PHAR 526: Physical Chemistry Diffusion

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PHAR 526 Physical ChemistryDiffusion

• R. Gary Hollenbeck
• Department of Pharmaceutical Sciences

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• Diffusion

Mass transfer of a dissolved substance from a
region of high concentration to a region of lower
concentration.
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• Diffusion

Flow of molecules through a barrier membrane
(not by active or facilitated transport) polymeri
c film stagnant solvent layer by molecular
penetration movement through pores movement
through a matrix Most diffusion occurs through
the path of least resistance.
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• Diffusion through a
• non-porous barrier

Requires dissolution of the permeating molecules
in the barrier material Usually slow (i.e.,
material has a small diffusion coefficient)
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• Diffusion through a
• porous barrier

Diffusion can occur through barrier as well as
solvent filled pores Diffusion coefficient
through solvent is usually higher A function of
porosity and tortuosity
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• Diffusion through a
• matrix

Diffusion can occur through or between the matrix
material Diffusion coefficient through solvent
is usually higher A function of porosity and
tortuosity
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• Characterizing the mass
• transfer process

J Flux dM/dt/A rate of material flowing per
unit cross-sectional area A g/sec/cm2
1 cm
J
1 cm
J Flux g/sec/cm2
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• Diffusion Across a Region

C1 higher concentration C2 lower
concentration x distance h thickness of the
region
C1
J
C2
Dimensional Analysis C g/cm3 x h cm
x
0
h
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• Ficks First Law

J is proportional to the concentration gradient.
C1
J
dC
J - D
C2
dx
D Diffusion Coefficient dC/dx Concentration
Gradient
x
negative sign because concentration decreases as
distance increases.
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• Dimensional Analysis

dC
J - D
dx
C1
J
C2
J g/sec/cm2 dC/dx g/cm3/cm g/cm4 D cm2
sec-1
x
0
h
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• Approach to Equilibrium

C1
dC/dx
dC/dx 0
C1
C2
C2
x
x
0
h
0
h
C1 gt C2
C1 C2
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• General Solution

Boundary
Solution C0
Pure Solvent
A
-3 -2 -1 0 1 2 3
x
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• General Solution

There is a gradual approach to what is usually
assumed to be a linear gradient.
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• Ficks Second Law

Examination of the concentration change with
respect to time at a point in the system rather
than the change in mass with respect to
distance.
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• Diffusion Cell

Donor Compartment
Receptor Compartment
Membrane
Flux In
Flux Out
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• Mass Balance

JIN
JOUT
A
?C
?x
For some time period ?t, ?J A ?t ?C A ?x
Difference between what can in and what went out
amount deposited in the compartment
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• Mass balance for an
• infinitesimally small volume

JIN
JOUT
A
dC
dx
For some time period dt, dJ A dt -dC A dx
dC/dt - dJ/dx Negative sign included to
account for a decrease in concentration as x
increases.
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• Ficks Second Law

Mass balance Ficks 1st Law Differentiating
the 1st Law with respect to time Substitution
from above
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• Ficks Second Law

This is a simplified expression considering only
concentration gradients in the x-direction.
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• Diffusion cell with sink conditions Two
  dimensional schematic

Sink Conditions Removal of material from the
receptor compartment (sometimes CR assumed to be
0) Examples Diffusion Experiments Large
receptor compartment or flow of fresh solvent
through the receptor compartment In vivo
absorption of drug from the gut
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• Diffusion cell with sink conditions Two
  dimensional schematic

Donor Compartment
Receptor Compartment
CD
Flow of solvent
CR
0
h
x
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• Diffusion cell with sink conditions Two
  dimensional schematic

Under the right conditions, after sufficient time
has passed, the system stabilizes for a period of
time, where CD CR and there is no change in
concentration with respect to time in the
membrane (i.e., dC/dt 0.) This is the steady
state condition.
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• Ficks Second Law

Note If dC/dt 0, then d2C/d2x 0 This is a
second derivative d(dC/dx)/dx 0 Therefore dC/dx
constant
Ficks second law confirms that when a diffusion
cell is at steady state, the concentration
gradient is linear.
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• Ficks Law

dC
J - D
dx
C1
J
By integration between the limits of C C1 at x
0, and C C2 at x h.
C2
C1 - C2
J - D
x
0
h
0 - h
C1 - C2
J D
h
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• Partitioning

Donor
Receptor
Membrane
C1
J
CD
C2
CR
x
0
h
Cmemb
C1
C2
Partition Coefficient K

Csoln
CD
CR
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• Partitioning

Donor
Receptor
Membrane
C1
J
CD
C2
CR
x
0
h
KCD - KCR
J D
h
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• Permeability

KCD - KCR
J D
h
D K
J
CD - CR
h
P Permeability DK/h
J
P CD - CR
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• Absorption

CD - CR
Absorption rate J A P A
P Permeability DK/h
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• Absorption

CD - CR
Absorption rate J A P A
If we have "sink conditions", CR 0
Absorption rate J A P A CD
Absorption rate is directly proportional to CD
and A.