78 APPENDIX Il WORLDWIDE EFFECTS OF ATOMIC WEAPONS ea 2c \' Old. 14 (25) (14) Here B and C are constants. 79 NUCLEATION RATE Clusters that grow beyond the size x* have free energy that decreases with the addition of single molecules. The rate at which nuclei are formed is the rate at which clusters with x > x* are formed. This rate is‘) The concentration of clusters of this size is NGA) =Nayep(- £2). 27 B I= FS Vs N(x). (15) The rate of nucleation (see below) is directly proportional to this quantity. The constant B is given as —(AH/R)(1/T, — 1/T), of course. The value of C is not so easy to obtain, but we can estimate its magnitude by several methods. Here S is the area of a cluster divided by x%, ie., the area of a cluster is given by Sx; F is the flux of single molecules as used above. Using (18), we have If surface-tension data are known, then 30 \* surface free energy = tro{ x, P= (16) where o is the surface tension and 1, is the volume of a single molecule of the cluster. Another method for the estimation of C may be helpful. Consider a small cluster Ta ? —— AH 27 G — 7) | 1 _ 17 (17) In this consideration we have neglected entropy considerations and so C is actually (20) 7 If we assume that the precipitation takes place rather rapidly after a time ¢, has vot—th, we can write for the nucleation rate I(t’) = Aexp E a asi oz Aexp |~ i (: — *)I, Cx = C(13%) — (3215 ee (22) which is the simple exponential approximation we proposedearlier, i-e., | I(t") « et’, and we see that smaller, by a factor that may be as small as one-half. However, the magniiude must be +? 4 T? AH 1 a 2= 57 PERT, fy (23) at, = T, — To, (24) or, with substitution, AH where 7, is the temperature of condensation, and Use of this value for C gives for Eq. (15): N(x*) = N(A) exp |- 4 (53) =a 4 — where A and + are constants defined by the equation. In the last expression, at/T, elapsed and define C= = 1.25 1.25 22. Fe \ ORT ND) exp 4(T?\sH cule surrounded by its twelve nearest neighbors. This collection of molecules has AH Ps If the temperature is changing linearly with time, say, T, — T = at, then has been neglected. binding, therefore, is approximately (42/78)oH per molecule. If we set AH el — &£ having a simple symmetry. A convenient number of molecules to take is thirteen in hexagonal close packing. Thirteen molecules make a cluster with one central mole- 12 bonds to the central moiecule and 5 each to the others. The total number of bonds is (12 + 12 X 5) = 36 bonds. In the interior of a large solid, these thirteen molecules would have % X 13 X 12 = 78 bonds. The missing heat of (19) (18) “ 4 T, 2 a AH Ma = 7) T= TF,KT, = —~[ ---—-- ——--— —. (25) 2