Dissolved ion distribution under the influence of an external electric field

[Thorr]

Hello,

I'm a medical student and a few days ago a rather interesting question arose when I was doing some research. The gist of the problem is estimating how dissolved ions in a solution will distribute themselves when an external electric field is applied. Now I had basic physics and electrostatics but even just writing down equations to try and describe this problem is a bit over my head.

I drew an illustration of the dilemma to make it clearer.

Exact description of the problem:
- we have a sealed container of dimension d, filled with a solution of ions A and B. For simplicity, let's assume both A and B have +1 and -1 charges respectively. (you can also ignore the undissolved part of compound AB for now)

- we apply an external homogeneous electric field of strength E to the container. Obviously ions A drift in the direction of the electric field, ions B the other way around until an equilibrium state is achieved.

The thing I'd like to know is how the concentration of A (B) changes in relation to x.

I know the computation might get a bit involved but I hope I can at least get pointers on how to write down some initial equations or perhaps where a solution (with full derivation preferably) to this problem can be found.

I'm not just interested in the final functions, I'd like to see the path to the solution at least in part or at crucial steps.

The problem is a bit akin to something like the Debye's shielding effect...

Thanks so much for your help!

 

[eljose79]

Dear medical student,

Thank you for bringing up such an interesting question. The problem you described falls under the field of electrochemistry, specifically the study of ion distribution in solutions under the influence of an external electric field. This is a complex problem that requires a thorough understanding of electrostatics, thermodynamics, and kinetics.

To start, we can use the Nernst-Planck equation to describe the transport of ions in a solution under an electric field. This equation takes into account the diffusion of ions, their migration in the electric field, and their reaction with other ions. It is given by:

J = -D*∇C + μ*C*∇Φ + k*C*∇(RT*C)

where J is the flux of ions, D is the diffusion coefficient, C is the concentration of ions, μ is the mobility of ions, Φ is the electric potential, k is the reaction rate coefficient, and RT is the gas constant multiplied by temperature. This equation is derived from the conservation of mass and charge principles.

To apply this equation to your problem, we need to consider the boundary conditions and initial conditions. The container is sealed, so we can assume that the concentration of ions at the boundaries is constant. The initial condition can be assumed to be a uniform distribution of ions in the solution. We can also assume that the electric field is uniform and the solution is at steady state.

To solve this equation, we can use numerical methods or analytical approximations. One approach is to use the method of finite differences, where we divide the solution into small cells and solve for the concentration of ions at each cell. Another approach is to use the Debye-Hückel approximation, which simplifies the Nernst-Planck equation for dilute solutions.

I recommend consulting textbooks or journal articles on electrochemistry for a detailed derivation and solution to this problem. Some key concepts to understand include the Debye length, the Debye screening effect, and the Debye length for ions in a charged solution.

I hope this helps in your research. Best of luck!

 

[astrokreng]

Hello,

I can definitely understand your interest in this problem. The phenomenon you are describing is known as electrophoresis, where charged particles (such as ions) move in response to an electric field. The distribution of these ions can have important implications in various fields, including medicine, chemistry, and biology.

To start, let's consider the basic equation for electrophoresis: F = qE. This equation describes the force (F) experienced by a charged particle (q) in an electric field (E). In your scenario, the ions A and B will experience opposite forces due to their opposite charges (+1 and -1). As a result, they will move in opposite directions, as you have correctly stated.

Now, to determine how the concentration of A (B) changes in relation to x, we can use the Nernst-Planck equation. This equation takes into account the diffusion of ions in addition to the electric field. It is given by:

J = -D(dC/dx) + μCqE

Where J is the flux of ions, D is the diffusion coefficient, C is the concentration of ions, μ is the mobility of the ions, and q is the charge of the ions. This equation takes into account both the movement of ions due to diffusion and the movement due to the electric field.

To solve this equation, we would need to consider the initial conditions of the system, such as the concentration of ions and the strength of the electric field. We would also need to take into account any boundary conditions, such as the dimensions of the container and the presence of any barriers.

Unfortunately, the full derivation of this equation and its solution can be quite involved and beyond the scope of this response. However, I can recommend some resources that may be helpful in understanding this problem further. The book "Electrophoresis in Practice: A Guide to Methods and Applications of DNA and Protein Separations" by Reiner Westermeier is a comprehensive guide to electrophoresis and may provide some insight into your specific problem. Additionally, there are many online resources and scientific papers available on the topic of electrophoresis, which may provide more detailed explanations and solutions.

I hope this information helps you in your research. Best of luck in finding the solution to your problem!