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An Introductory Physics Course with Peter Eyland
Lecture 21 (Kinetic Theory)
In this lecture the following are introduced:
• Microscopic views of matter
• Einstein and Brownian motion
• Kinetic Theory assumptions
• The microscopic picture of pressure
• Internal energy
• Molar mass and the Atomic Mass Unit
• The microscopic picture of temperature
• The MaxwellBoltzmann distribution for molecular velocities
Microscopic views of matter
Microscopic or "atomistic" pictures of matter started well before Lucretius (100 BCE), who wrote:
"However solid objects seem,
They yet are formed of matter mixed with void …
in which they're set, and where they're moved around...
Bodies, again, are partly primal germs of things, and partly unions deriving from the primal germs …
our eyes no primal germs perceive."
Much later came Daniel Bernoulli, Rudolf Clausius, James C. Maxwell and Ludwig Boltzmann.
Ludwig Boltzmann
Born: 20 Feb 1844 in Vienna, Austria. 
Boltzmann began his professional life when logical positivism gained ascendancy. Logical positivists held that
"metaphysical speculation is nonsensical; that logical and mathematical propositions are tautological; and that moral and value statements are merely emotive." The "arrow of time" is reversible in Newtonian mechanics and Boltzmann argued that the microscopic picture explained irreversibility. This was ridiculed by many of his contemories. W. Ostwald led the opposition to Boltzmann's ideas. However some, including E. Mach, thought the arguments were too violent. In 1906, Boltzmann committed suicide just before conclusions, about Brownian motion and radioactivity, confirmed his theories. It is speculated that continous ridicule led him to suicidal depression. 
Brownian motion

Brownian motion was first observed by Jan Ingenhousz in 1785, but was subsequently rediscovered by Robert Brown in 1828.
Brown used a microscope to observe smoke particles in a light beam.
He saw specks of light moving about erratically and apparently unpredictibly. 
Kinetic Theory Assumptions
1. Gases consist of large numbers of molecules (or atoms) that are in continuous, random motion.
2. The total volume of the molecules themselves, is negligible compared to the total volume in which the gas is contained.
3. Attractive and repulsive forces between gas molecules are negligible.
4. The duration of a collision is much less than the time between collisions.
5. No kinetic energy is lost in collisions so (as long as the temperature does not change) the average kinetic energy per molecule does not change with time.
6. The average kinetic energy per molecule is proportional to the absolute temperature.
Microscopic explanation of pressure
The pressure of a gas in a container comes from the force of repeated collisions of particles with the wall. The force they provide depends on the momentum change and the time between collisions with the walls.
Consider one particle with mass, m_{o}, moving with speed, v, in a spherical container of radius r.
The particle has momentum, p = m_{o}v and will hit the wall repeatedly, at the same angle θ.
The change in momentum, on hitting the right hand side, is given by.
The vector diagram corresponding to this is shown with arrows.
The vector difference is shown as a dotted arrow at right angles to the wall.
The size of the momentum change is given by the length of the dotted arrow, which is the base of the isosceles momentum triangle, so
The time between collisions is found from the distance between collisions (the base of the isosceles distance triangle) and the speed of the particle.
The force on the particle from the wall is then
The force on the wall is equal and opposite to the force on the particle.
The pressure on the spherical area of the wall due to this one particle is given by
Where V is the volume of the spherical container.
The total pressure due to N particles, having the same mass m_{o}, but moving with different speeds,
is the sum of the individual pressures, i.e.
Here is the mean (or average) of the squares of the speeds.
The square root of this mean square speed is usually written as v_{rms} and called the "root mean square speed".
The kinetic theory equation is then usually written as
The kinetic theory equation shows that macroscopic pressure is the average result of a many microscopic particles
colliding with the walls.
Internal Energy
Since the particles are points,
the total kinetic energy of the particles gives the Internal Energy of the gas ( U ), so…
The Internal Energy is,
Example
A cylinder contains 0.03 m^{3} of Argon gas at a temperature of 25^{0} C and a pressure of 1.2 MPa.
Find the internal energy of the gas.
Molar mass and the Atomic Mass Unit.
A gas with Avogadro's number of atoms (i.e. one mole) will have a mass called the Molar mass ( M ).
The mass of individual particles is
.
m_{o} will be very small and so a new unit is used, the atomic mass unit (symbol u ).
In the periodic table
the mass in "u" is the mass of one atom (averaged over the naturally occuring isotopes).
This number expressed in grams is the (gram) Molar mass.
Note that some atoms will naturally exist by themselves but other atoms naturally ocur linked together as molecules.
Example
The atomic mass of Argon is 39.95u. Find the mass of one Argon atom in kilogram.
Argon exists as single atoms.
(39.95 g of Argon gas will have Avogadro's number of atoms).
The microscopic picture of temperature
Combining the kinetic theory equation with the Universal gas equation, we get
Here N, is the total number of particles, and n, is the number of gram moles.
For a completely microscopic picture we only want to replace n.
Now the number of moles is the total number of particles divided by the number in one mole, i.e. so
The constant k_{B} is called the Boltzmann constant and has the unit of J.K^{1}.
In purely microscopic terms:
and from this emerges the microscopic picture of temperature.
The (macroscopic) Absolute Temperature is (microscopically) a measure of the average kinetic energy per particle.
Example
A cylinder contains 0.06 m^{3} of Krypton gas at a temperature of 127^{0} C and a pressure of 1.4 MPa.
The atomic mass of Krypton is 83.80u. Find
(a) the internal energy of the gas.
(b) the average kinetic energy per particle.
(c) the number of particles.
(d) the mass of the gas.
From the definition of internal energy
From the microscopic definition of temperature
From the internal energy definition
The mass of gas depends on the number of particles
Putting all these concepts together
Be sure you can distinguish the 3 different masses, the 3 different numbers
and the 2 different constants..
"R" is used macroscopically with "n" and
"k_{B}" is used microscopically with "N".
The MaxwellBoltzmann distribution
The idea of a mean square speed indicates that there is a variation in speeds.
The distribution of molecular velocities can be precisely calculated for a given temperature.
This is called the MaxwellBoltzmann distribution. Its formulation is not required for this course.
Computer simulations of velocity distribution:
David N. Blauch at Davidson.
The Rappe group
Summarising:
The microscopic view of matter has particles that can't be seen, heard, touched etc.
The high speed movement of these particles through the void between them produces macroscopic ("observable") effects.
The kinetic theory equation:
,
shows that macroscopic pressure is the average result of a many microscopic particles colliding with the walls.
The kinetic theory equation combined with the Universal Gas equation:
,
shows that Absolute Temperature is proportional to average kinetic energy per molecule.
The Internal Energy of a gas is the total kinetic energy of its particles:
The atomic mass unit is 1/N_{A} grams = 1.67 x 10 ^{27} kg
The MaxwellBoltzmann distribution gives the distribution of molecular velocities.
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