682
Ind. Eng. C h m . HPS 1990, 29, 682-690
Litz, L. M. A Novel Gas-Liquid Stirred Tank Reactor. Chem. Eng. Prog. 1985, 36-39. Mann, R. Gas-Liquid Contacting in Mixing Vessels. Industrial Research Fellowship Report; The Institution of Chemical Enpineers: Rugby, England, 1983; 108 pp. Musil, L.; Vlk, J. Suspending Solid Particles in an Agitated Conical-Bottom Tank. Chem. Eng. Sei. 1978, 33 ( 8 ) , 1123-1131. Narayanan, S.; Bhatia, V. K.; Guha, D. K.; Rao, M. N. Suspension of Solids by Mechanical Agitation. Chem. Erzg. Sei. 1969, 24 (21, 223-230. Nienow, A. W.Suspension of Solid Particles in Turbine Agitated Baffled Vessels. Chem. Eng. Sci. 1969, 23 (121, 1453-1459. Rieger, F.; Ditl, P. Suspension of Solid Particles in Agitated Vessels. Proc. 4th European Conference on Miring; BHRA Fluid Engineering: Cranfield, Bedford, England, 1982; paper H1.
Schwartzberg, H. G.; Treybal, R. E. Fluid and Particle Motion in Turbulent Stirred Tanks. Ind. Eng. Chem. Fundam. 1968, 7, 1-12. Ulbrecht, J. J., Patterson, G. K., Eds. Mixing of Liquids by Mechanical Agitation. Chemical Engineering;Concepts and Reviews Series; Gordon and Breach Science Publishers: New York, 1985; Vol. 1. Weisman, J.; Efferding, L. E. Suspension of Slurries in Mechanical Mixers. AIChE J. 1960, 6, 419-429. Zwietering, T. N. Suspending of Solid Particles in Liquids by Agitators. Chem. Eng. Sei. 1958, 8 ( 3 ) ,244-253.
Received for review December 2, 1988 Revised manuscript received August 2, 1989 Accepted November 20, 1989
Determination of Activity Coefficients via Microdroplet Evaporation Experiments Theresa M. Allen, Daniel C. Taflin, and E. James Davis* Department of Chemical Engineering, BF-10, University of Washington, Seattle, Washington 98195
It is demonstrated that activity coefficients of binary miscible solutions can be determined by means of microdroplet evaporation experiments in which a microdroplet is suspended in an electrodynamic balance. Weight loss and light-scattering measurements were used to determine the size and composition as functions of time. The light-scattering measurements include phase functions (scattering intensity as a function of the angle) and optical resonance spectra. The latter are extremely sensitive to the size and refractive index of the microdroplet. T h e microdroplet technique was applied t o binary pairs consisting of 1-bromododecane, 1,8-dibromooctane, hexadecane, and heptadecane t o obtain the activity coefficients of each component from the data. These results are compared with application of the Gibbs-Duhem equation to test the thermodynamic consistency of the experimental results. The design of processes for the production, separation, and purification of petrochemicals, natural products, pharmaceuticals, food products, and other chemicals requires accurate phase-equilibrium data. Techniques have been and are being developed to predict multicomponent thermodynamic properties from information on binary pairs, but as pointed out by Donohue et al. (19851, accurate data are needed for the development of new theories. Frequently infinite-dilution activity coefficients are used to fit binary vapor-liquid equilibrium data to expressions for the excess Gibbs energy, but traditional methods for determining such activity coefficients are of questionable quality (Lobein and Prausnitz, 1982). Measurement of activity coefficients by classical vapor-liquid equilibrium techniques is exceedingly time consuming if wide ranges of composition, temperature, and pressure are involved, so in recent years two techniques have been developed to measure infinite-dilution activity coefficients. For binary pairs in which both species are volatile, differential ebulliometry is particularly useful (Nicolaides and Eckert, 1978; Lobein and Prausnitz, 1982). The gas chromatographic technique used by Donohue and his co-workers to study alkane-alkane, alkane-aromatic, and aromatic-aromatic pairs is more suitable when the solvent is relatively nonvolatile compared to the solute. The development of the electrodynamic balance for the study of microdroplets has made it possible to develop a new method of measuring thermodynamic properties, and several investigators have used it to measure activities of water in electrolyte salt solutions. Rubel (1981) studied aqueous solutions of phosphoric acid, and Richardson and his colleagues examined aqueous solutions of LiBr (Rich0888-5885/90/2629-0682$02.50/0
ardson and Kurtz, 1984), of LiI (Kurtz and Richardson, 1984), of (NH,)$O, (Richardson and Spann, 1984), and mixed solutions of (NHJ2S04 and NH4HS04(Spann and Richardson, 1985). By operating an electrodynamic balance under vacuum, Richardson et d. (1986) measured the vapor pressure of sulfuric acid over concentrated aqueous solutions for the temperature range 263-303 K. Cohen et al. (1987a,b) measured water activities for several singleand mixed-electrolyte solutions. Tang and his co-workers measured the water activities of NaCl-H20 and KCl-H20 systems (Tang et al., 1986) and obtained vapor-liquid equilibrium data for dilute nitric acid solutions (Tang et al., 1988). In all of these studies, the aqueous solutions were brought into contact with humid air, and the equilibrium state of the droplet was determined by weighing it, using the dc voltage required to levitate the droplet to determine the weight. Nonequilibrium experiments can also be used to measure thermodynamic properties of microdroplets. Rubel (1981) used an electrodynamic balance to measure the evaporation rates of multicomponent oil droplets, obtaining an “effective vapor pressure” of the mixture. The first electrodynamic experiments with binary organic microdroplets designed to obtain thermodynamic information were made by Ravindran and Davis (1982), who showed that the evaporation of submicrometer droplets of dioctyl phthalate and dibutyl phthalate followed ideal solution behavior. Their results were substantiated by Rubel (1982) using droplets of the order of 100 pm. Ravindran and Davis selected the dioctyl/dibutyl phthalate system with the expectation that it would show ideal solution behavior because of the similarity of the C 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 683
dc CONTROL VOLTAGE LASER PORT I
I
PHOTODIODE
SOURCE
II
I
II
dc SOURCE
Figure 1. Cross section of the electrodynamic balances used in the experiments.
molecules, and their results as well as those of Rube1 agreed well with the assumption of unit activity coefficient. Their work led us to reexamine the suitability of droplet evaporation experiments for determining activity coefficients, using much better light-scattering techniques to follow size and refractive index changes. It is the purpose of this study to show that microdroplet evaporation measurements can be used to measure activity coefficients of both species of a binary solution using angular scattering and optical resonance light-scattering techniques to measure the size and composition simultaneously as evaporation proceeds.
Apparatus The electrodynamic balances used in these experiments had bihyperboloidal electrodes shown in cross section in Figure 1. An ac potential, typically 500 V at 20-100 Hz, was applied to the ring electrode to center the microdroplet. For a negatively charged droplet, a dc potential, +Vd0 was applied to the upper end cap, and -Vdc was applied to the lower end cap. No time-average vertical force is produced by the ac field, so the dc field was adjusted to levitate the microdroplet at the null point (the midplane) of the balance, following the vertical force balance, co4vdc/zo = mdg - 6aakUc
(1)
The minimum distance between the end cap electrodes, 2z0, was 19.0 mm, and the geometrical constant of the device, Co, was determined by stability measurements described by Davis (1985) to be 0.79 f 0.02 for one of the balances and 0.84 f 0.02 for the other. The constant depends on the configuration of viewing ports and the size of holes in the dc electrodes. In the absence of any convective velocity or gas flow in the balance chamber (Vc = 0), the droplet mass is given by md = COq vdc/gzO
(2)
Thus, for a microdroplet evaporating with constant surface charge, q, in a stagnant gas, the mass is directly proportional to the dc suspension voltage. When a gas flow is used to sweep out vapor, eq 1 must be applied to relate the mass to the levitation voltage. The gas velocity can be determined by the calibration procedure outlined by Davis et al. (1987). One of the balances was equipped with the automatic feedback control system shown in Figure 1, and the other was operated by manually adjusting the dc voltage as evaporation proceeded. For rapid evaporation, automatic
control is necessary, but in these experiments, manual control was adequate. A small droplet of known composition was introduced into the balance chamber through a hole in the top electrode by electrifying (2-5 kV of dc) a small diameter flat-tipped hypodermic needle containing the solution. The initial diameter of a droplet was of the order of 100 km. The charged droplet was trapped in the ac field and oscillated until the dc voltage was adjusted to balance the gravitational force. This required a few seconds. The microdroplet then remained stationary at the null point and was illuminated by a laser beam that was introduced through a port in the side of the ring electrode. A vertically polarized helium-neon laser (A = 632.8 nm) was used for some of the experiments, and an unpolarized helium-neon laser was used for others. A photomultiplier tube (PMT) was mounted at right angles to the laser beam to record the resonance spectrum, and a 512 element photodiode array (Reticon 512EC/17) was mounted on the ring electrode to record the phase functions (intensity versus scattering angle) of the scattered light. Details of the designs have been presented by Davis (1987) and Taflin et al. (1989). The 512 photodiode elements were read sequentially and sent to an A/D converter (IBM data acquisition and control adapter) and recorded on floppy disks on a microcomputer. The dc suspension voltage and the output of the P M T were recorded continuously on a dual-channel oscillograph.
Light Scattering The phase functions and resonance spectra were analyzed by comparing them with the Mie theory. Details of the Mie theory can be found in treatises by Van de Hulst (1981), Kerker (1969), or Bohren and Huffman (1983). Resonances are the natural modes of electromagnetic oscillation of a dielectric sphere in the electromagnetic field generated by the laser beam. Resonance spectra, resulting from changes in the size and/or refractive index, are measured by means of a photodetector mounted at a fixed angle. Phase functions (or angular scattering data) and resonance spectra both provide information from which size and refractive index can be determined, but the latter are much more sensitive to the optical parameters than the former. Suffice it to say here that for a nonabsorbing homogeneous sphere illuminated by a monochromatic source the intensity of the scattering light is a function of the polarization of the laser beam, the scattering angle (e), the nondimensional size ( a = 2aa/A), and the ratio of the refractive index of the sphere to that in the surrounding medium (m). Figure 2 shows three typical phase functions for an evaporating droplet of l,&dibromooctane (DBO) and hexadecane (HXD) obtained with unpolarized light. The three phase functions shown correspond to times of 72.4,1947, and 4673 s during the evaporation. Using the peak-counting method of Davis and Periasamy (1985) to interpret the data, the radii indicated on the figure were determined. Note that the intensity and the number of peaks in the angle range covered decrease with size. The number of peaks in a given range of angles is very sensitive to size but less sensitive to refractive index. It is estimated that the size can be determined to about f0.570 by peak counting. Figure 3b shows a resonance spectrum for the experiment corresponding to Figure 2. For this relatively large droplet ( a = 280 near the middle of the spectrum), the resonances are not as sharply defined as they are for smaller spheres; the spectrum appears to consist of broad undulations with sharp resonances superposed. The use
684 Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 1
100
80
L-
t v)
0 2
z w
2
c-
0
f
40
L5
n TIME 20
-I^
I
I
I
I
I
I
I
( c ) M I E THEORY FOR m.14352 0 40
35
55
50
45
SCATTERING
60
65
70
ANGLE
Figure 2. Phase functions for an evaporating microdroplet of DBO/HXD. 50
1 7c5
w W
I 23'
5
I
I
I
I
I
206
203 5
203
232 5
2r2
I
I 24'
5
a , DIMENSIONLESS SIZE
Ie
& 25
Figure 4. (a) Levitation voltage, (b) experimental resonance spectrum, and (c) calculated resonance spectrum for an evaporating BDD/HPD microdroplet.
> 0
- 0
using phase functions obtained during the time period associated with the spectrum. An estimate of the composition was made using the voltage data shown in Figure 3a, invoking eq 1 and writing the droplet mass as md = 4na3p/3. The result can be rearranged to give
L
pa2 = AVdc/a 4- B
* I-
iC)
I 282
5
where A = 3qCo/8~gzo and B = 9pUJ2g. Thus, the droplet density was determined from the measured size and levitation voltage. The mixture density can be approximated by
MIE THEORY
c 28'
23:
5
is0
(4) + X2/P2r1 to yield the composition of the droplet. Because of the large density difference between the two pure components used in this study, this procedure gave an accurate estimate of the composition to begin matching the resonance spectrum with theory. We note that the charge on the droplet can be determined by application of eq 3, for a plot of pa2 versus Vdc/a should yield a straight line with slope A. From knowledge of the geometrical constants Co and zo, the charge is calculated from the measured value of A. Taflin et ai. (1989) used this method to study the droplet fission that occurs when a droplet reaches the Rayleigh limit of charge. For the droplets involved in our activity coefficient experiC, ments, the charge was typically of the order of which corresponds to 6 X lo5 elementary charges. From the estimated conposition, the refractive index of the mixture was calculated by using a mixing rule based on the volume fraction, y , P =
t 281
279
(3)
5
2';
2:8
5
S I Z E FARANIETER
Figure 3. (a) Levitation voltage; (b) experimental resonance spectrum, and (c) calculated resonance spectrum for the droplet corresponding to Figure 2.
of unpolarized light yields a superposition of resonances associated with vertical and horizontal polarization. Figure 4 shows results obtained near the end of an experiment with 1-bromododecane (BDD) and heptadecane (HPD) using vertically polarized light. In this case, the droplet was smaller ( a = 203 near the middle of the spectrum), and the resonances are more sharply defined. The levitation voltages shown in Figures 3a and 4a change very little over the time spans associated with the figures. Structural or optical resonances, which were first examined experimentally and theoretically by Ashkin and Dziedzic (1977), are extremely sensitive to both size and refractive index, and each of these parameters can be determined to about 4 parts in IO5 (Chylek et al., 1983) by analysis of resonance spectra. This great sensitivity to the optical parameters makes it very difficult to interpret resonance data, particularly when both size and refractive index change as they do with evaporating binary microdroplets. Because the inversion of resonance data to obtain the size and refractive index is not trivial, we shall sketch the procedure used. A set of resonances such as those shown in Figure 3b were analyzed by first estimating the size by
(Xl/Pl
m =
mlYl+ m 0 2
(5)
This mixing rule was found to be the best of three interpolation formulas based on the pure component refractive indices that we examined. Figure 5 is a plot of the refractive index data we obtained for binary mixtures of DBO/HPD and BDD/HPD as a function of composition based on the mass fraction, x, volume fraction, y , and mole fraction, z. Interpolation based on the volume fraction is clearly the best choice of the three.
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 685 Table I. Physical Properties and Evaporation Constants for chemical Mi pi, kg/m3 1,8-dibromooctane 272.03 1463.5 1-bromododecane 249.24 1039.9 heptadecane 240.48 776.9 hexadecane 226.45 771.1
the Pure Components Used in the Experiments mi Vi, m3/ kmol -Sij, pm2/s lo1*&, kmol/(m-s) 1.4977 0.18588 0.389 1.046 1.4569 0.239 68 0.338 0.705 1.4352 0.309 54 0.0431 0.0651 0.293 67 0.132 0.225 11.4345 1200
f
0.6 0.4 0.8
N
x X-
0.2
-
0 1.43
to00
ri i .O 00
m,
I’ N
800
1
1-
i
DBO BDD
E,
INTERPOLATION
I
I
I
J
1.46
1.48
1.49
150
I 1.47 REFRACTIVE INDEX
I 1.45
I 1.44
1
X,MASS FRACTION y, VOLUME FRACTION 2 , MOLE FRACTION
-LINEAR
-
\
Figure 5. Interpolation of refractive index data for DBO/HPD and BDD/HPD binary mixtures based on mass fraction, volume fraction, and mole fraction.
By use of the estimated values of CY and m, the Mie theory was applied to compute the resonance spectrum over some relatively small range of sizes, typically for six resonances. The refractive index was assumed to change monotonically over this range. The computed and experimental resonances were then compared. If both the global structure and fine structure of the spectra did not match, small perturbations in CY and m were introduced, and the procedure was repeated. Once a close match was achieved, the next set of six resonances was processed, and this procedure was used to determine the size and refractive index over the entire course of the experiment. The theoretically computed spectra corresponding to Figures 3b and 4b are shown in Figures 3c and 4c, respectively. The refractive indices obtained from the matches are also shown. Figure 4 corresponds to a time late in the experiment when the droplet was depleted of BDD. The agreement between theory and experiment is particularly good for Figure 4, and it is quite acceptable for Figure 3. By perturbing CY and m from the “best fit” values, we estimate that CY and m can be determined to about 5 parts in lo4 when both change, but when only one of the parameters varies it can be determined to about 4 parts in IO5. It is much easier to obtain a match between theory and experiment when the refractive index does not change, but the large change in mass that occurs during an evaporation experiment introduces a new problem. That problem is the existence of nonvolatile contaminants in the original material. By use of spectroscopic grade chemicals with a purity of 99.99%, if all of the impurities are nonvolatile, a radius change from 30 to 3 km increases the contaminant concentration from 0.01 ‘30to lo%, an unacceptable level of contamination. As a result, data obtained near the end of an experiment are not useful. The presence of contaminants was explored by permitting binary droplets to evaporate for many hours after an experiment was completed. In some cases, a very low volatility residue was obtained, and the residue scattered light as a sphere. In other cases, the final product was crystalline, as evidenced by particle rotation that caused the particle to “twinkle” when observed through the microscopic eyepiece. There was no evidence that the impurities were surface active,
200 1 0
IO
20
30
40
50
I 60
TIME, MINUTES
Figure 6. Evaporation characteristics of the pure Components used in this study.
such as those examined by Taflin et al. (1988). Evaporation Theory A t atmospheric pressure the evaporation process for slowly evaporating species proceeds isothermally at the temperature of the surrounding gas and follows quasisteady-state behavior (Davis and Ray, 1977; Taflin et al., 1988). For two low-vapor-pressure species in a binary droplet, the two diffusional processes in the gas phase are independent because of the low concentration of the diffusing species, so we may write the flux of the ith component as
Ji = &zi?i/a
(6)
where the pure component evaporation constant, &, is defined by 4i = Dijp:/RT, and we have assumed that the concentration of the diffusing species far from the droplet surface is zero, which is achieved experimentally by removing vapor using an inert gas flow (nitrogen) through the balance chamber during the evaporation. The chamber was initially purged with nitrogen to remove any residual vapor. The mass flux is related to the droplet radius by a component balance written for the spherical droplet, which is (7) For pure component isothermal evaporation, Viand $+ are constants, so in this limiting case, eqs 6 and 7 can be equated and integrated to give a2 = ~
0 ’
24iVi(t - to) = ao2 - Si;(t - t o )
(8)
where -Sij is the slope of a plot of a2 versus t - to,and a. is the radius at time t,. Figure 6 shows such a plot of the data for the four pure components used in this study. The experiments were performed at 295 K. Table I lists the relevant physical properties of the pure components and the measured values of diand Si,. Based on the evaporation constant, the brominated compounds are more than an order of
686
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990
magnitude more volatile than heptadecane and are appreciably more volatile than hexadecane. These alkanes and brominated alkenes were selected for several reasons: (i) there is a significant weight change due to the density difference as the more volatile brominated compound evaporates to permit the composition to be determined gravimetrically, (ii) there is a sufficiently large refractive index change as evaporation proceeds to significantly affect the resonance spectrum, (iii) the volatilities of the chemicals are sufficiently different to produce large changes in composition, and (iv) the rates of evaporation are not too fast to permit accurate measurement of the experimental parameters. For evaporation of a binary microdroplet, the radius is a function of the diffusive fluxes of both species; that is,
'( ")
dt 3Vm = -4ra2(J1
+ J2)
The molar volume of the droplet is a function of its composition, and for an ideal solution vm= &V, i
For the binary systems studied here, the deviations of the molar volumes from ideality were found to be less than 1% based on measurements of the densities of the liquid mixtures, so we shall apply eq 10 below. Now eqs 6 , 9 , and 10 can be combined and rearranged to give the following expressions for da2/dt and dz,/dt: and
(2V2 -
V,)ZI2
- V2Z2.l (12)
If the binary solution is ideal (yl = y2 = l),eqs 11 and 12, which are coupled nonlinear first-order differential equations, can be solved numerically to yield a2 as a function of time and z1 as a function of time. For nonideal systems, these equations can be solved algebraically for the activity coefficients in terms of measurable quantities, u2, du2/dt, z1 and dz,/dt; that is, V2a2 dz1 -1 -da2 2 dt 3z,vm dt Y1 = (13) 41Vm and V,a2 dz, 1 da2 2 dt 3(1 - z ~ ) V ,dt Y2 = (14)
+----
42 vro
Here component 1 is considered to be the more volatile component. Note that application of eqs 13 and 14 requires differentiation of the data for u2 as a function of time and for z , as a function of time. These differentiations were carried out by spline-fitting the data, using cubic splines (Kreysig, 1979) of the form G(t) = a3t3+ a2t2+ alt
+ a.
(15)
where G ( t ) = a2(t)or z,(t). Two pieces of information are required to fit one of these functions to a segment of data-the slopes and values of the data at the end points of the appropriate segment. The slopes at the end points were obtained by a least-squares fit of a straight line through three to five points about the node. Each data
,5
-
t i _ _0
50
--
100
I50
TIME MINUTES
Figure 7. Radius (a) and composition (b) of a BDD/HPD binary microdroplet versus time. The solid lines are the predictions based on ideal solution behavior.
set was split into regions spanning about 0.2 mole fraction. The computed activity coefficients are extremely sensitive to the derivatives of a2 and z,. Of the probable error associated with the activity coefficients, about two-thirds is related to the term dz,/dt. Since analysis of the data to determine activity coefficients requires its differentiation, high precision is necessary. The use of optical resonances yields such high precision, for the resonance spectrum provides a continuous record of size and refractive index changes, and the resonances are extremely sensitive to both parameters. Experimental Results Figures 7 and 8 show the radii and compositions as functions of time for the binary pairs BDD/HPD and BDD/HXD, respectively. Also shown on the figures are the theoretical results obtained by solving eqs 11 and 12 under the assumption of ideal solutions (yl = y2 = 1). Both pairs show ideal solution behavior within the limits of experimental accuracy, but such is not the case for DBO/HPD and DBO/HXD. Figure 9 shows that for DBO/HPD the experimental data for both radius and composition do not agree with ideal solution theory, for the droplet evaporates faster than ideal solution theory predicts. Thus, the experimental data were used to determine the activity coefficients of both components in the two systems. T o illustrate the method of analysis of the data, let us examine a complete set of raw data for run 922. Figure 10 shows the levitation voltage as a function of time for the evaporation of a microdroplet of DBO and HXD with initial radius a. = 37.91 pm and initial composition z1 = zDBO = 0.691. The mole fractions determined from the voltage data and the resonance spectrum by the procedure outlined above are shown in Figure 11, and the corresponding a 2 versus time data are presented in Figure 12. By use of the spline-fitting methods outlined above to smooth the data and to permit differentiation, the activity coefficients of DBO and HXD were computed for numerous points in time. Figure 13 shows the results for two separate experiments, runs 915 and 922. For run 915 the
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 687 1001
'
'
0
i
-0.05 0
1
0
> Eo:
N -
'
'
1
100
300
200
Y)
TIME, MINUTES
Figure 10. Levitation voltage as a function of time for an evaporating microdroplet of DBO/HXD for run 922. 0.71
.
.
I
06 0
TIME, MINUTES
Figure 8. Radius (a) and composition (b) of a BDD/HXD binary microdroplet versus time. The solid lines are the predictions based on ideal solution behavior.
0
05
z 2 c
04
f
03
DEO/HXD DATA
0
W
dI
0.8
P
g
IDEAL SOLUTION THEORY 0
no1
(b) COMPOSITION
0.6
02
00 0
20
40
60
80
100
TIME, MINUTES
Figure 11. Mole fraction determined from the resonance spectrum for the microdroplet corresponding to Figure 10.
-
0
N
0.0 0
"
20
'
"
40
' 60
1500 I
EO
100
1
120
I
IDEAL SOLUTION THEORY
1
1000
5
~
I
( a i DROPLET S I Z E
1
30
N
E, (u
0
500
IU
0
20
40
60
EO
100
o DBO/HXD DATA
-
0""' 0
120
40
20
TIME, MINUTES
Figure 9. Radius (a) and composition (b) of a DBO/HPD binary microdroplet versus time. The solid lines are the predictions based on ideal solution behavior.
initial radius was 13.87 pm, and the initial composition was 0.429 mole fraction DBO. We note the great scatter in the data for z1 < 0.1, particularly for run 922, which underwent the largest size change during the evaporation. This scatter is due to impurities in the chemicals used. Because of the contamination, we have disregarded all of the data for z1 < 0.1. The activity coefficient of HXD is very nearly unity over the whole range of composition examined, but DBO shows considerable nonideality. The reproducibility of the results is reasonably good provided that we exclude the data affected by impurities near the end of a run.
60
80
I
100
TIME, MINUTES
Figure 12. Square of the radius as a function of time for the microdroplet corresponding to Figure 10.
Since the binary pair must also satisfy the Gibbs-Duhem equation, z1 d In y1 z2 d In y2 = 0 (16) we can examine the thermodynamic consistency of our results by applying it. Writing
+
In
y1 =
Kz~~/RT
(17)
and In y2 = K z 1 2 / R T (18) we obtained K via point-by-point fitting of In y1versus 22.
688 Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990
3
& w -0 b 0
0 0
4-
- GIBBS-DUHEM E 9
lL
U
DBO
A HXD
31 I
C
O
0
0' 0
On
01
02
04
03
2 1 , MOLE FRACTION
DBO
Figure 13. Activity coefficients of DBO and HXD determined from the data of Figures 11 and 12 and from data for run 915. The solid line represents the activity coefficients of HXD determined for each run by applying the Gibbs-Duhem equation. The Gibbs-Duhem results for both runs lie on the same line.
I- 2 0 -
z
0
I-
u
- GIBBS-DUHEM EQ.
h
' 1
0.0
0.10
0.20
0.30
0.40
z,, MOLE FRACTION DBO Figure 14. Activity coefficients of DBO and HPD determined from evaporation data. The solid line is the activity coefficient of HPC determined by applying the Gibbs-Duhem equation.
Note that eqs 17 and 18 automatically satisfy the GibbsDuhem equation. The activity coefficient yzwas calculated by using eq 16 to obtain the solid line plotted on Figure 13. The solid line consists of a superposition of the results for both experiments, for the calculated values of y2 for the two experiments overlap. The agreement between y2 calculated by using the Gibbs-Duhem equation and the results obtained from analysis of the experiments is quite good. Similar results were obtained for the system DBO/HPD, and they are presented in Figure 14. Again DBO is nonideal, but the straight-chain hydrocarbon, HPD, follows nearly ideal solution behavior. The results appear to be thermodynamically consistent, for application of the Gibbs-Duhem equation yields results for yz in reasonably good agreement with those extracted from the experimental data. Again we have excluded data associated with low concentrations of the volatile component because of the problem of contamination. Although it would be highly desirable to extend the computations to infinite dilution of the volatile component, that is not possible because of the effects of contamination. We have also excluded data near the start of an experiment because some convection was introduced into the balance chamber during the initial phase of the experiment. Furthermore, it was difficult to obtain a unique fit of the resonance spectra for the largest droplets. This is suggested in Figures 3 and 4, for the spectra of the larger droplets associated with Figure 3 are not as well fit as those in Figure 4. It is also clear from the figures that the use of polarized light is preferable.
Discussion These results indicate that the straight-chain hydrocarbon with one bromine atom substituted for hydrogen, 1-bromododecane,forms a nearly ideal solution with either hexadecane or heptadecane, whereas l,&dibromooctane with two bromine atoms is quite nonideal in solution with either hexadecane or heptadecane. This difference in behavior may be attributed to entropic effects associated with the different chain lengths of BDD and DBO compared with HXD and HPD and to the effects of bromination. Although there appears to be no other data in the literature for these specific chemicals, the effects of chain length are demonstrated in the literature (Gmehling, 1977) for the systems cyclohexane/dodecane and cyclohexaneIhexadecane. For the former system, the activity coefficients at infinite dilution are yl,m= 0.90 and y2,.,, = 0.91, and for the latter pair yl,- = 0.80 and y2,== 0.70. This suggests that chain length has a pronounced effect on the activity coefficients. For brominated systems, Gmehling lists data for the systems cyclohexane/l,2-dibromoethane for which the activity coefficients (at infinite dilution) at 293 K are reported to be 4.41 for cyclohexane and 3.51 for the dibromoethane. The system heptane/ butyl bromide is somewhat less nonideal, the activity coefficients at infinite dilution being 1.54 for heptane and 1.42 for the brominated compound at 323 K. For pentane/pentyl bromide, Gmehling reports infinite-dilution activity coefficients of 1.56 for pentane and 1.78 for pentyl bromide at 293 K. Thus, the activity coefficients shown in Figures 13 and 14 are consistent with results for other alkane/brominated alkene systems. We have attempted to fit the data of Figure 13 to the Wilson equation (Prausnitz et al., 1986) point by point to determine the two parameters, A12 and The activity coefficients derived from the Wilson equation for the excess Gibbs energy are given by In y1 = -[ln ( t l+ h12z2)]+
+ and
The parameters determined via point-by-point application of eqs 19 and 20 were not constant over the range of compositions examined (0.15 Iz1 I0.35). If we use average values of A12 and Azl determined in this way, we obtain the results shown in Figure 15. For Figure 15, A12 = 0.788 and A21 = 0.430. Neither y1 nor y2 are well-fitted by the Wilson equation, for the data show less variation with composition than do eqs 19 and 20. We should point out that problems resulting from initially small amounts of impurities can be avoided by not attempting to cover a wide range of compositions with a single experiment. A better protocol would be to start with a 10-pm ( a = 100 for X = 632.8 nm) droplet with a high concentration of the more volatile species, letting it evaporate to, say, 5 pm. Then perform a second experiment with a lower initial concentration of the more volatile component, and repeat the experiment until the desired composition range is covered. The advantage of smaller size droplets is that more unique resonance spectra are obtained, and the interpretation of a spectrum is much less
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 689 2.3 1
I
00
w
***
V
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a
. 0.5
h
0
0
a*
O o 0
1.5
O0
O0
O
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I
A 0
RUN922
A
RUN915
, 0.0 ! 0.1
I I
I 0.2
0.3
0.4
z,,MOLE FRACTION DBO
Figure 15. Comparison between the experimental results of runs 915 and 922 with fits of the Wilson equation.
time-consuming than the analysis of spectra for a > 100. Three or four experiments should be adequate to cover the range 0.05 Iz1 I0.95.
Conclusions A new technique has been applied to measure activity coefficients of both components of miscible binary mixtures. By following the evaporation of a levitated microdroplet using light-scattering techniques, the size and composition were determined with sufficient accuracy to permit activity coefficients to be calculated from the data. For the nonideal systems DBO/HXD and DBO/HPD, the activity coefficients determined for the less volatile component agree with those calculated from the Gibbs-Duhem equation based on the activity coefficients determined for the more volatile component. The method is not restricted to components with similar volatilities but should be applicable to systems involving one relatively nonvolatile component. A major advantage of this procedure is that by means of a single experiment activity coefficients can be determined over a fairly wide range of compositions. Although the droplet evaporation technique is particularly well-suited to low-volatility systems, it is by no means limited to chemicals will low vapor pressures. Taflin et al. (1988) showed that rapid evaporation rates of aqueous solution droplets can be followed by optical resonance methods, but for rapid evaporation, corrections must be made for the lowering of the interfacial temperature. It is important to avoid highly charged droplets, for as evaporation proceeds the Rayleigh limit of charge can be reached, and the droplet will undergo fission (Taflin et al., 1988). The Rayleigh limit is given by where u is the surface tension, and to is the dielectric constant of air. In these experiments the droplet charge was always substantially lower than that predicted by eq 21 based on the initial size, and no droplet fission occurred during an experiment. Furthermore, the droplet charge was not high enough to affect either the light-scattering characteristics or the physical properties of the droplets. In principle, the vapor pressure of a pure component is affected by surface curvature and by the charge on the surface. The ratio of the vapor pressure over a curved surface, po, to that over a flat surface, porn, is given by (Reist, 1984)
where t is the dielectric constant of the liquid. A sample calculation based on the typical conditions for our experiments indicates that the corrections due to surface tension and charge are insignificant. For a 20pm-diameter hexadecane droplet (a = 27.44 m N/m, p = 771.1 kg/m3), the Rayleigh limit of charge is computed to C. If the droplet charge is one-half the be 3.917 X Rayleigh limit, eq 22 yields po/pom= 0.999 63. Thus, the correction to the vapor pressure is less than 0.1 70. The correction due to surface tension in the absence of electrical charges is po/pom= 1.00065. It can be concluded that corrections are necessary only for submicrometer droplets. Furthermore, the surface charge is not sufficiently large to affect the electromagnetic scattering, as evidenced by precise matching of optical resonances obtained for pure components with the Mie theory (Taflin et al., 1989). It is preferable to operate with droplets in the lightscattering size range 5 5 a I100, for in this range the structural resonances are very well defined. For a > 100 the interpretation of resonance data is more time-consuming, for the morphology of individual resonances is less unique and great care must be taken to determine the size and refractive index that yield the best match of the experimental resonances. It is most desirable to have multiple independent methods of estimating the size and refractive index because of the great sensitivity of the resonances to these parameters. Accurate estimates of the parameters are needed to begin the iterative procedure used to match resonance spectra.
Acknowledgment We are grateful to the National Science Foundation for Award CBT-8611779, and acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research.
Nomenclature a = droplet radius, pm or m ai= coefficients in eq 15 A = constant in eq 3 B = constant in eq 3 C, = geometrical constant of the electrodynamic balance,
dimensionless = gas-phase diffusion coefficient, m2/s g = gravitational acceleration constant, m/s2 G ( t ) = third-order polynomial J = molar flux, kmol/(m2.s) K = constant in eqs 16 and 17 md = droplet mass, kg m = refractive index M = molecular weight p o = vapor pressure, Pa q = Coulombic charge on the microdroplet, C R = gas constant, J/(kmol.K) Si, = slope of a plot of a2 versus time, pm2/s t = time, s T = temperature, K U , = convective velocity of gas in the balance, m/s V = molar volume, m3/kmol Vd, = dc levitation voltage, V x = mass fraction y = volume fraction z = mole fraction zo = half the minimum distance between the dc electrodes, Dij
mm
Greek Symbols a =
light-scattering size, dimensionless
690 Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 y = activity coefficient, dimensionless
= dielectric contant, F / m co = dielectric constant of free space, F / m 0 = scattering angle X = wavelength of light, n m .ilz= constant in the Wilson equation = constant i n the Wilson equation p = viscosity of the gas phase, kg/(m.s) p = density, kg/m3 c = surface tension, N / m $ = pure-component evaporation parameter in eq 6, kmol/ (mas) t
Subscripts i = i t h component j = carrier gas m = molar q u a n t i t y 0 = initial value or constant m = refers either to infinite surface curvature or infinite dilution Registry No. 1-Bromododecane, 143-15-7; 1,8-dibromooctane, 4549-32-0; hexadecane, 544-76-3; heptadecane, 629-78-7.
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Received for review August 28, 1989 Accepted January 5, 1990
