How Long Until a Radioactive Daughter Isotope Reaches 97% of Its Equilibrium?

[Winzer]

Homework Statement

A parent isotope has $$\tau_\frac{1}{2}=\delta$$. Its decays through a series of daughters to a final stable isotope. One of the daughter particles has the greatest half life of $$\tau_\frac{1}{2}=\alpha$$-- the others are less then a year. At t=0 the parent nuclei has $$N_0$$ nuclei, no daughters are present.

How long does it take for the population with the greatest half life to reach 97% its equilibrium value?
At some t, how many nuclei of the isotope with the greatest half life are present, assume no branching.

Homework Equations

$$\frac{dN}{dt}=e^{-\lambda t}$$

The Attempt at a Solution

So for the first one:
Its just solving the diff eq above right? The daughter is in its eq. value or do we have to worry about decay from the other daughters?

the second one:
Basically plugging in t right for the solved diff eq with initial nuclei right?

Just checking, I feel like I'm missing something.

 

[fatra2]

Hi there,

You have the right equation: $$\frac{dN}{dt} = e^{-\lambda t}$$ But don't forget that the daughter nuclei also decay at a certain rate. Therefore, you need to consider the same equation for the long life daughter nucleus.

By the way, just a further comment, typically what half-life are you talking about here? Because, daughter nuclei with half-life of more than a few split second are normally considered into the decay chain.

Cheers

 

[Winzer]

the halflife(longest) for the daughter is 20yr. The parent is 10^4 yr.
So for the daughter nuclei(20 yr):
$$\frac{dN}{dt} = e^{-\lambda_1 t}- e^{-\lambda_2 t}$$
Where 2 is the daughter. Should 1 be the half life of the 1yr daughter?

 

[fatra2]

Hi there,

When the equilibrium is reach, the decay rate of the parent nuclei is the same as the decay rate of the daughter nuclei, and it is independant of the daughters formed in the process. Therefore, you would have: $$\frac{dN_1}{dt} = \frac{dN_2}{dt}$$

If you solve this simple equation, you have the time needed to reach equilibrium.

Cheers

 

[fatra2]

Hi there,

Your question really caught my attention, and with the half lives you gave me, I find that the system will reach equilibrium after 138.2 years.

Cheers