If a potential difference dV of a long conductor dx and section S with a resistivity \rho_e is considered you have with Ohm's law that the current is

I = \displaystyle\frac{S}{\rho_e dx}dV

so with

j=\displaystyle\frac{I}{S}

y

\kappa_e=\displaystyle\frac{1}{\rho_e}

with what

$ j =- \kappa \displaystyle\frac{ dV }{ dx }$

(ID 3877)

In the case of ion conduction, conductivity must include the sign of the charge, which is entered with the number of charges z divided by the absolute value of said number \mid z \mid. Therefore the conductivity is

$ \kappa =\displaystyle\frac{ z }{ \mid z \mid } \mu_e c $

where \mu_e is mobility and c the concentration of ions.

(ID 3876)

The flow density j is understood as the current I by section S, so

$ j =\displaystyle\frac{ I }{ S }$

(ID 3221)

The electron current is the dQ charge that passes through a S section in a dt time. If it is assumed that electrons or ions travel at a speed v the volume of these that will pass in time dt through the section S is the same to Svdt. If, on the other hand, the ion concentration is c and its charge is q the current will be

I=\displaystyle\frac{dQ}{dt}=\displaystyle\frac{Svdtc}{dt}=Svc

that is

equation/druyd>

(ID 3222)

The diffusion leads to the difference in concentrations dc over a distance dx generates a flow of particles j that is calculated by the so-called Fick's law :

$ j =- D \displaystyle\frac{ dc }{ dx }$

where D is the diffusion constant.

(ID 3878)

The diffusion constant D was modeled by Einstien and depends on the absolute value of the number of charges \mid z \mid, the mobility \mu_e, the universal gas constant, T the absolute temperature and F the Faraday constant that has a value of 9.649E+4 C/mol:

$ D =\displaystyle\frac{ \mu_e R_C T }{\mid z \mid F }$

(ID 3879)

If there is more than one type of ion, the actual concentration of the ions must be estimated, that is, add the concentrations weighted by the number of charges they have

$c_m=\sum_i\mid z_i\mid c_i$

where R is the gas constant, T the temperature, z the number of charges, F the constant Farday and the concentrations between both sides of the membrane c_1 and c_2.

(ID 3883)

In case of a type of load

$ c_m =\mid z_1 \mid c_1 $

where R is the gas constant, T the temperature, z the number of charges, F the constant of Farday and the concentrations between both sides of the c_1 membrane.

(ID 3884)

In case of two types of charges

$ c_m = \mid z_1\mid c_1 + \mid z_2\mid c_2 $

where R is the gas constant, T the temperature, z the number of charges, F the constant Farday and the concentrations between both sides of the membrane c_1 and c_2.

(ID 3885)

In case of three types of charges

$ c_m = \mid z_1\mid c_1 + \mid z_2\mid c_2 + \mid z_3\mid c_3 $

where R is the gas constant, T the temperature, z the number of charges, F the constant Farday and the concentrations between both sides of the membrane c_1, c_2 and c_3.

(ID 3886)

The equilibrium condition occurs when the flow due to the potential difference is equal to the flow due to the diffusion. That is why you have to

-\displaystyle\frac{z\mu_ec}{\mid z\mid}\displaystyle\frac{dV}{dx}=-\displaystyle\frac{\mu_eRT}{\mid z\mid F}\displaystyle\frac{dc}{dx}

for what you have

$ dV =\displaystyle\frac{ R_C T }{ z F }\displaystyle\frac{ dc }{ c }$

(ID 3880)

If the potential difference is integrated, the relationship of the potential difference corresponding to the limit in which the electric field is compensated with the Diffusion can be established:

$ V_m =-\displaystyle\frac{ R_C T }{ F }\ln\displaystyle\frac{ c_1 }{ c_2 }$

where R is the gas constant, T the temperature, z the number of charges, F the constant Farday and the concentrations between both sides of the membrane c_1 and c_2.

(ID 3881)