4/22/2008 5
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
Boyle’s or Mariotte’s law
4/22/2008 6
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.
4/22/2008 7
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
M
m
pV
tot
4/22/2008 8
1.2 Kinetic-molecular model of an ideal gas:
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
GOAL: to relate state variables (temperature, pressure)
to molecular motions. In other words, we want construct
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.
4/22/2008 9
For a single collision:
(the x-component changes sign)
A
F
p
v
m v
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
x
<F
x
>
means
"time average"
One of the keys of the kinetic-molecular model is to relate pressure
to collisions of molecules with any wall:
4/22/2008 10
Assume we have a very sparse gas (no molecule-molecule collisions!):
x
v
d
t
2
Pressure from molecular collisions proportional
to the average translational kinetic energy of molecules:
d
mv
vd
mv
t
mv
F
x
x
xx
x
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
molecule:
Time between collisions with wall:
round-trip time (depends on speed)
d
v
x
Area A
average"time"
means
2
2
22
2
3
2
1
xzyxtr
vmvvvmk
tr
k
V
N
p
3
2
macroscopic variable
microscopic
property
4/22/2008 11
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
mole
tr
K
where
is the total translational
kinetic energy of 1 mole
Compare this with the ideal gas equation (for 1 mole, n=1):
RT
pV
RTK
mole
tr
2
3
for n moles of gas→
nRTK
tr
2
3
We have arrived to a simple, but important result:
The average total translational kinetic energy of gas
is proportional to the absolute temperature
4/22/2008 12
kT
T
N
R
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
kTk
tr
2
3
So, the average translational kinetic energy of a single molecule is
which depends only on absolute temperature.
The temperature can be considered as
the measure of random motion of molecules
4/22/2008 13
§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
4/22/2008 14
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 number of molecules
in unit volume
the molecular mass
the density of the
gas at the height z
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