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

equation

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.

In case of a type of load

equation

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.

In case of two types of charges

equation

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.

In case of three types of charges

equation

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.

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

equation

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

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

equation

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 :

equation

where D is the diffusion constant.

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

equation

The difference in concentration $c_1$ and $c_2$ at the ends of the membrane results in the difference:

kyon

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:

equation

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

equation

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>

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:

equation

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.