Derivation of Ilkovič Equation

The Ilkovič equation is fundamental in polarography, describing the diffusion-limited current at a dropping mercury electrode (DME). Below is a step-by-step derivation of the equation for the average diffusion current:

\[ i_d (\text{avg}) = 607 n D^{1/2} m^{2/3} t^{1/6} c_0 \]

where \( n \) is the number of electrons, \( D \) is the diffusion coefficient (cm²/s), \( m \) is the mercury flow rate (mg/s), \( t \) is the drop time (s), and \( c_0 \) is the bulk concentration (mmol/L).

Step 1: Fick's Laws of Diffusion

Diffusion in electrolysis is governed by Fick's laws.

• Fick's First Law: The flux of ions is proportional to the concentration gradient:
  \[ \frac{dN}{dt} = -D A \frac{\partial c}{\partial x} \]
  The diffusion current is:
  \[ i = n F A D \frac{\partial c}{\partial x} \]
  where \( F = 96485 \, \text{C/mol} \).
• Fick's Second Law: For non-steady-state conditions:
  \[ \frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2} \]
  Boundary conditions: \( c(0, t) = 0 \), \( c(x \to \infty, t) = c_0 \), \( c(x, 0) = c_0 \).

Step 2: Cottrell Equation for Planar Electrode

For a stationary planar electrode, solve Fick's second law. The concentration profile is:

\[ c(x, t) = c_0 \cdot \erf\left( \frac{x}{2\sqrt{D t}} \right) \]

The gradient at the surface (\( x = 0 \)):

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

The current is:

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

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

Step 3: Adaptation to Dropping Mercury Electrode (DME)

The DME is a growing spherical drop. Mercury mass at time \( t \): \( m t \). Drop radius:

\[ r(t) = \left( \frac{3 m t}{4 \pi \rho} \right)^{1/3} \]

Surface area:

\[ A(t) = 4 \pi r^2 = 0.8515 \, m^{2/3} t^{2/3} \]

Substituting into the Cottrell equation:

\[ i_d(t) = n F \sqrt{\frac{D}{\pi t}} \, c_0 \cdot 0.8515 \, m^{2/3} t^{2/3}\] \[ i_d(t) = n F \sqrt{\frac{D}{\pi}} \, c_0 \cdot 0.8515 \, m^{2/3} t^{1/6} \]

Step 4: Ilkovič Correction for Expanding Surface

The growing drop thins the diffusion layer, increasing current. Ilkovič's correction factor is \( \sqrt{7/3} \):

\[ i_d(t) = \sqrt{\frac{7}{3}} \, n F \sqrt{\frac{D}{\pi t}} \, c_0 \cdot 0.8515 \, m^{2/3} t^{2/3} \]

Evaluating constants (with \( F = 96485 \), \( i_d \) in μA, \( c_0 \) in mmol/L):

\[ i_d (\text{max}) = 708 \, n \, D^{1/2} \, m^{2/3} \, t^{1/6} \, c_0 \]

Step 5: Average Current Over Drop Life

The measured current is averaged over drop time \( t \):

\[ i_d (\text{avg}) = \frac{1}{t} \int_0^t i_d(t') \, dt' \]

Since \( i_d(t') \propto (t')^{1/6} \), the integral gives:

\[ i_d (\text{avg}) = \frac{6}{7} i_d (\text{max})\] \[ i_d (\text{avg}) = 607 \, n \, D^{1/2} \, m^{2/3} \, t^{1/6} \, c_0 \]

This is the Ilkovič equation used in polarography.

Realated Topics:
Derivation of Randles Sevcik Equation
Derivation of Cottrell Equation
Nicholson-Shain Theory for Cyclic Voltammetry