Question

One way to improve insulation in windows is to fill a sealed space between two glass panes with a gas that has a lower thermal conductivity than that of air. The thermal conductivity k of a gas depends on its molar heat capacity CV, molar mass M, and molecular radius r. The dependence on those quantities at a given temperature is approximated by k∝CV/r2M−−√. The noble gases have properties that make them particularly good choices as insulating gases. Noble gases range from helium (molar mass 4.0 g/mol, molecular radius 0.13 nm) to xenon (molar mass 131 g/mol, molecular radius 0.22 nm). (The noble gas radon is heavier than xenon, but radon is radioactive and so is not suitable for this purpose.)

Estimate the ratio of the thermal conductivity of Xe to that of He.

Estimate the ratio of the thermal conductivity of  to that of .

	0.061
	0.17
	0.10
	0.015

Answer 1

The dependence of thermal conductivity of a molecule on its molar heat capacity, molar mass, and molecular radius is, \(k \propto \frac{C_{r}}{r^{2} \sqrt{M}}\)

Both the molecules are monoatomic. Thus, there specific heats are same.

The ratio of thermal conductivity of Xe to that of He is calculated as follows:

\(\begin{aligned} \frac{k_{x_{e}}}{k_{H e}} &=\frac{\frac{C_{r}}{r_{y}^{2} \sqrt{M_{y}}}}{\frac{C_{T}}{r_{H}^{2} \sqrt{M_{H e}}}} \\ &=\frac{r_{12}^{2} \sqrt{M_{H e}}}{r_{x}^{2} \sqrt{M_{x}}} \\ &=\frac{(0.13 \mathrm{~nm})^{2} \sqrt{4.0 \mathrm{~g} / \mathrm{mol}}}{(0.22 \mathrm{~nm})^{2} \sqrt{131 \mathrm{~g} / \mathrm{mol}}} \\ &=0.061 \end{aligned}\)