• Molecular motion in liquids 7. The mobilities of ions
8. Conductivities and ion-ion interactions
• Diffusion
9. The thermodynamic view 10. The diffusion equation 11. Diffusion probability 12. The statistical view
• Ch. 21 Molecules in motion Molecular motion in gases Molecular motion in liquids
Diffusion: migration of matter down a concentration gradient
Lecture 5
• Consider a substance which moves from a location where its chemical potential is to a location where its chemical potential
+ d.
• At constant T and p, the maximum non-expansion work that can be done per mole of the substance is .
• In a system in which the chemical potential depends on the position x,
d dw
x dx d
dw
T p,
• Work can be always be expressed in terms of an opposing force (F),
dx
dw F
x dx d
dw
T p,
dw Fdx
• By comparing two expressions, the thermodynamic force is written as
T
x p,
F
• There is not necessarily a real force pushing the particles down the slope of the chemical potential.
• The force may represent the spontaneous tendency of the molecules to disperse as a consequence of the 2nd law.
a
o RT
ln
• In a solution, the chemical potential is expressed as
where a is the activity of the solute.
• If the solution is not uniform, the activity depends on the position.
T T p
p o
T
p x
RT a x
a RT
x , , ,
ln
ln
F
T
x p,
F
• If the solution is ideal, a may be replaced by the molar concentration (c).
T
x p
RT c
,
ln
F or
T
x p
c c
RT
,
F
for ideal solutions.
for real solutions.
• Suppose that c decays exponentially along the length of a tubular container. The c falls to half its value in 10 cm.
where the is the decay constant.
• Calculate the thermodynamic force on the solute at 25 oC.
RT
c e RTc x
e c c
RT x
c c
RT
x o
T p x o T
p
, ,
F
x o
e c
c
cm
10
coe co
2 1
cm 10 2
ln 1
2 ln
cm
10
kN/mol m 17
1 . 0
K) (298 mol)
J/K 31 . 8 ( 2
ln
F (1J 1Nm)
T
x p
c c RT
,
F
• The Fick’s 1st law of diffusion, deduced from the kinetic model of gases, can be also applied to the diffusion of species in
condensed phases.
• Consider the flux of diffusing particles by a thermodynamic force arising from a concentration gradient.
• When the thermodynamic force (F) is matched by the viscous drag, the particles reach a steady drift speed (s) in a condensed phase.
dz D d J(matter) N
T
x p
c c
RT
,
F
• By dividing both sides by NA,
dx D d J(matter) N
dx D d J c
• The flux can be also expressed as:
sc J
area time
particles of
number
: ) matter
(
J
volume
moles of
number time
distance area
time
moles of
number
time area moles ofnumber
:
J
• By combining two expressions of J,
dx D d sc c
dx d c
s D c
T
x p
c c
RT
,
F
RT
s DF
RT s DF
• For an ion in solution under the field E, the force experienced by the ion is .
• So, the force experienced by one mole of ions is:
E ze
E E
F N
Aze zF
• However, the drift speed of ions is also expressed with the
mobility as:
s u E
RT D s zFE
• The drift speed of ions can be expressed as:
RT u zFD
Therefore,
zF
D uRT Einstein Relation
zF D uRT
• Since the ionic molar conductivity () and its mobility (u) are related as:
zuF
RT DF zuF z
2
2
Using the Einstein relation, for each type of ion
• Therefore, the limiting molar conductivity of a electrolyte is:
RT F D v z
RT F D v z
v
o v
m
2 2
2 2
RT D F
z v D
z
o v
m
2 2
2
Nernst-Einstein Equation:
Determination of D± from conductivity.
zF D uRT
• Since the mobility (u) is expressed as , where f is the
frictional coefficient, f
u ze
f kT e
fN eRT fF
eRT zfF
zeRT zF
D uRT
A
f
D kT Stokes-Einstein Equation
• If the frictional coefficient (f) is described by the Stokes relation ( ), f 6a
a D kT
6
f D kT
a D kT
6
• The Stokes-Einstein equation also applies to neutral molecules, because of no reference to the charge of the diffusing species.
• The special case of the Stokes-Einstein equation is used to estimate the diffusion coefficient of neutral molecules in solution by measuring viscosity.
• Using the mobility (u) of sulfate ion (SO4-) in aqueous solution, estimate the diffusion coefficient (D), the limiting ionic molar
conductivity (), and the hydrodynamic radius (a) of the sulfate ion.
From the table, uSO24 8.29108 m2/Vs
/s m 10
1 .
1 9 2
zF D uRT
From the Einstein relation,
From the , zuF zuF 16mSm2/mol
Using 0.891 cP (= 8.91 10-4 kg/ms) for the viscosity of water,
a D kT
6 220pm
6
D
a kT
• It is empirically observed that the product m is approximately constant for the same ions in different solvents. Walden’s rule
m D
1
and D
Therefore, m 1
• However, there are numerous exceptions of the rule due to the role of solvation.
• Different solvents solvate the same ion to different extents, so both hydrodynamic radius and the viscosity change with the
solvent.
a D kT
6
RT D F z v D z
o v
m
2 2
2
• Next Reading:
8th Ed: p.776 ~ 782 9th Ed: p.770 ~ 774
