KINETICS OF THE FORMATION OF NUCLEI AND STATISTICAL THEORY OF CONDENSATION BY R. BECKER Received 25th February, 1949 In the theory of the formation of nuclei it has generally been assumed that in every gas there exist, in addition to single molecules (2, per unit volume), aggregates containing 2 , 3, . . . . n molecules, and if n is not too small these aggregates can be regarded as spherical drops containing n molecules each. I t may now be asked, what is the equilibrium number n of drops containing n molecules ? Provided that the vapour is not supersaturated there is only one solution to this problem. I t is possible to treat the problem from a thermodynamic standpoint by considering the equilibrium, or alternatively, kinetically by an examination of the rate of formation and disappearance of the number of drops containing n molecules.This may occur by means of an aggregate A,+x taking up a molecule, or by an aggregate An+I losing a molecule. A t equilibrium 2, must be a constant. This kinetic method of approach is less exact than the thermodynamic method as it is necessary to make certain definite assumptions concerning the rates of evaporation and condensation. However, it is superior to the thermo- dynamic approach in that it is applicable to systems which are not in equilibrium. For example, it is possible to determine kinetically the rate of change of 2, with time, and furthermore, the method leads to the solution of the problem of the frequency of formation of nuclei as a function of the degree of supersaturation. independently developed a statistical theory of condensation which must be regarded as one of the most important advances in statistical mechanics made in recent times. Mayer attacked the problem in a more general way by assuming that N molecules occupy a volume V and that between any pair of molecules there is a potential energy of interaction vr which depends only upon the distance apart r and which rapidly tends to zero as r increases.In an original discussion of the general integral of state Mayer shows that this may be expressed as a sum of terms represented by the symbols m,, m2 . . . . ml . . . . ; this series indicates that ml clusters containing I molecules are present. Even if the physical meaning of the clusters, which arises in Mayer’s theory from purely mathematical considerations, is not altogether clear, it is important to note that at temperatures which are not too high and for values of I which are not too low, the rnz terms of Mayer are the same as the drop number 2, of the earlier theory.Thus, fundamentally &layer’s theory indicates a method of calculation of 2, and in particular for the derivation of a numerical factor common to all values of Zn, hitherto uncalculated on previous theories. An important confirmation of this concept in Mayer’s theory is obtained if the potential energy associated with the earth’s gravitational field is introduced into the general partition function. It appears from this that individual clusters behave as particles of mass I times the molecular weight in relation to their distribution at various heights; i.e., the large clusters tend to sink to ground level.55 nA, +A,, About twelve years ago Mayer 356 KINETICS OF THE FORMATION OF NUCLEI A particularly brilliant aspect of Mayer's method is in its application to the Einstein treatment of the condensation of helium based on Bose statistics. The condensation is generally described as a difficultly conceivable process in the moment space but Uhlenbeck showed that it could be regarded as a true formation of clusters, analogous to that occurring for ordinary vapours in Mayer's theory. Even if no energy other than kinetic energy is introduced into the partition function the application of the Bose statistics to helium gas shows that an effect exists which amounts to a tendency for mutual attraction between the atoms, comparable to a potential energy of interaction, and which leads ultimately to the formation of clusters distributed in a gravitational field in the normal way.Thus the Einstein condensation may be regarded as a particularly simple case of Mayer's theory which has the special advantage that all the integrals concerned may be completely evaluated. Unfortunately its significance is lessened by the fact that it cannot take into account the van der Waals' attraction between the atoms which is decisive for the observed condensation of helium.56 The contribution of the theory to this particularly important phenomenon, which is so clearly related to the existence of helium 11, is not so great therefore as had been hoped. The Chemical Equilibrium.The methods outlined above may now be considered in somewhat greater detail. We shall consider first the reaction nA, + A n as a chemical equilibrium according to van't Hoff's method. Consider an equilibrium box in which there are 2, single molecules per ~ r n . ~ and 2% drops each containing n molecules ; let this box be connected to a vessel containing n mols of A, at a concentration 2,". We shall consider the work W , gained by a single molecule, when n mols of A, at concentration 2," are reversibly and isothermally mixed with one mol of the drops A,, at a concentration 2%". If this process is carried out in the usual manner using semipermeable membranes, so that the reaction takes place within the equilibrium box, then This reaction may also be carried out by first compressing the n mols from the initial concentration 2," to a concentration Zsat.(that of the saturated vapour over a flat liquid surface), then condensing the vapour to a liquid, and finally by producing one mol of drops from the liquid, each drop having a surface area Fn. If in this last operation the drops are regarded as macroscopic entities, the work done is OF,, where CT = surface tension, and the total work gained a t this stage in the process is therefore 2 " kTn log -1 - OF,. &at. There now arises a difficulty which is characteristic of the whole problem. In the above considerations we have arrived at a stage where the drops, regarded as macroscopic entities, are situated adjacent to each other, whereas in the statement of the problem they were described as gas molecules existing a t a concentration 2,".Actually we have departed from a strict thermodynamic cycle by considering the individual drops as being formed from the liquid one by one. The following solution of the above difficulty may be suggested. If Vn is the volume of a drop, it is possible to consider the drops situated adjacent to each other as a gas of concentration r/Vn. IfR. BECKER 57 this is permissible then the work gained in reaching the concentration 2%" is - kT log Zn"V,, hence Z " &at. n log - 1 - 1OgZn"Vn As W = W' then If F, the pressure vapour pressure, is If we substitute Kn/n for the somewhat uncertain quantity 1'n-l then we have in which the factor Kn depends on n in a manner which is not accurately known.The total number of molecules is N N = XnZ,,. . ' (4 I This series converges only when p < 9,. For p > p,, however, it diverges. Zfi considered as a function of n has a minimum value and from this value the terms increase indefinitely. The minimum may be calculated in the following way. Let In = radius of a drop containing n molecules ; then (vfiq. = the specific n'i; and therefore, From this we have 4r 3 Va = - rn3 = n . vfiq. volume of the liquid) ; hence Ffi is proportional to Here p a is the equilibrium vapour pressure of a drop of radius rn, i.e., Therefore the series of numbers Zn in (I) approaches so closely to zero for p < 9, that ZnZ,, converges ; for p > p,, however, it may be seen from (3) that the series has a minimum value for that value of the drop size for which the vapour pressure f i n is just equal to the given vapour pressure 9.In the latter case therefore equilibrium is impossible and the kinetic theory must be used. The Kinetic Treatment. In the kinetic treatment we consider the growth and disappearance of the drop separately. Let a. = the number of single molecules arriving per sec. per cm.2 at the surface of the drop An, and 4% = the number of molecules which evaporate per sec. per cm.2 from the surface of a drop A,. The ratio a,/qn is equal to $ / f i n , i.e.,58 KINETICS OF THE FORMATION OF NUCLEI The number of processes A,, -+ An+r occurring per second is given on these assumptions as Z,, .F,+Iao, and the number of processes An+x -+An as Z n + I . Fn+I . qn+x. The excess of the latter number over the former is designated by J , where J = adin+= (..- z ~ + ~ ~ p":') For the equilibrium condition, J = 0, and an expression essentially the same as (I) is obtained again for 2,. If, however, p >poo then all 2, values for n > ?Lk may be placed arbitrarily equal to zero (i.e., any nuclei formed are removed). The quasi-stationary state where J is independent of n may then be considered. Thus it follows from eqn. (8) that by eliminating all the terms Z2, Z3 . . . . Znk a value for J is obtained, and this may be called the frequency of formation of nuclei. Thus for example one obtains where A = Q a F k , the work which must be done isothermally and reversibly to produce one critical drop. In this theory N atoms are con- sidered with a potential energy v(r) dependent only on distance and the corresponding partition function c is Mayer's Theory of Condensation.If the integration is carried out with respect to the momentum then A, the de Broglie wavelength at a temperature T , may be written as h = h(znmkT)-''s and the term e - n as I + f ( r ) . Then c = (I +fi2) (I +fi3) . . . (I +fj~) . . . . dr,. . . . d?". . (4) If therefore an The f ( r ) values only differ from zero for small values of r. integral such as J f12f23dr1dr2dr3 has to be evaluated, then the integration may be performed firstly for a given r1 from o to a over r1 and r2. Only the last integration over rl contains the volume factor V . In separating the terms in (4), all those may be taken out in which, for example, the first three particles form a cluster at E = 3.These terms will be the ones containing the factors fiZf23 or fi2fi3 or f12f13f23 and in which the indices I, 2, 3 do not otherwise occur. From the sum of all these terms a factor c may be split out where the term b3 is defined so that it no longer contains the volume. From the sum remaining after the elimination of this factor the terms containing f4, and in which the indices 4, 5 do not otherwise occur may be extracted. Proceeding in this manner may finally be split up into terms cm,, m,. . m l . . , where the individual sums are represented by : (5) m, clusters containing I atom, m2 ,, ,, z atoms, mL ,, S J > )R. BECKER 59 With b defined by { fi2 f23 . . . . fLI, i + . . . . . 1 dr1 . . . drz . * (6) for the contribution of such a series of terms, (Vl ! bl)"l 1 is obtained.I Now in general there are N ! II ~ possibilities of distributing the N The final value obtained for c is 1 l!mlml! particles in the clusters given by (5). therefore m l The Z is taken over all values of the series m,, m2, . . which satisfy the condition Z lml= N . If one of the terms in the sum is relatively so great that for thermodynamic purposes it may replace c, then the indices ml of this term give the most probable values Zl for the numbers ml of the clusters containing I molecules. 1 The appropriate calculation gives = VbiAll, . (8) where the value of the parameter A is given by the condition N = XZml, i e . , by N 1=1 N = VClbLA' . ' (9) Using this approximation log% = - 3NlogA + log N ! - N log A + VZbi A' 3 JV is obtained.From this it follows that the pressure p = - kT __ log c, or Hence the clusters introduced by ( 5 ) affect the pressure as independent particles of an ideal gas. Expression (9) is of particular interest to us because it gives the value of A as a function of the given density N/V of the substance. If A , is the convergence limit of the power series XZbiAz, then the finite sum of (14) assumes enormous values as soon as A > A,. For increasing values of NJV, therefore, A can only increase up to this limiting value. For further increases of N / V , A retains the constant value A , and hence, from (8), the term Zl/V has also a definite value, i.e., blAl,. We are then in the region of condensation, where an isothermal diminution of volume only causes an increase in the amount of condensate but does not give rise to any change in the vapour phase.Also, as may be seen from (IO), the pressure remains independent of the volume in this region. If the convergence limit A , is introduced for the series (9) in (8) and if where 21 is the concentration of the clusters containing I atoms, then60 KINETICS OF THE FORMATION OF NUCLEI From this equation a comparison may be made with the previous equilibrium eqn. (6) for spherical drops, by taking and From (8) it is immediately seen that the parameter A signifies the concen- tration of the single molecules. In this the potential energy is not at first considered. In the calculation of the partition function The Bose-Einstein Condensation. I for N helium atoms in a volume V , E, has the form I - [El2 + ......+ &7. 2m If 'p.(rl . . . . . . r ~ ) is the symmetrical eigenfunction belonging to the energy E, of the total system with 19 I2dY1 ...... drhr = I, J then ?, may be written in the form This sum, taken over all eigenvalues, may be accurately evaluated and gives the result The summation must be carried out over all the N ! pennutations P of the spaces q. In a particular permutation, for example, rl should be taken as rp1, r2 as rp,, etc. Thus, a single partial integral arising in this manner from (12) is similarly to eqn. (4a) of Mayer's theory discussed above. of length I which occur within it. may be easily evaluated. Defining the value of b again, this time by Each single permutation may be denoted by the number mi of the cycles In this case the appropriate integrals With this value of hl, the partition function given by (7) may be formulated, and hence the numbers of the corresponding clusters given by (8) can beR. BECKER 61 obtained. The saturation density may also be accurately calculated. For the series resulting from the combination of (9) with (13)~ i.e., I = > EblA1 = - A3 2 (Ah3)1, 1 1 V there is a convergence limit at Ah3 = I, and in this case it yields the limiting value This corresponds to the well-known fact that the Einstein condensation begins when the single atoms have only the volume A3 available to each of them. In this equation h = h(axmkT)-'/*, i.e., the de Broglie wavelength corresponding to the temperature T. 2-61 Institut fiir theoretische Physik der Universitat Gottingen, Gcttingen, Bunsevtstrasse 9. 1 Volmer, Kinetik der Phasenbildung (Dresden, 1939). 2 Becker and Doring, Ann. Physik, 1935, 24, 719. 3 Mayer, J . Chem. Physics, 1937, 5, 67 ; and subsequent volumes. 4 Mayer and Goeppert Mayer, Statistical Mechanics (New York, 1946). In particular 6 Born and Fuchs, Proc. Roy. SOC. A , 1938, 166, 391. 6 Kahn, Dissertation (Utrecht, 1938). Chap. 13, 14.