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

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