Chemistry 232

Transport Properties

Definitions Transport property.

The ability of a substance to transport matter, energy, or some other property along a gradient.

Examples. Diffusion - transport of matter along a

concentration gradient. Thermal conductivity - transport of

thermal energy along a temperature gradient.

Transport Properties Defined Examples (cont’d).

Viscosity - transport of linear momentum along a velocity gradient.

Electrical conductivity - transport of charge along a potential gradient.

Collisions With Walls and Surfaces Rate at which molecules collide

with a wall of area A

2

1

i

Aw

RTM2

pNZ

Effusion Rate at which molecules pass

through a small hole of area Ao, r

2

1

i

oAow

RTM2

ApNAZ r

Effusion (Cont’d) Effusion.

A gas under pressure goes (escapes) from one compartment of a container to another by passing through a small opening.

Effusion

The Effusion Equation Graham’s Law - estimate the ratio

of the effusion rates for two different gases.

Effusion rate of gas 1 r1.

2

1

1

oAo1w1

RTM2

ApNAZ r

,

Effusion Equation (Cont’d) Effusion rate of gas 2 r2.

2

1

2

oAo2w2

RTM2

ApNAZ r

,

Effusion Ratio Ratio of effusion rates.

2

1

2

1

2

1

1

oA

2

1

2

oA

1

2

MM

RTM2

ApN

RTM2

ApN

r

r

Migration Down Gradients Rate of migration of a property is

measured by a flux J. Flux (J) - the quantity of that property

passing through a unit area/unit time.

gradient of magnitude J

Transport Properties in an Ideal Gas

Transport of matter.

0z

dmatter dz

dNDJ

Transport of momentum.

Transport of energy.

0z

xmomentum dt

dvJ

0zTenergy dz

dTJ

D - diffusion coefficient.

=viscosity coefficient.

T -thermal conductivity coefficient.

Diffusion Consider the following system.

Z=0 +Z-Z

z

Nd

Nd(-)

Nd(z=0)

Nd(+)

Number Densities and Fluxes The number densities and the fluxes

of the molecules are proportional to the positions of the molecules.

0z

ddd dz

dNz0zNzN

v zN4

1zJ d

The Net Flux The net (or total) flux is the sum of

the J(LR) and the J(RL).

vdz

dN

2

1

LRJRLJJ

0z

d

Total

The Diffusion Coefficient To a first approximation.

0z

dmatter dz

dNDJ

vdz

dN

2

1J

0z

dTotal

v2

1D

The Complication of Long Trajectories Not all molecules will reach the

imaginary wall at z=0!

Ao

Collision 2/3 of all molecules will make it to the wall in a given time interval t.

The Final Equation Taking into account of the number of

molecules that do not reach the wall.

v3

1

v2

1

3

2D

Thermal Conductivity Consider the following system.

Z=0 +Z-Z

Number Densities and Fluxes Assume each molecule carries an

average energy, = kBT. =3/2 for a monatomic gas. =5/2 for a diatomic gas, etc.

z

(-)

(z=0)

(+)

0zB dz

dTzTkz

The Net Flux The net (or total) flux is the sum of

the J(LR) and the J(RL).

0zdB

Total

dZ

dz

dT Nk v

2

1

LRJRLJJ

zN v4

1J

The Thermal Conductivity Coefficient To a first approximation.

0zTenergy dz

dTJ

0zdBTotal dz

dT Nk v

2

1J

Nk v 2

1dBT

The Final Equation Taking into account of the number of

molecules that do not reach the wall.

Nk v 3

1

Nk v 2

1

3

2

dB

dBT

Viscosity Consider the following system.

Z=0 +Z-Z

Direction of flow

Number Densities and Fluxes Molecules traveling L R transport

linear momentum (mvx()) to the new layer at z = 0!

z

mvx

mvx(-)

mvx(z=0)

mvx(+)

0z

xxx dz

dvm)0z(mvzmv

The Net Flux The net (or total) flux is again the

sum of the J(LR) and the J(RL).

0z

xd

Total

xdZ

dz

dv mNv

2

1

LRJRLJJ

zmvN v4

1J

The Viscosity Coefficient To a first approximation.

0z

xmomentum dz

dvJ

m N v 2

1d

0z

xdTotal dz

dv mNv

2

1J

The Final Equation Taking into account of the number of

molecules that do not reach the wall.

m N v 3

1

m N v 2

1

3

2

d

d

Viscosities Using Poiseuille’s Law Poiseuille’s law Relates the rate of

volume flow in a tube of length l to Pressure differential

across the tube Viscosity of the fluid Radius of the tube

o

422

21

p l 16

rpp

dt

dV

Transport in Condensed Phases Discussions of transport properties

have taken place without including a potential energy term.

Condensed phases - the potential energy contribution is important.

Viscosities in Liquids Liquid layers flowing past one

another experience significant attractive interactions.

Z=0 +Z-Z

Direction of flow

The Viscosity Equation For liquid systems

RT

E visa

Ae

*,

E*a,vix= activation energy for viscous flow

A = pre-exponential factor

Conductivities in Electrolyte Solutions Fundamental measurement of the

mobilities of ions in solutions electrical resistance of solution.

Experimentally - measure AC resistance.

Conductance - G = 1/R. R = AC resistance of solution.

Resistance Measurements Resistance of sample depends on

its length and cross-sectional area

A

lR = resistivity of the solution.

1

RA

l = conductivity of the solution.

Units of conductivity = S/m = 1/( m)

Charge Transport by Ions Interpreting charge transport.

Amount of charge transported by ions. The speed with which individual ions

move. The moving ions reach a terminal

speed (drift speed). Force of acceleration due to potential

gradient balances out frictional retarding force.

Drift Speed Consider the following system.

Length = l

1 2

+

+

+

+

+

-

-

-

--

+ +

+ --

-

Forces on Ions Accelerating force

Due to electric field, Ef = (2 - 1) / l

Retarding force Due to frictional resistance, F`= f s

S = drift speed F = frictional factor - estimated from

stokes law

The Drift Speed The drift speed is written as follows

Jo

fJfJ

a6

eEz

f

eEzs

zJ = charge of iono = solvent viscositye = electronic charge =1.602 x 10-19 CaJ = solvated radius of ion

In water, aJ = hydrodynamic radius.

Connection Between Mobility and Conductivity Consider the following system.

Z=0 +Z-Z

+

+

+

+

+

-

-

-

--

+ +

+ --

- d+=s+t

d-=s-t

Ion Fluxes For the cations J+ = + cJ NA s+

+= Number of cations cJ = electrolyte concentration S+ = Cation drift speed

Ion Flux (Cont’d) Flux of anions J- = - cJ NA s-

- = Number of cations cJ = electrolyte concentration S- = anion drift speed

Ion Flux and Charge Flux Total ion flux

Jion = J+ + J-= S cJ NA

Note = + + -

Total charge fluxJcharge = Jion z e

= (S cJ NA) z e

= ( cJ NA) z e u Ef

The Conductivity Equation. Ohm’s law

I = Jcharge A

The conductivity is related to the mobility as follows

Fczu JF = Faraday’s constant = 96486 C/mole

Measurement of Conductivity Problem - accurate measurements of

conductivity require a knowledge of l/A. Solution - compare the resistance of the

solution of interest with respect to a standard solution in the same cell.

RA

l

AR

l*

*

The Cell Constant The cell constant, C*

cell = * R* * - literature value for conductivity of

standard solution. R* - measured resistance of standard

solution. Conductivity - = C*

cell R Standard solutions - KCl (aq) of

various concentrations!

Molar Conductivities Molar conductivity

M = 1000 / cJ

Note c in mole/l Molar conductivity - extensive

property Two cases

Strong electrolytes Weak electrolytes

Ionic Contributions The molar conductivity can be

assumed to be due to the mobilities of the individual ions.

Fuz

Fuz

Molar Conductivities (Cont’d) Molar conductivities as a function

of electrolyte concentration.

m

C1/2

Strong electrolytes

Weak electrolytes

Strong Electrolyte Case Kohlrausch’s law

21o

mm Aco

m = molar conductivity of the electrolyte at infinite dilution

A = molar conductivity slope - depends on electrolyte type.

Weak Electrolytes The Ostwald dilution law.

2om

moomm K

c11

K = equilibrium constant for dissociation reaction in solution.

Law of Independent Migration Attributed to Kohlrausch. Ions move independently of one

another in dilute enough solution.ooo

m

Table of o values for ions in textbook.

Conductivity and Ion Diffusion Connection between the mobility

and conductivities of ions.

2J2

omo

JzF

RTD

J

BoJ a6

TkD

DoJ = ionic diffusion coefficient at infinite dilution.

Ionic Diffusion (Cont’d) For an electrolyte.

oooJ

J

DDD

Essentially, a restatement of the law of independent migration.

ONLY VALID NEAR INFINITE DOLUTION.

Transport Numbers Fraction of charge carried by the

ions – transport numbers.

m

t

m

t

t+ = fraction of charge carried by cations.

t- = fraction of charge carried by anions.

Transport Numbers and Mobilities Transport numbers can also be

determined from the ionic mobilities.

uu

ut

u+ = cation mobility.

u- = anion mobility.

uu

ut