# Calculating Diffusion Velocity in Electrochemical System

• ussername
In summary, the conversation discusses the diffusion motion of an ion in an electrochemical system. The concentration of the ion is a function of both the x and y coordinates, and it is zero at y=0. The system is steady, so the partial derivative of concentration with respect to y is constant. The first Fick's law is used to calculate the diffusion molar flux and the diffusion linear velocity of the ion, which changes as the concentration changes with y. The diffusion velocity is then used to express the diffusion coefficient, which is constant along x. The conversation also mentions the concept of diffusion time and how it can be measured.
ussername
Let's suppose an electrochemical system with given coordinates:

We are interested in diffusion motion of ion ##i## in the direction of ##y## axis.
Concentration ##c_i## is a function of both ##x## and ##y##. Concentration ##c_i## at ##y=0## is zero.
The system is steady thus ##\frac{\partial c_i}{\partial y}(x)## is not a function of ##y## (##\frac{\partial c_i}{\partial y}(x)## is constant with ##y##).
The first Fick's law for ion ##i## is (for simplicity I don't write ##i## subscript anymore):
$$J_{y}^{dif}(x,y=d) = - D \cdot \frac{\partial c}{\partial y}(x,y=d) = - D \frac{c(x,y=d)}{d}$$
The diffusion molar flux is:
$$J_{y}^{dif}(x,y=d) = c(x,y=d)\cdot v_{y}^{dif}(x,y=d)$$
where ##v^{dif}## is the diffusion linear velocity of ion ##i##:
$$v_{y}^{dif} = -D \cdot \frac{1}{c}\cdot \frac{\partial c}{\partial y}$$
The diffusion velocity clearly changes because concentration changes with ##y##.
Now I can put the diffusion velocity into the first equation and express ##D##:
$$D = \frac{-c(x,y=d)\cdot v_{y}^{dif}(x,y=d)\cdot d}{c(x,y=d)}=-v_{y}^{dif}(x,y=d)\cdot d$$
This seems strange to me. Since both diffusion coefficient ##D## and length ##d## are considered constant, the diffusion velocity ##v_{y}^{dif}(x,y=d)## seems to be constant along ##x## independently of the functions ##c(x) \ at \ y=d## and ##\frac{\partial c}{\partial y}(x) \ at \ y=d##. Is it true?

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Delta2
I strongly object to the term 'diffusion linear velocity'. Diffusion is due to random (Brownian) motion of molecules and other particles so what exactly is moving with 'diffusion linear velocity'? how would you measure it?
It makes no sense.
What you do have is Flick's law and it is a differential equation. Add initial and boundary condition, solve it and you have all the answers.
By the way, the diffusion equation is linear.

I found this problem when I was trying to derive the relation for diffusion time of ion ##i##:
$$\tau^{dif} = \frac{d^2}{D}$$
In order to understand what ##\tau^{dif}## really means, it is useful to define ##v^{dif}##.

By the way ##v^{dif}## can be measured no worse than for example molar fluxes of diffusion, migration or convection. It can be measured when the diffusion motion prevails.

ussername said:
I found this problem when I was trying to derive the relation for diffusion time of ion ##i##:
$$\tau^{dif} = \frac{d^2}{D}$$

When considering random-walk diffusion (for example in one dimension), the equation
<x 2> = 2 D t
relates the mean square displacement <x 2> of diffusing particles with the elapsed time t. Here it is assumed that the particles start out at position x = 0 at time t = 0.
Roughly speaking, a particle with a diffusion coefficient D diffuses thus a distance d in a time t d 2/2D.

## What is diffusion velocity in an electrochemical system?

Diffusion velocity in an electrochemical system refers to the speed at which charged particles, such as ions, move through a solution due to a concentration gradient. It is an important factor in determining the overall rate of a chemical reaction in an electrochemical system.

## How is diffusion velocity calculated?

Diffusion velocity can be calculated using Fick's law of diffusion, which states that the diffusion flux (J) is equal to the diffusion coefficient (D) multiplied by the concentration gradient (∇C). This can be written as J = -D∇C, where the negative sign indicates that diffusion occurs from areas of high concentration to areas of low concentration.

## What factors affect diffusion velocity in an electrochemical system?

The diffusion velocity in an electrochemical system can be affected by several factors, including the concentration gradient, temperature, viscosity of the solution, and the size and charge of the particles involved. Additionally, the presence of an electric field can also affect the diffusion velocity.

## How does diffusion velocity impact the overall reaction rate in an electrochemical system?

Diffusion velocity plays a crucial role in determining the overall reaction rate in an electrochemical system. The faster the diffusion velocity, the quicker the particles can reach the electrode and participate in the reaction. Therefore, a higher diffusion velocity can lead to a faster reaction rate, while a lower diffusion velocity can result in a slower reaction rate.

## Can diffusion velocity be controlled in an electrochemical system?

While some factors that affect diffusion velocity, such as temperature and viscosity, can be controlled, others, like the concentration gradient, are not easily adjustable. However, the use of electric fields or specific electrode designs can be utilized to manipulate the diffusion velocity in an electrochemical system to optimize reaction rates.

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