SDontaneous Nucleation in Supersaturated Water Vapor H
WORTH H. RODEBUSH UNIVERSITY O F ILLINOIS. U R B A N A , ILL.
' G i b b s showed t h a t no stable equilibrium, such as has been assumed i n t h e derivation of t h e Thomson equation, can exist between t h e liquid drop and t h e vapor phase. When t h e drop becomes small enough, it behaves as a large molecule and below a certain critical size t h e translational and rotational entropies contribute a stabilizing factor which makes equilibrium possible. It i s assumed t h a t t h e equilibrium concentrations of these aggregates or clusters are present a t all times. T h e equilibrium concentration decreases w i t h increasing cluster size. When t h e supersaturation is increased t o a point where a considerable concentration of t h e critical size is present, condensation occurs immediately. T h e theory appears t o agree w i t h t h e reported facts for water vapor, b u t more reliable data are needed.
T
H E theory presented here is based entirely on the work of Gibbs (1). Certain data were lacking to Gibbs that are now available. Thus he gave the formula for the entropy of a monatomic gas) but did not know Planck's constant. Anyone could insert it in the formula and Tetrode did it first in 1912. Gibbs with uncanny insight defined the general conditions for phase equilibrium always with the specification of plane surfaces for the phase boundaries. On the other hand, some of his most interesting and involved discussions are concerned with the situation where the radii of curvature are small. This is the one situation where Gibbs did not achieve a clean-cut answer to a problem. It would be a rash person indeed who attempted a rigorous solution to the problem where Gibbs failed, and no such claim is made here. However, many data are available now t h a t were lacking t o Gibbs and we know pretty well what the answer should be. Therefore, if we can account for the observed facts by a thermodynamic treatment, making certain plausible assumptions, the effort seems justified. No proof of the plausibility of the assumptions can be offered. They are always open to question. The data in regard t o spontaneous nucleation in water vapor have been reviewed by Ruedy ( 4 ) . In order to get rapid spontaneous condensation in water vapor at 25" C. a four- t o sixfold supersaturation is required. This is in no sense a firm datum. It is difficult to assume that foreign nuclei are not present and there is need of further careful work. An attempt is being made to obtain reliable data at the University of Illinois. Water because of its structure in the liquid state must be a unique substance and it seem6 desirable t o begin work with a nonpolar liquid. I t is obvious that in order to observe the phenomenon of rapid condensation a considerable degree of supersaturation must be attained. Aerosol theory tells UB that for a particle t o be readily visible it must be of about 1 micron diameter and t h a t about lo8 particles per cubic centimeter are required to produce a dense fog. This means that about 0.5 X lo-* gram of water must be condensed, which is roughly three times the quantity of water present in the saturated vapor at 25" C. We have thus accounted for both the lower and upper limits of the degree of supersaturation required for spontaneous nucleation by a very elementary sort of reasoning. One might ask what remains to be explained? This is a very superficial point of view, however. There is reason t o believe that the threshold limit for spontaneous nucleation is defined within much narrower limits than would be set by the crude criterion discussed above. A vapor may be maintained at a considerable degree of supersaturation for an indefinite time without, except for the presence of sporadic nuclei, lune 1952
any condensation. With a slight increase in supersaturation, a fog of the concentration and particle size described above forms in a matter of microseconds. The phenomenon is sufficiently interesting to justify an attempt a t explanation. INVALIDITY OF THOMSON EQUATION
The liquid drop of radius > l o + cm. may be considered as a liquid phase differing from any portion of liquid phase only in that the escaping tendency is greater from curved surfaces. The Thomson equation
purports t o give the equilibrium vapor pressure. Gibbs does not mention the Thomson equation in his famous memoir on the equilibrium of heterogeneous substances. H e does point out that no true equilibrium is possible between the liquid drop and a vapor phase indefinite in extent. The equilibrium which is assumed as the basis for the derivation of the Thomson equation is an unstable equilibrium, which from the standpoint of thermodynamics is no equilibrium at all. If a drop.which is in pseudoequilibrium with the vapor gains a small increment of liquid by condensation, then the vapor pressure of the drop is reduced and drops begin to grow by the condensation of vapor and will increase without limit. Conversely, if the drop loses a small mass, its vapor pressure is increased and it decreases in size u$il it changes its character and is no longer recognizable as a drop. If the vapor phase is limited in extent, there is a sort of stabilization, in that the partial pressure of the vapor changes in the same direction as the vapor pressure of the drop and more rapidly. The conditions for this to hold are stated in Equations 1t o 4. Condition for Stability
L(2.I)(") dn" dn'
r RT
dn
n' - =
l / ~
n"
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=
In P ~ P O
nR
NUCLEATION-Theory,
Review
Equation 1 assumes the transfer of 6, moles from liquid drop to vapor phase, designated by ’ and ”. These become equal Then the ratio of the number of drops per cubic centimeter t o the number of molecules per cubic centimet.er is equal to p / p o , which is given by the Thomson equation. This is an excessively high concentration and does not constitute a true equilibrium because, unless the drops are all of the same size, the larger drops xi11 gro3.v a t the eupense of the smaller ones.
n
10
t
10-10
’
centration is t,he translational entropy. As the cluster increases in size, the translational entropy increases as the logarithm of the mass to the three halves power but the share of each molecule in the entropy deceases as l l n , where 72 i,? the number of molecules per cluster. In the drop containing a large number of molecules the share of the translat,ional entropy becomes vanishingly small. A similar stabilizing factor is found in the rotational entropy. Because of the conservation and quantizat.ion of angular monientum it is easier for molecules to condense upon a large rotating molecule. This is a situation very different from condensation upon a flat liquid surface, R-here the rate is independent of the. mass of the liquid phase. I g a i n , this effect is shared between all molecules and decreases as l l n n-here n is the number of molecule? per cluster. E Q U I L I B R I U M CONCENTRATION OF CLUSTERS
‘
10-20
1 0 --e
10-7
0
r
Figure 1. Concentration of Clusters (100 Molecules) as a Function of Radius and Degree of Supersaturation
It can therefore be stated categorically that the equilibrium. concentration of clusters at. a fixed vapor pressure decreases v i t h increasing cluster size until a critical size is reached which ha.e t,acitly been defined as the t’ransition point bet,ween cluster and drop. Any aggregat’e that exceeds this size must increase JT-ithout limit in size if the pressure is great enough, or disappear. by evaporation if the pressure is too small. Stabilization of a certain drop size ryit,h a certain pressure is inconceivable. It is important t o consider the mechanism by which the eyui-librium is to be maintained for the clusters. The various mechanisms are given in Equations 5 to 8.
Gibbs discusses the equilibrium between drop and vapor in terms of the Laplace equation
He shows that the work of forming a drop is given by the expression
w
= ,yu
- (p’
- p”)V
The second termis t v o thirds of the surface energy, by the geometry of the sphere. The second term appears in the Thomson equation; the surface energy is used below in equations for equilibrium. Gibbs points out that the work of forming the sphere constitutes a sort of stabilizing factor in that it represents an activation energy, but this does not correspond to any sort of equilibrium. STABILIZATION
O F T H E CLUSTER
As the drop diminishes in size, its properties must change in such a way that it can no longer be considered as a portion of the liquid phase. There are two reasons for the change in character. One is that the heat of vaporization, surface tension, and other physical properties must change; in fact, the concept of surface tension must eventually become invalid. But there is another change in character of a more radical nature. When the drop becomes small enough, the chemical potential begins to depend upon the concentration of the drops considered as large gas molecules in a vapor phase. It can no longer be considered a part of the liquid phase in any sense of the word. TT’hen this situation prevails the term drop is no longer appropriate and the term “cluster” is preferred. The transition from drop t o cluster is not sharply defined, but any aggregate that contains less than 100 molecules of water should probably no longer be considered as a drop. The limiting size of the cluster is an aggregate of two molecules. It is easy t o visualize how the chemical potential must depend upon the concentration of both species in an equilibrium between dimers and monomers. It is less obvious a t first thought for larger aggregrates. The quantitative measure of the con-
1290
(7)
Equation 5 is the niechanism that is assumed in the derivatioiL of the Thomson equation and it is undoubtedly the one that occurs most often. If we ask which mechanism is significant for equilibrium, the answer is that all are equally significant. It is true t h a t Equations 6 and 7 are improbable mechanisms because of the great heat of activation required and the improbability of a cluster’s exploding into single molecules. For purposes of equilibrium all niechanisms are equally significant and the equilibrium constant for any one mechanism can be calculated from the others. There remains the question whether the agglomeration mechanism which is so important under some conditions in the formation of aerosols could be of any significance under conditions of supersaturation. The answer is that the agglomeration reaction must be subject t o equilibrium and can give us nothing different, and that for these reactions to be rapid enough to require consideration, the concentrations would have t o be impossibly high. For purposes of calculation we shall choose the mechanism of Equation 7 . We know the thermodynamic propertie.: of the single molecule and of the cluster large enough to be called a drop. For the intermediate range it is necessary to interpolate. The interpolation leads into the no man’s land delimited by Gibbs, where the drop radius is of the order of lo-? cm. We must recognize that both the surface tension and the heat of vaporization cannot be assumed constant in this region. The concept of *urface tension must become invalid for some small size. As we cannot be sure in which direction the error caused by these itssumptions d l lie, we can only hope that it will not be large. We
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 44, NO. 6,
NUCLEATION-Theory, may therefore write the chemical potentials of liquid and vapor, assuming the fugacity equal to the pressure.
yu = y
x
x
v
3 v
-= Y 4rr3 r n
Actually, however, such a cluster is probably as well defined as the molecule (12) in the equilibrium in iodine vapor between molecules and atoms under conditions where the mean life of a molecule is very short. It remains to consider the effect of changing the partial pressure of water vapor on the cluster concentration, which is expressed as number of nuclei per cubic centimeter in Equations 14 to 16.
s
S'" = constant
+
3/2
R In M
AF"
+
3/2
R In I
~
+
n
log P'
The single prime refers t o the clusters considered as a gas and t h e double primes refer to the monomeric vapor. The surface 3 energy is computed to be -2 VL. If we had chosen to consider t h e mechanism of Equation 5, we should have used
Equating the potentials of vapor and clusters, we obtain Equation 13. The important part of this is the parentheses at the end. , The surface energy term is negative and the entropy term is positive. Each term is reckoned for 1 mole of water, and both vanish as r and n increase without limit. n varies as rs, so that if the terms are of the same order of magnitude the surface energy term predominates a t large r , while the entropy term should become greater a t small r. The point of essential interest is the minimum in the curve, where the derivative of the terms in the parentheses with respect to r vanishes. I n order to locate this minimum we tabulate values of the parentheses A for various values of r in Table I. For a spherical cluster containing more than 100 molecules and of symmetry number one, the total translational and rotational entropy is in the neighborhood of 100 entropy units and may be assumed t o be constant over the range of values of interest ( 2 ) . This assumption cannot cause serious error, since the entropy is t o be divided by n, the number of molecules per cluster. The greater uncertainty involved in the assumptions regarding surface tension and heat of vaporization have already been mentioned. In Table I the values of the terms in the parentheses and their difference, A, are given for different values of r. The calculations indicate that a minimum .exists somewhere in the neighborhood of radius 8.5 A., corresponding to approximately 100 molecules per cluster. There is no purpose in drawing the curve beyond the minimum in the direction of increasing radius because no equilibrium can exist -where the curve rises with increasing radius.
Table I .
Values for Parentheses in Equation 13 a t Different Radii
7A
n
10.0 9.0 8.5 8.0 7.0 5.0
(100)
133
3yV
's"
rR T
nR
1.56 1 73 1.83 1 95 2 23 3 11
0.22 0 51 0 61 0.73 1.09 3.00
(2- %)
1 AFO - log P' = P" + 23RT - 2!3 n = -1.505 1.505 - 0.53
(12)
S'"
1
-A 1.19 1.22 1.22 1.14 0.11
Review
=
(14)
10-53 P" = 23.75 mm.
= -1.10 P" = 60 mm.
log P' = 10-13
N
108 molecules per cc.
(15) (16)
For a saturation pressure of 23.75 mm. a t 25" C. the concentration is vanishingly small. If the pressure is increased to 63 mm., the concentration becomes -108 nuclei per cubic centimeter. As many of these nuclei will by pure statistical probability grow to a size beyond the minimum in the curve. condensation will certainly occur. It would be very rash to assert that condensation would occur at any particular degree of supersaturation, for the calculation given above can hardly be of the right order of magnitude. Furthermore, this calculation is for one single species of cluster ( n = loo), whereas because the minimum is very flat one needs to consider a t least ten or more species of clusters which will serve as nuclei simultaneously. The fact that the minimum is flat means that the critical cluster size is not sharply defined, but it also means that its approximate location is not in serious error. It is believed that the theory given is adequate t o explain the process of spontaneous nucleation. The results obtained by the thermodynamic approach must agree in general with the results given by the Becker-Doring theory. The Becker-Doring theory represents the kinetic approach, but the rate constants can be estimated very accurately. The major uncertainty in each approach lies in the estimation of the energy and other properties of a cluster which is so complex as to defy any sort of theoretical calculation and yet too small to be subject to direct experimental observation. NOMENCLATURE
p
= partial pressure
PO = saturation vapor pressure radius of drop volume of 1 mole of liquid R = molal gas constant T = degrees Kelvin y = surface tension u i= surface area n', dn' = moles in liquid phase n", dn" = moles in vapor phase X, = molecular aggregate of n molecules = Gibbs potential PO = molal free energy 8'' 2 molal entropy M = mass of drop I = moment of inertia of spherical drop n = (Equations 9 to 16) number of molecules per cluster r
V
i=
;=
$0
LITERATURE CITED
The translational entropy varies as the term a/z R In M , where M is the mass of the drop, and the rotational entropy var-ies as 3/2 R In I , where I is the moment of inertia. The moment of inertia is given for a sphere by the formula Mra where r is the radius of gyration, which varies inversely as the cube root of the density. It has been argued by some that the entropy of such a fugitive and ill-defined complex as a cluster cannot be cald a t e d for the puppose of defining thermodynamic equilibrium. lune 1952
(1) Gibbs, J. W.,"Collected Works," New York, Longmane, Green and Go., 1928. (2) Paul, M. A., "Prinoiplea of Chemical Thermodynamics," p. 665, New York, McGraw-Hill Book Co., 1951. (3) Rodebush, W. H., Chem. Revs., 44,269 (1949). (4) Ruedy, R., Can. J. Research, A22,77 (1944). RECEIYED for review December 21. 1951. ACCEPTED March 30, 1952. A preliminary account of the theory was published in Chemical Reviews (3).
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