Hello there, I am trying to mathematically model diffusion through hemisphere-shaped particles using Fick's second law. I know the formula for a perfect sphere (where solutes diffuse radially), however in my case diffusion happens from the flat surface of the hemisphere and not from the curved side. I tried Crank's "The mathematics of diffusion", but I only got more confused. Do you have any advice for approaching this kind of modelling? Would it make more sense to model the system as a semi-infinite slab? Thanks in advance!
One can corelate it with Fick’s first law of diffusion on corelating ‘chemical potential gradient’ with ‘concentration gradient’. Can anyone please explain with/without proper equations?
Me and my partner were discussing this today, and we actually didn't really reach a satisfying conclusion.
We tried to do some rearrangements of Fick's Law (that is, flux (mol * m^2 * s^-1) = permeation coefficient (m*s^-1) * ([X_o] - [X_i])) in order to calculate the time to equilibrium (that is, the time from t=0 until flux = 0) occurred. However, we fell short, presumably because neither of us has any high school courses in differential mathemathics beyond early college.
Is there a known way to do this (I assume there is, since the problem looks so deceptively simple), and if so, how does it work?
(We're assuming that we're operating under the unconditional equilibrium in which [X_o]=[X_i] at equilibrium. Obviously, if any ions beyond the one which we are calculating on are present, the situation would be very different.)
There should be a concentration gradient of gases between earth's atmosphere and the outer space... If so does it mean that our atmosphere composition is influenced by space's? Sorry for my bad english.
Consider a group of particles moving randomly in three dimensions across a cross sectional area A. Let the area A be in the yz-plane. There are N(r) particles in the region to the left of A and N(r+ dr) particles in the region to the right of A. In this case, r is the position vector as a function of x and y. At each point (x,y), there will be some amount of particles spanning the length of the height of A.
In region 1 (to the left of A), the probability of a particle to move through area A is 1/6 since there are three degrees of freedom in space and for each degree of freedom there are two directions the particle can move (up/down, left/right). In order for a particle in region 1 to move through area A, it must have a component in the x direction, so that has a probability of 1/3, and it must be moving to the right (as opposed to the left), so that has a probability of 1/2. Therefore, the probability it will pass through area A is 1/6.
Is that argument about the probability correct?
A picture of the problem is in the comments. From what I understand the flux at position zero is 10, concentration at the end of the pipe (L) is zero, and at time zero the concentration is zero. But I’m getting stuck solving the rest
Please explain with proper equations
