We are going to assume that the fissile product has one mode of decay and does not do anything naughty, such as undergo neutron capture. We can then write its rate of production as:

Where is the decay constant for fission product N, is the number of fissions per second and is the fraction of fissions which produce fission product N. I have shown how to calculate previously and is given in the data presented in Composition of Spent Fuel.

Again I am making the approximation that the rate of fission is constant.

 

When t=0

where N₀ is the amount of the fissile product at time t=0.

If we assume that there is none of that fissile product in the fresh fuel we get:

Although I have not been able to find this formula on the web, a similar formula is given in http://www.ornl.gov/info/reports/1961/3445605157576.pdf

Below is a plot of this function with arbitrary units, i.e. I have not put in ‘real’ values for any of the constants. You can see that if the fissile product has a short half-life compared with the time that it is in the reactor, the amount soon reaches an equilibrium – i.e. the rate of production equals the rate of decay. However, if the fission product has a longer half-life, the amount continues to rise.