Mathematically expressing field driven water autoionization

  • Thread starter Thread starter HelloCthulhu
  • Start date Start date
  • Tags Tags
    Field Water
AI Thread Summary
The discussion centers on the application of the Laplace equation to analyze the use of an electric field for water autoionization. The original poster seeks guidance on solving for voltage and adapting the equation for a parallel plate setup instead of a hemispherical tip. Responses emphasize that the Poisson equation will be relevant, with boundary conditions varying based on the geometry. The need to switch from spherical to Cartesian coordinates for parallel plates is noted. The poster expresses confusion regarding specific variables and their roles, particularly in relation to calculating the electric field necessary for water ionization. Comparisons to Paschen's Law are made, highlighting the relationship between electric field, voltage, and distance. The conversation also touches on the poster's lack of familiarity with advanced mathematical concepts, suggesting a foundational understanding of calculus and vectors is necessary for tackling electrostatics effectively.
HelloCthulhu
Messages
150
Reaction score
3
I recently read a paper on using an electric field to drive water autoionizaton. I'm trying to figure out how to use the Laplace equation on pg 9; 4th paragraph; to solve for voltage. I'm also interested in how this equation would change if I replaced the hemispherical tip with a parallel plate. Anyone strongly familiar with this subject matter? Any help is greatly appreciated!
fig 6.png
maths.png
data.png


https://www.sciencedirect.com/science/article/abs/pii/S0009261411011511
 
Chemistry news on Phys.org
I haven’t run through the math specifically, but it’ll be the Poisson equation no matter what, but the boundary conditions will change. So for parallel plates, you’ll want to look at the Laplacian in Cartesian coordinates instead of spherical coordinates as your starting point.
 
  • Like
Likes HelloCthulhu
Thank you for the response! I still have some questions about the variables. I know how to use Figure 6 to find the values for maximum interfacial field Ew/v or Et/w and water thickness rw/rt, but I'm still not sure what to do with (E), d, or how to solve for ϕw or ϕv.

maths 2.png


In the end, all I'm really trying to figure out is what the electric the field between the molecules and the electrode has to be in order to cause water to ionize. Some of the fundamentals seem comparative to Paschen's Law (Electric Field = voltage x distance). As for adjusting that equation for parallel plates I think I've found an explanation on what it should look like, but I'm still far too ignorant about electrostatics for it to make any real sense to me yet. Thank you for your input so far, I hope you can continue to help me with this.

http://jsa.ece.illinois.edu/ece329/notes/329lect07.pdf
 
It’s unclear how much background knowledge you have and if this problem is at the right level for you. For instance, I don’t see a ##d## being used as a variable in any of these images. The only ##d##’s I see denote derivatives.

To get the potential (aka the voltage), you would solve the Poisson/Laplace equation for ##\phi##, given the appropriate boundary conditions. I doubt there’s an analytical way to do it without a really simple functional form for ##\epsilon(E)##, but it should be possible to write a code to get a numerical solution. To get the electric field, you take the gradient of the potential (assuming the system is time-invariant).
 
  • Like
Likes HelloCthulhu
TeethWhitener said:
It’s unclear how much background knowledge you have and if this problem is at the right level for you. For instance, I don’t see a d being used as a variable in any of these images. The only d’s I see denote derivatives.
It's far beyond anything I have experience with. I've solved derivatives in the past, but wasn't familiar with second derivatives. To be honest, I'm in way over my head on these subjects. But even so I'd like to keep working on trying to understand them. Thank you so much for your help.
 
HelloCthulhu said:
It's far beyond anything I have experience with. I've solved derivatives in the past, but wasn't familiar with second derivatives. To be honest, I'm in way over my head on these subjects. But even so I'd like to keep working on trying to understand them. Thank you so much for your help.
I would advise you to learn single and multivariable calculus as well as vectors before trying to tackle E&M.
 
  • Like
Likes HelloCthulhu
It seems like a simple enough question: what is the solubility of epsom salt in water at 20°C? A graph or table showing how it varies with temperature would be a bonus. But upon searching the internet I have been unable to determine this with confidence. Wikipedia gives the value of 113g/100ml. But other sources disagree and I can't find a definitive source for the information. I even asked chatgpt but it couldn't be sure either. I thought, naively, that this would be easy to look up without...
I was introduced to the Octet Rule recently and make me wonder, why does 8 valence electrons or a full p orbital always make an element inert? What is so special with a full p orbital? Like take Calcium for an example, its outer orbital is filled but its only the s orbital thats filled so its still reactive not so much as the Alkaline metals but still pretty reactive. Can someone explain it to me? Thanks!!
Back
Top