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Peter's Index Physics Home Lecture 7 Course Index Lecture 9

A Semester of First Year Physics with Peter Eyland

Lecture 8 (Kinetic Theory)

In this lecture the following are introduced

• Kinetic Theory assumptions

• Molar Mass and the Atomic Mass Unit.

• The Microscopic explanation of Pressure

• Average Molecular Speeds in gases

• The Microscopic explanation of Temperature

• The Mean Free Path

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.

Molar Mass and the Atomic Mass Unit.

One mole of a gas (i.e. Avogadro's number of particles) will have a mass called the *Molar mass*, *M*.

The mass of individual particles is
.

Since* m _{0}* will be very small a new unit is used, the

In the Periodic Table the mass in "u" is the atomic mass and the mass in grams is the Molar mass.

Example

The atomic mass of Argon is 39.95u.
Find the mass of one Argon atom in kilogram.

The 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 _{0}*, moving with speed,

The particle has momentum, ** p** =

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 wall due to this one particle is given by

The total pressure due to *N* particles, having the same mass *m _{0}*, 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, so

is the square root of the mean square speed.

This kinetic theory equation is written

Note: *m _{0}* is the mass of the individual molecules

The kinetic theory equation shows that macroscopic pressure is the average result of a many microscopic particles colliding with the walls.

Average Molecular speeds in gases

The rms speed can be found from the temperature and type of gas

Example

Helium has a molar mass of 4 g. Find the rms speed of a sample of helium gas at 300 K.

Note that the mass has to be expressed in kg to get an answer in m.s^{-1}.

(Really, the proper S.I. unit should be the kg. mole = 6.02 x 10^{26})

*Average Molecular speeds in gases at 300K*

Gas |
Molar mass |
v |

Hydrogen |
2 |
1920 |

H |
18 |
645 |

Nitrogen |
28 |
517 |

Oxygen |
32 |
483 |

CO |
44 |
412 |

SO |
64 |
342 |

Even though molecules move at high speed their *diffusion* from one place to another is slow
because collisions send them off in randomised directions.

The Microscopic explanation of Temperature

For a microscopic picture of the Ideal Gas Equation, the idea of the gram mole has to be removed.

The constant *k _{B}* is called the Boltzmann constant and has the unit of J.K

Combining the Kinetic Theory Equation with the Ideal Gas Equation in purely microscopic terms, we get,

From this, the microscopic picture of temperature emerges.

Temperature is then a measure of the average kinetic energy per particle.

Example

Find the average kinetic energy per molecule for Nitrogen gas at 1600K

The Mean Free Path

Because of its random high-speed motion (400 -> 2000 m.s^{-1}), a molecule in a gas, at atmospheric pressure,
makes many collisions per second with other molecules.
The mean free path, λ, is the average distance travelled between collisions.

Now a molecule with effective diameter, *d*, and travelling with the average speed *v _{av}*
sweeps out, in one second, a kinky cylinder.

Now any molecules having their *centres* within a radial distance,* d,* of the cylinders axis will be struck.

If the number of molecules per volume is *N/V* then the number of collisions per second is

Since the mean free path, λ, is the average distance travelled between collisions.

The calculation is oversimplified because all the other molecules are taken as being at rest.

Qualitatively, since all the molecules are moving the chance of collisions will be increased.

Maxwell worked out a correction factor to include this and concluded that:

Example

Find the mean free path for Nitrogen molecules (diameter 0.29 nm) at standard temperature and pressure.

The Loschmidt number is the number of molecules in 1 cm^{3} at standard temperature and pressure, so

Example

Find the mean free path for Hydrogen molecules (diameter 0.24 nm) at 27^{0}C and 1 Atmosphere.

*Summarising:*

Kinetic Theory assumes small molecules with high speed and no interactions between the molecules except collisions.

The kinetic theory equation shows that macroscopic pressure is the average result of a many microscopic particles colliding
with the walls.

Temperature is the average kinetic energy per particle:

Maxwell's mean free path:

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