Chapter XIV Kinetic-molecular theory of gases – Distribution function
From this Chapter we will study thermal properties of matter, that is
what means the terms “hot” or “cold”, what is the difference between
“heat” and “temparature”, and the laws relative to these concepts.
We will know that the thermal phenomena are determined by internal
motions of molecules inside a matter. There exists a form of energy
which is called thermal energy, or “heat”, which is the total energy of
all molecular motions, or internal energy.
GENERAL PHYSICS II
Electromagnetism
&
Thermal Physics
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Chapter XIV
Kinetic-molecular theory of gases –
Distribution functions
§1. Kinetic–molecular model of an ideal gas
§2. Distribution functions for molecules
§3. Internal energy and heat capacity of ideal gases
§4. State equation for real gases
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From this Chapter we will study thermal properties of matter, that is
what means the terms “hot” or “cold”, what is the difference between
“heat” and “temparature”, and the laws relative to these concepts.
We will know that the thermal phenomena are determined by internal
motions of molecules inside a matter. There exists a form of energy
which is called thermal energy, or “heat”, which is the total energy of
all molecular motions, or internal energy.
To find thermal laws one must connect the properties of molecular
motions (microscopic properties) with the macroscopic thermal
properties of matter (temperature, pressure,…). First we consider an
modelization of gas: “ideal gas”.
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§1. Kinetic–molecular model of an ideal gas:
1.1 Equations of state of an ideal gas:
Conditions in which an amount of matter exists are descrbied by the
following variables:
Pressure ( p )
Volume ( V )
Temperature ( T )
Amount of substance ( m or number of moles n, m = n.M)
These variables are called state variables
molar mass
There exist relationships between these variables. By experiment
measurements one could find these relationship.
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Relationship between p and V at a constant
temperarure:
The perssure of the gas is given by
where F is the force applied to the
piston.
By varying the force one can
determine how the volume of the
gas varies with the pressure.
Experiment showed that
where C is a constant This relation is known as
4/22/2008 Boyle’s or Mariotte’s law 5
Relationship between p and T while a
fixed amount of gas is confined to a closed
container which has rigid wall (that means V
is fixed).
Experiment showed that with a appropriate
temperature scale the pressure p is
proportional to T, and we can write
where A is a constant.
This relation is applicable for temperatures in
ºK (Kelvin). Temperatures in this units are
called absolute temperature.
The instrument shown in the picture can use as a type of thermometer
called constant volume gas thermometer.
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Relationship between the volume V and mass or the number of moles n:
Keeping pressure and temperature constant, the volume V is proportional
to the number of moles n.
Combining three mentioned relationships, one has a single equation :
# moles temperature
pV n RT
pressure volume gas constant
pV
This equation is called “equation of state of an ideal gas ”.
• The constant R has the same value for all gases at sufficiently high
temperature and low pressure → it called the gas constant (or ideal-gas
constant).
In SI units: p in Pa (1Pa = 1 N/m 2); V in m3 → R = 8.314 J/mol.ºK.
• We can expess the equation in terms of mass of gas: mtot = n.M
mtot
pV RT
M
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1.2 Kinetic-molecular model of an ideal gas:
GOAL: to relate state variables (temperature, pressure)
to molecular motions. In other words, we want construct
a “microscopic model of gas”:
Gas is a collection of molecules or atoms which
move around without touching much each other
Molecular velocities are random (every direction
equally likely) but there is a distribution of speeds
From the microscopic view point we have the IDEAL Gas definition:
molecules occupy only a small fraction of the volume
molecules interact so little that the energy is just the sum of the
separate energies of the molecules (i.e. no potential energy from
interactions)
Examples: The atmosphere is nearly ideal, but a gas under high
pressures and low temperatures (near liquidized state) is
far from ideal.
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One of the keys of the kinetic-molecular model is to relate pressure
to collisions of molecules with any wall:
Pressure is the outward force per unit area F
p
exerted by the gas on any wall : A
The force on a wall from gas is the time-averaged momentum
transfer due to collisions of the molecules off the walls:
x)
p (mv v
means
Fx x
"time average" t t
For a single collision: x mvx
p 2 m v
(the x-component changes sign) x
If the time between such collisions = dt, then the average force on
the wall due to this particle is:
Fx
2 mv x
Fx
t
4/22/2008 t 9
Assume we have a very sparse gas (no molecule-molecule collisions!):
2d
Time between collisions with wall: v
t
x
round-trip time (depends on speed)
Area A
2
Average force: 2mv 2mv mv
(one molecule) Fx x x
x
(2d / vx ) d
t vx
d
Nm
Net average force: Fx vx2
(N molecules) d
Fx Nm 2 Nm 2
PRESSURE: p vx vx means
A Ad V " time average"
We can relate this to the average translational kinetic energy of each
molecule:
1
2
2 2 3
ktr m vx vy vz2 m vx
2
2
microscopic
property
Pressure from molecular collisions proportional 2N
to the average translational kinetic energy of molecules: p ktr
3V
macroscopic variable
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Consider 1 mole of gas:
1 mole = the amount of gas which consists the number NA of molecules
NA = Avogadro’s number = 6.02 x 1023 molecules/mole
mass of 1 mole in gam = molecular weight (e.g, O2:32g; H2 :2g)
• Applying the equation for pressure to 1 mole of gas we have
2 2 mole mole
pV NA ktr Ktr where Ktr is the total translational
3 3 kinetic energy of 1 mole
Compare this with the ideal gas equation (for 1 mole, n=1):
pV RT 3
RT
3
for n moles of gas→ K tr nRT
mole
K tr
2 2
We have arrived to a simple, but important result:
The average total translational kinetic energy of gas
is proportional to the absolute temperature
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For a single molecule the translational kinetic energy is
mole
Ktr 3 R 3 R
ktr T kT where we have denoted k
NA 2 NA 2 NA
The constant k occures frequently in molecular physics. It is called the
Boltzmann constant. It’s value is
8 .314 J / mol .0 K
k .381 J / 0K
1 10 23
6.022 23 / mol
10
So, the average translational kinetic energy of a single molecule is
3
ktr kT
2
which depends only on absolute temperature.
The temperature can be considered as
the measure of random motion of molecules
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§2. Distribution functions for molecules:
In the view point of a microscopic theory an amount of ideal gas is
an ensemble of molecules, in which
• The number of molecules is very large
• Every molecule has an independent motion
So, what we can know about them:
• The average properties: average kinetic energy, average speed,…
• Distribution of molecules according to any properties, for example:
• How many per cent, or probability of molecules having the speed v ?
• Probability of molecules at a height z in a gravitational field?
Distribution of molecules is given by distribution functions.
We will consider two such distribution functions:
• Distribution on the height (or potential energy) in a gravitational field
• Distribution on the speed (or kinetic energy) of molecules
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2.1 Distribution of molecules in a gravitational field:
• Consider an ideal gas in a uniform
gravitational fields, for example in the
earth’s gravity.
• Assume that the temperature T is the
same everywhere.
The equation of state
gives the pressure as a function of height z :
the density of the
gas at the height z
the number of molecules
in unit volume the molecular mass
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•The difference in pressure between z and z + dz is given by
n at z = 0
or
For the pressure
This is the distribution
This formula is called function on gravitational
“the law of atmospheres” potential energy of molecules
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2.2 Distribution of molecular speeds in an ideal gas:
• Boltzmann pointed out that the decrease in
molecular density with height in a uniform
gravitational field can be understood in terms of
the distribution of the velocities of molecules
at lower levels in the gas:
• Molecules leaving the level z = 0 with the
velocities less than vz in the equation
will fail to reach the height z.
Similarly,
The number of molecules
The difference Δ between the
n
per unit volume which have = number per unit volume at
the z-component of velocity
height z and that at height z+Δz
between vz and vz + Δ z v
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Differentiating the distribution function with respect to z we obtain
Since the number of molecules per unit volume at
temperature T with z-component of velocity must be proportional with
the factor
Boltzmann reasoned that this proportional relation should be the same
where or not the gas is in a gravitational field, and therefore we can write
where A is a constant, Pz (vz) is the
probability per unit interval of vz .
The constant A is determined from the condition
Making the replacement and applying
the formula
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The fraction of molecules with z-component of velocity between vz and
vz + Δ z is given by
v
Similarly we have for the distribution
functions for vx and vy :
We now can write the expression for the fraction of molecules in an ideal
gas at temperature T with x-component of velocity lying in the interval
vx → vx + Δ x ; y-component of velocity lying in the interval vy → vy + Δy ;
v v
z-component of velocity lying in the interval vz → vz + Δ z
v
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The function
is known as the Maxwell-Boltzmann
velocity distribution function.
In a velocity diagram, the velocity of
a single molecule is represented by
a point having coordinates (vx, vy, vz )
The number of molecules having velocities
in the “volume” element is
(N: the total number of molecules
of the whole system)
The number of molecules with speeds between v and v + Δ is thev
number of loints in the sperical shell between the radius v and v +Δ :
v
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Deviding by N we have the fraction of molecules in a gas at temperature T
with speeds between v and v + Δ :
v
The function P(v) =
gives the Maxwell-Boltzmann distribution function of molecular speeds.
Remark that the Maxwell-Boltzmann distribution function depends
on temperature. This dependence is shown in the picture.
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