- #1

ussername

- 60

- 2

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?

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?