Equilibrium radioactive decay question

[nateja]

Homework Statement

A sample of Mo-101, initially pure at time zero, beta decays to Tc-101 which then beta decays to Ru-101 (stable). The half-lives of Mo-101 and Tc-101 are nearly the same and for this problem are assumed equal (14.4 min). After a decay period of one half-life how many atoms of each isotope per initial atom of Mo-101 are present? This problem needs to be solved analytically with integrating factors or numerically with short time steps.

Homework Equations

N(t) = N(0)* e^(-lambda*t)
lambda = decay constant
tau = half-life = ln(2)/lambda

The Attempt at a Solution

My first attempt was to just integrate N(0)*e^(-lambda*t) from t = 0 to t = ln(2)/14.4 (the half-life) and then repeat the same process to get the number for Ru-101. I realized this was incorrect because A) the percent was really really small for both isotopes and B) this did not account for the fact that the Tc-101 nuclei were decaying at the same times the Mo-101 nuclei were decaying so I think i'd have to calculate them simultaneously... not sure how to do that though.

I am reading up on my integrating factors and equilibrium calculations so I will be back with a better approach in an hour or so.

 

[ehild]

Homework Statement

A sample of Mo-101, initially pure at time zero, beta decays to Tc-101 which then beta decays to Ru-101 (stable). The half-lives of Mo-101 and Tc-101 are nearly the same and for this problem are assumed equal (14.4 min). After a decay period of one half-life how many atoms of each isotope per initial atom of Mo-101 are present? This problem needs to be solved analytically with integrating factors or numerically with short time steps.

Homework Equations

N(t) = N(0)* e^(-lambda*t)
lambda = decay constant
tau = half-life = ln(2)/lambda

The Attempt at a Solution

My first attempt was to just integrate N(0)*e^(-lambda*t) from t = 0 to t = ln(2)/14.4 (the half-life) and then repeat the same process to get the number for Ru-101. I realized this was incorrect because A) the percent was really really small for both isotopes and B) this did not account for the fact that the Tc-101 nuclei were decaying at the same times the Mo-101 nuclei were decaying so I think i'd have to calculate them simultaneously... not sure how to do that though.

I am reading up on my integrating factors and equilibrium calculations so I will be back with a better approach in an hour or so.

The number of atoms of a decaying isotope is N(t) = N(0)* e^(-lambda*t). You get the number of Mo atoms by substituting t=14.4 minutes (half life-time). No need to integrate further. "t = ln(2)/14.4" is not time.
The number N_Ru of Ru-101 atoms increases in the rate as the Mo-101 decays, and decreases because its own decay. Write up the equation for dN_Ru/dt. Remember, the number of atoms decayed in dt time is λN. You have to solve the differential equation with the initial condition that the number of Ru atoms is zero at t=0.

ehild

 

[Bryson]

Note, that you have one isotope that decays into another, which in turn decays. This means your formula is incorrect as it does not include the fact that there are daughter and grand-daughter isotopes. (Chapter 6 - Attix)