Decay - Abundance of Multiple Isotopes

[newtonuke]

Homework Statement

An element Z has two isotopes Z1 and Z2 with decay constants of λ1 and λ2. The natural activity of natural occurring 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.

Homework Equations

Activity = Nλ
N = N₀e^-λt

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...

N_{less z1}(t)=N₀e^-λ1t ; N_{less 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 N_z1 = 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!

 

[Jhero]

Hello!

Your solution is on the right track, but there is a small error. The activity is actually used to determine the number of atoms of each isotope at any given time, not just the remaining amount. The activity is equal to the number of decays per unit time, and since we know the decay constants, we can use the equation Activity = Nλ to determine the number of atoms of each isotope present at any given time.

So, to determine the abundance of each isotope, we can use the equation:

Abundance Z1 = N1/Ntotal = (Activity * e^(λ1t))/(Activity * e^(λ1t) + Activity * e^(λ2t))

Similarly, for Z2:

Abundance Z2 = N2/Ntotal = (Activity * e^(λ2t))/(Activity * e^(λ1t) + Activity * e^(λ2t))

The activity cancels out in the numerator and denominator, and we are left with the abundance of each isotope as a function of time.

I hope this helps clarify things! Let me know if you have any other questions.