Reaction-diffusion equation

[barcafan]

Homework Statement

In a tube filled with a dissolved substance, there is a probability for each molecule to disappear in time dt equal to kdt. Write an equation for dc/dt for the case in which c is not spatially uniform.

Homework Equations

When concentration is spatially uniform dc/dt=-kc. k is a constant.

The Attempt at a Solution

Only hint the text gives is that the answer is said to be of reaction-diffusion form. This is the first part of a multi-part problem and as the latter parts don't use the non spatially uniform equation it makes me think it should be something I could just write down. Just not sure how to think about this question.

 

[Mapes]

Hi barcafan, welcome to PF. One way of getting this equation is to do a mass balance on a differential element. Since the concentration isn't uniform, mass could flow in or out of the element due to diffusion, besides leaving the element due to consumption by some reaction. Know what I mean?

To put it another way, you need to augment Fick's Second Law (which describes diffusion) with a reaction term.

 

[barcafan]

I know on a basic level that the second law says how a concentration will change with time, so that makes sense. Since I know the probability for a molecule to disappear (changing the concentration) is kdt is it as simple as adding this factor on to Fick's law?

 

[Mapes]

Yes, and making sure the units match.

 

[paranormal]

The reaction-diffusion equation is a mathematical model that describes the time evolution of a concentration field in a system where both reaction and diffusion processes are present. In this case, the concentration of the dissolved substance is not spatially uniform, meaning it varies throughout the tube. The equation for dc/dt in this case can be written as:

dc/dt = D∇^2c - kc

where D is the diffusion coefficient and ∇^2 is the Laplace operator. This equation accounts for the diffusion of the substance from areas of high concentration to areas of low concentration, as well as the reaction rate (k) which causes molecules to disappear over time. This equation is commonly used in fields such as chemistry, biology, and physics to model various processes, and can be solved numerically to predict the behavior of the system over time.