# Homework Help: Differential equation - radioactive decay

1. Oct 18, 2009

### phil ess

1. The problem statement, all variables and given/known data

Suppose that a given radioactive element A decomposes into a second radioactive element B, and that B in turn decomposes into a third element C. The amount of A present is initially a0. The amounts of A and B present at a later time t are a and b respectively. If k1 and k2 are the reaction rates for the two chemical reactions, find b as a function of t.

Hint: Consider k1 and k2 to be positive, so, for example, one of the equations that you need is da/dt = rate in − rate out = −k1a. Also, you may assume k1 =/= k2.

2. Relevant equations

rate = kAmBn?

halflife = ln 2 / lambda ?

3. The attempt at a solution

Well this isnt like any other radioactive decay problem Ive seen. Usually we're given initial conditions, or the half life, or something! I dont even know where to begin...

2. Oct 18, 2009

### Redbelly98

Staff Emeritus
Does the problem statement really say these are chemical reactions? I ask because they are not -- they are nuclear reactions.

Okay, for starters, do you understand that radioactive decay is described by an equation like
da/dt = -k1 a​
where k1 is the decay rate?

The next steps are:
1. Solve the differential equation for da/dt
2. Then write an equation for db/dt

You are given initial conditions: the amount of A is initially a0. There is no B or C present initially.

3. Apr 24, 2010

### Bachelier

Can someone finish up the problem and get an equation for db/dt?