Sand Equation: Transition Time in Chronopotentiometry

In electroanalytical chemistry, understanding dynamic processes under mass-transport control is essential for postgraduate competitive examinations like CSIR NET, GATE, and IIT-JAM. While controlled-potential methods like cyclic voltammetry monitor current as a function of potential, chronopotentiometry applies a constant current step to an unstirred electrochemical cell and monitors the resulting electrode potential over time. The fundamental relation governing this technique is the Sand equation, formulated by Henry Sand in 1901. It defines the transition time τ, which marks the exact moment when the surface concentration of the electroactive reactant drops to zero.

Principles of Chronopotentiometry

In a chronopotentiometric experiment, a stationary planar working electrode of area A is immersed in an unstirred solution containing an electroactive species O at bulk concentration C*. An excess of inert supporting electrolyte is added to suppress electrical migration, ensuring that mass transport occurs purely by linear semi-infinite diffusion. When a constant current I is applied at t = 0, species O is reduced at the electrode surface: O + n e⁻ → R.

Because the applied current is constant, the flux of reactant diffusing to the electrode remains fixed. As electrolysis proceeds, reactant molecules at the surface are consumed faster than diffusion can supply them from the bulk solution. Consequently, the surface concentration C(0,t) decreases continuously. At a critical time called the transition time τ, the surface concentration reaches zero: C(0,τ) = 0. At this instant, diffusion alone cannot support the imposed current, and the electrode potential shifts rapidly to a more cathodic value where another species or the solvent is reduced.

Derivation from Fick's Second Law

The mathematical foundation begins with Fick's second law for linear semi-infinite diffusion:

∂C(x,t)/∂t = D·[∂²C(x,t)/∂x²]

where D is the diffusion coefficient and x is distance from the electrode. The boundary conditions are:

• Initial condition: At t = 0, C(x,0) = C* for all x ≥ 0.
• Semi-infinite boundary: As x → ∞, C(x,t) = C* for all t > 0.
• Constant flux at surface: At x = 0, D·[∂C/∂x]_x=0 = I/(n·F·A) = j/(n·F), where j is current density.

Solving this partial differential equation using Laplace transforms gives the surface concentration:

C(0,t) = C* − [2·I·t^1/2]/[n·F·A·π^1/2·D^1/2]

Setting C(0,τ) = 0 at the transition time τ gives the Sand equation:

I·τ^1/2 = (n·F·A·π^1/2·D^1/2·C*)/2

The Sand Constant and Consecutive Transitions

The Sand equation establishes that the product I·τ^1/2 (or j·τ^1/2) is directly proportional to bulk concentration C* and independent of current. The term (I·τ^1/2)/C* is the Sand constant. Constancy of I·τ^1/2 across varying currents confirms pure diffusion control.

For a mixture of two reducible species (species 1 followed by species 2), species 1 continues to diffuse and reduce during the second reduction. As derived by Berzins and Delahay, the transition times obey:

(τ₁ + τ₂)^1/2 − τ₁^1/2 = (n₂·F·A·π^1/2·D₂^1/2·C₂*)/(2·I)

When both species have equal concentrations, equal diffusion coefficients, and n₁ = n₂, the right-hand side equals τ₁^1/2. Thus, (τ₁ + τ₂)^1/2 = 2τ₁^1/2, which squares to τ₁ + τ₂ = 4τ₁, yielding τ₂ = 3τ₁.

Worked Numerical Examples

Exam Relevance and Comparison