07 Transport Phenomena (1)

8/12/2019 07 Transport Phenomena (1)

1/49

Transport phenomena

Johnny Wong


2/49

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.


3/49

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


4/49

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


5/49

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

= (+

)

+


6/49


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.


7/49


= (+

)

+

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.


8/49

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.


9/49


= ln()= ln()= ln ln()

= + ln()

Ionic atmosphere lowers the

chemical potential. >

< 1 > =


10/49

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.


11/49


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


12/49


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


13/49



14/49


The Debye Huckel limiting law is only valid when: Very dilute conditions Complete dissociation Small ionic charge

Small ionic strength


15/49

Debye-Huckel limiting law 21

log IzzA

Small ionic charge, Good agreement

larger ionic charge, poorer agreement


16/49

Debye-Huckel Extended law

CI

BI

IzzA

2

1

2

1

1

log

2

1

log IzzA

Debye-Huckel extended law


17/49

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


18/49

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.


19/49

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.


20/49

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.


21/49

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.


22/49

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.


23/49

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


24/49

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


25/49

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


26/49

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

=


27/49

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).


28/49

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


29/49

Viscosity

Viscosity of a liquid is a measure of its frictionalresistance.


30/49

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


31/49

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


32/49

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


33/49

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.


34/49


=()()

8()

However, the experimentalmeasurements of the parameters

in the Poiseullesequation areconsidered to be very difficult.

One solution is to look for therelative viscosity instead.


35/49


=()()

8() ()

=()()

8() ()

=

Combine (i) and (ii) :

Pis proportional to density : =


36/49

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


37/49

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


38/49

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).


39/49


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


40/49


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.


41/49


The mobility suggests: Higher charge ions have larger mobilities.

= 6

* The sign of charge is disregarded to avoid complications.


42/49


The mobility suggests: Higher charge ions have larger mobilities. Ions with smaller ionic radius are more mobile

Really ?

= 6


43/49


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.


44/49


!!!

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.


45/49

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


46/49

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


47/49

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


48/49

Summary

Activity of ions: =

= (+)+ , < 1

= + , = 12

Molar Gibbs free energy of transport:

=[]

[]

Diffusion rate:

= , = 6

Viscosity:

= ()()

8(),

=

Summary


49/49

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.

Documents

07 Transport Phenomena (1)