Thermodynamics

CHE 205 (PHYSICAL CHEMISTRY I)

KINETIC THEORY OF GASES

Gas is one of the fundamental states of mater. Pure substances can exist mainly in three different forms;
solid, liquid and gas. The parameters required to describe the state of a gas are pressure (p), volume (V),
temperature (T) and number of moles (n). These parameters are known as state variables.

Between about 1850 and 1880, James Clerk Maxwell (1831–1879), Rudolf Clausius (1822–1888), Ludwig
Boltzmann (1844 –1906), and others developed the kinetic theory of gases. The kinetic-molecular theory
is the model that accounts for macroscopic gas behaviour at the level of individual particles (atoms or
molecules). They based the theory on the idea that all gases behave similarly as far as particle motion is
concerned. Using kinetic gas theory, it is possible to derive or explain the experimental behaviour of
gases. It is important to note that:

1. Gases are mostly empty space. The total volume of the gas molecules is negligibly small compared with
that of the container to which they are confined.
2. Gas molecules are in constant chaotic motion. They gas molecules collide frequently with one another
and with the walls of the container. This behaviour makes their velocities to constantly change.
3. Gas molecules flow freely and have relatively low densities.
4. Collisions of gas molecules are elastic. The colliding molecules exchange energy but do not lose any
energy through friction. Thus, their total kinetic energy is constant. There are no attractive forces that
would tend to make molecules stick to one another or to the container walls.
5. Gas pressure is caused by collisions of molecules with the walls of the container. As a result, pressure
increases with the energy and frequency of the collisions.
6. Gas volume changes significantly with pressure and temperature. Gases are compressible.
7. Gases mix in any proportions to form solutions; liquids and solids generally do not. For instance, Air
is a solution of 18 gases.

Distribution of Molecular Velocities

At any time, the distribution of velocities is influenced by molar mass (M) and temperature (T).

As shown in Figure 2A, gases with low molecular mass will have their molecules moving at higher speeds.
As presented in Figure 2A. For gases with a high molecular mass, the average speed is low and the
distribution of speeds is less broad.

At a low temperature, most molecules of gases move with a speed close to the most probable value, but
at a high temperature, there is a broad spread of velocities. Each particle has a molecular speed (c); most
are moving near the most probable speed, but some are much faster and others much slower. The curve
(Figure 2B) flattens and spread at higher temperatures; and the most probable speed (the peak of each
curve) increases as the temperature increases.

The increase in most probable speed occurs because the average kinetic energy (e) of the molecules,
which is related to the most probable speed (cmp), is proportional to the absolute temperature (T). That
is e  T or e = c T , where c is a constant that is the same for any gas.
1

,Generally, at a high temperature value, gases tend to have high average velocities and broad distribution
of velocities, that is, more molecules of the gases will possess high speeds and a few molecules will possess
low speeds.




Figure 1: Maxwell-Boltzmann distribution of velocities



A B




Figure 2: Effect of molecular mass (A) and temperature (B) on distribution of velocities




For fractions of molecules that have velocities in the range of c + dc is proportional to width of the range
f(c)dc, where f(c) is the distribution of speeds or velocities.



2

, 3

 M  2 2 − MC 2

f (c ) = 4   c e 2 RT
dc (1)
 2 RT 

Equation 1 is known as Maxwell-Boltzmann distribution of speeds. In Equation 1, M = molecular mass
of the gas; T = temperature in Kelvin (K); c = root mean square velocity (m s–1); R = gas constant (8.314 J
mol–1 K–1);  = 227 ; and dc = change in velocity.

It implies that from Equation 1 that:

2
− MC
(i) The presence of decaying exponential function ( e 2 RT
) implies that small fraction of molecules
has very high speed. The values of e − x becomes smaller whenever x2 is large (i.e. when c is large).
2
MC 2 − MC
(ii) The 2RT part of the e 2 RT
in the exponent is large when M is large. This signifies that the
exponential factor will tend towards zero when M is large. Therefore, large molecules will not be
able to possess very high speeds.
MC 2
(iii) The factor 2RT is small when T is high. Therefore, a great fraction of the molecules will possess
high speeds at high temperature values.
2
2 − MC
(iv) The value of tends c e 2 RT
towards zero as c goes to zero. Therefore, the fraction of the molecules
with very low speed will be very small.
3

 M 2
(v) The 4   factor ensures that a zero (0) will be obtained when fractions over the entire
 2 RT 
range of speeds are added up.

Average and Root Mean Square Velocities

If n represents the molecules of a gas having individual velocities c1, c2, c3, ………, ci, then

c1 + c2 + c3 + ...... + ci 1 N
C= =  ci (2)
N N i =1

8RT
Similarly, C = (3)
M

In Equations 2 and 3, C represents the average velocity of the gas (m s–1) and M is the molar mass (kg
mol–1).

It is known that pV = 13 nN Amc 2 (4)

pV = nRT (5)
3