Solving I-128 Atom Equations: A Homework Challenge

[6Stang7]

Homework Statement

In an experiment, the isotope I-128 is created by bombarding I-127 with neutrons. This creates 1.5E6 I-128 atoms per second. Initially there are no I-128. The half-life of I-128 is 25 minutes. Find the equation that represents the number of I-128 atoms as a function of time.

Homework Equations

dy/dt=creation rate-decay rate

The Attempt at a Solution

I set up dy/dt=(1.5E6)t-ay.
a is the decay rate in e^(-at), and y(t) is the number of I-128 after time t.
I tried solving this as a first order linear equation and got a horriable answer. Where am I going wrong?

 

[Dick]

The solution may be a little messy, but I don't think it it's horriable. The general solution to the ode is C*e^(-at) as you know. Just find a particular solution and add it. It should just be a linear function. Post your solution if you really need more advice.

 

[6Stang7]

 

[Dick]

I would have solved the ode by first solving y'+ay=0, getting Ce^(-at). Then found a particular solution to y'+ay=kt by observing a linear function will work, setting y=At+B, substituting it in and solving for A and B, getting (kt/a-k/a^2) and then adding it to Ce^(-at). But that's not that different from what you did.

 

[6Stang7]

I think I have it, I just want it to get check. At first, my thinkging was that in 25 mins 2.25E9(1.5E6*1500seconds) I-128 atoms are created. If I double 2.25E9, I get 4.5E9. Once there are that many I-128 atoms, then in 25 minutes, half of 4.5E9 will decay; however, in 25 mins 2.25E9 will have been created. However, thinking about that now, I see an error in my logic. The only way 2.25E9 atoms can be created in 25mins is if there is no decay rate. That being said, limit to the number of atoms that will exist after a very long time must be less then 4.5E9. I worked the problem over again this morning with fresh eyes and got this. Does it look good? Seems correct.

 

[Dick]

This looks much better. The creation rate is constant - not an increasing function of time. Sorry, I didn't catch that. Thinking more about the mechanics of the solution than the actual problem.