Is there a relationship between electrical conductivity and thermal conductivity?

Asked by:
Darell Hayes

Answer

There is a relationship for metals and it is known as the Wiedemann-Franz law. Metals are
good electrical conductors because there are lots of free charges in them. The free
charges are usually negative electrons, but in some metals, e.g., tungsten, they are
positive 'holes.' For purposes of discussion, let's assume we have free electron charges.

When a voltage difference exists between two points in a metal, it creates an electric
field which causes the electrons to move, i.e., it causes a current. Of course, the
electrons bump into some of the stationary atoms (actually, 'ion cores') of the metal and
this frictional 'resistance' tends to slow them down. The resistance depends on the
specific type of metal we're dealing with. E.g., the friction in silver is much less than
it is in iron. The greater the distance an electron can travel without bumping into an
ion core, the smaller is the resistance, i.e., the greater is the electrical
conductivity. The average distance an electron can travel without colliding is called the
'mean free path.' But there's another factor at work too. The electrons which are free
to respond to the electric field have a thermal speed a sizable percentage of the speed of
light, but since they travel randomly with this high speed, they go nowhere on average,
i.e., this thermal speed itself doesn't create any current.

The thermal conductivity of this metal is, like electrical conductivity, determined
largely by the free electrons. Suppose now that the metal has different temperatures at
its ends. The electrons are moving slightly faster at the hot end and slower at the cool
end. The faster electrons transmit energy to the cooler, slower ones by colliding with
them, and just as for electrical conductivity, the longer the mean free path, the faster
the energy can be transmitted, i.e., the greater the thermal conductivity. But the rate
is also determined by the very high thermal speed'the higher the speed, the more rapidly
does heat energy flow(i.e., the more rapidly collisions occur). In fact, the thermal
conductivity is directly proportional to the product of the mean free path and thermal
speed.

Both thermal and electrical conductivity depend in the same way on not just the mean free
path, but also on other properties such as electron mass and even the number of free
electrons per unit volume. But as we have seen, they depend differently on the thermal
speed of the electrons'electrical conductivity is inversely proportional to it and thermal
conductivity is directly proportional to it. The upshot is that the ratio of thermal to
electrical conductivity depends primarily on the square of the thermal speed. But this
square is proportional to the temperature, with the result that the ratio depends on
temperature, T, and two physical constants: Boltzmann's constant, k, and the electron
charge, e. Boltzmann's constant is, in this context, a measure of how much kinetic energy
an electron has per degree of temperature.

Putting it all together, the ratio of thermal to electrical conductivity is:

( ^{2} / 3 ) * ( (k/e)^{2} ) * T

the value of the constant multiplying T being: 2.45x10^{-8} W-ohm-K-squared.
Answered by:
Frank Munley, Ph.D., Associate Professor, Physics, Roanoke College

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