07 Transport Phenomena (1)
Transcript of 07 Transport Phenomena (1)
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Transport phenomena
Johnny Wong
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Transport phenomena in biological system
Study the transport of ions across membranes
Develop basic understanding on ionic strengthDistinguish between the passive transport and the
active transportusing the concept of Gibbs freeenergy.
Understand the molecular motions in liquids (diffusion)Use mathematic equation (Ficks laws) to analyze
diffusion
Use the Piosenllesequation to calculate the absoluteand relative viscosity of liquids
Study the ion mobility and admire the design of ion
channels.
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Electrolytic/non-electrolytic solutions
Non-electrolytic solution
No significant intermolecular interactionbetween solute molecules Solute molecules move independently of one
another Idea behavior under dilute conditions.
Electrolytic solution (e.g. NaCl) Long-range Coulombic interactions between ions. Solute molecules form clusters Their movements are not independent of one
another Non-ideal behavior even under dilute conditions.
Coulombic interaction
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For dilute non-electrolytic solutions (ideal)
Activity of a solute J is equal to the itsmolality : =
The chemical potential is : = ln
For electrolytic solutions (non-ideal) Activity of a solute J is proportional to the
its molality :
= where is the activity coefficient
The chemical potential is : = ln()
Electrolytic/non-electrolytic solutions
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Mean activity coefficient
The activity coefficients of cation and anion are
denoted by:Cation: +Anion:
Since cation and anion always occur together insolution, we cannot measure of activities ofcation and anion separately.
It is more appropriate to use the mean activitycoefficient, .
For a salt MX (e.g. NaCl)
= (+)
For a salt MpXqin general
= (+
)
+
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Mean activity coefficient
The activity coefficients of Na+and SO42-ions in 0.01 m Na2SO4(aq)
are 0.98 and 0.84 respectively. Calculate the activities of the two ions.
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Mean activity coefficient
= (+
)
+
We dont know these numbers !!!
The activity coefficients of cation and anion cannot bemeasured separately.
The equation above is just a definition of the mean activity
coefficient. To estimate the value of , we need to use the Debye-Huckel
law.
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Debye-Huckel limiting law
In dilute electrolytic solution, ions are
not evenly distributed.
Due to the Coulombic interaction, thecation tends to attract anions andexpulse cations. (same for anion).
As a result, the cation is surrounded byan atmosphere of anions; and the anionis surrounded by an atmosphere ofcations.
Each ion is in an atmosphere ofopposite charge, its energy is lowerthan in uniform ideal solution.
Therefore, the its chemical potential is
lower than in ideal solution.
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Debye-Huckel limiting law
= ln()= ln()= ln ln()
= + ln()
Ionic atmosphere lowers the
chemical potential. >
< 1 > =
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Debye-Huckel limiting law - Limitation
The activity coefficient of an electrolyte in water is always
smaller than 1 (i.e. < )under very dilute conditions.
Under very dilute conditions, monovalent electrolytes behavemore ideally.
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Debye-Huckel limiting law
According to the Debye-Huckel law, the mean activity coefficient
is given by the following equation :
2
1
log IzzA
)SOfor2andNafor1(e.g.
iontheofnumberschargetheareand
2
4
zz
zz
Iis the ionic strengthof the solution and it is related to thecharge numbers and the molarities of the ions :
bzbzI
22
2
1
0.509)isC,25at(for water
solventoftypeon thedependsandconstantaisWhereo
A
A
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Debye-Huckel limiting law
bzbzI
22
2
1
However, we also need to include allthe ionic species in the solution,not just the ions we are interested. Therefore, the expression above hasa general form :
i
ii bzI2
2
1
For example, for an aqueous solution containing NaCl and CaCO3,
the ionic strength of this solution should be expressed as :
23
2
23
2
44
2
1
22112
1 2222
COCaClNa
COCaClNa
bbbb
bbbbI
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Debye-Huckel limiting law - Limitation
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Debye-Huckel limiting law - Limitation
The Debye Huckel limiting law is only valid when: Very dilute conditions Complete dissociation Small ionic charge
Small ionic strength
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Debye-Huckel limiting law 21
log IzzA
Small ionic charge, Good agreement
larger ionic charge, poorer agreement
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Debye-Huckel Extended law
CI
BI
IzzA
2
1
2
1
1
log
2
1
log IzzA
Debye-Huckel extended law
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Transport phenomena in biological system
Matters enter and leave the cell through the cell membrane. Passive transport ( < )a spontaneous movement of
species down concentration gradient or membrane potentialgradient.
Active transport > movement of species againstgradients. Driven by ATP hydrolysis
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Passive transport
outside
inside
outsideinsidem
a
aRTG ln
:celltheofoutsideandinsideebetween thenergyfreeGibbsofdifferenceThe
inside
outside
outsideinside
outsideinside
outside
inside
m
AA
AAa
aa
a
aRT
G
][][
][][
1tcoefficienactivityset theWe
0ln
0
:whenfavorable
amicallythermodynisprocessTransport
The transport process follows the concentration gradient.
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Passive transport ionic
)JCorV:(unit
differencepotentialmembranetheis)molkC485.96(
constantsFaraday'is
0][
][ln
:whenfavorable
amicallythermodynisionsofprocessTransport
1-
1-
A
outside
inside
m
eNF
F
zFA
ARTG
inside
inside
outside
outside
When an ion is passing through a membrane, it needs to crossthe membrane potential difference
= arises
from differences in Coulomb repulsions on each side of themembrane.
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Active transport non-ionic/ionic
0][][ln
and][][ outsideinsideoutside
zFAARTG
AA
outside
inside
m
inside
inside
inside
outside
outside
When the transport is against the gradients, the process is
thermodynamically unfavorable.
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Active transport non-ionic/ionic
inside
inside
outside
outside
The process can be made possible bya large amount of energy input (e.g.ATP hydrolysis)
0
0][
][ln
,0][
][
lnAlthough
][
][ln
m
ATP
r
outside
inside
outside
inside
ATP
r
outside
inside
m
G
GzFA
ART
zFA
A
RT
GzFA
ARTG
When the transport is against the gradients, the process is
thermodynamically unfavorable.
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Molecular motion in liquids
The ink molecule keeps being collidedby its neighbors solvent molecules and
moves in a series of short jumps calledrandom walk.
The discussions in previous slides focused on , thespontaneity.
In the following slides, we will discuss the kinetic aspects oftransport phenomena.
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Molecular motion in liquids
The process of migration of molecules by means of therandom walkis called diffusion.
If there is an initial concentration gradient in the liquid,then the rate at which the molecules spread out isproportional to the concentration gradient:
Rate of diffusionconcentration gradient
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Ficks first law
The rate of diffusion is measured as flux,J. Flux is defined as the number of molecules passing through
an imaginary window in a given time interval, divided by thearea of the window and the duration of the interval.
dx
dcDJ
J
:asexpressedallyMathematic
intervaltimewindowofarea
indowthrough wpassingparticlesofnumber
Ficks first law of diffusion
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Ficks first law derivation
Supposed there is an imaginary window at In a given time interval , the number ofmolecules passing through this window isproportional to the area, the window thicknessand the time.
()()() = 1
2
Similarly :
()()() = 1 2
The net flux is :
1 2 1 2
= 1
2 1
2
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Ficks first law derivation
We now express the two concentrations in terms
of the concentration at the window itself, (): 12 =
12
12 =
12
: = ()
Then the flux is : 1 2 1 2 =
12
12
=
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Ficks first law
dxdcDJ
:lawfirstsFick'
)(mtcoefficiendiffusion
gradientsionconcentrat
12
sD
dx
dc
Large concentration gradientsfast diffusion
Diffusion is only spontaneous
when the concentrationgradient is negative.(diffusion is always from highconcentration regions to lowconcentration regions).
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The diffusion coefficient
Diffusion is slower in high
viscosity() liquids. 1 Diffusion is faster in higher
temperatures.
This relationship can be expressedby the Stokes-Einstein relation:
=
6
is the Boltzmann constantis the radius of solute molecule
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Viscosity
Viscosity of a liquid is a measure of its frictionalresistance.
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Viscosity
or
visocsitytheasknownalso,tcoefficienabyrelatedareblesfour variaThese
1and,
:motionrelativetheresisting(F)forcefrictionalThe
dv
dx
A
F
dx
dvAF
dx
dvAF
10P)(smkg:unitSI
P)10(cPcentipoiseor(P)Poise:Unit
1-1-
-2
The viscosity is defined as :the force of resistance per unit area which will maintain unit velocitydifference between two layers of a liquid at a unit distance from each other
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Viscosity At 20 oC
Glycerine
Non-polar organic solvents (e.g. benzene, CCl4,ether) are veryrunny. Weak intermolecular interaction (Van de Waals force)
Polar solvents (e.g. water, ethanol, Glycerine) are very viscous. Strong hydrogen bonds.
dv
dx
A
F
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Determining the Viscosity The absolute viscosity of a liquid can be found using the
Ostwald Viscometer
The viscosity is given by thePoiseullesequation:
= ()
()8()
Pressure difference Capillary radius
Time Length of the capillary Volume of liquid
= is the difference between the twomarks on the viscometer
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Determining the Viscosity
=()()
8()
=
()8
The flow rate strongly depends onthe radius ()!!!
Aorta rupture: = 120 = 15996 = 0.5 = 0.005 = 10 = 0.1
=0.004kg m-1s-1
=
(15996)(0.005)8(0.1)(0.004) = 0.010
3 = 10 L
This accounts for the tremendous blood loss in aorta rupture.
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Determining the Viscosity
=()()
8()
However, the experimentalmeasurements of the parameters
in the Poiseullesequation areconsidered to be very difficult.
One solution is to look for therelative viscosity instead.
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Determining the Viscosity
=()()
8() ()
=()()
8() ()
=
Combine (i) and (ii) :
Pis proportional to density : =
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Determining the Viscosity The relativeviscosity of a liquid can be found using the
Ostwald Viscometer
If the viscosity of one of the liquids is known, we cancalculate the absolute viscosity of the other.
capillaryhethrough tflowingliquidfor thetime:
liquidofdensity:
:viscosityrelativeThe
22
11
2
1
t
t
t
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Determining the Viscosity The absolute viscosity of a liquid can be found using the
relative viscosity if one of the liquids is water
3-
1-1-3
111
111
mkg0.1298K)(atsmkg10891.0
:viscosityabsoluteThe
water
water
waterwater
water
waterwaterwater
t
t
t
t
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The mobility of ions
speedDrift
6Friction
s
rsF
lE
zeEF
electric
The cation is pulled towards the negative electrode by a force, As the ion moves through the solvent it experiences a frictionalretarding force, .
The ion reaches the drift speed, s, when the accelerating force equals tothe decelerating force (think about a free falling object).
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The mobility of ions
The frictional force is given by:
=6
The electric force is given by:
= , =
(The sign of charge number is disregarded to avoid complications)
lE
zeEF
electric
speedDrift
6Friction
s
rsF
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The mobility of ions
When the accelerating force is balanced by the retarding force
=
6=
= 6
It follows that the drift speed is proportional to the strength of theapplied field.
= , = 6 , = 1.602 10
Where is called the mobility of the ion, the unit is m2s-1V-1.
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The mobility of ions
The mobility suggests: Higher charge ions have larger mobilities.
= 6
* The sign of charge is disregarded to avoid complications.
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The mobility of ions
The mobility suggests: Higher charge ions have larger mobilities. Ions with smaller ionic radius are more mobile
Really ?
= 6
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The mobility of ions
The deviation of the ion mobility is due to the hydrodynamic radius. When an ion migrates, it carries its hydrating water molecules with it.
Smaller ions have higher charge densities and stronger electric fields.They are more extensively hydrated and carry more H2O moleculesaround.
Therefore, smaller ions have larger effective radii.
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The mobility of ions
!!!
Proton is the smallest ion, its hydrodynamic radius should be the
largest, and therefore the smallest mobility. But in reality, its mobility is exceptionally high.
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Proton hopping
The proton on one H2O molecule
migrates to its neighbors, and soon along the chain.
Not really a diffusion becausethis is not a motion of a singleproton.
The proton at the end might notbe the same proton at thebeginning of the migration.
This is called the Grotthusmechanism.
W t h l A i
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Water channel - Aquaporin
Water molecules line up inside the channel.
Proton hopping may occur. Proton is transported and the water channel becomes a proton
pump! It will collapse the cell membrane potential.
Water channel Aquaporin
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Water channel - Aquaporin
The pore area of aquaporin contains two positively charge amino
acid residues. Expels protons
Water molecule must re-adjust its orientation when passingthrough the channel. Prevent proton hopping
Summary
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Summary
Activity of ions: =
= (+)+ , < 1
= + , = 12
Molar Gibbs free energy of transport:
=[]
[]
Diffusion rate:
= , = 6
Viscosity:
= ()()
8(),
=
Summary
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Summary
Drift velocity and Ion mobility:
= , = 6
The ion mobility depends on the effective radius. Smaller ions have highercharge density and carry more hydrating water molecules. They have largereffective radius.