# Homework Help: Radioactive Decay Calculations

1. Apr 23, 2010

### Lissajoux

1. The problem statement, all variables and given/known data

${}^{241}_{95}$Am produces $\alpha$ particles, which interact with ${}_{4}^{9}$ Be to produce neutrons.

The Am sample emits $\alpha$ particles at a rate of $70.0s^{-1}$ on June 16, 1996.

What was the activity on June 16, 2004?

2. Relevant equations

The half life for ${}^{241}$Am :

$$T_{1/2}=432.2y$$

Also know that:

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

3. The attempt at a solution

I know this calculation isn't particularly tricky, but I just can't figure it out at the moment

I've calculated the wavelength:

$$\lambda = \frac{ln(2)}{T_{1/2}} = \frac{ln(2)}{1.363\times 10^{10}}} = 5.086\times 10^{-11}m = 0.05nm$$

Then guessing just need to calculate what the value of N is, and can multiply this with the value of $\lambda$ to get A.

Now N, or rather N(t), is just the number of nuclei remaining in the sample after a time t.

I'm not sure, but is this just.. 241?! :shy:

I'm getting a bit confused with this, really shouldn't be I know , but a bit of guidance would be great.

2. Apr 23, 2010

### nickjer

First note: $$\lambda$$ isn't wavelength in this problem, it is called the decay constant. So check your units.

You don't know your initial or final N. But you know the initial dN/dt. Knowing the initial dN/dT you can plug this into your 2nd equation to get the initial N. Then you are right... Solve for the final N and get the final dN/dt.

A simpler way would be to take the ratio of both dN/dt's, like so:

$$\frac{\frac{dN_f}{dt}}{\frac{dN_i}{dt}} = \frac{N_f}{N_i} = \frac{N_0 e^{-\lambda t_f}}{N_0 e^{-\lambda t_i}} = e^{-\lambda (t_f-t_i)}$$

3. Apr 23, 2010

### Lissajoux

Ah yes of course it's the decay constant not the wavelength, silly me!

Right so something like this:

$$\frac{dN}{dt}=70.0s^{-1}$$

Have calculated $\lambda$ as: $5.086\times 10^{-11} s^{-1}$

So can rearrange this equation:

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

Therefore:

$$N = 1.376 \times 10^{12}$$

.. which is the initial N.

Correct?

Then I need to calculate final N, which I can do from:

$$N(t)=N_{0}e^{-\lambda t}$$

Using the calculated value of $\lambda$ and t value of .. 8? Or $252.288\times 10^{6}$. I.e. I assume it's the latter and that I need to convert this too to seconds?

Once I've found this value of N, can use $A=-\lambda N$ to get the activity at this time, right?

Hopefully I'm getting there with this question somewhat.

4. Apr 23, 2010

### Rajini

Hello nickjer,
is it not correct to take 70 per sec as initial value?
just curious to know.

5. Apr 23, 2010

### nickjer

Remember your t is in units of years. So convert them to seconds. You are right about plugging the time back in to get the final N. Then you get the activity from this.

Or you can just use the final formula I gave in the last post. Just solve for the decay constant in units of years (very easy to do). Then plug it in, and keep your t in units of years. Since you know the initial dN/dt, the final dN/dt just comes out from it.

Edit: It is correct to take 70 per sec as the initial rate, dN/dt. Just make sure all the units work out correctly.

6. Apr 23, 2010

### Lissajoux

So if I stick in $t=252.288\times 10^{6}$ into $$N(t)=N_{0}e^{-\lambda t}$$ I can find the final value of N, or rather the value of N at the later time stated in the question. Then can use this to find A at the time required but using $A=-\lambda N$

7. Apr 23, 2010

### nickjer

Yes, that is correct. Just make the solution positive in the end.

8. Apr 23, 2010

### Rajini

Okay,
then better use this:
$$N(t)=N_0\exp\left(\frac{-t\ln 2}{T_{1/2}}\right),$$.
take N0 as 70 s-1.
t is 8 yrs and T1/2 is 432.2 yrs

9. Apr 23, 2010

### nickjer

That formula can be confusing with your notation. The same formula can be seen in my first post at the end.

10. Apr 23, 2010

### Lissajoux

The minus sign simply denotes that the activity is decreasing, so I can ignore it in my actualy calculations because the value I calcualte must be positive, right. If that makes sense.

11. Apr 23, 2010

### nickjer

The minus sign just means the number of particles is decreasing. You can ignore it because they just want magnitude of the rate.

12. Apr 24, 2010

### Lissajoux

Right so I've got that $N(t)\le N_{0}$ , so that $N(t)\approx 0.987 N_{0}$

Then from this value, that $A$ at time $t=8y$ is 69.1.

Do I have to round this to an integer value i.e. 69? I assume not, because A before was 70.0, and this is average decay rate so doesn't have to be whole number of nuclei right? So I'd have that $A=69.1 s^{-1}$

.. correct?

13. Apr 24, 2010

### Rajini

Yes, if you use the formula that i posted you get 69.1 per sec.
Also you no need to round off. But you mentioned that the solution is 241 ??[no unit!]
Do you understand the formula. ?

14. Apr 24, 2010

### Rajini

Here please note that this formula is something different. Here 't' is not same for both initial and final states. So i guess this formula is something different and may not be applicable for you.
EDIT: Since you have 3 unknowns factors, $$t_f,t_i,N_f$$.

15. Apr 24, 2010

### nickjer

Ummmm... You do know them:

$$t_f$$ = June 16, 2004
$$t_i$$ = June 16, 1996

So you are left with:

$$t_f - t_i = \Delta t = 2004 - 1996 = 8$$ yrs

It is just the change in time.

16. Apr 24, 2010

### Lissajoux

Sorry Rajini you've now confused me.

Is this correct what I have calculated:

$$N(t) \approx 0.987 N_{0}$$

Then from this value:

$A$ at time $t=8y$ is $A=69.1 s^{-1}$

17. Apr 24, 2010

### Rajini

Sorry!
Nickjer, I overlooked some thing.
soln. is correct 69.1 per sec. (I did not confuse you). Offcourse you can use the formula which i wrote or Nickjer's one.

18. Apr 25, 2010

### Lissajoux

I also need to find out how many Am nuclei were in the same at each date. How do I go about this? can't figure it out.

19. Apr 25, 2010

### nickjer

Use the 2nd equation from your first post. The equation with $$A$$.