CONDENSATION OF WATERIN
A
433
CLOUDCHAMBER
Condensation of Water in a Cloud Chamber by Welby G. Courtney Thiokol Chemical Corpwatbn, Denville, New Jersey 07801
(Received March 6 , 1967)
The condensation of water vapor in a cloud chamber was investigated by measuring pressure as a function of time. A dense mist formed and the pressure-time curve was reproducible, suggesting that homogeneous nucleation occurred in the present work. Condensation a t high temperature (259°K) could be adequately explained with the classical nucleation theory combined with a small accommodation coefficient for particle growth. However, nucleation at low temperature (228") was significantly faster than the maximum of 0.4 particle/cma predicted by the classical theory of nucleation, and the classical theory therefore is suspect. The present results were compared to several models of condensation kinetics obtained by combining collision-frequency growth kinetics with the recent 1020 revised nucleation theory amended for various types of monomer loss. Results faintly suggested that ultra-small water clusters are about as stable as predicted by macroscopic cluster thermodynamics. Compression of the bulk gas due to heat conduction from the hot walls was appreciable, both during and after expansion. Previous work, which ostensibly supported the classical nucleation theory, omitted compression during expansion and therefore is suspect.
I. Introduction Recent studies of homogeneous nucleation from the vapor phase have presented a disconcerting diversity of conclusions. It has been generally accepted in the past' that Sc, data2 observed in cloud chambers were in satisfactory agreement with the classical nucleation t h e ~ r y . ~However, ,~ Lothe and Pound5 and others6,' recently suggested that the theoretical nucleation rate I , should be about 10'8 particles/cma sec faster than predicted by the classical theory and that a lo1*disparity existed between theory and S c r data. Oriani and Sundquist8 essentially concurred with the lo1* revised theory but concluded that the theory was in reasonable agreement with So, data. Wes have suggested that I , should be up to about lom greater than the classical theory but that the true I , for water nucleation is unknown because considerable vapor probably is lost into clusters. However, recent light scattering, lo drop concentration," and temperature12 data on the condensation of water in a cloud chamber seemingly were in good agreement with the classical nucleation theory (or a slightly modified versionla) combined with reasonable particlegrowth theories. This paper" presents pressure data obtained during water condensation in a cloud chamber. Results a t high temperature could be adequately explained by a classical nucleation combined with slow particle growth, but nucleation at low temperature was considerably faster than predicted by the classical theory.
11. Experimental Section A . General Remarks. Control of the rate of condensation to a factor of, say, 2 requires controlling the nucleation rate to about IO* and the supersaturation to about 0.29. The temperature must be known to about 0.3" in order to obtain the vapor pressure and thus the supersaturation to such an accuracy. (1) A. J. Barnard, Proc. Roy. Soc. (London), A220, 132 (1961). (2) For example: (a) C. T. R. Wilson, Phil. Trans. Roy. Soc. London, A189, 263 (1897); (b) M. Volmer and H. Flood, 2. Physik. Chem., A170, 273 (1934); (c) A. Sander and G Damkohler, Naturwksenehcrften, 31,460 (1943); (d) L. A. Madonna, C. M. Scuilli, L. N . Canjar, and G. M. Pound, Proc. Phye. Soc. (London), 78, 1218 (1961). (3) R. Becker and W. Doring, Ann. Phyaik, 24, 719 (1935). (4) J. Frenkel, "Kinetic Theory of Liquids," Dover Publications, New York, N. Y., 1955. (5) J. Lothe and G. M . Pound, J. Chem. Phye., 36, 2080 (1962). (6) J. P. Hirth, Ann. N . Y . Acad. Sci., 101, 805 (1963). (7) J. Feder, J. P. Hirth, J. Lothe, K. C. Russell, and G. M. Pound in "Heterogeneous Combustion," H. G. Wolfhard, et al., Ed., Academic Press Inc., New York, N. Y., 1964, p 667. (8) R. A. Oriani and B. E. Sundquist, J . Chem. Phye., 38, 2082 (1963). (9) W. G. Courtney, J . Phya. Chem., 72,421 (1968). (10) W. G . Courtney, J . Chem. Phya., 38, 1448 (1963). (11) E. F. AUard and J. L. Kassner, Jr., &id., 42, 1401 (1965); J. L. Kassner, Jr., and R. J. Schmitt, ibid., 44, 4166 (1966). (12) D.Stachbrska, ibid., 42, 1887 (1965). (13) W. G. Courtney, ibid., 35, 2249 (1961). (14) Details are given by W. G. Courtney, Thiokol Chemical Corp., Reaction Motors Division, Denville, N. J., 1965, Report TR-5515. Other raw data are given by W. G. Courtney, Texaco Experiment Inc., Richmond, Va., 1961-1963, Report TM-1250, 1340, and 1377. Volume 72,Number 2 February 1968
WELBYG. COURTNEY
434 ~~~~
~~~~~~
Table I : Nominal Experimental Conditions
-
Initial conditions Inert
Pi,
Ti,
Tssturstor,
gas
atm
O K
O K
Ar
1
300
290 273 263 290 290 290 290 263
300 300 300 300 300
'/2
'/a
Na He
1 1 1
A popular experimental technique for studying condensation kinetics is the Wilson expansion chamber, where a vapor-inert gas mixture is cooled by physically expanding the mixture. It is customary to assume that the expansion process is sufficiently rapid to be adiabatic, and the final temperature and pressure (and thus supersaturation) after expansion are usually calculated by
E
1.22
1.38 1.50 1.27 1.35 1.43 1.24 1.53
Find conditions Tf,
Pf,
O K
torr
S1
530 440 400
7-9 9-12 11-15
260 240 230 280 240 260 260 230
TRANSCORMER
$?
240
7
130 460 530 400
7 7-9 7-9 11-15
DUU-BEAM OSClLLOYOPE
n
PHOTO ELECTRIC CELL
PIEZOELECTRIC CRYSTAL
Tf = TiEl-7 Pf = PiE-7 (1) where E = V f / V i and is the physical expansion ratio, Vi and Vf are the initial and final chamber volumes, and Pi and P f are chamber pressures. However, the hot walls of the chamber immediately begin to heat the adjacent gas layer, thereby reheating the main body of gas by a slow conduction process" and also by a rapid compression afterls and during16the expansion process, due to adiabatic expansion of the hot gas in the boundary layer. This nonadiabaticity makes the final supersaturation after expansion be somewhat smaller than calculated by eq 1. For example, the classical Volmel-Flood cloud-chamber work used a pneumatically driven chamber with a volume/surface ( V / S ) ratio of 2 cm. Barnardl concluded that their expansion time was about 20 msec and that compression during expansion caused their actual Tf to be only 0.2" higher than calculated. The actual So, then typically would be 4.9, instead of the nominal value of 5. However, Hazen16 similarly used a pneumatic drive and his shortest reported expansion time was 100 msec. His measured final pressure after expansion was about 5 torr higher than the adiabatic value. The corresponding Ti was about 0.6" higher than expected, and the temperature increase due to compression during expansion now is appreciable. B. Equipment and Technique. In the present work, the general technique was to saturate an inert gas at elevated temperature with water vapor, cool the wet gas to a known temperature, and thereby establish the concentration of water vapor before expansion, rapidly expand the gas by a known E in a piston-driven The Journal of Phydcal Chemistry
Figure 1. Schematic diagram of expansion chamber.
expansion chamber, measure the chamber pressure during expansion and condensation, and compare the pressure-time result to theoretical models for nucleElr tion and growth kinetics. Table I summarizes conditions examined in this work. The chamber (Figure 1) was similar to that used by Mills" and involved a steel cylindrical chamber with a flat glass top and a flat aluminum bottom piston driven by a 750-lb spring. The chamber was 5 in. in diameter and 3.7 in. high before expansion. A Neoprene 0 ring was used between the piston and chamber wall with Apiezon M grease as lubrication. Gas leakage during the piston stroke was negligible. The space below the piston was evacuated to decrease expansion time. Steel bars were welded to the '/z in. thick middle plate to prevent a slight overexpansion as the piston hit bottom. Expansion ratios up to 1.53 could be obtained. As E increased, the expansion (15) E.J. Williams, Proc. Cambridge Phil. SOC.,35, 512 (1939). (16) W. E.Hazen, Rev.Sci. Instr., 13, 247 (1942). (17) R. G. Mills, ibid., 24, 1041 (1953).
CONDENSATION OF WATERIN
A
CLOUDCHAMBER
time increased from about 4 to 8 msec and V / S increased from 1.2 t o 1.52 cm. Chamber pressure during expansion and condensation was measured piezoelectrically by a 1.5 X 1.5 X 0,125-in. cylindrical shell of doped barium titanate suspended directly in the chamber by rubber bands. The piston bounced twice for 5 msec after hitting bottom, and attempts to eliminate this bounce by adding cushions were uiisuccessful. The voltage trace also contained a high-frequency oscillation after the piston hit bottom. Frequency was about 1300 cps with Ar and Nz and 3000 cps with He. Amplitude was about 0.1 v and was considerably greater when the crystal was more rigidly mounted in the chamber. These oscillations were assumed to be due to mechanical noise in the crystal and not aerodynamic pressure waves in the chamber and were ignored. Thermostat temperature and chamber temperature before expansion were measured to 0.3" with ironconstantan thermocouples. Chamber pressure before expansion was measured to 0.5 torr by a manometer. Light scattering sit 90" was used as a qualitative indication of condemation and no attempt was made to eliminate a 120-c noise. A dense mist, resembling tobacco smoke, was always observed and the rain associated with ions or impurities was never found in this work. The experimental technique was in general rather casual compared to the techniques of' other workers, in that the chamber was often exposed to room air and ordinary distilled water was used. Considerable trouble was initially encountered in o'btaining reproducible results because of a memory effect due to previous expansions.l8 Results became reproducible (voltage traces for repeat runs could be superimposed) when the chamber was purged with new gas between runs. A minimum expansion ratio to give a homogeneous mist throughout the chamber (i-e., S c r ) was briefly examined by visually examining the 90" scattered light. This S,, was quite subjective, for a less intense incident light bea,m required a larger Sora Momentary wisps of mist weire observed immediately after expansion when a water-argon mixture saturated at room temperature was expanded by ratios less than the minimum ratio, but not with other conditions or systems. A dc field up to 500 v/cm had no effect on results. A higher voltage allowed condensation to occur with subminimum expansion ratio, presumably because ions had been introduced into the chamber. C. Pressure Calibration. The piezoelectric pressure gauge was calibrated by expanding the various gases and comparing the voltage output, v, measured immediately after expansion with the theoretical pressure change, APtheory, assuming adiabatic expansion, where
435 6,
1
I
I
1
Ar 5-
Dry A Wet A
N,
Ha
0
0
a
4 -
-
-
-
-
-
0
100
200
300
400
b~~~~~~~~ torr
Figure 2. Pressure calibration.
Values of y were taken as 1.670 for Ar, 1.660 for He, 1.404 for Nz, and 1,330 for HzO and were assumed independent of temperature. Richarz' formulalg was used to calculate y for a mixture. Figure 2 plots typical results. For Ar and Nz, v was proportional to APtheory up to about 200 torr (E to l.2), but was slightly low for higher E. The results for He were slightly lower than the other results at low E and appreciably lower at high E. Results obtained with dry and wet gases were essentially identical if E was too small to cause condensation. The disparity from proportionality is attributed t o compression during expansion. The disparity a t the higher expansion ratios would correspond to the longer expansion times and the greater temperature difference between the wall and cold gas. The greater disparity with He would be due to its greater thermal conductivity. The linear range corresponded to a pressure calibration of
AP = 6 4 . 0 ~ or
P, = Pi
- 64.0~
(3) where P , is the chamber pressure in torr a t time t and the voltage in volts measured at t. Equation 3 is used to calculate the pressure in the chamber during and after expansion from the experi(18) Condensation in the used gas could be obtained with expansion ratios much smaller than were required with fresh gas, even though the used gas was held recompressed for 10 min. (19) R. Richarz, Ann. Physik, 19, 639 (1906). Volume r d , Number B
February 1068
WELBYG. COURTNEY
436 mental v-t traces. Absolute accuracy in pressure was about 5 4 and 1 7 torr in the high- and low-temperature work, respectively. These accuracies are due t o an accuracy of 10.03 v (A2 torr) in a voltage measurement and the limitations of the pressure calibration. Attempts to obtain an independent measurement of pressure with a strain gauge or direct measurement of temperature with a tungsten resistance thermometer were unsuccessful.
111. Results The disparity between wet and dry P-t curves measured with the same E is t o be explained in terms of condensation kinetics. The dry P-t curves were used to investigate compression, after and during expansion, and t o obtain the volume-time history, V-t, of the chamber. The V-t data were then combined with condensation theory to calculate a theoretical P-t curve which was compared to the experimental P-t curve. A . Compression after Expansion. Williams considered the boundary-layer compression effect after an instantaneous expansion and concluded that the pressure P a t time t is
P
=
pi
+ mt‘/’
(4)
420v I
I
400
380
I
I
where
where Ti is the temperature before expansion, Pr, T I ,Sf, and Vt are the pressure, temperature, surface area, and volume in the chamber immediately after expansion, h = (K//~P)~’’ and is the thermal diffusivity of the gas, K = thermal conductivity, p = specific heat at constant pressure, and p = gas density. Units are cgs, OK, cal, and torr. Typical P-t data obtained after expansion are given in Figure 3. Results were reasonably linear, although the intercept often was less than the pressure at zero time, presumably because the heat conduction during the bounce was less since the gas temperature was higher. The measured values of m for Ar and N2 were similar and were about one-third the values for He for similar conditions. The experimental values for m had considerable scatter but were about twice as large as the theoretical values. Such an increase in m might be expected because of an increase in h., for the theory assumes a simple diffusional heat transfer in the boundary layer and some turbulent transfer should also occur. B. Compression during Expansion. Hazen examined compression during expansion for small expansions in a constant-area chamber (i.e., he used E to 1.125 and a moving diaphragm) and used average values of the temperature during expansion. I n the present work, expansion ratios up to 1.5 were The Journal of Physical Chemistry
250
I
I
0
0.1
I
I
I
0.2
0.3
0.4
TIME a/!
I
sec 112
Figure 3. Compression after expansion; legend in Figure 2.
used, and the use of average values becomes less appropriate. Also, our piston motion exposed new surface during the expansion, and the boundary layer of this new surface must be considered. Hazen’s approach was extended to include a variable surface area assuming S , = So bt (the initial cosine-type piston motion was ignored). The volume of the boundary layer during expansion then is
+
where Tw is the wall temperature, Tb,$is the temperature of the bulk gas at time t after the start of the exto 8/15 as t increases, pansion, X decreases from and T , - T b , lwas assumed to be a quadratic function of t.20 The volume change due to the boundary layer is still small compared to the change due to piston motion, and the measured P-t curve can be used to (20) The factor X decreases to l assumed to be a cubic function of
B / as ~ ~t t.14
increases, if T ,
-
Tb,i is
CONDENSATION OF WATERIN
A
437
CLOUDCHAMBER
calculate the surface area of the chamber at T. With the present chamber, SO= 793 cmz and b = 27,000 cmz/sec. Taking X = 2/a, the pressure increase due to the boundary layer at t then is
(TwTb,t
) (793 + 18,000t)t"z
(7)
We will assume average values of h and will take h ~ =? 1 and hHe = 2.8 cm sec-"* (=2htheoly)in view of the results of the previous section. A typical experimental 8Pbl of 13 torr for Ar at t = 6 msec (E = 1.5) was about twice the theoretical value of 7 torr calculated by eq 7, but agreement is satisfactorily within experimental error. Experimental and theoretical values for He of 13 and 16 torr a t 4 msec ( E = 1.3#4)and 29 and 23 torr at 6 msec (E = 1.5) agree more closely than could be expected. The above results suggest that the calibration line in Figure 2 miglht better have been drawn with a slightly smaller slope, say AP = 6311, to allow for conduction during expansion. This was not done in order to present a conservative interpretation of the data. C. Condensation. Typical experimental and theoretical results14 are given in Figures 4-6. The raw voltage-time (upper) and scattering-time (lower) traces for dry and wet runs are shown, with pressure, temperature, supersaturation with respect to liquid 8 1 , and particle concentration c, included. The solid lines for P , T , and S I correspond to conditions in the chamber, if condensation were prevented and serve as base lines. Experimental P-t data during condensation are given as large dots in the pressure graph and are to be compared to the results for no condensation (solid line) and for several theoretical models (broken lines and points). Qualitatively, condensation occurred within 50 m e c when 81 was about 8 a t 257-259" (Figures 4 and 5) and about 12 a t 228" (Figure 6). The high-temperature results roughly agree with the Wilson So, of 7.9 at 257". The low-temperature result is appreciably higher than the So, of about 7 observed by other workers.2c,d After .the piston hit bottom, the rate of pressure rise during condensation in He was about twice that obtained in Ar or N2 (e.g., 4 compared to 2 torr/ msec). These rates are much greater than the rates due to the compression effect and suggest that particle growth kinetics is faster in He, i.e., that diffusion processes control particle growth in Ar and N2 (and perhaps also in He) and that the accommodation coefficient for particle growthz1is less than unity. Quantitative interpretation of the present results requires including the variation of chamber volume with time; i.e., the present chamber was a highly transient system and the supersaturation typically went from 8 to 9 to 8 in '/2 msec (e.g., Figure 4) as
TIME. m..c
Figure 4. Condensation in Ar at 257" (legend in text): Pi = 741 torr; Ti = 293.5K"; Tthermostat = 292.5%; E = 1.228.
the piston hit bottom and recoiled upward. Our approach was to: (1) write theoretical equations for (21) The coefficient for particle growth is defined as the number of molecules which hit and stay on unit surfaee of a growing particle divided by the number of molecules which collide with unit surface.
Volume 78, Number 8 February 1968
WELBYG. COURTNEY
438
8'
:...
10'
'
,L-
1
'
1
.
1
'
.
TIME, nwec
Figure 6. Condensation in Ar a t 228°K (legend in text): Pi = 746.4 torr; Ti = 296.2'K; Ttherrnastat = 263.2'K; E = 1.504. 0
2
4
b
8
10
'I2
TIME, msee
Figure 5. Condensation in He a t 259' (legend in text): Pi = 743.5 torr; Ti = 296.7'K; Tthermoetgt = 291.3'K; E = 1.256.
supersaturation SI and temperature T in terms of condensation kinetics and volume V ; (2) deduce V as a function of t from the dry P-t curve by eq 1; (3) solve the equations for 8 1 and T (and also several supplementary equations) simultaneously on a computer; and (4)compute the resulting theoretical value for P at t by the ideal gas law and compare it to the experimental value. Kinetic assumptions were: nucleation of liquid particles, rapid freezing of nuclei, The Journal of Physical Chemistry
collision-frequency-controlled growth of spherical solid particles, no agglomeration between particles, a closed quiescent system, and temperature equilibration between vapor and growing particles. Accommodation coefficients an and a. were used to adjust nucleation growth kinetics. Nucleation kinetics were taken as the classical theory, the 1 0 2 0 revised theory assuming no monomer loss, and the 1 0 2 0 theory assuming several types of monomer loss. The relation between the true supersaturation S in the system and the presumed supersaturation S due to monomer loss was discussed in ref 9. The true supersaturation S a t 263" approximately corresponded to
CONDENSATION OF WATERIN S = (1.67s
4-2.3)'12
A
439
CLOUDCHAMBER
(for C , = lO-'(Cp)maoro)
(84
IV. Discussion
+ 2.6)'12
Since the experimental P-t results were reproducible over a 1-year period and since a dense mist was always (for Co = 5 X 10-2(Cp)maaro) (8b) formed, we conclude that nucleation in the present S = (0.35s 2.75)'la (for C , = lO-'(Cg)maoro) ( 8 ~ ) work was homogeneous and heterogeneous. The present chamber undoubtedly contained the usual Le., S N '/z& Equation 8 will be used here at 257impurities (dust, ions, etc.) and some condensation 259". Monomer loss is negligible with the classical undoubtedly occurred on the impurities during expannucleation theory. Several of the theoretical models sion. However, the present expansion rate apparently considered here are summarized and their respective was sufficiently high (10"/msec) that the wet gas symbols used in the figures are given in parentheses reached the conditions for homogeneous condensation (a, = 1) : (1) classical nucleation, no monomer loss, before the condensable vapor was depleted owing to ap = 1 (circles); (la) classical (solid nucleiz2), no growth of heterogeneously nucleated particles. Such monomer loss, a. = 1 (squares); (2) classical nucleaan argument is also used in wind-tunnel experiment^.^^ tion, no monomer loss, a, = 0.1 (triangles); (3) 1020 The momentary wisps of mist in Ar at a submarginal nucleation, no monomer loss, au = 1 (dotted line); expansion suggest local cold zones of high supersatura(4) 1 0 2 0 nucleation, eq 8a, a, = 1 (short dashes); tion due to turbulent expansion waves from the piston ( 5 ) lozonuclealion, eq 8a, a, = 0.1 (omitted); (6) lozo motion (500 in./sec), but such waves seemingly were nucleation, eq 8b, a. = 1 (long dashes); and (7) lozo of secondary influence in the present work. nucleation, eq 8c, all = 1 (omitted). Theoretical The memory effect probably is due to heterogeneous results which predict no condensation usually are nucleation onto ions formed during the previous exomitted from the figures except for the 8 1 and c, graphs. pansion by the piston motion along the wall (triboThe computation was started 2 msec before the piston electrification). Such ions would cause heterogeneous hit bottom, in order to include any condensation during nucleation near the wall during the first expansion but expansion.2a*2 4 The time increment usually was 0.1 apparently do not have sufficient time to diffuse msec and 50 successive points were calculated. throughout the chamber during the short expansion At low temperature (228", Figure 6), a dense mist time. A memory effect due to ions has been reported was observed while the lozotheory (no. 3) predicted by other workersaZ6 lo9 particles/cma to form, and the classical theory The present results are only semiquantitative in predicted a total of only 0.05 liquid or 0.4 solid nuclei/ view of the limited accuracy of the pressure measurecm3 (no. 1 and la) to form. Similar results were obments and the pressure calibration. Also, interpreserved and predicted with He. Since a, = a, = 1 tation of P-t results is inherently limited by the negives the fastest possible nucleation that the classical cessity of introducing a growth theory. Nevertheless, theory permits, the observed dense mist cannot be a dense mist was observed, and the inability of the explained by the classical theory and the classical classical theory to predict at low temperature more theory of nucleation therefore is suspect. The difthan 0.4 particle/cm3 via solid nucleation (400 parference between wet and dry P-t data at low temper* ticles in the present chamber) indicates, in our opinion, ture was small, and the present low-temperature P-t that the classical theory is void.27 data were abandoned as being too insensitive to nuThe compression during expansion suggests that a cleation and growth kinetics. For example, the theocalibration of AP = 63v would be more accurate than retical P-t curves with 1020 nucleation and a, = 1 and the value used here of AP = 64v. The supersatura0.01 were within experimental error. tions reported in Figures 4-6 then are overestimated At high temperature (257-259"), condensation was because the true AP is slightly less than reported here. much slower than predicted by the 1020theory (no. 3) At 259", the overestimate in AP is about 3 torr, which and somewhat slower than predicted by the classical corresponds to the temperature after expansion being theory (no. 1). The P-t data (e.g., Figure 4) were 0.6" higher and 8 1 being 0.3 lower than the values used bracketed with the lozo theory amended by (22) At temperatures below about 233O, formation of solid nuclei by monomer loss monomer loss (too fast) and 5 X direct sublimation is faster than liquid nucleation according to the (too slow) with a, = a. = 1 (no. 4 and 6). The input data used in ref 9. model with a. = 0.1 (no. 5 ) probably is more realistic (23) B. J. Mason, Proc. Phys. SOC. (London), B64, 773 (1951). and also was close to the experimental P-t curve. (24) The 1 0 2 0 theory predicted that considerable condensation should have occurred with Nz,even earlier than 2 msec before the piston Further speculation seems unwarranted, until inhit bottom. dependent information on monomer loss and a. be(25) P. P. Wegener and A. A. Pouring, Phge. Fluids, 7, 352 (1964). comes available. Curiously, classical nucleation with (26) For example: G. R. Evans, N. E. Fancey, J. R. Norbury, and A. A. Watson, J. Sci. Instr., 41, 770 (1964). a, = 0.1 (no. 2) gives good agreement with the high(27) The surface free energy of clusters could very well be greater temperature P-t data in Ar and Nz, while a, = 0.5 than the macroscopic values used here, but the rate of classical would probably fit the He data. nucleation then would be even less.
S
= (1.38s
+
Volume 73,Number 8 February 1988
440
WELBYG. COURTNEY
here. At 228", the overestimate is 5 torr, which corresponds to the final temperature being 1. l ohigher and S1 being 1.8 lower than used here. Condensation then would have occurred at milder conditions than reported here, and the disparity in Figure 6 between the observed condensation rate and the rate predicted by the classical theory would be more acute, e.g., the classical 1,would decrease by IO4 using the 63v calibration. The high-temperature P-t data could probably be sufficiently fitted by lozo nucleation modified by a type of monomer loss and an ag less than unity. Such a fit would imply that particle growth involves a diffusion envelope and that ultra-small clusters containing less than 40 or so molecules are only lo-* as stable as predicted by macroscopic thermodynamics. However, the monomer-loss problem prevents a detailed interpretation of the present data in terms of particle growth kinetics, for a small variation in monomer loss causes a large variation in the theoretical P-t results. The present data therefore remain unanalyzed but are believed to be considerably more accurate than has hitherto been reported for cloud chambers because of inclusion of compression during expansion. We concluded earlierl0 that our light-scattering results were a semiquantitative verification of a slightly modified versionla of nucleation kinetics, but compression during expansion and piston bounce had not been included. Kassner, et al.,ll expanded an H20-He mixture in a pneumatically driven cloud chamber, assumed adiabatic expansion, and typically measured a nucleation rate of about 180 particles/cm3 sec when S1 nominally was 5 at 268", which agreed with the amended1* classical nucleation rate. However, their chamber had a 100-msec expansion time. For their expansion, V / S 2 cm, y 1.65, T , - Tf 25") T f 268", and Pf 540 torr. Taking h 2.8 cm sec-l'a, the pressure increase, due to compression during their expansion, was about 14 torr according to eq 7, assuming a constant surface area. The temperature
-- -
The Journal of Physical Chemistry
-- -
after expansion therefore is about 2' higher than expected adiabatically, the actual S1 is about 4.3, and the experimental nucleation rate now is lo5 faster than predicted by the classical theory. Stachdr~ka'~ expanded an ethyl alcohol-air mixture in a cloud chamber and measured temperature-time. The time after expansion to obtain an essentially constant temperature was compared to an approximate condensation-time equation, wherein Frenkel's nucleation kinetics was combined with Hazen's r2-t growth kinetics, and Stach6rska concluded that his experiments confirmed Frenkel's classical theory. However, his T-t data after expansion closely correspond to the results to be expected from compression 1.4, h 1, T , Tf after expansion. Taking y 18", and assuming a V / S of 1.3 (his V / S ratio was not mentioned), the theoretical temperature increase due to compression after expansion would, according to eq 4, match the typical value of 2.3" measured 140 msec after expansion (case c in his Figure 2). Future experimental work in condensation either should use materials for which monomer loss is expected to be negligible or should be devoted to a direct measurement of cluster thermodynamics, for until the monomer loss problem is clarified further, studies of condensation kinetics seem useless unless data for correlation purposes are desired. The present results (and also recent expansion-nozzle work26*28) indicate that the classical nucleation theory with ag = 0.1-0.5 provides a good correlation for water condensation a t 260-300°K, but not at lower temperatures.
- -
-
-
Acknowledgments. The equipment was designed' and constructed by Mr. Wendall J. Clark and experimental work was done by S. W. Finley, 11, W. Hall, and K. Renalds at Texaco Experiment Inc., Richmond, Va. Computer work was done by Mr. George A. Brown. Part of this work was supported by the Office of Naval Research. (28) For example: P.G. Hill, J . Fluid Mech., 25, 593 (1966).
