Kinetic energy of gases
In the ideal gas model, the gas molecules are considered
to move randomly with a wide range of speeds within
a certain volume (Figure 1.2). The size ofthe molecules
is neglected and the molecules are assumed not to inter-
act except during the collision process. Pressure arises from
the collisions of thegas molecules with the walls of the
container. Since billions of collisions arise every second
the pressure is constant with time. With this kinetic model the temperature
of a particle can related to its motion. For an ideal gas with nmoles and
a molecular weight of M, the average velocity, ,can be related to the
product of the pressure and volume:
(1.17)
The reader is invited to follow the derivation of this relationship in
Derivation box 1.1. The form of this relationship can be understood if we
consider the temperature, T, to reflect the energy of the system, realizing
that the kinetic energy, KE, is proportional to the square of the velocity:
(1.18)
Since the temperature Tis proportional to energy and hence the velocity
squared, it should not be a surprise that the product PVcan also be written
in terms of the velocity squared. The factor of 1/3 in eqn 1.17 arises because
not all of the molecules are moving in the same direction. The root mean
square velocity can be determined by solving eqn 1.17 for the velocity:
(1.19)
Real gases
Ideal gases are assumed to be particles with no size and interactions other
than the ability to collide with each other. Although many gases behave in
a nearly ideal way, none are perfectly ideal and their properties deviate
from those predicted based upon the ideal gas law. These deviations
provide insight into the interactions between molecules and the reactions
that they can undergo in chemical reactions. The simplest approach is
to assume that these effects are small and that the ideal gas law can be
(^9) rms=
3 RT
M
KE= m
1
2
92
PV==nRT nM
1
3
92
9
Figure 1.2The
molecules of an ideal
gas move randomly
within the enclosure.