Thermal Conductivity - Simple Kinetic Picture

Simple Kinetic Picture

In this Section we will derive an expression for the thermal conductivity. Consider a gas with a vertical temperature gradient. The upper side is hot and the lower side cold. There is a downward energy flow due to the fact that the gas atoms, going down, have a higher energy than the atoms going up. The net flow of energy per second is the heat flow H. The heat flow is proportional to the number of particles that cross the area A per second. This number is proportional to the product nvA where n is the particle density and v the mean particle velocity. The magnitude of the heat flow will also be proportional to amount of energy transported per particle so with the heat capacity per particle c and some characteristic temperature difference ΔT. So far we have

The unit of H is J/s and of the right-hand side it is (particle/m3)•(m/s)•(J/(K•particle))•(m2)•(K) = J/s, so this is already of the right dimension. Only a numerical factor is missing. For ΔT we take the temperature difference of the gas between two collisions

where l is the mean free path. Detailed kinetic calculations show that the numerical factor is -1/3, so, all in all,

Comparison with the one-dimension expression for the heat flow, given above, gives as the final result

The particle density and the heat capacity per particle can be combined as the heat capacity per unit volume

so

where C_V is the molar heat capacity at constant volume and V_m the molar volume.

For an ideal gas the mean free path is given by

where σ is the collision cross section. So

The heat capacity per particle c and the cross section σ both are temperature independent so the temperature dependence of k is determined by the T dependence of v. For a monatomic ideal gas, with atomic mass M, v is given by

So

This expression also shows why gases with a low mass (hydrogen, helium) have a high thermal conductivity.

For metals at low temperatures the heat is carried mainly by the free electrons. In this case the mean velocity is the Fermi velocity which is temperature independent. The mean free path is determined by the impurities and the crystal imperfections which are temperature independent as well. So the only temperature-dependent quantity is the heat capacity c, which, in this case, is proportional to T. So

with k₀ a constant. For pure metals such as copper, silver, etc. l is large, so the thermal conductivity is high. At higher temperatures the mean free path is limited by the phonons, so the thermal conductivity tends to down with temperature. In alloys the density of the impurities is very high, so l and, consequently k, are small. Therefore alloys, such as stainless steel, can be used for thermal insulation.