Butler-Volmer Equation

核心方程：电流反应速率(current due to diffusion)

这里，是电流密度(current density)，是Faraday Constant，
是反应速率（reaction rates）。

这里，是rate constant，是物质的浓度。

注意：电流密度的三个成因(diffusion, migration, convection)

2.速率常数（reaction rate）的定义

depends on the activation energy for reaction, that is, the energy
barrier for reaction. These expressions are results of experimental
observations, rather than theoretical derivative.

can be represented by the potential difference between the
electrode and the electrolyte .

Because cannot be measured, usually,
is used by measuring two electrode/electrolyte potentials with respect
to the same reference electrode/electrolyte potential.

usually, is the Nernst potential, that is, without current flows.
Thus , is called standard electrochemical rate constant.

的具体来历，请参考《伏安法教程》的第42页。可以认为是过渡态处自由能变化是反应态处自由能变化的缩小系数。

电流的表达式

Pls note that it is not necessary of here before referencing to the same standard potential. They are rate constants at the electrode potential .

定义 (exchange current density)

这里和分别是还原物和氧化物的bulk浓度。此方程的定义，目的是为了联系平衡时的 Nernst potential () 和当前的真实势 ()，以便定义 overpotential ()。

Butler-Volmer equation

上面两个方程相除 and with (rate constant at ),

当和时

Derivative directly from

It has , where $η$ is over-potential. At equilibrium and by Nernst Equation,
we have and . At last, we have

Substituting the above equation and into Equation, We have (Note：查先生的书有另一套推导方法，而且比这个精细！)

with

Tafel lines

When

$η$

When

$η$

We also can directly use the expression for in section 3, if we can express
with activation energy barrier related to the equilibrium free energy of
reactants and products at the