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