# Radioisotopes with the greatest activity?

1. Jul 4, 2006

### danago

"You are studying two radioisotopes, each of which contain the same number of undecayed atoms at the instant the activity of each is measured. Sample X has half life of 8 seconds, while sample Y has a half life of 1600 years. Which sample has the greatest activity? Explain."

Im not sure. I would have said the 1600 year one, since it takes much longer for the activity to decrease, but im not sure. Any confirmation on this?

2. Jul 4, 2006

### Hootenanny

Staff Emeritus
Activity is the rate of decay of radioactive nuclei and is represented thus;

$$A = - \frac{dN}{dt} = -\lambda N$$

where $\lambda$ is the decay constant and is related to the half life thus;

$$T_{\frac{1}{2}} = \frac{\ln 2}{\lambda}$$

So, combining the two equations we obtain;

$$A = - \frac{\ln 2}{T_{\frac{1}{2}}} \cdot N$$

Where T1/2 is the half life. Can you use the above to now answer your question?

3. Jul 4, 2006

### danago

Thanks for the reply. With that last equation, can i say that as the half life approaches infinity, activity approaches zero? Or am i completely off track?

4. Jul 4, 2006

### nrqed

You are not off track. And that makes sense since an infinite half life means that the element is perfectly stable, which then implies that there are no decays and hence no activity.

5. Jul 4, 2006

### danago

So the answer to the question would be the 8 second substance then?

6. Jul 4, 2006

### Hootenanny

Staff Emeritus
I would agree with that

7. Jul 4, 2006

### danago

yay :)

Thanks very much :)

8. Jul 4, 2006

### Hootenanny

Staff Emeritus
My pleasure.

9. Jul 4, 2006

### nrqed

As Hootenany's fromula shows you, the activity depends on two factors: the number of particles present and the half life.

If you have a trillion nuclei with a half life of 8 seconds and a trillion nuclei with a half life of 1600 years, in which sample will you observe more decays per second? Here the numbers of nuclei are the same so it leaves only the effect of the half life.

The one with the half life of 8 seconds decays much faster than the other one, right? So just using simple intution, which simple would you expect to have a larger activity? (and then you can confirm this with the equation)