Question 2.170: A certain closed cell foam used as an insulating material in...
A certain closed cell foam used as an insulating material in houses is manufactured in such a way that the cells are initially filled with a polyatomic gas of molecular weight ∼60. After several years the gas diffuses out of the foam and is replaced by dry air (mean molecular weight ∼ 30 ). Assuming that the insulating property arises largely from the thermal conductivity
of the gas.
Discuss the factors which influence the thermal conductivity of the gas. For each factor make an argument for whether the insulating ability increases or decreases. What is the overall effect upon the insulating ability?
The thermal conductivity is \kappa \sim \lambda \bar{v} n C_{v}, \text { where }\lambda is the mean free path, \overline{ v } the mean speed and n the number density of the gas molecules and c_{v}, is the thermal capacity per molecule. We have \bar{v} \propto \sqrt{T / A}, where A is the molecular weight. n \lambda \propto 1 / \sigma, \text { where } \sigma is the cross section of a molecule, being \propto A^{2 / 3} So
\kappa \propto A^{-2 / 3} \sqrt{T / A}=\sqrt{T} / A^{7 / 6} .Thus the molecular weight is the most important factor. The overall insulating ability decreases when the poly-atomic gas is replaced by dry air.