# Math Help - Isotopic Decay: derive r/ship b/n effective conc. & initial/additional doses

1. ## Isotopic Decay: derive r/ship b/n effective conc. & initial/additional doses

Hi Forum Members,

I am examining how the logarithmic decay of an isotope varies based on several variables:

$C_{t}=C_{o} e\frac{-0.693t}{t_\frac{1}{2}} + C_{add'l}$

where

$C_{t}$ = concentration at time = t
$C_{o}$ = initial concentration at time t = 0
$C_{add'l}$ = concentration of additional exposure
$t_\frac{1}{2}$ = isotope half-life

To try and make this a little clearer, I have also attached a snapshot of an excel s/sheet which performs this calculation for me. In this example:

$t_\frac{1}{2}$ (isotope 1) = 3 days
$t_\frac{1}{2}$ (isotope 1) = 8 days

Thus, for isotope 2: you will see how the concentration of isotope 2 decays from an initial 400 units from day 0 to 308.44 units of day 3. On day 4, a fresh exposure of 150 units is made, at which point the concentrations become additive and then that total decays further.

Note that exposure to 400, 150 and then 125 units at 4 day intervals gives rise to an effective concentration of $C_{t}$ = approx. 430 units on days where t = 4, 8, 12 and 16.

So my question is this:

If someone told me that exposure to an isotope with $t_\frac{1}{2}$ = 8 days at 4 day intervals gave rise to approx. 430 units on days 4, 8, 12, and 16, is there a way in which I could work backwards to determine the numbers in red (ie, 400, 150 and 125 units) ?

In other words, I need to derive a relationship to solve for $C_{o}$ and $C_{add'l}$ in the equation above, with the known constants of t and $t_\frac{1}{2}$.

Ultimately I need to be able to convert such an expression into a formula for an excel spreadsheet so I would greatly appreciate a layman's response.

Many thanks,

CJ.