Today’s Lecture
Ideal Gas
• Ideal Gas Law
• Kinetic Theory of an Ideal Gas
• Mean Free Path
Gas
Gas is matter in a rarefied state.
The molecules are moving freely most of the time, and only once
in a while suffer short term collisions.
The macroscopic state of a gas in thermodynamic equilibrium is
determined completely by its temperature, pressure, and volume.
The ideal gas law
P is the pressure, V is the volume, T is the absolute temperature…
N is the total number of molecules in the gas and
kB is Boltzmann’s constant, kB = 1.38×10-23 J/K
Ideal Gas Law
Gas in a cylinder under a piston
Pressure, P, is given by
P = Patm + mg / A
Where m is the total mass of the
piston and the lead and A is the
area of the piston.
We can:
• add or remove the lead shots to
change the pressure of the gas;
• tune the temperature of the
thermal reservoir.
http://phet.colorado.edu/web-pages/simulations-base.html
Ideal Gas Law
Doubling the temperature,
number of molecules, pressure?
Keeping the volume and the
number of particles constant,
but doubling the temperature?
N is normally very big, while kB is a very
small number… kB=1.38X10-23J/K
N = nNA
NA = 6.022X1023 – Avogadro number, number
of molecules in 1 mole of a substance;
n is the number of moles in the gas.
R = nNA =8.31 J/mol⋅K
Universal Gas Constant
Ideal Gas - example
What is the volume occupied by 1.00 mol of an ideal gas
at standard temperature and pressure (STP), where T=0oC
and P=101.3kPa=1.013x105Pa (1atm)?
The ideal gas law
An Avogadro's number of gas molecules occupies 22.4L at STP!
Fun with the Ideal Gas Law??
A 3L flask is initially open in a room at 1atm and 20oC. The flask is
closed and immersed in boiling water. After the flask has reached
equilibrium it is opened and the air allowed to escape. It is then closed
and cooled back to room temperature.
(a) What is the maximum pressure reached in the flask?
(b) How many moles escape when the flask is opened?
(c) What is the final pressure in the flask?
A Non-Ideal Gas Law
What happens if we reduce T to zero.
Is volume of the gas, V, going to become zero?
Not necessarily, since it maybe that the
pressure, P, becomes zero at T = 0.
First let’s set P ≠ 0. Then what?
By the ideal gas law we would have V = 0, which cannot be true.
We can correct for it by a term equal to the total volume of the gas
molecules when totally compressed (condensed) nb, where n is the
number of moles.
P(V - nb) = nRT
Now at T = 0 and P ≠ 0 we have V = nb.
Conditions for Which the Ideal Gas Law Breaks Down?
1. The molecules occupy a significant fraction of the volume.
Collisions are more frequent.
There is less volume available for molecular motion.
2. There are long range attractive – Van der Waals forces between the
molecules, which become more important as the density grows.
Real gas – Van der Waals equation.
Introduces corrections to the
ideal gas law to take into account
some of these effects.
n2a
2
V
V − nb
2
na
( P + 2 )(V − nb) = nRT
V
Attractive force between pairs of molecules. Goes
as (n/V)2 square of the concentration.
Less volume available for motion, because of n moles
of the gas.
Real Gas – Van der Waals Equation.
For nitrogen a =.14Pa.m6/mol2 and b = 3.9x10-5m3/mol. If 1.0 mole of
nitrogen is confined to 2.00L and is at P=10atm what is Tideal and TVdW?
Under these conditions the temperature only changes by ~1%.
Kinetic Theory of an Ideal Gas
Kinetic energy is the only form
of molecular energy that is
important and it is preserved
in the collision events.
Implicit in these assumptions is that
the velocity and density distribution of
the molecules is:
homogeneous and isotropic !
Collisions of Gas Molecules
with a Wall
L
Freaction
Small volume, V=LA, adjacent to
wall where L is less than the
mean free path, λ.
As a result of a collision with the wall
the momentum of a molecule changes by
This change in momentum is due to an
impulsive force of the wall on the molecule.
The reaction force of the molecule on the
wall is the negative of this.
Collisions of Gas Molecules with a Wall
Assume that in a time Δt every molecule (atom) in the original
volume, V=LA, within the range of velocities
will collide with the wall.
This means that Δt is given by:
The “reaction force” of a molecule on the wall is the negative
of the average rate of change in the momentum of gas
molecules in the volume V that collide with the wall in the time
Δt. This force of the molecules on the wall with a velocity close
to vx is:
Collisions of Gas Molecules with a Wall
The total force on the wall is the sum of the average rate of
momentum change for all molecules in the volume V=LA
that collide with the wall:
Here we have divided by 2 since
only ½ of the molecules in our
volume have a positive velocity
toward the wall.
L
Collisions of Gas Molecules with a Wall
We do the sum by noting that the total number of molecules in
the volume V is (N/ Vtot)V=NLA/Vtot, where N and Vtot are the
total number of molecules and the volume under consideration.
Remembering Pascal’s law
dividing by A yields the pressure
everywhere.
L
Here, for notational convenience, we
have dropped the tot subscript.
Kinetic Theory of an Ideal Gas
L
Velocity squared of a molecule:
The average of a sum is equal to
the sum of averages.
All the directions of motion
(x, y, z) are equally probable.
Remember homogeneous
and isotropic!
v = vx + v y + vz
2
2
2
2
Kinetic Theory of an Ideal Gas
Combing these results yields:
From the ideal gas law
The average (translational) kinetic energy per molecule is
See my notes
Physical meaning of the absolute
temperature is a measure of the
average kinetic energy of a molecule.
The rms speed of a
molecule – thermal speed:
Maxwell-Boltzmann Distribution
Maxwell-Boltzmann distribution for
nitrogen gas at temperatures 80K and
300K. Note that the thermal speed is not
the same as the most probable speed.
The areas under both curves and equal,
hence they correspond to the same
number of total molecules. Here v is the
speed (not velocity) of the gas molecules
(atoms).
N ( v ) Δv
Number of molecules having speeds in an interval of width Δv
around v. It is proportional to Δv, the total number of molecules,
N, and to the height of the distribution curve.
Kinetic Theory of an Ideal Gas
We already found that
Could we have derived the ideal gas law? YES!
Consider again the Maxwell Boltzmann distribution of velocities,
which is consistent with “homogeneous and isotropic” requirement.
To do this integral we merely have to integrate by parts a single time.
Kinetic Theory of an Ideal Gas
Performing the integration we find
Since
We have the ideal gas law:
Elastic collisions with the wall and the Maxwell Boltzmann
velocity distribution is all we need to derive Ideal the Gas Law!
Kinetic Theory - Solar Corona
The solar corona is a hot gas (2 x 106 K) surrounding the Sun.
The pressure is .03Pa. Find the density of particles and
velocity of the Helium ions, compare to Earth’s atmosphere.
For the Sun’s corona:
For the Earth’s atmosphere:
More Examples of Kinetic Theory
A student dormitory room at 1atm and 20oC measures 3.0m x 3.5m x 2.6m.
(a) How many air molecules are in the room?
(b) What is the total translational kinetic energy of all of these molecules?
(c) What is the kinetic energy of the student’s 1200kg car traveling 90km/hr ?
Collisions Between Molecules
In a time dt a molecule of radius r will collide
with any other molecule within a cylindrical
volume of radius 2r and length vdt.
There are N molecules in a volume V, so the
number of molecules with their centers
inside this differential volume is:
The rate of collisions is found by dividing by dt:
This result only applies when all of the other
molecules are stationary. Taking into account that
all of the molecules are moving leads to the result:
Mean Free Path/Time
In a time dt a molecule of radius r will collide
with any other molecule within a cylindrical
volume of radius 2r and length vdt. The rate of
collisions is:
The mean time between collisions, tmean, is the reciprocal of
dN/dt. Taking into account the ideal gas law yields:
The mean free path between
collisions is λ=vtmean, or
Mean Free Path/Time
(a) Estimate the mean free path, λ, of an air
molecule at 27oC and 105Pa. Assume that the
molecules are spheres with r = 2x10-10m.
The mean free path is:
(b) Estimate the mean free time for a nitrogen molecule which has a thermal
speed of v=517m/s at 300K.
The mean free path seems small but it is almost 300 times the molecular radius.
Mean Free Path/Time
(a) Now estimate the mean free path, λ, of an
He atom in the solar corona at 2x106K and
.03Pa. Again assume that the atoms are
spheres with r = 2x10-10m.
The mean free path is:
(b) Estimate the mean free time for this He atom which has a thermal speed
of v=1.1x105m/s at 2x106K.
The mean free path is LARGE in this rarified gas.
Mean Free Path/Time
What about the mean free path of a point
particle (e.g. an electron) in a gas?
For the analysis of a point particle, the radius
of the cylinder is reduced to r versus 2r. This
is simply a statement that a point particle does
not take up as much space (doh!).
For this case the mean free path is:
(c) Find the mean free path of an electron in the Sun’s corona.
The mean free path is simply increased by a factor of 4!
Summary of Kinetic Theory
Physical meaning of the absolute
temperature is a measure of the
average kinetic energy of a molecule.
From this we can express the
pressure of an ideal gas as
Maxwell Boltzmann Distribution
Mean free path, λ