Solving a Radioactive Decay Problem: Na(t) and Nb(t)

[blaksheep423]

Homework Statement

Consider a radiactive decay problem involving two types of nuclei, A and B, with populations Na(t) and Nb(t). Suppose that type A nuclei decay to form type B nuclei, which then also decay, according to differential equations:

Homework Equations

dNa/dt = - Na/Ta

dNb/dt = Na/Ta - Nb/Tb

where Ta and Tb are the decay constants. Na(0) = 100 and Nb(0) = 0.

The Attempt at a Solution

I solved the equation for Na(t) and got 100*e^(-t/Ta), but I'm not sure about Nb. Intuitively, I think it has to be something like:

100*(1 - e^(-t/Ta))*e^(-t/Tb)

but I know this is wrongany advice/ideas?

 

[blaksheep423]

...anyone?

 

[vela]

Homework Statement

Consider a radiactive decay problem involving two types of nuclei, A and B, with populations Na(t) and Nb(t). Suppose that type A nuclei decay to form type B nuclei, which then also decay, according to differential equations:

Homework Equations

dNa/dt = - Na/Ta

dNb/dt = Na/Ta - Nb/Tb

where Ta and Tb are the decay constants. Na(0) = 100 and Nb(0) = 0.

The Attempt at a Solution

I solved the equation for Na(t) and got 100*e^(-t/Ta), but I'm not sure about Nb. Intuitively, I think it has to be something like:

100*(1 - e^(-t/Ta))*e^(-t/Tb)

but I know this is wrongany advice/ideas?

Just plug it in and solve! ;)

Your equation for N_b,

$$N'_b(t)+\frac{1}{T_b} N_b(t) = 100e^{-t/T_a}$$

is a basic first-order, inhomogeneous differential equation. You can solve it using an integrating factor or the method of undetermined coefficients.