# Deriving the Bateman equation of Nuclear Decay Chains

by Elariel
Tags: bateman, chains, decay, deriving, equation, nuclear
 P: 1 1. The problem statement, all variables and given/known data Derive Bateman equation for a decay chain a->b->c->d where each decays with a given mean life let decay constant be L, where L=1/mean life Na(0)=No, Nb(0)=Nc(0)=Nd(0)=0 2. Relevant equations Want to derive Nb(t)={(No)(La)/(Lb-La)}*{exp[-La*t]-exp[-Lb*t]} extend for Nc(t) 3. The attempt at a solution dNa(t)/dt=-La*Na(t) Na(t)=No*exp[-La*t] dNb(t)/dt=-Lb*Nb(t)+LaNa(t) dNb(t)/dt=-Lb*Nb(t)+La{No*exp[-La*t]} this is a none homogenous differential equation. I can't find a way to solve it. dNc(t)/dt=-Lc*Nc(t)+LbNb(t) I'm really not sure where to go from here. If anyone could lend a hand it would be greatly appreciated.
 Sci Advisor HW Helper Thanks P: 24,980 Ok, so what you are really asking is how to solve y'(t)+L*y(t)=f(t). You know the homogenous solution is exp(-L*t). Guess the solution will be of the form y(t)=g(t)*exp(-L*t). Put this guess into your original equation and get: g'(t)*exp(-L*t)-L*g(t)*exp(-L*t)+L*g(t)*exp(-L*t)=f(t). So g'(t)=exp(L*t)*f(t) and you can just integrate to get g(t).

 Related Discussions Programming & Computer Science 5 Advanced Physics Homework 3 Introductory Physics Homework 4 Introductory Physics Homework 3 High Energy, Nuclear, Particle Physics 1