Derivation of Randles-Sevcik Equation

Derivation of the Randles-Sevcik Equation

The Randles-Sevcik equation describes the peak current in cyclic voltammetry for a reversible, diffusion-controlled electrochemical reaction at a planar electrode:

\[ i_p = (2.69 \times 10^5) \, n^{3/2} A D^{1/2} v^{1/2} c_0 \]

where \( n \) is the number of electrons, \( A \) is the electrode area (cm²), \( D \) is the diffusion coefficient (cm²/s), \( v \) is the scan rate (V/s), and \( c_0 \) is the bulk concentration (mol/cm³).

Step 1: Context and Fick's Second Law

In cyclic voltammetry, the potential is swept linearly (\( E = E_0 \pm v t \)), causing a peak current when the potential reaches the redox potential. The current is diffusion-limited, governed by Fick's Second Law:

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

Boundary conditions for a reversible reaction:

• At the electrode (\( x = 0 \), \( t > 0 \)): Concentration follows the Nernst equation.
• 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: Nernstian Boundary Condition

For a reversible redox reaction (\( \text{O} + n e^- \leftrightarrow \text{R} \)), the surface concentrations of oxidized (\( c_O(0, t) \)) and reduced (\( c_R(0, t) \)) species follow the Nernst equation:

\[ \frac{c_O(0, t)}{c_R(0, t)} = \exp\left( \frac{n F}{R T} (E(t) - E^0) \right) \]

where \( E(t) = E_i - v t \) (for a reductive scan), \( E^0 \) is the standard potential, \( R = 8.314 \, \text{J/(mol·K)} \), and \( T \approx 298 \, \text{K} \). Assuming only the oxidized species is initially present (\( c_R(x, 0) = 0 \)), the total concentration is conserved: \( c_O(0, t) + c_R(0, t) = c_0 \).

Step 3: Solving the Diffusion Equation

Solve Fick's Second Law for both species. For the oxidized species:

\[ \frac{\partial c_O}{\partial t} = D_O \frac{\partial^2 c_O}{\partial x^2} \]

Assume \( D_O \approx D_R = D \) for simplicity. Use a dimensionless variable to account for the time-dependent potential. Define:

\[ \xi = x \sqrt{\frac{n F v}{R T D}} \]

The current is related to the flux at the electrode:

\[ i_p = n F A D \left. \frac{\partial c_O}{\partial x} \right|_{x=0} \]

The solution involves a convolution integral, solved numerically or via approximations (e.g., by Randles and Sevcik).

Step 4: Peak Current Calculation

The peak current occurs when the surface concentration gradient is maximized. Using a semi-empirical approach, the flux at the peak is found to be proportional to \( \sqrt{D v} \). The analytical solution yields:

\[ i_p = n F A c_0 \sqrt{\frac{n F D v}{R T}} \cdot \chi_{\text{max}} \]

where \( \chi_{\text{max}} \approx 0.446 \) is a dimensionless constant from the numerical solution of the voltammetric response. Substituting constants (\( F = 96485 \), \( R = 8.314 \), \( T = 298 \)):

\[ \sqrt{\frac{F}{R T}} \approx \sqrt{\frac{96485}{8.314 \cdot 298}}\] \[\approx 6.24 \times 10^3 \, \text{C}^{1/2} \text{mol}^{-1/2} \text{K}^{-1/2} \]

Thus:

\[ i_p = (0.446) n^{3/2} F A c_0 \sqrt{\frac{D v}{R T / F}} \]

Converting units (\( c_0 \) in mol/cm³, \( i_p \) in A, \( A \) in cm², \( D \) in cm²/s, \( v \) in V/s) and evaluating constants gives:

\[ i_p = (2.69 \times 10^5) \, n^{3/2} A D^{1/2} v^{1/2} c_0 \]

Step 5: Application

The Randles-Sevcik equation is used in cyclic voltammetry to determine diffusion coefficients, concentrations, or electron transfer numbers by measuring peak current as a function of scan rate. Unlike the Cottrell equation (for chronoamperometry), it accounts for the dynamic potential sweep.

Related topics:
Derivation of Ilkovic Equation
Derivation of Cottrell Equation
Nicholson-Shain Theory for Cyclic Voltammetry