Flow, deformation, and transport - Physical Processes in Earth and Environmental Sciences - page 192

In this thought experiment the container has its right hand
wall as an elastic membrane. Individual gas molecules are
shown approximately to scale so that the average separation
distance between neighbors is about 20 times molecular
radius. The individual molecules all have their own instanta-
neous velocity, u , but since the directions are random the sum
of all the velocities, Σ u , and therefore the average velocity must
be zero. This is true whether we compute the average velocity
of an individual molecule over a long time period or the
instantaneous average velocity of a large number of individual
molecules.
The arrows denote instantaneous velocities. Nevertheless the
gas molecules have a mean speed, u, that is not zero. This is
because although the directions cancel out the magnitudes of
the molecular velocities, that is, their speeds, do not. In such
cases we compute the mean velocity by finding the value of
the mean square of all the velocities and taking the square
root, the result being termed the root-mean-square velocity, or
u rms in the present notation.
This is NOT the same as the mean speed, a feature you can
easily test by calculating the mean and rms values of , say, 1, 2,
and 3.

u

_

u rms = ( u 2 ) 0.5

The internal energy, E , of any gas is the sum of all the
molecular kinetic energies. In symbols, for a gas with N
molecules:

E = N (0.5 mu 2 rms )
Or we may alternatively view the molecular velocity as a
direct function of the thermal energy:
u 2 rms = 2 E/mN
Fig. 4.143 Molecular collisions and the internal thermal energy of a gas. One molecule is shown striking the elastic wall, which responds by
displacing outward, signifying the existence of a gaseous pressure force and hence molecular kinetic energy transfer.

or flux of individual molecular momentum is the origin of
gaseous pressure, temperature, and mean kinetic energy
(Fig. 4.144). These properties arise from the mean speed
of the constituent molecules: every gas possesses its own
internal energy, E , given by the product of the number of
molecules present times their mean kinetic energy. In a
major development in molecular theory, Maxwell calcu-
lated the mean velocity of gaseous molecules by relating it
to a kinetic version of the ideal gas laws, together with a
statistical view of the distribution of gas molecular speed.
The resulting kinetic theory of gases depends upon the
simple idea that randomly moving molecules have a proba-
bility of collision, not only with the walls of any container,
but also with other moving molecules. Each molecule thus
has a statistical path length along which it moves with its
characteristic speed free from collision with other mole-
cules: this is the concept of mean free path . Since gases are
dilute the time spent in collisions between gas molecules is
infrequent compared to the time spent traveling between
collisions. Thus the typical mean free path for air is of
order 300 atomic diameters and a typical molecule may
experience billions of collisions per second. Similar ideas

have informed understanding of the behavior, flow, and
deformation of loose granular solids, from Reynolds' con-
cept of dilatancy to the motion of avalanches (Section 4.11).

4.18.3 Heat flow by conduction in solids

In solid heat conduction, it is the molecular vibration
frequency in space and time that varies (Fig. 4.145). Heat
energy diffuses as it is transmitted from molecule to mole-
cule, as if the molecules were vibrating on interconnected
springs; we thus “feel” heat energy transfer by touch as it
transmits through a substance. In fact, all atoms in any
state whatsoever vibrate at a characteristic frequency about
their mean positions, this defines their mean thermal
energy . Vibration frequency increases with increasing
temperature until, as the melting point is approached,
the atoms vibrate a large proportion of their interatomic
separation distances. Conductive heat energy is always
transferred from areas of higher temperature to areas of
lower temperature, that is, down a temperature gradient,
d T /d x , so as to equalize the overall net mean temperature.