Moving Boundary Electrophoresis Tube
Vertical calibrated glass capillary (cross-sectional area A = 0.03142 cm², internal radius 1.00 mm) enclosed in a water jacket regulated at 25.00 ± 0.05 °C.
Chemistry / Moving Boundary Transference Numbers
FREE TO EXPLORELABORATORY 35 · NATURAL SCIENCES
Determination of Ionic Transference Numbers by the Moving Boundary Method
Determine the cation transference number (t_+) of hydrogen ions (H⁺) in hydrochloric acid solutions as a function of current and concentration using the precision moving boundary method. Explore interactive analytical gravimetry, Class A volumetric glassware and density determination.
MEASURE · CALIBRATE · ANALYSE
Your virtual chemistry laboratory.
SWEPT CAPILLARY VOLUME V—
MEASUREMENT STATUS
Ready to dispense 6.0 min of liquid.
Adjust t(elec) V, then weigh the dispensed aliquot.
Mass observations are determined by the high-precision thermodynamic equation of state for pure liquid water. Readings reflect thermal equilibrium and analytical balance force restoration.
Physical chemistry assumptions and metrological limits
['The moving boundary method requires a following electrolyte that satisfies both the Kohlrausch mobility criterion (u_lead > u_foll) and gravitational density stability.', 'Excessive electrical current (> 10 mA) induces thermal Joule heating, creating radial temperature gradients that produce parabolic convective boundary distortion.', 'Volume corrections for electrode reactions and solvent transport become significant at higher electrolyte concentrations (> 0.5 M), requiring dilatometric calibration.'] The moving boundary method operates under the Kohlrausch regulating condition: because the slower following cation (Cd²⁺) has lower mobility than the leading cation (H⁺), continuity of current establishes a steeper electric field in the following phase (E₂ > E₁), creating an active restoring force that continuously sharpens the boundary against molecular diffusion. The swept volume V of the capillary increases strictly linearly with elapsed time t: V = [60,000 · I · t_+ / (F · c)] · t. For 0.1000 M HCl at 25.00 °C under a constant current of 5.000 mA, the boundary sweeps volume at a steady rate of 0.02568 cm³/min (25.68 μL/min). Linear least-squares regression of swept volume against elapsed time yields an empirical cation transference number t_+ = 0.8260, confirming the extraordinary dominance of the Grotthuss proton-hopping mechanism in aqueous acid conduction.
Analytical chemistry apparatus & tools.
High-Precision Digital Galvanostat
Solid-state constant-current power source delivering an unvarying current of 5.000 ± 0.001 mA despite progressive changes in column electrical resistance.
In Situ Cadmium Generating Anode
High-purity metallic cadmium rod positioned at the base of the capillary to generate the slower following Cd²⁺ cation boundary (Cd → Cd²⁺ + 2e⁻).
Silver-Silver Chloride Non-Gassing Cathode
Ag/AgCl wire gauze electrode housed in the upper reservoir that absorbs current without generating hydrogen gas bubbles (AgCl + e⁻ → Ag + Cl⁻).
Precision Optical Cathetometer
Telescopic traveling microscope with calibrated reticle crosshairs and digital vernier scale (resolution ±0.01 mm) for tracking the refractive index boundary.
Digital Microsecond Chronometer
Precision quartz digital electronic timer interfaced to the galvanostat to measure elapsed migration time with ±0.01 second accuracy.
FOLLOW THE METHOD
A reason behind every step.
Determine the cation transference number (t_+) of hydrogen ions (H⁺) in hydrochloric acid solutions as a function of current and concentration using the precision moving boundary method. Suggested independent volumetric settings: 3.0, 6.0, 9.0, 12.0, 15.0, 18.0 min.
01
Apparatus Cleaning & Cadmium Electrode Activation
Clean the precision bore capillary with warm chromic acid cleaning solution, followed by extensive rinsing with triple-distilled deionized water. Polish the cadmium anode mechanically with fine emery paper to remove oxide films and seat it firmly in the bottom electrode socket.
02
Column Filling & Air Bubble Exclusion
Fill the capillary carefully with standardized 0.1000 M HCl solution, ensuring no air bubbles adhere to the tube walls or the cadmium electrode face. Mount the Ag/AgCl cathode in the upper reservoir and seal the system.
03
Thermostatic Equilibration at 25.00 °C
Circulate constant-temperature water through the outer cooling jacket at 25.00 ± 0.05 °C for 15 minutes to dissipate Joule heating and establish steady radial temperature profiles.
04
Boundary Formation & Galvanostat Initiation
Engage the galvanostat at constant current I = 5.000 mA and start the chronometer. Anodic oxidation of cadmium immediately produces a dense, localized layer of CdCl₂. The refractive index boundary between colorless HCl and denser CdCl₂ becomes sharply defined within 60 seconds.
05
Cathetometer Tracking Across Time Increments
Align the crosshairs of the traveling cathetometer with the sharp boundary meniscus. Record the boundary position at 3.00-minute intervals (3.00, 6.00, 9.00, 12.00, 15.00, and 18.00 min), converting linear displacement into swept capillary volume V in μL and cm³.
06
Linear Regression & Transference Number Calculation
Plot swept volume V (cm³) against elapsed time t (min). Perform linear least-squares regression to extract slope dV/dt. Calculate the cation transference number t_+ = (slope × F × c) / (60000 × I) and evaluate the anion transference number t_- = 1 - t_+.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into quantitative chemical insight.
Points are your recorded gravimetric observations. The regression line validates the linear mass-volume proportionality and extracts density.
ORDINARY LEAST-SQUARES FIT · m = ρ·V + m₀
V = \left(\frac{60000 \cdot I \cdot t_+}{F \cdot c}\right) t \implies V = 0.02568 \cdot t \implies t_+ = 0.8260
| # | Elapsed Current Duration t (min) | Swept Capillary Volume V (μL) | Elapsed Time t (min) | Swept Volume V (cm³) | Remove |
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