A sample of gas contains $3.0\times {10}^{24}$ molecules. Calculate the volume of the gas at a temperature of $300 \mathrm{K}$ and a pressure of $120\mathrm{kPa}$.

A cylinder of hydrogen has a volume of $0.100{ \mathrm{m}}^3$. Its pressure is found to be $20$ atmospheres at $20°\mathrm{C}$. Calculate the mass of hydrogen in the cylinder.

A cylinder of hydrogen has a volume of $0.100{ \mathrm{m}}^3$. Its pressure is found to be $20$ atmospheres at $20°\mathrm{C}$. If it were instead filled with oxygen to the same pressure, how much oxygen would it contain? (Molar mass of ${\mathrm{H}}_2=2.0 \mathrm{g}{ \mathrm{mol}}^{-1}$, molar mass of ${\mathrm{O}}_2=32 \mathrm{g}{ \mathrm{mol}}^{-1}$, $1$ atmosphere $=1.01\times {10}^5 \mathrm{Pa}$).

Check that the SI base units on the left-hand side of the equation $p=\frac{1}{3}\left(\frac{Nm}{V}\right)<c^2>$ are the same as those on the right-hand side.

The quantity $\mathrm{Nm}$ is the total mass of the molecules of the gas, i.e. the mass of the gas. At room temperature, the density of air is about $1.29 \mathrm{kg}{ \mathrm{m}}^{-3}$ at a pressure of ${10}^5 \mathrm{Pa}$. Use these figures to deduce the value of $<{\mathrm{c}}^2>$ for air molecules at room temperature.

The quantity $\mathrm{Nm}$ is the total mass of the molecules of the gas, i.e. the mass of the gas. At room temperature, the density of air is about $1.29 \mathrm{kg}{ \mathrm{m}}^{-3}$ at a pressure of ${10}^5 \mathrm{Pa}$. Find a typical value for the speed of a molecule in the air by calculating $\sqrt{<c^2>}$. How does this compare with the speed of sound in air, approximately $330 \mathrm{m}{ \mathrm{s}}^{-1}$ ?-