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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 )

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 :
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 m
3
→ R = 8.314 J/mol.ºK.
• We can expess the equation in terms of mass of gas: m
tot
= n.M
pV RT

n
pressure volume
# moles
gas constant
temperature
pV
RT

x
x x
x
p (mv )
F
t t
 
 
 
x
x
mvp 2


t
mv
F
x
x


2
 Pressure is the outward force per unit area
exerted by the gas on any wall :
 The force on a wall from gas is the time-averaged momentum
transfer due to collisions of the molecules off the walls:
 If the time between such collisions = dt, then the average force on
the wall due to this particle is:
t
F

2
)/2(
22



2
xx
v
d
Nm
F 
22
xx
x
v
V
Nm
v
Ad
Nm
A
F
p 
 Average force:
(one molecule)
 Net average force:
(N molecules)
PRESSURE:
 We can relate this to the average translational kinetic energy of each

3
2

macroscopic variable
microscopic
property
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N
A
= Avogadro’s number = 6.02 x 10
23
molecules/mole
mass of 1 mole in gam = molecular weight (e.g, O
2
:32g; H
2
:2g)
Consider 1 mole of gas:
1 mole = the amount of gas which consists the number N
A
of molecules
• Applying the equation for pressure to 1 mole of gas we have
mole
trtrA
KkNpV
3
2
3
2


N
K
k
AA
mole
tr
tr
2
3
2
3

A
N
R
k 
For a single molecule the translational kinetic energy is
where we have denoted
The constant k occures frequently in molecular physics. It is called the
Boltzmann constant. It’s value is
KJ
mol
KmolJ
k
023
23
0
/10381.1
/10022.6
./314.8

• 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 number of molecules
in unit volume
the molecular mass
the density of the
gas at the height z
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n at z = 0
•The difference in pressure between z and z + dz is given by
For the pressure
or
This formula is called
“the law of atmospheres”
This is the distribution
function on gravitational
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

probability per unit interval of v
z
.
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 v
z
and
v
z
+ Δv
z
is given by
Similarly we have for the distribution
functions for v
x
and v
y
:
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
v
x
→ v
x
+ Δv
x
; y-component of velocity lying in the interval v

number of loints in the sperical shell between the radius v and v +Δv:
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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 + Δv :
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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T
2
> T > T
1
At higher temperature the distribution curve is flatter,
and the maximum of the curve shifts to higher speed.
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2.3 Average speeds of molecules:
Using the distribution function of molecular speeds one can calculate
average values of molecular speeds. There are three types of average
values:
2.3.1 The average speed of the molecules < v >:
• Definition:
• Result:



0
)( dvvvPv
The auxilary integral:
m

The molecular kinetic energy is proportional to
the absolute temperature
What is the energy of an amount of gas ?
kTk
tr
2
3

Now we consider an ideal gas in aspects of energy
3.1 Internal energy of ideal gas:
The kinetic-molecular model states that
The internal energy of an ideal gas is of the sum of the kinetic energies
of all molecules
For a single molecule the kinetic energy consists of two parts: the
translational and rotational