# Decay - Abundance of Multiple Isotopes

1. Sep 16, 2009

### newtonuke

1. The problem statement, all variables and given/known data
An element Z has two isotopes Z1 and Z2 with decay constants of λ1 and λ2. The natural activity of natural occuring element Z is A. What is the abundance of each isotope?

Assume that these decay constants are long - the abundance of the element over time does not change significantly.

2. Relevant equations
Activity = Nλ
N = N0e-λt

3. The attempt at a solution

I don't understand how activity comes into play here. It seems to me that the given decay constants determine the abundance of each isotope and the activity is just a scalar for the total amounts of each. Is that correct?

So...

Nless z1(t)=N0e-λ1t ; Nless z2(t)=N0e-λ2t

But these quantities give quantities of Z remaining, not Z1 and Z2, so to get daughter quantities, we manipulate the equation to Nz1 = N0-N0e-λ1t = N0(1-e-λ1t)

This would give the number of atoms of each isotope at time (t). To determine the abundance of each, we would just put the number of atoms for each respective isotope in the numerator and the total number of atoms, N0, in the denomenator.

Abundance Z1 = N0(1-e-λ1t)/N0 and likewise for Z2. The N0 cancels and we have abundance Z1 = 1-e-λ1t

This answer seems a bit weird and I didn't use the activity at all. Please provide any thoughts. Thanks!

Last edited: Sep 16, 2009