Derivation for Time to Max Radioactivity Transient Equilibrium

1. Mar 20, 2013

matthewt

Hi,

I'm having a bit of difficulty deriving the time to max activity for the case of transient equilibrium for a parent-daughter.

This is where I want to get to , tm = (1 / (λ1-λ2)) * 1n(λ1/λ2)

I believe there is an alternative equation for tm as well expressed in terms of half-life.

I would be gratefeul if someone could walk me through a derivation,

BW,
Matt

2. Mar 20, 2013

voko

Do you mean that you have some parent substance, with a given half-life time, which produces some daughter substance, with another given half-life time (which then decays into an inactive substance), and you need to find out the time it gets for the mix to attain maximum activity?

3. Mar 20, 2013

matthewt

that's right. For the case of a Mo-99 and Tc-99m generator, at t=0, let the activity of Tc-99m be zero. The activity of Tc99m will increase until it reaches a maximum value, but will then start to decline as per transient equilibrium. I think using the bateman equations you can derive an expression for Tmax, the time it takes for the Tc-99m to reach it's max activity (which is ~ 24 hours). A derivation of that equation is what i'm after.

thanks,
matt

4. Mar 20, 2013

voko

Do you understand how the single-step decay equation is obtained and solved?