*E*_{k} = *nN*_{A}*mv*^{2}/2 =
*nMv*^{2}/2

In this expression *m* is the mass of a single molecule, so the molar
mass *M* is the product *mN*_{A}. Since the kinetic energy
is also 3*nRT*/2, the square root of the square of the mean
velocity, known as the root-mean-square velocity *v*(rms), of the
molecules of the gas is proportional to the square root of its molar mass.
The root-mean-square velocity, like the
actual distribution of velocities embodied in the Maxwell law, is a function
only of the absolute temperature.

*v*(rms) = (the square root of)3*RT*/*M*

Example. Let us calculate the root-mean-square velocity of oxygen molecules at room temperature, 25

*v*(rms) = (the square root of)3(8.3143)(298.15)/(0.0319998) = 482.1 m/s

A speed of 482.1 m/s is 1726 km/h, much faster than a jetliner can fly and faster than most rifle bullets.

The very high speed of gas molecules under normal room conditions would indicate that a gas molecule would travel across a room almost instantly. In fact, gas molecules do not do so. If a small sample of the very odorous (and poisonous!) gas hydrogen sulfide is released in one corner of a room, our noses will not detect it in another corner of the room for several minutes unless the air is vigorously stirred by a mechanical fan. The slow diffusion of gas molecules which are moving very quickly occurs because the gas molecules travel only short distances in straight lines before they are deflected in a new direction by collision with other gas molecules.

The distance any single molecule travels between collisions will vary from very short
to very long distances, but the average distance that a molecule travels between collisions in
a gas can be calculated. This distance is called the **mean free path**
*l* of the gas molecules. If
the root-mean-square velocity is divided by the mean free path of the gas molecules, the result
will be the number of collisions one molecule undergoes per second. This number is called the
**collision frequency** *Z*_{1} of the gas molecules.

The postulates of the kinetic-molecular theory of gases permit the calculation of the
mean free path of gas molecules. The gas molecules are visualized as small hard spheres. A
sphere of diameter *d* sweeps through a cylinder of cross-sectional area
(pi)*d*^{2} and length *v*(rms)
each second, colliding with all molecules in the cylinder, as shown in the
Figure below.

Figure is not available.

The radius of the end of the cylinder is

*Z*_{1} = (pi)*d*^{2}*v*(rms)(the square root of)2

This total number of collisions must now be divided by the number of molecules which are
present per unit volume. The number of gas molecules present per unit volume is found by
rearrangement of the ideal gas law to *n*/*V* = *p*/*RT*
and use of Avogadro's number, *n* = *N*/*N*_{A};
thus *N*/*V* = *pN*_{A}/*RT*. This gives the mean
free path of the gas molecules, *l*, as

(*v*(rms)/*Z*_{1})/(*N*/*V*) = *l*
= *RT*/(pi)*d*^{2}*pN*_{A}(the square root of)2

According to this expression, the mean free path of the molecules should get longer as the temperature increases; as the pressure decreases; and as the size of the molecules decreases.

Example. Let us calculate the length of the mean free path of oxygen molecules at room temperature, 25

*l* =
(8.3143 kg m^{2}/s^{2}K mol)(298.15 K)/3.14159(370 x
10^{-12} m)^{2}(101325 kg/m s^{2})
(6.0225 x 10^{+23} mol^{-1})((the
square root of)2),

so *l* = 6.7 x 10^{-8} m = 67 nm. The utility of SI units
and of the quantity calculus in this example should be obvious.

*Z*_{1} =
(pi)*d*^{2}*pN*_{A}(the square root of)2/*RT*
= *v*(rms)/*l*

For oxygen at room temperature, each gas molecule collides with another every 0.13
nanoseconds (one nanosecond is 1.0 x 10^{-9} s), since the collision
frequency is 7.2 x 10^{+9}
collisions per second per molecule.

For an ideal gas, the number of molecules per unit volume is given using
*pV* = *nRT* and *n* = *N*/*N*_{A} as

*N*/*V* = *N*_{A}*p*/*RT*

which for oxygen at 25^{o}C would be
(6.0225 x 10^{+23} mol^{-1})(101325 kg/m s^{2})/(8.3143
kg m^{2}/s^{2} K mol)(298.15 K)
or 2.46 x 10^{+25} molecules/m^{3}. The number of collisions
between **two** molecules in a volume, *Z*_{11},
would then be the product of the number of collisions each
molecule makes times the number of molecules there are,
*Z*_{1}*N*/*V*, except that this would count
each collision twice (since two molecules are involved in each one collision).
The correct equation must be

*Z*_{11} =
(pi)*d*^{2}*p*^{2}*N*_{A}^{2}*v*(rms)
(the square root of)2/2*R*^{2}*T*^{2}

If the molecules present in the gas had different masses they would also have different
speeds, so an average value of *v*(rms) would be using a weighted average of the molar masses;
the partial pressures of the different gases in the mixture would also be required. Although
such calculations involve no new principles, they are beyond our scope. However, the
number of collisions which occur per second in gases and in liquids are extremely important
in chemical kinetics, so we shall return to this topic in other sections.

*v*(rms)_{1}/*v*(rms)_{2} =
(the square root of)(*M*_{2}/*M*_{1}) =
(the square root of)(*d*_{2}/*d*_{1})

This equation gives the velocity ratio in terms of either the molar mass ratio or the ratio of
densities *d*. The ratio of root-mean-square velocities is also the ratio of the rates of effusion, the
process by which gases escape from containers through small holes, and the ratio of the rates
of diffusion of gases.

This equation is called **Graham's law of diffusion and effusion**
because it was observed by Thomas Graham (1805-1869) well before the
kinetic-molecular theory of gases was developed. As an empirical law, it simply stated that the rates of diffusion and of effusion of
gases varied as the square root of the densities of the gases. Graham's law is the basis of
many separations of gases. The most significant is the separation of the isotopes of uranium
as the gases ^{238}UF_{6} and ^{235}UF_{6}.
Fluorine has only one isotope, so the separation on the basis
of molar mass is really a separation on the basis of isotopic mass.

Example. The ratio of root-mean-square velocities of

[NEXT: Equations of State for Gases]

[PREVIOUS: Kinetic-Molecular Theory Explains the Gas Laws: Partial Pressure]

Copyright 1995 James A. Plambeck (Jim.Plambeck@ualberta.ca). Updated September 26, 1995.