Law of Nernst-Einstein

In absence of force

Let us consider the movements on an axis X (for example by projection on this axis).

In absence of force, the defects migrate by chance, by jumps of a site to a nearby site. These jumps are possible thanks to thermal agitation.

Per unit of time, a species has a probability Γ_I of making a jump towards a site I close. The mean velocity of the particles is null (case similar to the Brownian Movement); the quadratic average of < displacements; X ²> during a time T is not it not null and one a:

= T \ cdot \ sum_1^n \ Gamma_i \ cdot \ delta \ xi_i

if δξ_I is the algebraic length (positive or negative according to the direction of reference) of the jump I .

See the detailed articles Diffusion of the matter and Law of Fick.

Effect of a force

When the species is subjected to a force, that breaks the symmetry of the jumps, the probabilities of two opposite jumps is not equal any more. To simplify, one considers only one species, and a movement in a given direction. If Γ₊ is the probability that the particle moves a length +δ X per unit of time, and Γ_- the probability that it moves a length - δ X , then the range < X > after a time T is worth:

= T \ cdot (\ Gamma_+ - \ Gamma_-) \ cdot \ delta x

What makes it possible to define the average Speed v :

v = \ frac {} {T} = (\ Gamma_+ - \ Gamma_-) \ cdot \ delta x

This movement under the effect of a force creates a gradient of concentration. However, the random diffusion tends to level the concentrations, and thus is opposed to the “forced” migration, one thus has two flows:

a flow J ₁ created by the force
J ₁ = v · C , where C is the concentration of the species;
a flow J ₂ opposite which follows the law of Fick
$j_2 = - D \ cdot \ frac {\ partial C} {\ partial X}$ where D is the Coefficient of diffusion of the species.

Total flow is thus worth:

j = v \ cdot C - D \ cdot \ frac {\ partial C} {\ partial X}

Stationary mode

If one waits “sufficiently a long time”, one reaches a stationary mode: flows J ₁ and J ₂ are compensated, one has a constant gradient of concentration. There is thus J = 0, that is to say, if C ^∞ ( X ) is this constant concentration:

v \ cdot c^ \ infty = D \ cdot \ frac {\ partial c^ \ infty} {\ partial X}

Let us suppose now that the force is conservative (the most frequent case). It thus derives from a Potentiel η:

F = - \ frac {\ partial \ eta} {\ partial X}

With dynamic balance, the particles are distributed according to a Statistique of Maxwell-Boltzmann:

c^ \ infty (X) = c_0 \ cdot \ exp \ left (- \ frac {\ eta} {kT} \ right)

where K is the Boltzmann constant and T is the absolute Température. By introducing this into the preceding equation, one obtains:

v \ cdot c_0 \ cdot e^ {- \ frac {\ eta} {kT}} = - \ frac {D} {kT} \ cdot \ frac {\ partial \ eta} {\ partial X} \ cdot c_0 \ cdot e^ {- \ frac {\ eta} {kT}}

what gives us the law of Nernst-Einstein

Friction

This law resembles a law of fluid Frottement. At the time of a movement at low speed in a nonturbulent fluid, one can estimate that the force of friction is proportional to the speed, and thus that one reaches a stationary mode where speed is proportional to the force (it is the principle of the Parachute):

v = B · F

where B is the mobility of the species ( German Beweglichkeit in ).

The law of Nernst-Einstein thus gives us:

B = \ frac {D} {kT}

from where one deduces the law from Einstein :

Applications

Chemical potential

The force F_c resulting from the chemical Potentiel μ can be written, with a dimension:

F_c = - \ frac {\ partial \ driven} {\ partial X}

and thus the equation of Nernst-Einstein becomes:

v = - \ frac {D} {kT} \ cdot \ frac {\ partial \ driven} {\ partial X}

Electric field

If a particle carries Z elementary charges E , then it undergoes the force F_e (electrostatic Force or Force of Coulomb):

\ vec {F_e} = Z \ cdot E \ cdot \ vec {E}

The electric field E drift of a potential V , which is written with a dimension:

E = - \ frac {\ partial V} {\ partial X}

thus the law of Nernst-Einstein becomes:

v = - \ frac {partial Dze} {kT} \ cdot \ frac {\ V} {\ partial X}

Let us consider the flow of loads j_el , also called density of electric current. There is

j_el = Z · E · J = Z · E · C · v

being C concentration of charge carriers, and then

j_ {el} = \ frac {partial Dz^2e^2c} {kT} \ cdot \ frac {\ V} {\ partial X}

One can make a parallel with the Loi of Ohm connecting this density of electric current j_el to the gradient of potential:

j_ {el} = - \ partial sigma \ cdot \ frac {\ V} {\ partial X}

σ being the electric Conductivité, which gives us

\ sigma = \ frac {Dz^2e^2c} {kT}

See too

Random links: