Derivation of Cottrell Equation

Cottrell Equation Derivation

The Cottrell equation describes the diffusion-limited current at a planar electrode under non-steady-state conditions, a key step in polarography (e.g., Ilkovič equation derivation). The equation is:

\[ i_d = n F A \sqrt{\frac{D}{\pi t}} \, c_0 \]

where \( n \) is the number of electrons, \( F = 96485 \, \text{C/mol} \), \( A \) is the electrode area (cm²), \( D \) is the diffusion coefficient (cm²/s), \( t \) is time (s), and \( c_0 \) is the bulk concentration (mol/cm³).

Step 1: Fick's Second Law of Diffusion

The Cottrell equation is derived from Fick's Second Law, which governs non-steady-state diffusion:

\[ \frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2} \]

where \( c(x, t) \) is the concentration (mol/cm³) at position \( x \) and time \( t \), and \( D \) is the diffusion coefficient. Boundary and initial conditions:

• At the electrode (\( x = 0 \), \( t > 0 \)): \( c(0, t) = 0 \) (complete depletion due to limiting potential).
• Far from the electrode (\( x \to \infty \), \( t \geq 0 \)): \( c(x, t) = c_0 \).
• Initially (\( t = 0 \), \( x \geq 0 \)): \( c(x, 0) = c_0 \).

Step 2: Solving Fick's Second Law

Use a similarity transform or Laplace transform to solve the partial differential equation. Let's use the Laplace transform for clarity. Define the Laplace transform of \( c(x, t) \):

\[ \bar{c}(x, s) = \int_0^\infty c(x, t) e^{-st} \, dt \]

Transform Fick’s Second Law:

\[ s \bar{c}(x, s) - c(x, 0) = D \frac{\partial^2 \bar{c}}{\partial x^2} \]

With \( c(x, 0) = c_0 \), this becomes:

\[ \frac{\partial^2 \bar{c}}{\partial x^2} - \frac{s}{D} \bar{c} = -\frac{c_0}{D} \]

The general solution is:

\[ \bar{c}(x, s) = A e^{-\sqrt{s/D} x} + B e^{\sqrt{s/D} x} + \frac{c_0}{s} \]

Apply boundary conditions: As \( x \to \infty \), \( \bar{c} \to c_0 / s \), so \( B = 0 \). At \( x = 0 \), \( \bar{c}(0, s) = 0 \), so:

\[ A + \frac{c_0}{s} = 0 \implies A = -\frac{c_0}{s} \]

Thus:

\[ \bar{c}(x, s) = \frac{c_0}{s} \left( 1 - e^{-\sqrt{s/D} x} \right) \]

Step 3: Concentration Gradient at the Electrode

The current depends on the concentration gradient at \( x = 0 \). Differentiate:

\[ \frac{\partial \bar{c}}{\partial x} = \frac{c_0}{s} \cdot \sqrt{\frac{s}{D}} e^{-\sqrt{s/D} x} \]

At \( x = 0 \):

\[ \left. \frac{\partial \bar{c}}{\partial x} \right|_{x=0} = \frac{c_0}{\sqrt{D s}} \]

Inverse Laplace transform (\( \mathcal{L}^{-1} \{ s^{-1/2} \} = \sqrt{\pi t} \)):

\[ \left. \frac{\partial c}{\partial x} \right|_{x=0} = \frac{c_0}{\sqrt{\pi D t}} \]

Step 4: Current from Fick's First Law

Using Fick's First Law, the flux at the electrode is:

\[ J = -D \left. \frac{\partial c}{\partial x} \right|_{x=0} = -\frac{D c_0}{\sqrt{\pi D t}} = -\frac{c_0 \sqrt{D}}{\sqrt{\pi t}} \]

The current is:

\[ i_d = n F A J = n F A \cdot \frac{c_0 \sqrt{D}}{\sqrt{\pi t}} = n F A \sqrt{\frac{D}{\pi t}} \, c_0 \]

This is the Cottrell equation, showing current decreases as \( t^{-1/2} \) due to diffusion layer growth.

Step 5: Application to Polarography

The Cottrell equation assumes a planar, stationary electrode. In polarography, the dropping mercury electrode (DME) is spherical and grows, requiring corrections (e.g., Ilkovič equation) for surface area growth and convective effects.