The Simulation of Aerosol Growth by Coagulation Chung H. Ahn and J. W . Gentry* Department of Chemical €ngineering, University of Maryland, College Park, Maryland
A mathematical model was developed to describe the growth of aerosols generated by condensation of alkaline halide vapors. A computer code employing the invariant functional form model of Ahn was written to solve the mathematical model. The calculation showed that the simulated particle size distributions were not sensitive to the assumed initial values of the distribution. They also showed that at higher oven temperatures the particles became significantly larger, but the number concentration increased only slightly. The simulations were compared with available experimental data.
Introduction Perhaps the single most significant property of aerosols characterizing their behavior is particle size. The collection efficiency of filters, cyclones, and electrostatic precipitators is primarily determined by the particle size. The chemical reactivity of aerosols, their vapor pressure, their dominant mode for the acquisition of electrical charge, as well as the rate of growth by coagulation are functions of particle size. Since aerosols are not uniform in size, it is necessary to specify the particle size distribution as well as the mean particle size. In this paper, experiments measuring the change in the particle size distribution of particles generated by condensation of alkaline halide vapors in a combustion tube are reported. The simulation of this growth process is modeled, and numerical calculations for the mean particle diameter and distribution parameters are compared with experiment. In the study of particle growth, a time-dependent simulation of the particle size distribution was made. This calculation allowed the evaluation of the effect of flow rate, temperature, and. the temperature profile on the particle size distribution. Model In developing the model it is necessary to find a compromise between a model which is an accurate portrayal of the aerosol generation process and a model which is suitable for mathematical solution. For the model used in this study the following assumptions were made. (1) The particle size distribution function is accurately described by a log normal function with time-dependent parameters. ( 2 ) The temperature, gas velocity, and particle concentration profiles are uniform in the radial direction and have no angular dependence. (3) The total mass of the aerosol generated by this process is proportional to the temperature of the combustion tube wall which is the temperature of the furnace. (4)Aside from thermal coagulation the only mechanism accounting for particle growth or removal is gravitational settling. The model employed in this study is analogous to a continuously stirred tank (CSTR). The height of the system was (*/2)dct where d,, is the diameter of the combustion tube. (5) The rate of coagulation is given by the Muller equation. Furthermore, the assumption of a time-invariant functional form is made. (6) The particle size distribution leaving the furnace is calculated from the mass collected on each stage of an Andersen cascade impactor. The assumption of ideal stages for the impactor is made, as well as the assumptions that the particles were spherical and had a homogeneous composition. The first of these assumptions is primarily based upon
the fact that the log normal distribution has been successfully applied in correlating experimental data (Friedman, 1952; Huber, 1965; Lindauer and Castleman, 1971). Also the standard procedure for the interpretation cf experimental measurements taken with an Andersen impactor is predicated upon the assumption that the distribution is log normal. The fraction of particles having a radius between r and r + br for a log normal particle size distribution is
where the median particle radius (rp) given by
and standard deviation (S) is given by
S L 2 L w (In
y/Yp)2f(Y)gv
(3)
It should be noted that for the log normal distribution the mass median particle radius (rm) and area median particle radius (rA) can be expressed as a function of the standard deviation parameter and the median particle radius. Specifically these relations are ym
= Y, e x p ( I . 5 ~ ' )
(4)
and
v A = Y, exp(S2) (5) Another property of the log normal distribution is that the total mass in the distribution ( M , ) is given by the relation
4 M - - n-p,~,~iV,exp ( 2 . 2 5s') t-3 is the density of the solid and Nt is the total
where ps number of particles in the distribution. Undoubtedly the temperature, gas velocity, and particle concentration do have a radial dependence. Specifically, the gas velocity profile in the tube could be approximated by Poiseuille flow. In addition, the particle concentration does not have angular symmetry, for the gravitational force acts perpendicular to the direction of flow. The primary justification for simplifying the model is that it reduces the dimensionality of the problem from 4 to 2 . An approximate method for accounting for the variation of velocity within the tube can be made by subdividing the cross section of the tube into annuli, each of which has a constant velocity. The growth of the particles within each of the annuli is unaffected by the particles in the adInd. Eng. Chem.,
Fundam., Vol. 13, NO, 4,1974 361
jacent annuli. The velocity in each annulus is taken as the mean velocity calculated from the Poiseuille equation. Xumerical calculations indicated that this correction made very little difference (Fan, 1974). Previous studies with alkaline halides have shown that larger particles are more prevalent a t higher temperatures. (Espenscheid, et al., 1964; Gentry, 1969; Jacobsen, 1964). This result was confirmed by the electron micrographs of sodium chloride particles obtained by Matijevic, et al. (1963). The most likely explanation for this phenomenon is that since the vapor pressure of the aerosol increases with temperature, the total mass of the aerosol distribution is also a n increasing function of temperature. Specifically, the model described in this paper makes use of the assumption that the total particle mass of the distribution is directly proportional to the vapor pressure. Secondly, the temperature of the walls of the combustion tube and the contents of the combustion tube are at the temperature of the furnace wall. This is the temperature of the combustion tube corresponding to zero flow conditions. The vapor pressure of lithium bromide crystals can be correlated according to the expression
lo, are given by Ahn (1973). It should be noted that equations of type (9) can be written for every particle radius rl. In general, the values of bS/dt and $ r p / 6 t calculated for any two pairs will be different. The criterion suggested by Ahn and Gentry was to choose these values so that the objective function, @, defined by
is a minimum. The calculational procedure is to use the values of S and r p a t time tn with eq 10 to find the values of 6S/6t and 6rp/6t which give the minimum least-squares ’deviation. The values of the parameter a t time t.v+1 which is equal to tn + At are given by
and
As mentioned above, the coagulation coefficient used in the Muller equation included the Fuchs and Cunningham corrections. With these inclusions the Muller equation has the form (Muller, 1928a,b; Zebel, 1958)
Combining eq 6 and 7 yields the equation
where the superscript O indicates the initial values of the particle concentration, the mean particle radius, and the standard deviation parameter. A* is a proportionality constant. In the model reported here, the only mechanism considered for particle removal was gravitational settling. Theoretical calculations indicated that 3-5% of the mass of the total distribution was removed by this means. Thermal coagulation was the only mechanism considered for particle growth. Both the Cunningham correction for deviation from Stokes’ law and the Fuchs (1964) correction for noncontinuum effects were included. The Muller equation has been widely employed in describing coagulation processes for aerosols. (Lindauer and Castleman, 1970, 1971; Takahashi and Kashara, 1968). Unfortunately, closed solutions exist for only a few idealized coagulation kernels (Swift and Friedlander, 1964; Zebel, 1958), not including Brownian motion, evaporation, or gravitational settling. Ahn and Gentry (1972) suggest that an economical, approximate solution can be obtained by assuming a constant functional form for the particle size distribution function. They selected the log normal function because of its widespread use in correlating experimental data and because of its mathematical convenience. Their calculational procedure is to choose the values of the parameters r p ( t ) , S ( t ) , and N ( t ) which will give the smallest least-squares deviation for the Muller equation. The basic procedure employed in this method is to express the Muller equation and the continuity equation in terms of the time derivatives brp/6t, &Slat, and 6Nt/6t. Using the continuity equation to eliminate the bLVt/6t yields the equations
zoj--I
-6-s- I ‘jdt
-6Y= o 2j
6t
where lo,, II,, and Zz, are complicated integrals. The details of the derivation and the expressions for lo,, Zl,, and 362
Ind. Eng. C h e m . , Fundam., vol. 13, ~ 0 . 41974 ,
Y
’
Y’
2
6r’ - N ( Y , t ) J e 0
(y)
A ( # , Y)N(Y‘, t ) 6 -~
where C ( r ) is the Cunningham-Stokes correction for very small particles and
A(Y’,Y”) =
[’
+
‘(”)
Y’
+
$ (1 + C(v”))] x
with the Fuchs correction 20defined as
The particles leaving the furnace were collected with a n Andersen cascade impactor. The data from the impactor consisted of the mass collected on each stage and the temperature and velocity of the carrier gas. The data were analyzed under the assumption that each stage in the impactor was ideal; that is all particles larger than a characteristic diameter were collected and no particle smaller than the characteristic diameter was collected. The characteristic diameter is taken as the particle diameter where the probability of collection is 50%. Several investigators have reported procedures for correcting this assumption. Theoretical Calculations By carrying nut calculations similar to those carried out by Ahn and Gentry, a number of interesting characteristics of growth can be shown. These include the effect of the temperature profile on the rate of growth, whether increasing the mass concentration (higher generation tem-
-a
-
zan2 yazo-
0 T * 0 T * 6
T:
qf+qx+c, Qx + c a
c.
6
N(O1 = IO*Pf\RTICLES/CC
B(O) * 1.75 no) = 0.9
K,.15: -e , , , ,,
0.10-
,
,
,
, , , ,
,
, ,,
,
,
TIME (SECONDS)
T I ME
Figure 1 . Piumber median radius as a function of time and tem-
perature profile.
5x16 01
1
02
05
10
20
,
1
50
1
1
1
0.1
100
0.2
perature profile. perature) results in larger particles, a higher particle concentration, or both. These calculations would allow qualitative comparisons regarding the effect of flow rate on particle size. The primary difference between the simulations presented here and the earlier calculations is that temperature and viscosity are functions of temperature. Since the time is taken as the quotient of the distance from the combustion boat divided by the gas velocity, it is necessary to know or measure the temperature profile within the combustion tube. The variation in temperature alters the Muller equation in three important ways. First, the thermal coagulation coefficient is proportional to T / q . Secondly, the Fuchs correction has a temperature dependence of t T / v . Thirdly, the rate of gravitational settling is proportional to 7 - l . Underlying these calculations is the basic assumption that the coagulation process is given by the Muller equation. The oven temperature affects the particle size distribution in two ways. The higher the oven temperature, the greater the mass of particles contained in the distribution. Secondly, the temperature profile within the tube is altered. It is this second effect that is illustrated in Figures 1 and 2. In Figure 1 the number median diameter is plotted as a function of time for three different temperature profiles. In Figure 2 the particle concentration is plotted for the same three temperature profiles. The temperature profiles were chosen to model the experimental profile. The profiles were parabolic, linear, and uniform. As is apparent from Figure 1, there is no significant difference in the simulations for the three temperature profiles for times less than 20 sec. From Figure 2 , it is evi-
IO
TI ME
TIME (SECONDS)
Figure 2. Particle concentration as a function of time and tem-
(
SECONDS)
Figure 3. Ratio of particle concentration at different temperatures as a function of time.
2.0
50
10.0
20 0
(SECONDS)
Figure 4. Ratio of particle radius at different temperatures as a
function of time. dent that there are small changes in the number concentration after 2 sec. (The experimental residence time is 2-5 sec.) On the basis of these figures, it can be concluded that the temperature profile does not play an important role in the particle size distribution, although slight errors can arise in the number concentration. The more significant temperature effect is the increase in particle mass that occurs a t higher oven temperature. Equation 8 allows comparisons to be made at different temperatures. However, this equation does not specify whether a higher oven temperature means a higher initial concentration or larger particles. The importance of determining how the excess mass a t higher temperature is distributed between higher initial concentrations and larger initial radii can be assessed from Figures 3 and 4. In Figure 3 the ratio of the number concentration a t temperature T2 (or 5”s) to the number concentration a t temperature T I is plotted as a function of time for two cases. In the first case, the initial particle concentration ratio is unity, and the greater mass in the distribution generated with the higher oven temperature (7’2) is manifested by a larger initial radius. In the second case, the initial ratio of the mean particle radii is unity, and the distribution a t the higher temperature has a correspondingly higher initial particle concentration. The calculations were carried out for two different temperature ratios of
T2 -
TI
689°C 642°C
and
A T =666°C TI 642°C Ind. fng. Chern.,Fundam., Vol. 13, No. 4 , 1974
363
'"OF
0
5x16 01
05
02
2.0
1.0
100
5.0
TIME (SECONDS)
X
i0.251
0.1
A 0
i
0.5
1.0
2.0
50
iao
20.0
TIME (SECONDS)
Figure 6. Kumber median radius as a function of time and standard deviation parameter ( E = e x p ( S / v 2 ) ) .
In Figure 3, the dashed line represents the simulations a t the higher temperature. The figure suggests two conclusions. First, the particle concentration for a given total distribution mass approaches an asymptotic value. Secondly, the particle concentration tends to an asymptotic value which is a slight function of the initial concentration. The importance of the first of these conclusions is that the assumed values of parameters of the initial particle size distribution are not important in determining the final parameters of the distribution. The second conclusion implies that the excess mass in the distribution a t higher temperatures is present in larger particles rather than higher concentrations. These conclusions are reinforced by the results presented in Figure 4, where the ratios of the number median radius are plotted as a function of time. Again the solutions for the limiting cases of equal initial concentrations and initial number median diameters apparently converge to a single value. This convergence occurs within 2 sec, which is shorter than the residence times of the experiments. Calculations of the standard deviation parameter for the two different limiting cases in Figures 3 and 4 indicated that there was no perceptible difference in the standard deviation parameter. These calculations indicate that approximately 10% of the excess mass is distributed in higher concentrations whereas 90% is in larger particles for distribution with greater mass. There was no apparent effect of temperature on the relative distribution of the excess mass. Another indication of this conclusion is the simulation presented in Figures 5 and 6. In the figures the particle concentration and number median radius are presented as 364
80
IX)
I60
m2+028090
(MHUTES)
Figure 7. Comparison of simulation for particle concentration with experiment.
a function of time for different values of the standard deviation parameter but the same initial values of number concentration and number median radius. From eq 8, it is evident that the total mass concentration is proportional to e~p(2.255'~). Also, Ahn showed that the standard deviation parameter approached a constant value ( a result that was confirmed in this study). It is evident from Figures 5 and 6 that almost all the additional mass is transformed into larger particles, while the number concentration shows no difference.
B(O1 : 1.4 BIOI = 1.5 BIOI 1.6 B(01 = 1.7
0.2
40
TIME
Figure 5. Particle concentration as a function of time and standard deviation parameter ( B = e x p ( S / v 2 ) ) .
0
0
WrrHouT NCHS CORRECTlON mM FKHS C O R R E C W W A D E EXPERIMENTAL DATA
Ind. Eng. Chern., Fundarn.,Vol. 13, No. 4 , 1974
Comparison with Experiment The question remains as to how well the simulations compare with experiment. The simulations were compared with experimental measurements of Ahn (1973), for lithium bromide aerosols and Ranade, et al. (1973), for lead chloride aerosols. Although the aerosols were generated by the same method, the two experiments were significantly different. The Ahn experiment had very short residence time (less than 10 sec), whereas the Ranade experiments had a duration of over 1 hr. The mass median diameter was measured by Ahn using an Andersen cascade impactor. The lead chloride particles were sized with an electron microscope. Notwithstanding these differences, there were striking similarities. Both experiments showed a relatively high standard deviation parameter (between 0.9 and 1.1) rather than the theoretical value from the simulation of approximately 0.6. In Figure 7 , the number concentration for the lead chloride particles is presented as a function of time. The initial particle concentration, radius, and standard deviation were reported. Using these data, two curves were generated, one including the Fuchs correction while the second neglected this effect. There were no adjustable parameters or unknown variables in the simulation. Two conclusions are apparent-inclusion of the Fuchs correction is significant and agreement with theory is excellent. It is interesting to note that in this experiment the particles were so small that gravitational settling was not significant. The prediction that the excess mass of particles results in larger particles rather than higher concentration was qualitatively confirmed. In the experimental design of Gentry and Ahn (1974), measurements were taken a t four flow rates and four temperatures. These data were correlated with the relation Y, = a,
+ bT
where the coefficient a, is dependent on the flow rate. The result of the experiments were that b was essentially
FLOW RATE 0.65 C N
FLOW RATE 0.43 CFM B ( 0 ) ' 1.75
m
W O l = 1.75 N(O) - 0 . 3 2 1 X l d ~ ( O ~I 0 . 0 5 6 p x EXPERIMENTAL DATA
N(0) = 0 . 3 2 1 XlO'
r(o) =O.O56p A : EXPERIMENTAL DATA
1.00
1.02
1.06
1.04
IT
1.08
1.10
1.12
116
1.14
RATE 0.54CFM B ( 0 ) = 1.75 N(0) = 0 . 3 2 I X I O *
Fu;w
-i
-028-
i9
0 M P E R M N l Z L DATA
30.24-
rio)
\
3
0.C66 p
&om 30.16-
o . 1 2 h i J 2
'
1104
l k '
'
IT
1
1.02
I.O*
'
~
1.04
1
~
LO6
I/T x
0.31 -
z
1
x lo3 ( I P K )
Figure 8. Comparison of simulated values of mass median diameter as a function of temperature with experiment.
K
a12 1.00
I.'lO
'
1.;2
'
I.;.?
'
11.;
x 10' (II'K)
Figure 9. Comparison of simulated values of mass median diameter as a function of temperature with experiment. independent of the flow rate, and equal to 0.004 p/"K. This compares very favorably with the theoretical value of 0.0043 r/"K for lithium bromide. T h e theoretical value of 0.0043 is derived as follows. From eq 8 one has the ratio of the total mass generated a t two temperatures Tz and T I
If the concentration and standard deviation parameter are taken to be independent of the generation temperature, the equation reduces to
The coefficient B1 can be obtained from the vapor pressure of lithium bromide. The variation in temperature was less than 5%, so that as a first approximation
1
t
1.08
l
1.10
1
1.12
1
I14
)
I
I.
18 ( I P K )
Figure 10. Comparison of simulated values of mass median d i ameter as a function of temperature with experiment. However, from the results described in Figures 3 and 4, it is apparent that the distribution parameters after 2-5 sec are not strongly dependent on the initial concentration but approach an asymptotic value. As a consequence the simulations for all flow rates and temperatures in the study are dependent upon only one variable estimated directly from the experiment. Once the initial total mass is set for one temperature, the simulation variables are determined for all experimental conditions. The mass ratios a t different temperatures are set by the ratios of the vapor pressure. Comparison of the simulation with the experimental data is given in Figures 8, 9, 10. As is apparent from these curves the simulation gives good agreement with the experimental data.
Conclusions A procedure for estimating the particle size distribution for aerosols generated by condensation was developed using the approximation of an invariant particle size distribution. The computer algorithm based on this method proved to be accurate and efficient. Simulations based on the algorithm indicated that the temperature profile in the generation chamber did not play an important role in the particle size distribution. Several tests were carried out for particle distributions having the same initial total mass but different particle size distribution parameters. The resulting particle size distribution approached an asymptotic distribution independent of the initial distribution. The simulation predicts that for distributions resulting from experiments conducted a t higher oven temperatures, the excess mass will result in larger particles and not higher particle concentrations. Comparisons of the predicted distribution were made with the experimental measurements of Ranade and Ahn. The simulation gave good agreement with the measurements for particle concentration of Ranade and the measurements of the mass median diameter by Ahn. Also, the experiments of Ahn supported the contention that larger particles and not higher concentrations will result a t higher oven temperatures. Quantitative agreement of the mass median diameter with temperature was satisfactory. Literature Cited
A more stringent test is to compare the predictions of the model as a function of flow rate and temperature with the measured values of the mass median diameter. In making the comparison, it is necessary to estimate the initial value of the distribution parameters. These values occurring a t the midpoint of the oven cannot be measured directly.
Ahn, C. H., Ph.D. Thesis, University of Maryland, 1973. A h n , C. H., Gentry, J. W.. Ind. Eng. Chem., Fundam., 11, 483 (1972). Espenscheid. W . F.. Matijevic, E., K e r k e r , M . , J . Phys. Chem.. 68, 2831
(1964).
Fan, K. C., unpublished data, 1974. Friedman, S., Chem. Eng. Progr., 48, 1 1 8 (1952). Fuchs, N. A.. "The Mechanics of Aerosols," Pergamon Press, London, 1964. Gentry, J. W.. Ph.D. Thesis, University of Texas at A u s t i n , 1969.
Ind. Eng. Chem., Fundam., Vol. 13, No. 4 , 1974
365
1
~
~
Gentry, J. W.. Ahn. C. H.. Atrnos. Environ., 8, 765 (1974). Huber, R. S., "An Approximate Solution to the General Equation for the Coagulation of Heterogeneous Aerosols," AI-AEC-MEMO-12880 (1965). Jacobsen. S.. M.Sc. Thesis, University of Texas at Austin, 1964. Lindauer, G. C.. Castleman, A. W., Jr., Nucl. Sci. Eng., 42, 58 (1970). Lindauer. G. C., Castleman, A. W.. Jr., Nucl. Sci. Eng., 43, 212 (1971a). Lindauer, G. C., Castelman, A. W.. Jr.. J. AerosolSci., 2, 85. (1971b). Matijevic, E.. Espenscheid. W., Kerker, M.. J. Colloid Sci., 18, 91 (1963).
Muller, H., KolloidBeih., 26, 257 (1928a). Muller, H., KolloidBeih.. 27, 223 (1928b). Ranade, M. B., Wasan, L3. T., Davies. R., AlChE J., 20, 273 (1974) Swift, D. L., Friedlander, S.K., J. ColloidSci., 19, 621 (1964) Takahashi, K., Kashara, M.,Atrnos. Environ., 2, 4441 (1968). Zebel. G.. KolloidZ.. 157. 3 7 (1958).
Received for reuiew December 6, 1973 Accepted July 22, 1974
Distribution Coefficient Correlations for Predicting the Extraction Equilibria of Lutetium-Ytterbium and Lutetium-Thulium Binary Rare Earth Mixtures between Di(2-ethylhexyl) Phosphoric Acid in Amsco and Aqueous Hydrochloric Acid Solutions Norman E. Thomas and Lawrence E. Burkhart* Ames Laboratory, USAEC, and Department of Chemical Engineering, lowa State University, Ames, lowa 50070
Equilibrium data were obtained for the extraction of the binary rare earth mixtures lutetium-ytterbium and lutetium-thulium from 5 M HCI-HpO solutions by 1 M di(2-ethylhexyl) phosphoric acid in Amsco odorless mineral spirits. The separation factors for the above binary mixtures were substantially constant with respect to mixture composition, the arithmetic averages for and PL",Y~being 6.96 and 1.82, respectively. Empirical correlations for predicting total distribution coefficients for the Lu-Yb and Lu-Tm mixtures were developed. These correlations are comparable to one developed earlier for Yb-Tm mixtures and have direct applications in the design and optimal control of heavy rare-earth fractionation columns in which di(2-ethylhexyl) phosphoric acid in Amsco and HCI in water are employed as liquid solvents.
Introduction For the economical design and control of commercial solvent extraction processes for the fractionation of multicomponent rare earth mixtures, it is essential to have a t hand reliable correlations for predicting the complex phase equilibria involved. In an earlier study (Thomas, et al,, 1971), equilibrium data were successfully correlated for the extraction of the binary rare earth mixture ytterbium-thulium from 5 M HCl-HzO solutions by 1 M di(2ethylhexyl) phosphoric acid [(CsH1,0)2PO(OH), hereafter designated as HDEHP] in Amsco odorless mineral spirits. It is the purpose of this paper to describe a comparable method of correlating equilibrium data for the extraction of the binary rare earth mixtures lutetium-ytterbium and lutetium-thulium from aqueous hydrochloric acid solutions by 1M HDEHP in Amsco. Experimental Procedures The rare earth oxides Lu203, Yb203, and Tmz03 used in this study were obtained from the Rare-Earth Separations Group of the Ames Laboratory of the U. S. Atomic Energy Commission and had purities, as determined by emission spectroscopy, of greater than 99.9% with respect to the presence of other rare earths. The HDEHP, obtained from Union Carbide Corp., was 98.8% monoacidic and was used without further purification. The diluent, Amsco odorless mineral spirits (a nonpolar aliphatic hydrocarbon composed predominately of isoalkanes with a boiling range from 178 to 199"C), was obtained from the American Mineral Spirits Co. Throughout this study a 1 M solution of HDEHP as monomer in 366
Ind. Eng. Chern., Fundarn., Vol. 13, No. 4 , 1 9 7 4
Amsco (1 M HDEHP-Amsco) was employed as the organic solvent. The hydrochloric acid was of reagent grade purity. Experimental procedures have been described previously (Thomas, et al., 1971; Thomas and Burkhart, 1974) for: (a) the preparation of rare earth chloride stock solutions and initial aqueous feed solutions, (b) the equilibration of aqueous and organic phases, (c) the recovery of extracted rare earths by back-extraction, (d) the measurement of aqueous-phase acidities, and (e) the determination of total rare earth concentration in the equilibrated phases. The compositions of the rare earth mixtures present in the initial aqueous feed solutions, the equilibrium aqueous phase, and the equilibrium organic phase were determined as follows. Aliquots of the organic back-extract, the initial aqueous phase, and the equilibrium aqueous phase, each of which contained approximately 3-4 mmol of rare earth chloride, were pipetted into 250-ml beakers, diluted to approximately 75 ml with distilled water, and then adjusted to pH 2.0 by the addition of either NH40H or HC1. The samples were then heated to boiling on a hot plate, and the rare earths precipitated as the oxalates by the addition of saturated oxalic acid. The oxalates were filtered through highly retentive filter paper and washed several times with a solution of 0.32 M "03-0.11 M H2C204H2O. The washed oxalate samples were placed in porcelain crucibles, dried for 6 hr at 110°C in an automatic drying oven, preheated for 2 hr in a 400°C oven, and finally ignited to the oxides by heating for a t least 12 hr a t 800°C in an electric muffle furnace. Carefully weighed samples of the freshly ignited oxides, normally 300 mg, were each dissolved in 1 to 2 ml of 72%
