恒电势电流曲线和浓度分布（i-t curve）

Test condition

1. No stirring;
2. The concentration of reactant at the electrode surface is always 0.
   Half reaction:
   $$O+ne\rightarrow R \tag{a}$$
   The equation of the limiting current versus concentration will be
   $$\dfrac{\partial C_O}{\partial t}=D_O \dfrac{\partial ^2 C_O}{\partial x^2} \tag{1}$$
   Initial condition:(Concentration is homogeneous)
   $$C_O(x,0)=C_O ^* \tag{2}$$
   Boundary conditions:
   $$C_O(\infty,t)=C_O ^* \tag{3}$$
   $$C_O(0,t)=0 \quad (t>0) \tag{4}$$

Apply Laplace transform to solve Eq.(1)

Laplace transform function

$$F (s)=\int_0^\infty f(t)e^{-st}dt \tag{5}$$
Apply Laplace transform on the left part of Eq(1)

L{ ∂CO∂t}=∫∞0∂CO∂te−stdt=COe−st∣∣∣∞0+s∫∞0COe−stdt=CO(x,∞)e−s∞−CO(x,0)e0+s∫∞0COe−stdt=−C∗O+s∫∞0COe−st