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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.
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.
J  magnitude of gradient
Transport Properties in an Ideal
Gas

Transport of matter.
J matter

 dN d 
 D 

 dz z 0
D - diffusion coefficient.

Transport of energy.
J energy
 dT 
 T 

 dz z 0
T -thermal conductivity
coefficient.
Transport of momentum.
J momentum
 dv x 
  

 dt  z 0
=viscosity coefficient.
Diffusion

Consider the following system.
Nd(-)
Nd
-Z
Z=0
Nd(z=0)
+Z
Nd(+)
z
Number Densities and Fluxes

The number densities and the fluxes of
the molecules are proportional to the
positions of the molecules.
 dN d 
N d z   N d z  0   z 

 dz z 0
1
J z   N d z  v
4
The Net Flux

The net (or total) flux is the sum of the
J(LR) and the J(RL).
J Total  J L  R   J R  L 
1

2
 dN d 

 v 
 dz  z 0
The Diffusion Coefficient

To a first approximation.
 dN d 
J matter  D 

 dz z 0
1  dN d 
J Total   
v 

2  dz  z 0
1
D  v
2
The Complication of Long
Trajectories

Not all molecules will reach the
imaginary wall at z=0!
Collision
Ao
 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.
2 1

D   v 
3 2

1
 v
3
Thermal Conductivity

Consider the following system.
-Z
Z=0
+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.

dT  
 z   k B T  z   
 dz z 0 


(-)
(z=0)
(+)
z
The Net Flux

The net (or total) flux is the sum of the
J(LR) and the J(RL).
J Total
1
J Z  v N d  z 
4
 J L  R   J R  L 
1
 dT 
   v kB N d 

2
 dz  z 0
The Thermal Conductivity
Coefficient

To a first approximation.
 dT 
J energy  T 

 dz z 0
J Total
1
 dT 
   v k B N d 

2
 dz  z 0
1
T   v k B N d 
2
The Final Equation

Taking into account of the number of
molecules that do not reach the wall.
21
T    v
32
1
  kB v
3

kB Nd  

Nd 
Viscosity

Consider the following system.
Direction of flow
-Z
Z=0
+Z
Number Densities and Fluxes

Molecules traveling L  R transport
linear momentum (mvx()) to the new
layer at z = 0!
 dv x 
mv x z   mv x (z  0 )  m 

 dz z 0
mvx(-)
mvx(z=0)
mvx
mvx(+)
z
The Net Flux

The net (or total) flux is again the sum of
the J(LR) and the J(RL).
1
J Z  v N d mv x z 
4
J Total  J L  R   J R  L 
1
 dv x 

v mN d  

2
 dz  z 0
The Viscosity Coefficient

To a first approximation.
 dv x 
J momentum   

 dz  z 0
J Total
1
 dv x 

v mN d  

2
 dz  z 0
1

v Nd  m
2
The Final Equation

Taking into account of the number of
molecules that do not reach the wall.
2 1

   v Nd  m 
3 2

1

v Nd  m
3
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
dV p  p r

dt
16  l po
2
1
2
2
4
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.
Direction of flow
-Z
Z=0 +Z
The Viscosity Equation

For liquid systems
  Ae
E a*,vis
RT
E*a,vis= 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
l
R 
A
l
1


RA 
 = resistivity of the solution.
 = conductivity of the solution.
Units of conductivity = S/m = 1/( m)
Charge Transport by Ions

Interpreting charge transport.

The moving ions reach a terminal speed
(drift speed).
• Amount of charge transported by ions.
• The speed with which individual ions move.
• Force of acceleration due to potential gradient
balances out frictional retarding force.
Drift Speed

Consider the following system.
+
-
1
-
+
+
- +
+
+
+
+
Length = l
2
Forces on Ions

Accelerating force

Retarding force
• Due to electric field, Ef = (2 - 1) / l
• 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
z J eE f
z J eE f
s 

f
6 o a J
zJ = charge of ion
o = solvent viscosity
e = electronic charge
=1.602 x 10-19 C
aJ = solvated radius of ion
In water, aJ = hydrodynamic radius.
Connection Between Mobility
and Conductivity

Consider the following system.
+
-
-
+
+
- +
+
+
+
+
-
d+=s+t
-Z
Z=0
d-=s-t
+Z
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  c J NA
Note   = + + -

Total charge flux
Jcharge = 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
  zuc J F
F = 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.
l

RA
 
*
l
R A
*
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.
  z u F
  z  u F
Molar Conductivities (Cont’d)

Molar conductivities as a function of
electrolyte concentration.
m
Strong electrolytes
Weak electrolytes
C1/2
Strong Electrolyte Case

Kohlrausch’s law
m    Ac
o
m
1
2
om = molar conductivity of the
electrolyte at infinite dilution
A = molar conductivity slope - depends on
electrolyte type.
Weak Electrolytes

The Ostwald dilution law.
c o m
1
1
 o 
2
o
m
m K m 
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.
        
o
m
o
o
Table of o values for ions in textbook.
Conductivity and Ion Diffusion

Connection between the mobility and
conductivities of ions.
k BT
D 
6 a J
o
J
D
o
J
RT
 2
F z J
o
m

2
DoJ = ionic diffusion coefficient at infinite dilution.
Ionic Diffusion (Cont’d)

For an electrolyte.
J


 o  o
o
DJ
D D
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.
  
t 
m
  
t 
m
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.
t 
u
u u
u+ = cation
mobility.
t 
u
u u
u- = anion
mobility.