Date: Tue, 24 Mar 1998 14:10:44 -0600 (CST)
From: Dave Rusin <rusin>
To: allyshel@email.unc.edu
Subject: Re: Math in physical chemistry question
Newsgroups: sci.math

In article <6eu2m5$kfv$1@fddinewz.oit.unc.edu> you write:
>I am currently working on a calculation of the rate of phosphorescence in
>an organic sample.  The machine is an antique spectrofluorometer and I
>have to calculate the rates from a time versus signal ratio at specific
>emmission and excitation wavelengths. 
>
>My question is this...
>to acocunt for instrument response, the relating equation is in a form as
>follows
>
>r(t)=e^(-k2*t) _  e^((k1-k2)*t)
>     --------	 ------------
>	k2	     k1-k2
>
>where r(t) is the signal
>t is time
>k2 is what I want to know, and k1 is a constant

This isn't my area of expertise, so I'd be happy to hear of what you hear
from someone else, but here's my take on it.

In principle, one reading will suffice to compute  k2: given numerical
values for  a=k1, b=r(t), and  c=e^(-t), we need only solve one equation
for one unknown  x=k2:
	b = c^x / x - c^(x-a) / (x-a)
This can be solved e.g. with Newton's method.

Here you'll face two difficult problems: first, the numerical solution  x  is
going to be difficult to compute with precision in extreme cases, e.g.
a near 0, c near 0 or 1, etc. Oh, software exist to do this, certainly, 
but the answers are in the extreme cases sensitive to the accuracy of
values of  a, b, and  c; a small error in their measurement will throw
off the calculations of  x.

Second, the value of  x  computed from one measurement -- even if quite
accurate (numerically) and stable (experimentally) may not agree with the
value of  x  computed from another experiment. In principle there is no
problem if the values of  r(t)  lie _exactly_ along a curve of the
form you've proposed, for some  k2; but are you sure there is no
deviation from this model -- no secondary effect dominated but not
obliterated by the proposed equation for  r(t)? If the data do not lie
exactly along a curve of the form shown, then you'd need to do some
least-squares computations (say) to determine the "best"  k2  to fit the
data.

On the other hand, if you do have many observed values of  a, b, c  then
you may be in _better_ shape. For example, with sufficiently large values of
x  (relative to  a), the equation is nearly

	b = c^x (1 - c^(-a)) / x
which can be written in the forms

	b' = c^x / x    or

	b'' = x - log(x)/log(c)

If  b''  is sufficiently large compared to  c, this is then roughly

	x = b'' + log(b'')/log(c)

and further asymptotic terms can be added in. 

I'm sorry I can't be more helpful; the topic of "model fitting" is rather
delicate in general; what constitutes a good suggestion depends a lot on the
kind of data which are given and the intended qualities of the fit.

dave