1
Physical Chemistry
Lecture 1
Distributions and Transport
Processes
Macroscopic characterization
with distribution functions
A macroscopic system contains
a LARGE number of particles,
not all of which have the same
set of microscopic properties
To give a macroscopic system’s
state requires a
distribution
function
,
F
, that describes the
“amounts” of properties, either
macroscopic or microscopic, as
a function of independent
variables
The equilibrium state is
described by a unique
distribution function for each
property
A Gaussian distribution function,
, of
the two coordinates
x
and
y
Particles in a Box
Example distribution
function for particles in a
box, showing two regions
Left side has more particles
than the right
This particular distribution
can be considered bimodal
Real distribution functions
are more complex functions
of position
Equilibrium particle
distribution under no
outside constraints
Particle density is uniform
(i.e. a constant, independent
of position)
Two equilibrium distributions
Two other simple
equilibrium distributions
Thermal equilibrium
No external constraints
Temperature is
independent of position
Mechanical equilibrium
No external constraints
Pressure is independent of
position
Not all equilibrium
distributions are
constants, independent of
the variable
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Boltzmann’s distribution: the
speed distribution at equilibrium
An equilibrium distribution
that depends on the
variable
Boltzmann distribution of
speeds
Only kinetic energy
Compromise between
minimal energy and
maximal entropy
Normalized
Depends on speed (
v
),
mass (
m
), and temperature
(
T
)
kT
mv
v
kT
m
v
F
D
2
exp
2
4
)
(
2
2
2
/
3
3
1
)
(
1
)
(
1
0
3
x
x
D
D
dv
v
F
dv
v
F
kT
mv
kT
m
v
F
x
x
D
2
exp
2
)
(
2
1
3D speed distribution functions at three different
temperatures
Calculating average molecular
properties
Averages are integrals of
properties weighted by the
distribution function
Integral must be carried out
over all possible values of
the independent variable
Examples
Average speed in one
dimension
Average speed in three
dimensions
Average temperature at
thermal equilibrium
For a constant distribution,
the average is the single
value of the temperature,
T
0
2
exp
2
)
(
2
1
1
,
x
x
x
x
x
D
x
D
ave
dv
kT
mv
v
m
kT
dv
v
F
v
v
0
3
,
3
)
(
)
(
dv
v
F
v
f
f
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 Fall '10
 Staff
 Physical chemistry, Thermodynamics, pH, Statistical Mechanics, Distribution function, Fick

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