Lecture 5 Diffusion of interacting species

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????? ??Lecture 5Diffusion of interacting
species

• ??? ????

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Interactions between diffusing molecules

• These interactions can strongly affect the
  apparent diffusion coefficients
• chemical interactions affecting diffusion
• solute-solute interactions
• especially in strong electrolytes
• solute-solvent interactions
• includes solvation, spinodal decomposition
• solute-boundary interactions
• in porous solids with fluid-filled pores

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solute-solute interactions
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Diffusion of strong electrolytes

• Electrolytes
• 0.1 M sodium chloride solution passes an electric
  current one million times more easily than water
  does (i.e., sodium ions and chloride ions move
  freely through the solution)
• The large ion size relative to electrons means
  that such a solution passes current ten thousand
  times less easily than a metal does.
• Ionic diffusion coefficient (Table 6.1-1 p.143)
• value 10-5 cm2/sec, measured largely in
  electrochemistry
• several methods, including tracer diffusion
  determinations
• The proton (H) and the hydroxyl (OH-) are fast
  and big fat organic ions are slow.

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Diffusion coefficient of electrolytes

• We know that the diffusion coefficient Cl- gt Na
  , what is its value for NaCl?
• Analogy to an old grandfather and a young girl.
• The diffusion of a large, fat cation (the
  positive ions) and a small, quick anion (the
  negative ions) will be dominated by the slower
  ion.
• We consider that a sodium ion in dilute solution
  follows the Ficks law (avoid the complicated
  reference velocity)

But the electric field will affect diffusion?
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The ion velocity is proportional to the sum of
the forces acting on it
Ionic charge
Faradays constant
Electrostatic potential
Chemical forces
Electrical forces
Ion mobility
ui physical property of the ion
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The Nernst-Plank equation
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Single strong 1-1 electrolyte

• Ionizes completely, producing equal numbers of
  cations and anions.
• The concentrations of anions and cations may vary
  through the solutions, the concentrations and the
  concentration gradients of these species are
  equal everywhere because of electroneutrality
• The ion fluxes

Current density
Magnitude of the ionic charge
1 and 2 refer to cation and anion, respectively
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The basic flux equation for each ion
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When there is no current
The harmonic average diffusion coefficient of the
electrolyte, dominated by the slower ions.
When the solution is well mixed
The arithmetic average of the ion diffusion
coefficients, called the transference number,
is dominated by the faster ion.
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Diffusion of hydrogen chloride what is the
diffusion coefficient at 25ºC for a very dilute
solution of HCl in water? What is the
transference number for the proton under these
conditions?
The slow ion dominates!
The faster protons carry 82 of the current.
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Non-1-1 electrolytes
The basic flux equation
The constraints on concentration and flux at zero
current
When there is no electrostatic potential
(Harned and Owen, 1950)
The total electrolyte flux and the total
electrolyte concentrations
The flux for a single non-1-1 electrolyte
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Diffusion of lanthanum chloride What is the
diffusion coefficient of 0.001M lanthanum
chloride?
or
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Diffusion of lanthanum chloride in excess sodium
chloride How will the result of the previous
example be changed if the lanthanum chloride
differs through 1M NaCl?
The diffusion of dilute LaCl3 in concentrated
NaCl is dominated by the diffusion of the
uncommon ion, La3.
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Diffusion versus conductance

• Diffusion coefficient can be difficult to
  measure. How about in the solutions of
  electrolytes?
• The conductance of a single electrolyte in
  solution is most easily measured in a conductance
  cell.

(Evans and Matesich, 1973)
The electrical resistance of the stirred solution
is measured with a Wheatstone bridge and a
rapidly oscillating AC field or fixed maximum
voltage, so that the solution remains homogeneous
throughout the experiment. The resistance is
inversely proportional to the current through the
cell, but the current, in turn, is proportional
to the ion fluxes
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The proportional constant Kcell in this relation
is a function of the electrode area, the
electrode separation, and the cell shape. It is
found by calibration of the cell, most commonly
with a potassium chloride solution.
How to apply to the diffusion coefficient
measurement?
The equivalent conductance ? can be accurately
measured. In many practical problems, the ion
transport is well described by assuming that ? is
a constant.
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?
?i is the equivalent ionic conductance. It is a
function of the ion mobility and the charge.
Therefore, it is closely related to the ionic
diffusion through the mobilities
T 298K
We can then use this equation to measure the
ionic conductance, and then the ionic diffusion
coefficient.
1-1 electrolyte
Compared to the real case for D
Harmonic mean
Arithmetic mean
The two cases will be the same only when both
ions have the same mobility. But...
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Estimate the diffusion coefficient of CaCl2 from
conductance measurement. The equivalent ionic
conductance at infinite dilute is 59.5 for Ca2
and 76.4 for Cl-.
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Diffusion of associating solute

• Solutes associate to form aggregates.
• Studied as early as Arrhenius (1884) suggested
  that materials like acetic acid partially
  dissociated in water.
• Consider diffusion of potassium chloride across
  two thin membranes. The first membrane is a thin
  film of water. At steady state

B.C.
How about the second thin membrane? The second
membrane consists of a chloroform solution of a
macrocyclic polyether.
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The dielectric constant of the second membrane is
low, the potassium and chloride ions are largely
associated as ion pairs The ions are stuck
together with electrostatic glue
B.C.
The diffusion coefficient is that of the ion
pairs, not the average of the ions.
The flux is now proportional to the square of the
potassium chloride concentration.
Diffusion coefficient is a function of
concentration may result from solute association.
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Weak Electrolytes

• Produce solutions of a cation, an anion and
  molecule in equilibrium with each other

Mass balance on the ion (species 1) and on the
molecules (species 2) at steady state
The rate of formation of the molecules
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B.C.
Function of concentration
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Diffusion of acetic acid What is the diffusion
coefficient of the acetic acid molecule if the
apparent diffusion coefficient of acetic acid is
1.80 x 10-5 cm2/sec at 25C and 10M? The pKa of
acetic acid is 4.756.
The pKa of a weak acid HA is defined as
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Detergent sodium dodecylsulfate (SDS)

• Detergent molecules remains separate at low
  concentration but then suddenly aggregate.
• The aggregates are called micelles.
• The oil-bearing particles are captured in the
  micelles (i.e., an ionic hydrophilic skin
  surrounding an oily hydrophobic core).

Many monomers form one size micelle
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Steady-state diffusion in such a system of
monomer and micelle
For monomer
For micelle
Critical micelle concentration
Works for nonionic detergent and ionic
detergents at high ionic strength!
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Isodesmic aggregation

• Aggregate gently, resulting in a slow and steady
  deviation from the unaggregated limit.
• A stacking of dye molecules can form this type
  aggregation.
• When the ease of stacking is the same for all
  sizes in the stack, this aggregation is called
  isodesmic.

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Summary for solute-solute interaction

• Our goal derive the apparent diffusion
  coefficient for...
• strong electrolyte electrostatic
• weak electrolyte dimer
• dyes and detergents large aggregates

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Solvation

• Solute-solvent interaction is considered.
• Solute and solvent combine to form a new species,
  which is that actually diffusing. For water, it
  is called hydration.
• Assumption
• solutes flux is proportional to chemical
  potential gradient
• diffusion coefficient in dilute solution is given
  by Stoke-Einstein equation

D0 is a new diffusion coefficient, ? is the
solvent viscosity, R0 is the solute radius, and
?1 is an activity coefficient.
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Hydration can affect the solute radius
molar volume of water
the number of water molecules bound to a solute,
the hydration number
true solute radius
Hydration decreases diffusion.
Hydration can affect the concentration
(Scatchard, 1921)
Hydration increases diffusion.
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The author of the textbook, E.L. Cussler was not
too sure about the hydration theory. He mentioned
the ideas of water structure for several ions
and the jump mechanism for proton to describe
the diffusion coefficient of the ions.

• Water structure ion can either destroy or
  enhance the water structure. Because of this, the
  viscosity of the water changes, and the motion of
  the ions is affected.
• Proton motion refer to page 166.

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Diffusion near critical or consolute points

• Solute and solvent are on the verge of a phase
  separation.
• Solute and solvent are not randomly distributed,
  but tend to form small clusters of molecules of
  one species (i.e., solute and solvent do not like
  each other)
• Consolute
• points represent the temperature and composition
  where two liquids become miscible in all
  properties.
• Near the upper consolute point and the lower
  consolute point, the binary diffusion coefficient
  drops to zero.

Why?
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First explanation diffusion flux is proportional
to chemical potential gradient
chemical potential per molecule in terms of
activity ?1c1
At the consolute point, ,
D 0
Not exactly fit the experimental observation in
temp.
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Second explanation Modified Ficks law near the
consolute points
The linear form of Ficks law include higher
terms. Similar to the flow of non-Newtonian
fluids which require higher terms than that in
Newtons law of viscosity.
This expression fits the zero diffusion
coefficient near the consolute points and the
temperature dependence. However, it does not show
fit the experimental results in the concentration
profile in a free-diffusion experimental data.
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Third explanation Long-range fluctuations
dominate behaviour near the consolute point

• When fluctuations of concentration and of fluid
  velocity couple, diffusion occurs
• ordinary condition the concentration
  fluctuations are dominated by motion of a single
  molecule
• near the critical point the concentration
  fluctuations exist even when the average fluid
  velocity is zero. (similar to a turbulent eddy
  diffusion, but without flow)

the average size of a cluster, (function of
thermodynamic factor)
Best of the three explanations
(Cussler, 1980)
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Phase separation

• Cool to its equilibrium condition is a two-phase
  mixture.
• Separation begins as soon as the solution is
  cooled below its phase boundary or its binodal.
• The region just below this boundary is
  metastable, waiting for events that cause phase
  separation.
• The phase separation begins with nucleation of
  small droplets of the new phase.
• These droplets grow with time.

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Spinodal decomposition

• The original solutions is rapidly quenched (??).
• The equilibrium condition drops suddenly through
  the binodal and below the spinodal curve.
• The spinodal curve is the lower boundary of the
  metastable region.
• Below this curve, the solution is unstable and
  phase separation is immediate
• no small dust particles are needed to nucleate
  the phase separation
• the separation is spontaneous and fast

(Mazurin, 1984)
? is the interfacial influence
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Mass balance
If ?i is positive, the concentration fluctuations
decay and the solution stays homogeneous if it
is negative, the concentration fluctuations grow
over time and the phase separation proceeds.
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In the shortest negative time, the fluctuation
grows fasted (i.e., )
Eliminate ?
Characteristic length the size of the molecules
or clusters (maybe 10-9 nm)
With reasonable values into this equation, we
have ?min -10-8 sec (very fast!!)
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solute-boundary interactions
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Solute - boundary interaction

• When a solute diffuses through small pores, its
  speed may be affected by the size and the
  chemistry of the pores.
• Diffusion in a porous media (p.173)
• the effects of longer pores and smaller areas are
  often lumped together in the definition of a new,
  effective, diffusion coefficient Deff

Diffusion coefficient within the pores
Tortuosity (26, average about 3)
Void fraction
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Diffusion in periodic composite

• Assumption
• diffusion take places both in the interstitial
  region between the spheres and through the
  spheres themselves (p.173)

Maxwell, 1873
Void fraction of the spheres in the composite
materials
Diffusion coefficient in the interstitial pores
Diffusion coefficient through the spheres
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The diffusion coefficient of KCl through a
protein gel is 6x10-7 cm2/sec. However, the gel
is not homogeneous, because it contains water
droplets about 10-2 cm in diameter that are
separated by only 2x10-2 cm. The diffusion in
these water droplets is about 2x10-5 cm2/sec.
What is the diffusion in the homogeneous gel?
D 5 x 10-7 cm2/sec
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Diffusion in a cylindrical pore

• Several effects possible in a straight
  cylindrical pore is shown in P.176.
• The flux through the largest pores tends to be
  fastest, and that through the smallest pores
  tends to be slowest.
• High flux often with low selectivity and vice
  versa.

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A gas flowing in a small cylindrical pore will
obey the Hagen-Pouiseville law
pore diameter
pressure drop across the pore
gas viscosity
gas viscosity
Flux from convection
How can we tell the difference?
By selectivity!
Compared to flux from diffusion
permeability
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Pores affect the transportation

• Knudsen diffusion
• The diffusing solute interacts only with the pore
  wall
• Dominated by the distance between molecular
  collisions and the pore diameter
• Hindered diffusion
• The solute size is less than the pore size but
  comparable to it and the solvent size is much
  smaller
• Sieving
• Both solute and solvent are of sizes comparable
  to the pore
• Transport of linear and branched alkanes into
  zeolites
• Osmosis membrane
• Chemical effects

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Knudsen diffusion
mean free path
The Knudsen number, Kn is defined as
pore diameter

• Small Kn
• effective coefficients and tortuosities are
  capable to describe the transportation and
  Knudsen diffusion is not important
• for example, liquid (l a few angstroms)
• Large Kn
• diffusion is dominated by collisions with the
  boundary
• for example, gas
• air at room temperature and pressure, gt 600
  angstroms
• hydrogen at 300C and 1 atm, gt 2000 angstroms

collision diameter of the diffusing species
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Apply the ideas from kinetic theory of rigid
spheres
molecular velocity
pore diameter (instead of mean free path)
The Knudsen diffusion coefficient is independent
of pressure and of the molecular weight of the
second species.
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Hindered diffusion as a rigid sphere in a
solvent continuum that fills the pore
The solutes transport is retarded by the viscous
drag of the solvent, which is affected by the
proximity if the pore wall.
Rankin equation
(Gaydos and Brenner, 1978)
solute diameter divided by the pore diameter
Stokes-Einstein diffusion coefficient
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Chemical effect

• Capillary condensation
• a small effect resulted from the increased vapor
  pressure of a liquid inside a pore
• it exists only when a surface tension exists
• not work for separation of air (c.p. 155K)at
  room temperature, but works for separation of CO2
  (c.p. 304K) and methane at room temperature
• Surface diffusion
• important, especially for gases
• When the adsorption is physical, the adsorption
  energy is less than kBT, and the adsorbed solutes
  are highly mobile. When the adsorption is
  chemical, the adsorption energy is greater than
  kBT, and the adsorbed solutes are more tightly
  bound.

Ds surface diffusion coefficient ( 10-5
cm2/sec, like liquid ?) l diaphragm
thickness ? diaphragm cross-sectional area
like solid?