Date post: | 31-Dec-2015 |
Category: |
Documents |
Upload: | fredericka-mclaughlin |
View: | 31 times |
Download: | 1 times |
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