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 Maxwell-Boltzmann 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. Died: 5 Oct 1906 in Duino (near Trieste), Italy

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." In the scientific world it meant that you were not allowed to believe in things you couldn't see, hear, touch, taste, or smell. Boltzmann believed in particles that couldn't be seen, heard, touched etc. He said the movement of these microscopic (really "nanoscopic") particles through the void between them produced macroscopic ("observable") effects. 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. Using kinetic theory ideas, Albert Einstein (1879 - 1955) made predictions about the movements of these particles, which were later proved by experiment.

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_ov 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 m³ of Argon gas at a temperature of 25⁰ 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 m³ of Krypton gas at a temperature of 127⁰ 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 Maxwell-Boltzmann 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 Maxwell-Boltzmann 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 Maxwell-Boltzmann distribution gives the distribution of molecular velocities.