Final expression for the thermal conductivity

Having obtained an expression for the mean free path A, we write the formula for the coefficient of thermal conductivity by combining Eqs. (30.21) and (30.17)  

This equation leads to the interesting conclusion that the thermal conductivity is independent of the pressure. This lack of dependence on pressure is a result of two compensating effects. By Eq. (30.17), Kj is proportional to N and to X but X is inversely proportional to N so that the product NX is independent of pressure. At lower pressures fewer molecules cross the surface in one second, but they come from a larger distance (X is larger at lower p) and so carry a proportionately greater excess energy. Experiment confirms that is independent of pressure. 

If Cy is independent of temperature, then everything on the right of Eq. (30.24) is constant except c , which is proportional to Therefore, should increase as This is also confirmed experimentally. 

In this derivation of the expression for we have assumed that the pressure is high enough so that X is much smaller than the distance separating the two plates. At very low pressures where X is much larger than the distance between the plates, the molecule bounces back and forth between the plates and only rarely collides with another gas molecule. In this case the mean free path does not enter the calculation, and the value of depends on the separation of the plates. At these low pressures the thermal conductivity is proportional to the pressure, since it must be proportional to A, and X does not appear in the formula to compensate for the pressure dependence of N. 

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