1. ## Explanation for a series of numbers (radioactive Decay)

Hello Again!
Thank you to all who helped with my last problem! i sincerely appreciate it!
I have done all the maths to calculate the radioactive decay of the three substances (a,b & c), but i have been asked by my tutor to describe the results over time (paying particular attention to limits). Can anybody help explain what he might mean by this or perhaps share some opinions?

-corey

2. That is a typical nuclear beta-decay reaction of the form...

$\displaystyle A \rightarrow B \rightarrow C$

If we indicate with $\displaystyle m_{a} (t)$, $\displaystyle m_{b} (t)$ and $\displaystyle m_{c}(t)$ the mass of the elements as a function of time and $\displaystyle t_{a}$, $\displaystyle t_{b}$ and $\displaystyle t_{c}$ their 'half life time' and consider that C is 'stable' [i.e. $\displaystyle t_{c} = \infty$...] , we obtain...

$\displaystyle m_{a} (t) = m_{a} (0) \cdot 2^{-\frac{t}{t_{a}}}$

$\displaystyle m_{b} (t) = \{m_{b} (0) + m_{a} (0) (1-2^ {-\frac{t}{t_{a}}}) \} \cdot 2^ {-\frac{t}{t_{b}}}$

$\displaystyle m_{c} (t) = m_{c} (0) + \{m_{b} (0) + m_{a} (0) (1-2^ {-\frac{t}{t_{a}}}) \} \cdot (1- 2^ {-\frac{t}{t_{b}}})$

From the diagrams it seems to be...

$\displaystyle m_{a} (0)= 800 g$

$\displaystyle m_{b} (0) = 100 g$

$\displaystyle m_{c} (0) = 0 g$

The numerical value of $\displaystyle t_{a}$ and $\displaystyle t_{b}$ can be extrapolated from the diagrams...

A very interesting problem!...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. thanks alot! I think i am missing something however, how does this relate to limits? thats the piece i am struggling with

(+rep to you btw)

4. Very easy!... at the time $\displaystyle t=0$ the global mass is $\displaystyle m_{a} (0) + m_{b} (0) + m_{c} (0) = 900 g$. At the time $\displaystyle t=\infty$ the global mass is the mass of C alone and for the law of canservation must be $\displaystyle m_{c} (\infty) = 900 g$...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$