Thermal conductivity of a gas
Let the upper and lower plates in Fig. 4.18 be at rest but at different temperatures,
so that there is a temperature gradient rather than a velocity gradient in
the gas. Note that it is difficult to prevent conductive heat flow in a gas
from being masked by convection currents. The gas layer must be thin, and
the upper plate must be at a higher temperature than the lower to consider
pure conduction. If 
is the temperature gradient
normal to a surface within the gas, the thermal conductivity λ is defined
by the equation
|
(4.147) |
where H is the heat flow or heat current per unit area and per unit time
across the surface. The negative sign is included because if
is positive the heat current is downward and is negative.
From the molecular point of view, we consider the thermal conductivity of
a gas to result from the net flux of molecular kinetic energy across a surface.
The total kinetic energy per mole of the molecules of an ideal gas is simply
its internal energy u, which is turn equals cvT. The average kinetic
energy of a single molecule is therefore cvT divided by Avogadro’s
number NA and if we define a “molecular heat capacity”
as 
, the average
molecular kinetic energy is T
We assume as before that each molecule crossing the surface made its last
collision at a distance of 
above or below the
surface, and that its kinetic energy corresponds to the temperature at that
distance. If T0 is the temperature at the surface S-S, the kinetic
energy of a molecule at a distance below the
surface is
|
(4.148) |
The energy transported in an upward direction, per unit area and per unit time, is the product of this quantity and the molecular flux Φ:
|
(4.149) |
In the same way, the energy transported by molecules crossing from the above is
|
(4.150) |
The net rate of transport per unit area, which we identify with the heat current H, is
|
(4.151) |
And by comparison with Eq. (4.147) we see that the thermal conductivity λ is
|
(4.152) |
Thus the thermal conductivity, like the viscosity, should be independent
of density. This is also in good agreement with experiments down to pressures
so low that the mean free path becomes of the same order of magnitude as dimensions
of the container.
The ratio of thermal conductivity to viscosity is
|
(4.153) |
And
|
(4.154) |
Where M is the molecular weight of the gas. Therefore the theory predicts that for all gases this combination of experimental properties should equal unity. Some figures are given in Table (4.4) for comparison. The ratio does have the right order of magnitude, but we see again that the Hard-sphere model for molecules is inadequate.
Table 4.4 Values of the thermal conductivity λ, molecular weight
M, viscosity 
and specific heat capacity cv
of a number of gases
