256
FRIEDLANDER
Expressions for g(t) and v’(t) can be obtained by substituting
Eq. 8 into Eqs. 1 and 3:
g(t) = Ne if va ()|
vtt)=
(9)
+
oJ,* ¥ d(v/v*)
(10)
N, Jp (v/v*) ¢ dlv/v*)
Substituting these expressions in Eq. 8 and noting that Jo @ d(v/v*) and
> w/v*t) w d(v/v*) are dimensionless constants, one finds
n(v,t) =
vy (“X=)
(11)
The function %, differs from ¥. It is a dimensionless size-distribution
function, and its argument
VN. _
a
(12)
is the ratio of the particle volume for a given size range to the average
volume ¢/N,,. Substitution of Eq. 12 into Eqs. 1 and 3 showsthat
(13a)
fy ban) dn =1
and
SS nda) an = 1
|
(13b)
SELF-PRESERVING FORM AND SMOLUCHOWSKI KINETICS
The self-preserving form Eq. 11 must satisfy the equation for the
rate of change of the distribution function withtime. If the Smoluchowski
expression, Eq. 5, is accepted as the proper kinetic equation, Eq. 11
can be substituted to give the following result:
4
a
roy
~
~
1
x [A+ - iN) + (n —7)%
nO
a
i dN.
dip
wae
[Be+1 a4ef W@) vn ~ 7)
xan fo (9) [1 + 9% I |a
(14)