Molecular Motion
KineticMolecular Theory
The ideal gas equation
pV = nRT
Has been presented as an empirical observation. Does it have meaning?
The KineticMolecular Theory ("the theory
of moving molecules"; Rudolf Clausius, 1857)
 Gases consist of large numbers of molecules (or atoms,
in the case of the noble gases) that are in continuous, random
motion
 The volume of all the molecules of the gas is negligible
compared to the total volume in which the gas is contained
 Attractive and repulsive forces between gas molecules
are negligible
 The average kinetic energy of the molecules does not
change with time (as long as the temperature of the gas remains constant).
Energy can be transferred between molecules during collisions but not lost because the
collisions are perfectly elastic
 The average kinetic energy of the molecules is proportional
to absolute temperature (A result of Thermodynamics).
At a given temperature the molecules of all
gases have the same average kinetic energy
Pressure
 The pressure of a gas is caused by collisions (momentum transfer)
of the
molecules of the gas with the walls of the container.
 The magnitude of the pressure is related to how hard
and how often the molecules strike the wall
Absolute Temperature
 The absolute temperature of a gas is a measure of the average
kinetic energy of its' molecules
 If two different gases are at the same temperature, their
molecules have the same average kinetic energy
 If the temperature of a gas is doubled,
the average kinetic energy of its molecules is doubled
Molecular Speed
 Although the molecules in a sample of gas have an average
kinetic energy (and therefore an average speed) the individual molecules
move at various speeds, i.e. they exhibit a DISTRIBUTION of speeds; Some
move fast, others relatively slowly. Collisions can change individual
molecular speeds but the distribution of speeds remains the same.
 At the same temperature, lighter gases move on average faster than
heavier gases.
 At higher temperatures at greater fraction of the molecules
are moving at higher speeds. This is important for activated chemical
processes, reactions.
 !The average kinetic energy, e,
is related to the root mean square (rms) speed u
The average of a distribution must be taken (defined) in a
specific way. In general the mean, the root mean square and the
most probable value in a distribution are all different.
A Note on Distributions a simple numerical
example:
Suppose we have four molecules in our gas sample. Their
speeds are 3.0, 4.5, 5.2 and 8.3 m/s.
 The root mean square speed is:
The rms speed as well as the entire distribution of speeds of gas
molecules are a function of temperature. Below, the blue line is a cold gas and the
red line is a hot gas. Note that the rms speed, u as well as the entire
speed distribution changes with
temperature for a given gas.
The rms speed for a given speed distribution (which is determined by the
temperature and molecular weight of the gas) is greater in magnitude
than the most probable
speed or the mean speed.
Trick question: What is the mean velocity of the molecules in a gas at any
temperature.
Gas Laws and Kinetic Theory
 At constant temperature, the average kinetic energy
of the gas molecules remains constant
 Therefore, the rms speed of the molecules, u, also
remains unchanged
 If the rms speed remains unchanged, but the volume increases,
there will be fewer collisions with the container walls
over a a given time:
Therefore, the pressure will decrease (Boyle's
law)
 An increase in temperature means an increase in the average
kinetic energy of the gas molecules, thus an increase in u
 At constant volume, the greater speed will mean more
collisions per unit time and an increase in pressure

If, instead, we allow the volume to change to maintain constant
pressure, the volume must increase with increasing temperature to maintain constant
pressure, which is just
(Charles's law)
The Ideal Gas Equation of State follows directly from the Kinetic Theory
of Gases. Here is a PseudoDerivation
Molecular Effusion and Diffusion
Kineticmolecular theory states that the average kinetic energy of
a mole of molecules molecules is proportional to absolute
temperature, and the proportionality constant is R, the universal gas
constant
(1/2)Mu^{2} = (3/2)R T
 At a given temperature,
all gases have the same average kinetic energy and for a three dimensional gas
this value is (3/2)RT. (what is the molar kinetic energy of a two dimesional
gas trapped in the surface of a metal?)
 The rms velocity, u, in m/s, is simply
where
M is the molar mass in kg/mole, R is the gas constant in J/K^{.}mole,
and T is the absolute temperature in K.
Numerical Example:
Calculate the rms speed, u, of an N_{2}
molecule at room temperature (25°C) Be careful of your UNITS!
T = (25+273)K = 298K
M = 28 g/mol = 0.028 kg/mol
R = 8.314 J/mol K = 8.314 kg m^{2}/s^{2}
mol K
u = 515 m/s
Note: this is equal to 1,150 miles/hour!
Effusion
The escape of a gas through a tiny pore or pinhole
in its container is called EFFUSION.
The effusion rate, r, has been found to
be inversely proportional to the square root of its molar mass: (Why?)
Thus, comparison of the effusion rates of two gases with different
masses will follow the relation:
This effect was observed in the 19^{th} century by Graham and is sometimes
called GRAHAM's LAW
A note on Rates and Times
The effusion time (the time it takes for a certain amount of gas
to escape a vessel)
is inversely proportional to the effusion rate (the amount of gas
effusing from the hole per unit time). Be careful that you understand whether it is a
rate or a time that you are tyring to find out.
Gas may effuse, but for this to happen a molecule must pass through a
pore or pinhole and escape to the outside. In effect, a molecule must
'collide' with an escape hole. The number of such collisions will be linearly
proportional to the average speed
of the molecules in the gas and thus the effusion rate. The effusion
time should the be inversely proportional to the average speed
The ratio of effusion rates, r_{i}, for two gases labelled by i, are
proportional to the ratio of the RMS speeds of the gases, u_{i}
Diffusion
Similarly to effusion, the process of diffusion is the
spontaneous intermingling (mixing) of dissimilar gases (fluids) that are
initially spatially separated. If you put a drop of ink in a glass of water and
you see the ink gradually spread out to fill the glass, this is diffusion
 The relative rates of diffusion of two gases is also determined by
the ratio of their average (rms) speeds
 The speed of molecules is quite high, but the rates of diffusion are
slower than molecular
speeds due to molecular collisions
 At the density of the atmosphere at sea level,
each gas molecule experiences collisions at a rate of about 10^{10}
(i.e. 10 billion)
times per second
 Due to these collisions, the direction of a molecule
of gas in the atmosphere is constantly changing, and the diffusion rate is much
reduced from the instantaneous speed of the molecule
The average distance traveled by a molecule between collisions
is called the mean free path
 The higher the density of gas, the smaller the mean free
path (more likelyhood of a collision)
 At sea level the mean free path is about 60 nm
 At 100 km altitude the atmosphere is less dense and colder than
where we live at the surface of the earth,
the mean free path is about 0.1 m (about 1 million times longer than at
sea level)
See some multiple choice
problems relating to gases.
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PJ Brucat // University of Florida