(24) S. Evans, R. G. Pritchard, and J. M. Thomas, unpublished work, Aberystwyth, 1976. (25) C. J. Powell, Surf. Sci., 44, 29 (1974). (26) C. C. Chang, Surf. Sci., 48, 9 (1975). (27) J. M. Morabito, Surf. Sci., 49, 318 (1975). (28) B. A. Joyce and J. H.Neave, Surf. Sci., 34, 401 (1973) and references
therein.
(29) S. Evans, J. Pielaszek, and J. M. Thomas, Surf. Sci.. 5 5 , 644 (1976)
and references therein. (30) S. Evans and J. M. Thomas, Proc. R Soc. London, Ser. A , 353, 103 (1977) (31) G.-Brown, Ed., “The X-ray Identification and Crystal Structures of Clay Minerals” Mineral Society, London, 2nd ed., 1961. (32) A. Barrie and C. R. Brundie, J . Nectron Spectrosc., 5 , 321 (1974). (33) H. Bennett and R. A. Reed, “Chemical Methods of Silicate Analysis”. Academic Press New York, N.Y., 1971. (34) R. F. Reilman. A. Msezane, and S. T. Manson, J . Electron Spectrosc., 8 , 389 (1976).
(35) R. G. Hayes, Chem. Phys. Lett., 38, 463 (1976). (36) J. M. Adams. S. Evans, and J. M. Thomas, J . Phys. C: Solid State Phys., 6, L382 (1973). (37) N. Beatham and A. F. Orchard, J . Nectron Spectrosc., 9, 129 (1976). (38) C. D. Wagner, in “ E m o n Spec+~’oscopy”, D. A. Shirley, Ed., North Holland, Amsterdam, 1972, p 861. (39) K. Siegbahn, U. Gellus, H. Slegbahn, and E. Olson, Phys. Left. A , 32, 221 (1970). (40) R. Belcher and A. J. Nutten, “Quantitative Inorcwnk Anatysis”, Butterwcnth, London, 2nd ed., 1960, p 86. (41) A.P.I. Research Project 49: Reference Clay Minerals (ColumbiaUniversity) Preliminarv ReDort No. 8 (1951). (42) I.Adams,’J. M‘. Thomas, and G : M. Bancrofi, Earth Planet. S o . Lett., 16, 429 (1972).
RECEIVEDfor review May 12, 1977. Accepted July 25, 1977. We thank the Science Research Council for fimancid support.
Sample-Loss Mechanism in a Constant-Temperature Graphite Furnace Ray Woodriff,’ Momir Marinkovlc,’ R. A. Howald, and I s a a c Elierer Department of chemistry, Montana State University, Bozeman, Montana 597 15
A constant temperature furnace, used for the study of sample loss, has an advantage of essentially constant temperature over the duration of the observatlon pulse. I n addltlon, because of the enclosed nature of the furnace and the absence of rapld heating of the graphlte tube, the convectlon loss of the sample vapor Is small. I t has been found that one group of elements closely follows a simple dlffuslonal law, whlle for other elements a considerable devlatlon Is observed. Devlatlons are linked to the sample vapor redeposklng In the heater tube for those elements whlch form compounds wlth carbon at elevated temperatures. I n the case of gas phase dlffudon controlled sample loss, a new procedure for expanslon of the useful analytlcal range Is proposed.
Atom liberation from the distribution in a graphite furnace has been recently studied by several workers (1-4). Such studies are important for the interpretation of atomic absorption measurements in these furnaces. It has been found that reduction by carbon is the probable mechanism for free atom formation for most elements (5-7) at least at lower temperatures. Modification of the inside of the graphite tube by reaction with carbide-forming elements has been used to prevent carbide formation by other elements and to improve the limits of detection for these other elements (8-10). Tt is clear that carbide formation and decomposition on the graphite can affect the height and shape of absorption peaks observed in graphite tube furnaces; however, these effects have not been studied previously. In this paper, a simple technique for the study of the diffusion mechanisms for sample loss from graphite furnaces is described. Cases of both simple diffusion and diffusion complicated by carbide formation have been found and interpreted. For the cases of simple diffusional loss, a new procedure for the expansion of the useful analytical range is proposed. ‘Present address, Chemistry Laboratory, Boris K i d r i c Institute, Beograd, Yugoslavia.
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ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
THEORY In a well-designed tube atomizer, the sample loss can be limited to diffusion through the apertures and to diffusion through the porous walls if the heater tube is made of graphite. If the sample loss is controlled by diffusion through the apertures of the tube, and the evaporation time is short compared to the mean residence time of the analyte atoms, the total amount of analyte present in the tube at the time t , M,, is given by the exponential equation:
M,
=
1
M , exp(--), c
la
where a is the length of the graphite tube, M , is the initial weight of analyte, t is the time elapsed after the sample is vaporized, and T, is the time constant for gas diffusion through the apertures, i.e., mean residence time of the analyte atoms in the tube ( 1 1 ) . As discussed in the Appendix, this single term is adequate to represent our measurement on Zn and Cd, since they were made at time sufficiently long that the material used diffused to the end of the furnace tube. Under such conditions, t >> 0, the concentration in the gas phase is given by (see Appendix):
C, = C ,,
sin ( m / a ) exp(-t/ra)
(2)
Inserting the expression into the differential equation for diffusion (12) gives the relation between 7, and the effective diffusion coefficient for transport in the gas phase, D.
r , = a2/ri2D
(3)
For materials which react with the graphite tube, there are two additional effects to be considered. The first effect is diffusion out through the graphite wall which provides an additiond mechanism for removal of atoms from the furnace tube and can be characterized by a time constant
(4) where b is the wall thickness of the graphite tube and D‘ is an effective diffusion coefficient in graphite. This would give
-
Direction of Observat on
5
431 t - 5 O s i
Flyure 1. Recorder traces for Cd 326.1 nm line
a time constant less than r,. The second effect is the amount of material stored in the walls. It takes additional time for the two diffusion processes to remove atoms from the wall along with those in the gas phase. The two largest time constants for this coupled process, T~ and T ~ (see , Appendix) depend upon T,, T b , and the equilibrium constant for distribution between graphite and the gas phase, K , in a complex manner which can be expressed by the transcendental equation (see Appendix)
where A is the cross-sectional area of the graphite tube. As explained in the Appendix, this equation is valid for the two largest T, roots, i.e., T~ and T ~ . For certain elements, only one time constant is required to adequately represent the experimental behavior, and Equation 1can be used with 72 substituted for T. The symbol T~ is used here since in the limit as T b increases, it is the second , approaches 7,. root of Equation 5 , T ~ which
M t = M , exp(-t/.r2)
(6)
If we define M , as a mass of analyte vapor present in the heater tube that gives absorption equal to a, and t u as a time which elapses from the moment of complete vaporization of the analyte to the moment when absorption attains the value of a, then Equation 6 can be written as: I
M,
=
(7)
M , exp(-L") 72
After taking the logarithm and rearranging we obtain:
t, = 72 logM, 0.434
+d
where
(p-72 0.434 log M a
2
Flgure 2. Constant temperature furnace (schematically). (1) Heater tube, (2) side tube. (3) pedestal, (4) crucible, (5)heater tube holder, (6) thermal insulation
(9)
From Equation 8, it follows for a given a (and therefore M a ) that there is a linear relationship between t , and log M , provided that evaporation is short compared to the residence time and analyte vapor loss is governed by gas phase diffusion. Three applications of Equation 8 can be proposed: (a) it can be used as a criterion for the applicability of the diffusion loss mechanisms we propose, (b) the residence time can be evaluated from the slope of the curve (t, vs. log M,), and (c) if the vapor loss follows the gas phase diffusional mechanism, the above relationship can be used for construction of straight line analytical curves and thus can extend the useful analytical range of furnace atomic absorption spectrometry. For example, a sample of 1500 ng Cd which gives 100% absorption for a considerable time as shown in Figure 1, can be analyzed without dilution by measuring the time required to reach 40% absorption.
For those elements which require two time constants to describe their diffusion out of the furnace, one has
M t = M o C exP(-t/T 1 ) + M o ( ~
c)exP(-t/72)
(10)
This equation follows from the standard procedure of representing the solution of a differential equation as a sum of products of functions of position only and of time only. The fractions of the first two modes, C and 1 - C a t t = 0, are functions of the initial conditions and of the equilibrium constant, K . For the particular time when t = t , and M = Ma, one can divide the equation by M,, M a , giving:
A plot of -log M , vs. t u in this case should be a curved line, with the same two time constants and the same fraction C as a plot of log M , vs. t. The technique developed here of measuring t, as M , is varied, gives accurate points over a wider range of concentration values than one can obtain for direct measurements of M with the same optical system.
EXPERIMENTAL The furnace to be used to check Equation 8 must fulfill the followingtwo conditions: (a) sample vapor loss due to convection must be eliminated or minimized, and (b) temperature of the heater tube over the duration of the observation pulse must essentially remain constant because the residence time depends on temperature. Both conditions are almost ideally fulfilled with the constant-temperature furnace developed at Montana State University ( 1 3 J 4 ) . In addition, the analyte has a long residence time which makes the measurement more precise. As a mattes of convenience,a schematic drawing of the constant-temperature furnace is shown in Figure 2. The heater tube 1 and the side tube 2 are kept continuously at high temperature by resistance heating with the three-phase system (15). This makes possible a uniform temperature distribution along the heater tube and side tube. A sample-containing graphite crucible 4 is introduced into the furnace by means of the pedestal 3. The crucible is heated by radiation and conduction rapidly eriough that a short evaporation time compared to the residence time is obtained (16). The pedestal makes a seal with the constrict,ion in the side tube, and sample vapor diffuses through the end of the heater tube and, eventually, through the heater tube walls. Graphite felt insulation 6 reduces the radiation and conduction loss which makes possible the use of a relatively small power supply. For this study, a heater tube was made of POCO FXI graphite. The tube length was 13 cm, the outer diameter was 9.27 mm, and the wall thickness was 0.67 mm. As a primary source, conventional hollow cathode lamps with modulated power supply were used. Signals were amplified with an Ithaco lock-in amplifier and recorded with a strip chart recorder having a chart speed of up to 5 s/inch.
RESULTS AND DISCUSSION A recorder trace for the Cd 326.1 nm line for different weights of analyte is shown in Figure 1. The value of to,.,is represented by the distance from the rising to the falling part of the recorder trace at the 40% absorption level. A possible constant error oft, does not change the shape or slope of the curve obtained by plotting Equation 8, but only displaces i t along the t , axis. ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
2009
70t
A n a l y t e weight / n g
Flgure 3. Plot of to vs. log Mo. Dots represent experimental points while solid lines represent calculated curves
Plots of Equation 8 for various elements show that these elements can be divided into two groups. The first group comprises elements for which the plot gives a straight line over two orders of magnitude, while the second group comprises the elements for which no straight line portion of the plot is observed. Figure 3 shows plots of several elements from both groups. The temperature of the heater tube was 1900 K for Ag, Cd, and Zn, and 2400 K for Cu, K, and Mn. These are reasonable choices of temperature for analytical work with these elements as the time constants for Cu, K, and Mn are excessively long at the lower temperature. For Ag, Cd, and Zn, straight lines are obtained over more than two decades. Deviation of the experimental points for larger weights of analyte for these elements may be caused by back diffusion from the furnace envelope to the heater tube. The plot of two Cd lines of different sensitivities, or plots of lines of different elements show that curvature occurs approximately at the same height in a plot like Figure 3 in all the cases. This supports the conclusion that curvature occurs due to back diffusion. In the furnace used, the heater tube was surrounded with graphite felt as insulation to diminish the heat losses due to radiation and conduction. However, graphite felt also decreases the diffusion rate of the analyte escaping from the heater tube walls toward the cold outside envelope of the furnace and analyte atom concentration builds up until the analyte diffuses back to the heater tube. However the analyte eventually diffuses through the felt insulation to the cold walls (in about 10 min) and the furnace becomes useable a g g n for the next sample. it should be noted that a long cleaning time is necessary only for extremely high amounts of analyte. Smaller analyte samples can be run at much shorter intervals depending on the residence time. For the elements Cu, K, and Mn, no straight line is obtained when t , is plotted vs. iog M,. This can be interpreted as an indication that, for these elements, sample loss from the heater tube does not follow the exponential law of Equation 1. A possible explanation of this behavior is the assumption that these elements form compounds with graphite that have lower vapor pressure than the elements themselves. After evaporation of the sample from the crucible, part of the analyte vapor becomes absorbed by the heater tube forming a compound with graphite, while the remainder diffuses out of the crucible and the heater tube. Reevaporation (and probably multiple absorption and evaporation) of the absorbed analyte followed by diffusion accounts for the longer residence time and for the observation of the complex coupled diffusion process with two important time constants represented in Equation 10. The following experiment confirms the above assumptions. The sample-containing crucible is inserted into the furnace and after the maximum absorption is attained, the crucible 2010
ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
Flgure 4. Absorption pulse shape for Cd 326.1 nm line recorded with linear system. The sudden change In the solld line at point B Is due to the removing of the sample crucible. The dashed line represents the pulse shape which would be obtained If the cruclble had been left In the furnace
Flgure 5. Absorption pulse shape for Cu 324.8 nm line recorded with linear system. The sudden change in the solid llne at point B is due to removing of the sample crucible. The dashed llne represents the pulse shape which would be obtained if the crucible had been left In the furnace
Table I. Constants Used for the Calculation of the Curve in Figure 3 Equi-
Time constant in seconds
Ele-
ment K
5.11 5.36 4.76
Cu Mn ~~
Tb
71
15.2 16.3 19.4
16.8 18.25 20.0
7 2
5.59 5.75 4.62
libri- Paramurn eter from con- Equation stant 10 K
C
3.07 3.72 1.49
0.132 0.148 0.0483
~
and pedestal are rapidly removed allowing the argon to flow through the side tube and sweep the analyte atoms out of the heater tube. Figure 4 shows that the argon flow rapidly sweeps out the zinc atoms while an appreciable concentration of copper atoms (Figure 5) remains long after the crucible is removed, presumably because of slow vaporization of the compound formed in the graphite walls. These assumptions can also be checked by a comparison of the points in Figure 3 with a curve calculated on the basis of Equation 10. It is possible to estimate the time constants T~ and T~ from the curves. For a quantitative fit of the observed r2 values with Equation 10 the fraction C must also be determined. This is most easily done graphically by finding the time a t which two straight lines for the two time constants, 71 and 7 2 , intersect. Values obtained for il,T ~and , C are given in Table I. With a computer programmed for Equation 8, one can find values for T, and 7b which will give the observed time constants r1 and T ~ .The values of obtained for K, Cu, and Mn are, as expected, of the same order of magnitude as the time constants observed for Zn and Cd. All the r,, values correspond to values of the effective diffusion coefficient D, of the order of magnitude expected at these temperatures, Le., around 3 cm2 s-l. It is even possible to fit the value for C by adjusting
Table 11. Residence Times Evaluated from the Tail of the Absorption Signal Analyte weight , mg 1
5.1
3 10
5.1
30 100
300 1000
3000 10000 30000 a
Ag 328
5.3 5.6 6.5 9.4
Cd 326.1
5.2 5.2 5.5 6.1 6.8 7.7 9.0
Zn 307.6
4.2 4.0 4.2 4.4 5.0
Residence tirrda/@ Cu 324.8 Cu 249.2 6 9.8 14.2 15.7 17.6 20.0
K 404
5.0 8.9 6.3 10.7 16.2 19.2
13.4
19.4 32.6
Mn 403 4.8 7.4 10.9 19.7 36.5 46.0
Defined as time required for absorption to fall from 70 to 35.8%.
the value chosen for the equilibrium constant, K. The values of K are also listed in Table I, although this experiment is not the method of preference for the determination of the equilibrium constants. Residence times can also be evaluated from a single recorder trace. For the diffusion process, described by Equation 1,the plot of the log A vs. t is a straight line, the slope of which is determined by the mean residence time of the analyte atoms. For the linear recording system used in this work, the residence time is equal to the time required for the absorption to change from 0.70 to 0.358, i.e., the time required for absorbance to fall from 0.523 to 0.192, i.e. to 36.8% (l/e). Residence times evaluated in this way are shown in Table 11. Residence times for Ag, Cd, and Zn remain reasonably constant for low weights, while for higher weights a small increase is observed, presumably due to back diffusion. Contrasted with this, residence times evaluated for Cu, K, and Mn strongly depend on the weight of analyte. Plots of log A vs. t give a straight line for Ag, Cd, and Zn, while for Cu, K, and Mn, a considerable deviation from the straight line is observed. These results confirm the conclusion that analyte vapor loss for Ag, Cd, and Zn follows Equation 6 which is not the case for Cu, K, and Mn. The layer compounds formed by potassium in graphite are well known and have been studied a t temperatures up to 600 K (17). Calculations using the heat of formation indicate that the vapor pressures of the known compounds reach 1 atm at 1200 to 1600 K, but this is not inconsistent with our observation of some solubility of potassium in graphite even at 2400 K. The existence of manganese carbides stable a t high temperatures is clear from the Mn-C phase diagram (18). On the other hand we have not found any unequivocal evidence in the literature for stable copper carbides at high temperatures. However, neither does the literature rule out the possibility of an appreciable solubility of copper in graphite, sufficient to explain the behavior shown by copper atoms in our furnace illustrated in Figures 3 and 5. T h e possibility of using a linear relationship of t , vs. log M , as the analytical curve has been studied. The plot offers an extension of the useful analytical range to over two orders of magnitude. A modification of the furnace to minimize back diffusion may make possible still further extension of the linear portion of the analytical curves. The relation between the standard deviation o f t , and the standard deviation of M , is obtained by differentiating Equation 8:
Equating the differential with the standard deviation we obtain:
RSD = F
/T
Table 111. Precision of Zn Determination Zn, mg 6 , mg RSD in % 15 150 1500
0.25 0.15 0.24
2.6 1.5 2.5
where 6 is the standard deviation of the variable t , and RSD is the relative standard deviation of the variable M,. The precision of the Zn determination is presented in Table 111. Since the standard deviation of t , remains roughly constant over a wide range of analyte weights, it follows that the relative standard deviation of M , remains approximately constant as well. From Equation 13, it is apparent that longer residence times of material in the furnace will result in smaller standard deviations, i.e., in more precise results. For plotting of the analytical graphs according to Equation 8, measurements oft, are always made at the same absorption level (in the example included in this work, i t is 40% absorption). As a consequence, the linearity of the analytical graph ( t , vs. log M,) is not influenced by the various effects which are well known in atomic absorption spectroscopy causing nonlinear detector response. The curvature of the analytical graphs obtained by plotting t, vs. log M , depends solely on the deviation from the exponential law of diffusion represented by Equation 1. It is interesting to point out that it is possible to construct the analytical curve from only one standard, provided the analytical curve is a straight line. From the recorder trace, t , and 7 2 can be determined, which is enough for calculation of the constant d in Equation 8, and the plotting of the analytical curve. Similar possibilities exist in atomic absorption and atomic emission flame spectroscopy provided that the analytical curve is a straight line and passes through the origin.
APPENDIX One set of general solutions to the differential equations describing diffusion out of a graphite furnace are products of functions of the position coordinates with exponentials of the form exp(-t/~,).Any particular solution can be expressed (12) as a sum of such terms
where C, is the gas phase concentration and Fi(x) is a function of the position coordinates x . For the period before any material has diffused to the ends of the tube, five or six terms in the sum will be required. However the experimental data reported here are for times a t which material has already reached the ends of the tube so that t >> T ~ and , so the third term is negligible and only the terms with T~ and 7 2 are of importance. ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
2011
We assume that the concentration is uniform across the gas phase. Solving the differential equation for diffusion then gives (12) Cgi =
Cgoi
sin ( n i n x / a )e x p ( - t / T i )
(A21
where Cg0i is a constant related to the concentration at t = 0, a is the length of the tube, and n, is an integer. One must remember in using this equation that a sum of terms of this form with different n values may be required to satisfy the boundary conditions. For equilibrium between the walls and the gas phase concentrations at the inside surface, z = b, the x dependence must carry over into the walls. The full expression for the concentration in the walls, C, is (12): m
c=
c, = i=x
C,i =
x BK sin [ ( n z / b ) ( T b / T i ) ” 2 ]
sin ( & n x / a )
i= 1
1
exP(-t/Ti
1
(A3)
where z is the distance through the graphite wall of thickness b. We have used T b , defined as 7 6 = b2/r2D‘ (see Equation 4 in text) to eliminate the effective diffusion coefficient within the w d s , D’from the equation. K is the equilibrium constant for distribution of atoms between the walls and the gas phase, and therefore, using Equation A3, we obtain, assuming the single term with T , is adequate to represent our measurements
K = C, ( z = b ) / C , = BK sin
[n(Tb/q)’ ” ] ~ g o i
(A41
and = B sin
,C ,
C,
=B
sin
[n(Tb/?-j)1’2]
[n(Tb/T$/’]
(A51
sin ( q n x / a )
(A61
exP(-t/Ti)
T h e concentration gradient in the wall is:
dC,/dz = B K ( n / b ) ( T b / T i ) ’ ” cos (?TZ/ b ) (7 b /Ti) ‘1 sin (q‘i?x/ a )
’
(A7 1
exp(-t/Ti)
dC, / d z , = b = K ( n / b ) ( T b / T i ) ’ ” 1/2 c cotan [ n ( ? - b / T i ) 1 g
(A81
which gives a net flow from the wail into the gas. The complete equation for the time dependence of C, considering both diffusional flows is:
dC,/dt
= bd2Cg/dxZ+ (4n/A)”’D‘
dCw
/&z=
b
(A5)
where A is the cross-sectional area of the tube, and the factor ( 4 ~ / A ) l allows /~ for the surface to volume ratio for a cylinder. Substituting for the derivativatives in this expression by using Equations A6 and AS, and then dividing by C, gives the transcendental equation:
2012
ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
This equation has a steady state solution, T , = a,provided material is continuously supplied, and there are an infinite number of successively smdler time constants, T ~ 7,2 , 73, etc. , correspond to n, = 1, The two largest: r 1 and T ~ normally giving Equation 10 cited in the text. The smaller time constants are normally negligible after 1 or 2 seconds, especially if the sample is introduced in the center of the tube (which eliminates the contributions from n, = 2). The experimental curves for the decrease of absorbance with time can all be adequately fitted with two time constants, r1and T ~ r;2 is sufficient by itself for zinc and cadmium which do not form compounds with graphite. The fact that the single time constant, r2is sufficient for those elements which do not form compounds with graphite can also be predicted from Equation 10 (see text) since in this case C approaches zero as K approaches zero, and one is left with the equation M = M , exp ( - t / r 2 ) which is Equation 5 in the text.
LITERATURE CITED G. Tessari and G. Torsi, Anal. Chem., 47, 842 (1975). D. J. Johnson, B. L. Sharp, T. S. West, and R. M. Dagnall. Anal. Chem., 47, 1234 (1975). F. T. M. T. Maessen and F. D.Posma, Anal Chem., 48, 1439 (1974). R. E. Sturgeon, C. L. Chakrabarti. and C. H. Langford. Anal. Chem., 48, 1792 (1976). C. W. Fuller, Analyst(London). 99, 739 (1974). W. C. Campbell and T. M. Ottway, Talanta, 21, 837 (1974). R. W. Stone, Ph.D. Thesis, Montana State University, Bozeman, Mont., 1974. H. M. Ortmer and E. Kantuscher, Talanta, 22, 581 (1975). T. H. Runnels, R. Merryfield, and H. 8. Fisher, Anal. Chem., 47, 1258 (1975). K. C. Thompson, R. G. Gooden, and D. R. Thomerson. Anal. Chim. Acta, 74, 289 (1975). B. V. L‘vov: “Atomic Absorption Spectrochemical Analysis”, Elsevier. New York, N.Y., 1970. W. J. Moore: “physical Chernisby”, Prentice Hall, knglewood Cliffs, N.J.. 1955, 2nd ed, p 448: H. S.Carslaw and J. C. Jaeger, “Conduction of Heat in Solids”. Oxford, New York. N.Y., 1947. R. Woodriff and G. Ramelow, Specfrochim. Acta, Part B , 23, 665 (1968). R. Woodriff and R. Stone, Appl. Opt., 7, 337 (1968). R. Woodriff and M. MarinkoviB, to be published. M. MarinkoviE and R. Woodriff, App. Spectrosc., 30, 458 (1976). S . Aronson, F. J. Salzano, and D. Bellafiore, J . Chem. Phys., 49, 434 (1968). R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, and K. K. KelleX, “Selected Values of the Thermodynamic Properties of Binary Alloys , American Society for Metals, Metals Park, Ohio, 1973.
RECEIVED for review November 15, 1976. Accepted August 17, 1977. Support of this work by National Science Foundation Grant No. CHE-7615180 and by the Energy Research and Development Administration under Contract No. E(49-18)-1811 is gratefully acknowledged.
