# Homework Help: 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?