Chemistry · Lesson

Electrophoresis and membrane-based separation

Add signed electrophoretic and solvent velocities before predicting detector arrival. Compare that field-driven separation with a membrane stream balance and condition-specific rejection.

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By the end, you can

Define the coordinate direction in an electrophoretic transport calculation.
Calculate arrival only when the total drift points toward the chosen detector.
Use form fractions in a fixed-EOF rapid-exchange model.
Distinguish feed-based concentration rejection from the fraction of solute passed through a membrane.

Charge-driven motion is relative to a moving medium

Electrophoresis separates constituents through their field-dependent motion in a medium. Electrophoretic mobility μ_ep relates a constituent’s electrophoretic velocity to electric field E under specified conditions: v_ep=μ_ep E. The mobility depends on form, charge, friction and medium; a charge sign alone does not determine an arrival time.

Electroosmotic flow, abbreviated EOF, is field-driven solvent motion associated with charged interfaces. Define its mobility μ_EOF so v_EOF=μ_EOF E in the same coordinate convention. With no additional pressure flow in the present model, laboratory-frame velocity is v_total=(μ_ep+μ_EOF)E. The two contributions must be added with their signs.

Choose the positive axis from injection toward a specified positive-side detector and give E a positive value in that axis. For detector distance L_eff>0 and constant v_total>0, drift arrival time is t=L_eff/v_total. If the stipulated drift is negative, it moves away from that detector; a negative quotient is not an arrival time. Effective injection-to-detector distance need not be the total capillary length.

An anion can arrive when solvent motion is stronger

An original one-dimensional model assigns μ_ep=−1.00×10⁻⁸ m² V⁻¹ s⁻¹, μ_EOF=+3.00×10⁻⁸ m² V⁻¹ s⁻¹ and E=+3.00×10⁴ V m⁻¹. Electrophoretic velocity is −0.000300 m s⁻¹ while EOF velocity is +0.000900 m s⁻¹. Total velocity is +0.000600 m s⁻¹.

At L_eff=0.300 m, the assigned constant-drift arrival time is 500 s. Over that time the electrophoretic contribution is −0.150 m and the EOF contribution +0.450 m, summing to the detector distance. The given anionic mobility opposes the solvent flow but does not prevent positive arrival.

The assumptions include constant medium, temperature, field and mobilities, with no adsorption, pressure flow or diffusion broadening. They illustrate a signed balance, not an apparatus setting or a universal electrode labeling convention.

A stronger opposing mobility reverses total drift

In a separate model, assign μ_EOF=+1.50×10⁻⁸ m² V⁻¹ s⁻¹, E=+2.00×10⁴ V m⁻¹ and L_eff=0.200 m. Give a cation μ_ep=+2.00×10⁻⁸, a neutral form μ_ep=0 and an anion μ_ep=−2.00×10⁻⁸ in the same mobility units. These are independent hypothetical forms with given mobilities.

The cation’s total velocity is +0.000700 m s⁻¹ and its arrival is 285.714 s. The neutral form is carried with EOF at +0.000300 m s⁻¹ and arrives at 666.667 s. The anion’s total velocity is −0.000100 m s⁻¹, so there is no positive-side drift arrival in this model.

A common cations-before-neutrals-before-anions ordering is therefore conditional on the chosen direction and sufficient EOF. Reversing polarity, locating a detector elsewhere or adding other transport changes the problem. The neutral form can move with solvent even though its specified electrophoretic mobility is zero.

Check total drift before reporting an arrival time

Given form	μ_ep / 10⁻⁸ m² V⁻¹ s⁻¹	Total velocity / mm s⁻¹	Positive-side drift arrival / s
Cation	+2.00	+0.700	285.714
Neutral	0	+0.300	666.667
Anion	−2.00	−0.100	No positive-side drift arrival

Original signed model: positive axis from injection toward a detector 0.200 m away, E=+2.00×10⁴ V m⁻¹ and μ_EOF=+1.50×10⁻⁸ m² V⁻¹ s⁻¹. Given cation, neutral and anion mobilities are constant; pressure flow, diffusion and adsorption are excluded. The anion moves away from the specified detector, so no negative arrival time is reported.

Speciation can change drift while other effects are held fixed

For rapidly exchanging forms, an effective electrophoretic mobility can be represented by a fraction-weighted average when the model supports it. Use symbolic HA and A⁻, with assigned ideal pK_a=6.00, standard concentration 1 mol L⁻¹ and activity coefficients one. The ion fraction is f_A=10^(pH−pK_a)/(1+10^(pH−pK_a)).

Stipulate μ_HA=0, μ_A=−4.00×10⁻⁸ and a deliberately fixed μ_EOF=+2.00×10⁻⁸ m² V⁻¹ s⁻¹. Then μ_ep,eff=f_A μ_A and μ_total=μ_ep,eff+μ_EOF. At pH 5, 6 and 7, total mobilities are respectively +1.63636×10⁻⁸, zero and −1.63636×10⁻⁸ m² V⁻¹ s⁻¹.

At half ionization the opposing drift contributions cancel in this invented example. Zero mean drift does not mean that diffusion stops. In a real capillary, pH can also change wall charge, EOF and the individual form mobilities; the fixed-EOF assumption isolates only one effect. These values establish no universal pH–arrival rule.

A selective barrier poses a different transport question

A membrane can separate streams through condition-dependent transport across a barrier. Size, charge, solubility, diffusion and material interactions can contribute, depending on the membrane and process. A nominal cutoff or a method label is not an absolute identity filter. Membrane transport need not be driven by the electric-field mechanism above.

Define a feed-based apparent concentration rejection R_bulk=1−C_p/C_f, where C_p and C_f are the chosen permeate and bulk-feed concentrations of the conserved solute. This operational ratio is not automatically intrinsic membrane-surface selectivity. Concentration polarization can make the local surface composition differ from the bulk feed.

Original membrane studies examine particular materials, feeds, operating conditions and outcomes. Incomplete rejection in a cooling-tower effluent study or changed permeance in a nanostructured membrane study supports attention to those conditions; it does not supply a universal rejection, pressure trend or purity guarantee. The numerical balance here is independently invented.

Concentration rejection and passed inventory differ

Use a hypothetical steady feed Q_f=1.00 L h⁻¹ with C_f=10.0 mg L⁻¹, permeate Q_p=0.600 L h⁻¹ with C_p=0.500 mg L⁻¹, and no solute accumulation, reaction or adsorption. Retentate flow is Q_r=Q_f−Q_p=0.400 L h⁻¹. Conserved solute requires Q_fC_f=Q_pC_p+Q_rC_r.

Feed carries 10.0 mg h⁻¹ and permeate carries 0.300 mg h⁻¹. Retentate therefore carries 9.70 mg h⁻¹, giving C_r=24.25 mg L⁻¹. The feed-based concentration rejection is 1−0.500/10.0=95.0%, while the fraction of entering solute passed to permeate is 0.300/10.0=3.00%.

The 5.00% concentration ratio differs from the 3.00% passed fraction because the streams have different flow rates. Neither number means that the solute was chemically destroyed. A real system needs the actual stream definitions, steady-state or accumulation terms and relevant sampling and uncertainty evidence.

Worked examples

Add opposing contributions with signs

Use μ_ep=−1.00×10⁻⁸, μ_EOF=+3.00×10⁻⁸ m² V⁻¹ s⁻¹, E=+3.00×10⁴ V m⁻¹ and distance 0.300 m.

The two velocities are −0.000300 and +0.000900 m s⁻¹.
Their sum is +0.000600 m s⁻¹, so t=0.300/0.000600=500 s.
Signed displacements −0.150+0.450 m close the 0.300 m distance.

An opposing anionic contribution can coexist with positive detector arrival.

Reject a negative quotient as an arrival time

Use μ_EOF=+1.50×10⁻⁸ and anion μ_ep=−2.00×10⁻⁸ m² V⁻¹ s⁻¹ with E=+2.00×10⁴ V m⁻¹.

Total mobility is −0.500×10⁻⁸ m² V⁻¹ s⁻¹.
Total velocity is −0.000100 m s⁻¹.
The drift points away from the positive-side detector; no positive-side drift arrival is assigned.

Determine direction before dividing distance by velocity.

Find cancellation at half ionization

Use pH=pK_a=6.00 with μ_HA=0, μ_A=−4.00×10⁻⁸ and fixed μ_EOF=+2.00×10⁻⁸ m² V⁻¹ s⁻¹.

The ideal ion fraction is 1/2.
Effective electrophoretic mobility is −2.00×10⁻⁸ m² V⁻¹ s⁻¹.
Adding EOF gives zero mean drift mobility.

Cancellation is conditional on the given values and does not stop diffusion.

Close the steady stream inventory

Use feed 1.00 L h⁻¹ at 10.0 mg L⁻¹ and permeate 0.600 L h⁻¹ at 0.500 mg L⁻¹.

Retentate flow is 0.400 L h⁻¹; feed and permeate solute rates are 10.0 and 0.300 mg h⁻¹.
Retentate rate is 9.70 mg h⁻¹, so concentration is 24.25 mg L⁻¹.
R_bulk=95.0%, whereas the entering-solute fraction passed is 3.00%.

A concentration ratio is not the same as a stream-inventory fraction.

Try it yourself

Keep the first anion model’s total velocity +0.000600 m s⁻¹ but locate the detector 0.150 m away. Find the constant-drift arrival time.
Show the answer and reasoning
250 s.
The positive distance is half as large while the positive velocity stays fixed.
In a new steady balance, feed remains 1.00 L h⁻¹ at 10.0 mg L⁻¹, while permeate is 0.500 L h⁻¹ at 1.00 mg L⁻¹. Find retentate concentration and entering-solute fraction passed.
Show the answer and reasoning
19.0 mg L⁻¹ and 5.00%.
Retentate flow is 0.500 L h⁻¹; permeate carries 0.500 mg h⁻¹ and retentate 9.50 mg h⁻¹. Divide by the corresponding flow or entering solute rate.
What can be inferred, and what cannot, when the specified electrophoretic and EOF contributions sum to zero?
Show the answer and reasoning
Mean drift cancels within the model; molecular diffusion is not thereby absent.
Cancellation describes the mean velocity terms. Diffusive spreading is a separate process not removed by setting that mean to zero.
A stream model reports 95.0% feed-based concentration rejection. Why does this not mean that 95.0% of the entering solute was destroyed?
Show the answer and reasoning
The ratio compares chosen concentrations, while conserved solute is distributed among streams.
Flow rates are needed for passed and retained inventory fractions. The stipulated model includes no chemical destruction, adsorption or accumulation.

What to carry forward

Use one signed coordinate convention for field and mobilities.
Include solvent motion before predicting detector arrival.
A negative drift toward an opposite direction is not a negative arrival time.
Fixed-EOF speciation examples do not predict every capillary.
Membrane concentration rejection and passed inventory require different denominators.

References

Teaching resources and reference material supporting this lesson’s explanations and examples.