- #1

Dario56

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In the textbook: Electrochemical Systems by Newman and Alyea, 3rd edition, chapter 11.9: Moderately Dilute Solutions, equation for the mole flux of the component ##i## is given by: $$ N_i = - \frac {u_i c_i} {z_i F} \nabla \bar\mu_i\ + c_i v \tag {1}$$

where ##u_i## is the ionic mobility, ##c_i## is the concentration, ##\bar \mu_i## is the electrochemical potential and ##v## is the velocity of the streaming fluid (assuming a low concentration of the component ##i## as situation becomes more complicated at higher concentrations).

One thing to note is that I wrote ionic mobility ##u_i## a bit differently compared to the textbook. I defined it as a terminal velocity of the ion in the unitary electric field, so its measuring unit is ##[\frac {m^2}{Vs}]## in my case. In the textbook they defined it also as a terminal velocity in the unitary electric field, but divided by ##z_i F##. This is something to keep in mind to avoid confusion.

Basically, equation (1) states that the mole flux of the component has three contributions: diffusion, migration and convection. Diffusion and migration are accounted for in the first term (electrochemical potential gradient) and convection is accounted for in the second term.

My problem is with the first term ##- \frac {u_i c_i } {z_i F}\nabla \mu_i## which I know how to derive and I will do this here as I don't understand something about the equation (1).

To start, we define diffusion mole flux from as the 1st Fick's law: $$N_{i,dif} = -\frac {D_i c_i} {RT} \nabla \mu_i \tag {2}$$

where ##D_i##is the diffusion coefficient and ## \mu_i ## is the chemical potential of the component ##i##.

Migration mole flux is given by: $$N_{i,mig} = -u_i c_i \nabla \phi \tag {3}$$

where ##\phi## is the electric potential.

If there is no convection, we can write total mole flux of the component as: $$N_i = -(\frac {D_i c_i} {RT} \nabla \mu_i + u_i c_i \nabla \phi) \tag {4}$$

We recall the definition of electrochemical potential ##\bar \mu_i##: $$ \bar \mu_i = \mu_i + z_i F \phi \tag {5} $$

If we assume that system is in thermodynamic equilibrium than electrochemical potential gradient and net mole flux vanish: $$ \nabla \bar \mu_i = 0 \tag {6} $$ $$ N_i = 0 \tag {7}$$

From equations (5) and (6), we can write: $$\frac {\nabla \mu_i}{\nabla \phi} = -z_i F \tag {8}$$

From equations (4) and (7), we can write: $$ \frac {\nabla \mu_i}{\nabla \phi} = - \frac {u_i RT}{D_i} \tag {9}$$

By equating (8) and (9), we get the Einstein equation, which relates diffusion coefficient ##D_i## and ionic mobility ##u_i## in the thermodynamic equilibrium: $$D_i = \frac {u_i RT}{z_i F} \tag {10}$$

If we know substitute the equation (10) into equation (4), we can write: $$N_i = -u_i c_i(\frac {1}{z_i F} \nabla \mu_i + \nabla \phi) = - \frac {u_i c_i} {z_i F} \nabla \bar\mu_i\ \tag {11}$$

After the derivation of the first term in the equation (1), I can get to my point. Namely, that the first term in the equation must be equal to zero since it is impossible to write this term in that form (via gradient of the electrochemical potential) without using Einstein equation. We saw earlier that Einstein equation is only valid in the thermodynamic equilibrium and therefore equation (6) applies.

where ##u_i## is the ionic mobility, ##c_i## is the concentration, ##\bar \mu_i## is the electrochemical potential and ##v## is the velocity of the streaming fluid (assuming a low concentration of the component ##i## as situation becomes more complicated at higher concentrations).

One thing to note is that I wrote ionic mobility ##u_i## a bit differently compared to the textbook. I defined it as a terminal velocity of the ion in the unitary electric field, so its measuring unit is ##[\frac {m^2}{Vs}]## in my case. In the textbook they defined it also as a terminal velocity in the unitary electric field, but divided by ##z_i F##. This is something to keep in mind to avoid confusion.

Basically, equation (1) states that the mole flux of the component has three contributions: diffusion, migration and convection. Diffusion and migration are accounted for in the first term (electrochemical potential gradient) and convection is accounted for in the second term.

My problem is with the first term ##- \frac {u_i c_i } {z_i F}\nabla \mu_i## which I know how to derive and I will do this here as I don't understand something about the equation (1).

To start, we define diffusion mole flux from as the 1st Fick's law: $$N_{i,dif} = -\frac {D_i c_i} {RT} \nabla \mu_i \tag {2}$$

where ##D_i##is the diffusion coefficient and ## \mu_i ## is the chemical potential of the component ##i##.

Migration mole flux is given by: $$N_{i,mig} = -u_i c_i \nabla \phi \tag {3}$$

where ##\phi## is the electric potential.

If there is no convection, we can write total mole flux of the component as: $$N_i = -(\frac {D_i c_i} {RT} \nabla \mu_i + u_i c_i \nabla \phi) \tag {4}$$

We recall the definition of electrochemical potential ##\bar \mu_i##: $$ \bar \mu_i = \mu_i + z_i F \phi \tag {5} $$

If we assume that system is in thermodynamic equilibrium than electrochemical potential gradient and net mole flux vanish: $$ \nabla \bar \mu_i = 0 \tag {6} $$ $$ N_i = 0 \tag {7}$$

From equations (5) and (6), we can write: $$\frac {\nabla \mu_i}{\nabla \phi} = -z_i F \tag {8}$$

From equations (4) and (7), we can write: $$ \frac {\nabla \mu_i}{\nabla \phi} = - \frac {u_i RT}{D_i} \tag {9}$$

By equating (8) and (9), we get the Einstein equation, which relates diffusion coefficient ##D_i## and ionic mobility ##u_i## in the thermodynamic equilibrium: $$D_i = \frac {u_i RT}{z_i F} \tag {10}$$

If we know substitute the equation (10) into equation (4), we can write: $$N_i = -u_i c_i(\frac {1}{z_i F} \nabla \mu_i + \nabla \phi) = - \frac {u_i c_i} {z_i F} \nabla \bar\mu_i\ \tag {11}$$

After the derivation of the first term in the equation (1), I can get to my point. Namely, that the first term in the equation must be equal to zero since it is impossible to write this term in that form (via gradient of the electrochemical potential) without using Einstein equation. We saw earlier that Einstein equation is only valid in the thermodynamic equilibrium and therefore equation (6) applies.

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