, H, N, 0). The presence of fluorine, corresponding to a concentration of HF currently used for decomposing silicates has practically no effect on the atomization of silicon. Under the conditions in the acetylene flame fluorine is present almost exclusively as HF(g) in the equilibrium mixture. Similarly, we confirmed the conclusion [19] that most of the chlorine is bound in the form of HCl(g). Table 4 shows the silicon distribution (%wt) of the important components at temperatures of 2800 and 3000K. The atomization of silicon is negligible at the subcritical composition of wet flame (C/O = 0.94). The main component is SiO(g) (see also Ref. 5). The condensed phase of Si(f) is important at the supercritical composition of wet flame. DE GALAN and SAMAEY (171 have utilized the incandescent flame and found the experimental value of p = 0.055, which is in excellent agreement with our predicted fl value of 0.054. Under the given conditions in a wet acetylene flame, the assumption of complete vaporization of silicon into the gas phase is never fulfilled. 4. CONCLUSIONS The calculations have confirmed the relation between the concentration of C(s) and the atomizing efficiency of the acetylene flame. The analytical qualities of wet [I91 Ref. IO, p. 72.
314
+t = < O.OS%wt.
acetylene flames are defined by the C/O ratio and temperature. However, all major elements of the flame must be included in an accurate description of the atomizing medium. Our results from heterogeneous equilibrium calculations for titanium, zirconium, silicon and aluminium have shown that, depending on the operating conditions used, the formation of a condensed system (generally metal-oxide-carbidenitride) in the flame is thermodynamically probable. This fact acts as a limiting factor in the atom vapour formation of elements at the temperatures in an acetylene flame. It is possible to find a relationship between the position of the element in the Periodic Table and the nature of the condensed phase. In the present study calculated /I values are in excellent agreement with the experimental data of DE GALANand SAMAEYand those of WILLIS.Thus, the equilibrium model would be useful for numerical optimization of flame spectrometric measurements. Acknowfedgunenr-The authors thank Prof. Dr. ROBERT HOLUB of the Prague Institute of Chemical Technology for suggestions on the methods of chemical equilibria calculation. The help of Dr. R. J. DECKER.University of Zimbabwe, with the improvement of the English style of the manuscript is gratefully acknowledged. APPENDIX Computation of chemical equilibrium compositions There are in principle two approaches to the solution of the chemical equilibrium of a system [l3]. The first approach sets out from a description of the overall chemical conversion of given system by means of a set of chemical reactions, and the corresponding standard changes of free energy or equilibrium constants. Assuming R to be number of linearly independent reactions, this procedure converts to the solution of a non-linear set of R equations for R unknown variables (degrees of conversion). In the second approach, the system is considered as a whole, its thermodynamic properties being expressed by the overall free energy of the system. Calculation of the chemical equilibrium is equivalent to the problem of finding the minimum of the free energy of the system dependent on the numbers of moles of the constituents ni subject to the constraint of maintaining the validity of the balanced relations. We shall now expound some methods which use the second approach. Let us consider a closed system containing N constituents formed from M different elements, and distributed among (1 phases at a given temperature, T. and pressure, P. Not only a neutral molecule but also a radical can be regarded as a constituent. Provided an electron is additionally defined as one of the set of elements, ions may also be considered to be independent constituents. Only ideal systems will be specifically considered, although the method may also be applied to those non-ideal systems where the activity coefficients are known functions of the composition. The total free energy of the system divided by the term RT, is defined [20,2l] by the expression:
1201 F. P. BOYNTON, 1. Chem. Phys. 32, 1880 (1960). (211 F. J. ZELEZNIK and S. GORDON. fnd. Eng. Chrm. 60, 27 (1968).
Condensed
phases in flame spectromelry
31.5
where
The ~qujIibrium ~nsti~ueu~ values ~7 are those which minimize the total free energy of eqn (I), subject to the constraint of the conservation of all of the etemeotr. These limitations are expressed as
Xn a ~mogcoous system D = 1. ~en~e~n~tb we shall omit the superscript o. The summations in eqn (1) and (2) can also be omitted. By applying the welt known method of Lagrangian muhipiiers to eqn (f) with canstraints (2) we obtain a set of M + t equations for unknown Lagrange multipliers fIi and the total number of moles n in the system: N
I;a&-$4 j=f,2,...,M &=1-
a-
where yr are molar fractions given by the relations: (4)
The set (3) of not&tear equations cart be solved by, e.g. Newton’s method with mducti~~ parameter.We chose the set of Lagmnge multipliers R$@,i = 1,2,. . . , M @rst ~pF~~ation~ $0 that for the term in parentheses in eqn (4) the fortowing r&&ion hoids:
We dcvetop the feft-hand side of the eqn (3) into a Taylor series to the point we have t&en as the first approximation. i.e. neglecting second- and hiir-order terms of the series. This leads to a set of linear equations:
where Antl’“=rr,
ITi@second approximation
-Rf’
k==I , 2 t.“t
M
of lT* values is obtain& from the relation
where e is the reduction parameter. ‘L”bereduction parameter rE (0,1) is &en chosen so as to make the second approximation also satisfy the inequalities (5). In other words, e guatautees the validity of the condition y, E @,I) i = 1, 2, . . . , N thmghaut the numerical process. The whote procedure is repeated until the values of max fA&f is smaller than some predetermined number (usually IO* is taken).
Here, we shalt restrict ourselves to a description of the method of cafcuiation published by STOREYand VANZECJCEREN fI4& which we used. The method is called a steepest descent search for the minimum of the free energy of the system.
ii6
1. I.mr;
We shall assume pure condemed eyn (I) the following
Cl
ph:lses Only. Assming
ni. that condensed
phases do not mix. we obtain from
relations: L1
i=1.2
,:=cC+&E,,ln>
.___. N.
introducing a search parameter A which characterizes the rate of progress in the selected direction and also a set of variables .$ defined by n: =exp(fT).
i=
I,2 ,___. N
(7)
we may write eqn (1) as follows
(8) Similarly, eqn (2) assumes the form:
(9)
where n4 will always be non-negative in virtue of the definition (7). At a given suitable composition (specified by a set n;‘), the values of de/d,4 are taken to be those which minimize dr/dr\, while satisfying eqn (9) and an additional subsidiary condition
W) is chosen for convenience of scaling. The use of Lagrange’s method of multipliers leads to the set of equations:
where ll is the Lagrange multiplier from constraint (10). Multiplying eqn (11) by o&‘,
summing over i and
a and using eqn (Y), leads to the set of linear equations for the TI,:
(12) Once the f’f, have been found, the required dt:ldh specified by the & given by
are given by eqn (11). The next set of values n4 is
where the sign of dA and its magnitude are chosen so as to lead to a smaller value r. The computation is halted when no further reduction in f can be made.
List of Symbois constitution coetlicient, denotes the number of moles of element j in constituent totat number of gram-atoms of element j = (G’jIRT) + In P dimensionless partial molar free energy of constituent i index denoting gaseous state free energy in standard state number of elements total number of moles in a system number of moles of constituent i number of constituents in a system total pressure in a system gas constant absolute temperature molar fraction of constituent i
i
Condensed
phases in flame spectrometry
Greek Letters a
superscript
denotes
dimensionless Kronecker’s
a phase a. or number
function
delta (unit diagonal
matrix
of dimension
final difference reduction
parameter
search parameter =Inn, Lagrangian
multipliers
of phases
of free energy
in Newton’s
method
g x a)
317