A Binomial Implication for the General First Order Linear Rate
Equation.
Douglas Eagleson, 2005
See the bottom of this webpage for the note-
http://llef.tripod.com/eagelson
The first equation, No 1, is the contemporary usage of the general
first order linear rate equation
for a simple two compartment model. Here the radioactive aspect is
taken advantage.
A number of atoms of the daughter, n=2 is related to the initial number
of atoms of the
parent, n=1. l1 is lamda the decay constant of the parent while l2 is
lamda of the daughter.
No.1:
(Number of atoms of n=2) = (Initial number of n=1) *l1*(
(exp(-l2*t)/(l1-l2)) + (exp(-l1*t)/(l2-l1)) )
This equation relates the number of actual atoms of origin from the
parent.
A noncontemporary usage is to relate the activity of the parent to the
daugther in this
exact atom correspondence fashion.
An activity unit may be substituted as the dependent and independent
variable.
(Activity of n=2) is the activity of the daughter.
(Activity of i=1) is the initial activity of the parent at time zero.
t- is the time since time zero.
(li=1)- is lamda for the parent to daughter fraction.
No 2:
(Activity of n=2 ) = (Activity of i=1) * (li=1) * (
(exp(-l2*t)/(l1-l2)) + (exp(-l1*t)/(l2-l1)) )
A function to test this equation is the decay function and it has been
well tested.
A special discovery? is this claim.
A binomial function is the decay usage. And the differential may equal
the fractional change
relation of this function. Meaning the lamdas and the time function
part comprise the decay fraction
and this may equal the differential of the activity.
A characteristic of the binomial function is its element or unit as
opposed to the number,
A unit atom change comprises the unit of the function.
And when the time is short the decay is negligible, making the unit of
difference not
exceeded. Causing the fractional change in the short period to equal
the true rate.
An implication is the usage. A set of many parents may be the system.
And the set
of many distinct daughters is to be counted. And the ratio of the
activities at the start of the count, when time is the time in the
equation above, directly relates the fraction present at time
zero!!!!!!!!
A rate function of the second order is given. Equation one and two are
second order differential
functions.
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