# How to model mass diffusion with half life

I am in the process of writing a model for the diffusion of a chemical in an aqueous medium. Typically, to do this, one would use the diffusion equation

$$\frac{\partial}{\partial t}U=\gamma \nabla ^2 U$$

where $$U(\mathbf{x},t)$$ is the concentration of the chemical at position $$\mathbf{x}$$ and time $$t$$, and $$\gamma$$ is a positive constant that represents the rate at which the spacial curvature of the concentration function is reduced.

The quirk in the model I am interested in producing is that I would like to include the half life of the chemical in my partial differential equation. To do this, I have concidered including a term to model the removal of the chemical from the system at a rate proportional to to the concentration of the chemical. This modified PDE looks like

$$\frac{\partial}{\partial t}U=\gamma \nabla ^2 U - \delta U$$

where $$\delta$$ is a positive constant between $$0$$ and $$1$$.

Is this a reasonable way of modeling chemical degradation? How would I go about determining the relation between the degradation rate $$\delta$$ and the half life $$\lambda$$ of the chemical?

Your equations look about right (of course, depending on the degradation reaction – the form is correct if it is really a spontaneous decay). However, for dimensional reasons $$\delta$$ must have units of inverse time, so it cannot be a "number between 0 and 1", but is rather a positive inverse time.
To connect the half life and the decay rate you can consider the uniform case $$U(x, t) = U(t)$$, which makes $$\Delta U = 0$$, so you just end up with the equation: $$\partial_t U(t) = - \delta U(t).$$ The solution of this equation is $$U(t) = U(0) e^{-\delta t}$$, so the half-life time $$T_{\frac 1 2}$$ is connected to the decay rate by: $$e^{-\delta T_{\frac 1 2}} = \frac 1 2.$$