The ideal gas equation
pV = nRT
Has been presented as a compliation of empirical observation,
i.e. the historically significant Gas Laws, but does The Ideal Gas equation
have some deeper, more fundamental meaning?
The Kinetic-Molecular 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. Usually there is a great distance between each other,
so the molecules travel in straight lines
between abrupt collisions at the walls and between each other. These collisions randomize the motion of the molecules. Most
of the collisions between molecules are binary, in that only
two molecules are involved.
The volume of the molecules of the gas is negligible compared
to the total volume in which the gas is contained. A common bond
length between atoms is about 10-10 m or 1 Angstrom. Small molecules
are therefore on the order of 10 Angstroms in diameter, or less than 10-24
Liters in Molecular Volume, quite tiny indeed!
Remember, however that there
can be a great many molecules in the sample of gas, perhaps on the order
of a mole, or 6 x 1023. So that when concentrations of
molecules exceed about 1 mol/liter, then the approximation that the volume
of ALL the molecules in the container is much less than the volume of the
container itself, fails. In the case of an ideal gas, we will assume
that molecules are point masses, i.e., the volume of a mole of gas molecules (as if they were at rest) is zero, so molecular and container volumes
never become comparable.
Attractive forces between gas molecules are
We know that if these forces were significant, the molecules would stick
together. This happens when it rains and gaseous water molecules
stick together to form a liquid. Water vapor is a condensible gas,
and this shows us that gas molecules are sticky, but at a high enough temperature
they form only a permanent gas, because their stickiness can be considered negligible.
We will assume that in an ideal gas, molecular attractive forces are not
just small, but identically zero.
The Molecular picture of Pressure
The average kinetic energy of the molecules does not change
with time. The molecules bounce and bounce
but, on average, do not slow down 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 (not sticky)
The average kinetic energy of the molecules is proportional
to absolute temperature (A result of Thermodynamics). At
a given temperature the molecules of all species of gas, no matter what
size shape or weight, have the same average kinetic energy.
The pressure of a gas is manifested at the boundary of the
vessel it is confined in, and 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.
Check out this
of molecules in motion in a gas.
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 absolute temperature of a gas is doubled, the
average kinetic energy of its molecules is doubled.
If the temperature approaches absolute zero,
the kinetic energy of the molecules approach zero and they 'stop'
Molecular Speed, Averaging, and The Maxwell-Boltzmann
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 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.
The following graph shows the Distribution of Speeds for Gases, i.e.the
fraction of the sample of gas molecules that have a given speed is shown
by the height of the curve above the speed axis. There are no molecules
exactly at rest.
At higher temperatures at greater fraction of the molecules
are moving at higher speeds. This is important for activated chemical processes,
The average kinetic energy, e,
is related to the root mean square (rms) speed u
Where does this number lie on our Graph of the speed distribution?
To find out, the root mean square 'average' of the distribution must be
taken (defined) in a specific way. It is defined exactly how it sounds.
First your square all the speeds, then average those
numbers, and take the square root of that average. The mean
of a distribution is just the average of all the numbers in the distribtion.
The most probable value of a distribution is also exactly what it sounds
like, the speed of that the largest fraction of molecules are travelling.
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 mean (simple average)
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
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
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
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
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 (i.e. the number and strength of 'hits' per wall), which is just Charles's law
The Ideal Gas Equation of State follows directly from the
Kinetic Theory of Gases. Here is a Pseudo-Derivation
Molecular Effusion and Diffusion
Kinetic-molecular 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)Mu2 = (3/2)R
T = Molar Kinetic (translational) Energy of the gas
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
M is the molar mass in kg/mole, R is the
gas constant in J/K.mole, and T is the absolute temperature
Calculate the rms speed, u, of an N2
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 m2/s2
u = 515 m/s
Note: this is equal to 1,150 miles/hour!
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 19th century
by Graham and is sometimes called GRAHAM's LAW
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 calculating.
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 be inversely proportional to the average speed
of the molecules or proportional to the square root of the ratio of the
The ratio of effusion rates, ri,
for two gases labelled by i, is proportional to the ratio of the
RMS speeds of the gases, ui
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 average distance traveled by a molecule between collisions with another molecule
is called the mean free path
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 1010 (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 higher the density of gas, the smaller the mean free
path (more likelyhood of a collision). The larger the molecules, the smaller the mean free path.
The mean free path depends on the number density of the gas molecules and their size --- and nothing else
At sea level the mean free path of atmospheric gases is about 60 nm
At 100 km altitude, the atmosphere is less dense
than where we live at the surface of the earth, and the mean free path is about
0.1 m (about 1 million times longer than at sea level)
Syllabus || Staff ||
Operations || TOP
PJ Brucat ||
University of Florida