Measurement of Transference Number by Moving Boundary Method

The Moving Boundary Method is based on the direct observation of migration of ions under the influence of an applied electrical potential gradient. This analytical method provides exceptionally high accuracy and is widely preferred in modern laboratory studies for precision transport measurements. The apparatus consists essentially of a long, standardized vertical glass tube fitted with specialized electrodes at both terminal ends.

To demonstrate the mechanism, the tube is filled with a solution of cadmium chloride (CdCl₂) at the lower terminal portion (serving as the indicator electrolyte) and hydrochloric acid (HCl) occupying the upper section (serving as the principal electrolyte). These fluids are layered carefully so that a sharp, distinct boundary line is formed between them, visible due to the marked differences in their respective optical refractive indices.

A platinum cathode is dipped into the HCl solution at the top, while a solid cadmium stick anode is introduced at the bottom base. Upon passing a direct electric current through the closed apparatus, electrochemical reduction causes hydrogen gas to evolve at the top cathode, driving the migration of H⁺ ions upward toward that same negative cathode electrode. Concurrently, cadmium undergoes oxidation at the bottom anode, generating Cd²⁺ ions that track directly behind the migrating acid front.

As the faster-moving H⁺ ions travel upwards, they are continuously replaced from underneath by the slower Cd²⁺ indicator ions, forcing the observable refractive boundary line to migrate upward. By measuring the precise linear distance (l) traversed by this boundary alongside the total quantity of electricity (Q) passed through the circuit, the exact transport number of the H⁺ cation is calculated.

In generalized applications designed to find the transport value of any target cation (A⁺), the primary electrolyte solution (AX) is systematically placed in the upper portion of the tube, while a indicator layer of a secondary electrolyte (BX)—sharing a common anion (X^–)—is placed directly underneath it. Crucially, the indicator electrolyte (BX) must be chosen such that the relative mobility/velocity of the B⁺ follower ion is strictly less than the velocity of the primary A⁺ lead ion.

Calculation of Transference Number by Moving Boundary Method

Let c represent the original concentration of the primary A⁺ ions expressed in gram-equivalents per milliliter (g-eq/mL) of solution.

If the linear distance across which the refractive boundary moves is defined as l (cm), and the uniform internal cross-sectional area of the vertical tube is defined as s (cm²):

Then, the net number of equivalents of A⁺ ions migrating upward through the observed volume is calculated as:

Equivalents of A⁺ = s × l × c

Let n represent the total quantity of electricity passed through the electrolyte solution expressed in Faradays. The precise fraction of total current safely sustained by the moving A⁺ ions corresponds directly to n × t_A⁺.

By balancing these structural relationships, we establish:

n × t_A⁺ = s × l × c

t_A⁺ = (s × l × c) / n      [Equation 1]

Since the Faradaic equivalent is represented as n = Q / F (where Q is the aggregate quantity of charge passed in coulombs, and F represents Faraday's constant equal to 96,500 coulombs), the baseline expression can be rearranged into a direct experimental working format:

t_A⁺ = (s × l × c × F) / Q      [Equation 2]

Using Equation 1 and Equation 2, the true transference value (t_A⁺) of a given cation can be evaluated dynamically using the experimental values of cross-sectional area, boundary displacement distance, baseline solution concentration, and net coulombs passed.

Related Topics: 1. Measurement of Transport Number by Hittorf's Classic Method
2. Measurement of Transport Number by Concentration Cell EMF Method