76 WORLDWIDE EFFECTS OF ATOMIC WEAPONS The radius of any particle increases linearly with time: R(t”) = BU — 1), (3) where Cis Thepesca at the time Is APPENDIX II 77 W(x) = (Hsotta — Meas) + surface free energy, (9) where peniia — Hean iS the difference in free energy for one molecule between a large solid and the gaseous state in question. If we can approximate the vapor c prea ay GME UIir UE WRITER TAC ay p = const exp [ — (4SH/RT)], t (10) it can be shown that where I(t’) dt’ is the number of nuclei that were formed in the time interval d?’. If a gas:is chilled at a constant rate, the rate of nucleation also increases with time. We shall see below that a reasonable first approximation is I(t’) = lest (-o <?’ <4), (5) where |, is the nucleation rate at a time that is set equal to zero, and a is a constant. Substitution of Eqs. (3) and (5) into (4) allows an analytical integration (from / = —c tof’ =f): (6) In this integration, 8 was taken to be a constant, which means that this formula can only be used when there is no scrious depletion of pascous single molecules of the condensate. At such a time the particle-size distribution has a particularly simple form. The number of particles having a value of R in the interval R to R + dR were those formed at the time f -- R/B to (tf — R/B) + dR/f. Thus N(R ana i(r— i) p R\adR RAR 1,6 a dR (a ~ 4 2) melar~ SR) (7)7 of, more simply, N(R) « exp | ~(a/B)R]. The big part'-" of ne" AH 1 1 (; — i): (11) where AH is the heat of vaporization and 7, is the saturation temperature for the system; K is the gas constant in this formula (it was used as patticle radius above). The work required to form a cluster of x molecules is divided into two terms. The first term, AH ( 1 1 vey = 473 BoacGeaty. a! (R) 1. RT (Msotia Henn) = R (8) efore, have the distribution given by (8). Since the rate il in time, it can be argued that the entire distribution same function. The development ofa distribution funcdepletion has not been considered as yet. RAT, Ty? is negative for all temperatures below the saturation temperature and varies directly as the cluster size; the second term, surface free energy « x", is always positive and varies as the surface area of the cluster. For increasing surface area, the sum of these terms goes through a maximum. In the condensation theory it is assumed that all clusters having positive free energy and that are smaller than the cluster of maximum free energy are in equilibrium. It can be shown that Noy = NC) ep] — BO, (12) where N(x) is the number of clusters of size (x) per unit volume and N(1) is the number of uncombined condensable molecules per unit volume. The maximum value of the function WO) _ SH (7. - 1)s 4+ Cx = —Bx + Cx (13) FREE ENERGY OF bon “10. ister of x condensable molecules with respect to the i.€., variety of products. For example, Al,O, yields the gaseous products AIO, Al,O, and O,. This method is also applicable in such cases.