Kinetic Theory and Real Gases: Quick Revision

A brief note on Kinetic Theory and Real Gases. Read it before a test. Try each question at the end before you open its answer.

Kinetic theory

• Assumptions: tiny molecules, random motion, no forces except in collisions, elastic collisions, average kinetic energy proportional to $T$.
• $PV = \tfrac{1}{3}Nmu_{\text{rms}}^2$. For one mole, $u_{\text{rms}} = \sqrt{3RT/M}$ with $M$ in kg/mol.
• Average kinetic energy: $\tfrac{3}{2}kT$ for a molecule and $\tfrac{3}{2}RT$ for a mole, the same for every gas.
• At the same $T$, the lighter gas is faster: $u \propto 1/\sqrt{M}$.

The Maxwell distribution

• Fraction of molecules between $u$ and $u + du$ is $f(u)\,du$. The area under the curve is $1$.
• $u_{\text{mp}} = \sqrt{2RT/M}$, $\bar{u} = \sqrt{8RT/\pi M}$, $u_{\text{rms}} = \sqrt{3RT/M}$. The ratio is $1 : 1.128 : 1.225$.
• A higher $T$ moves the peak right and lowers it. The fast tail grows a lot. A lighter gas has a wider curve.

Real gases

• $Z = \dfrac{PV_m}{RT}$. $Z < 1$: attraction wins. $Z > 1$: size wins. $Z \to 1$ as $P \to 0$.
• van der Waals: $\left(P + \dfrac{an^2}{V^2}\right)(V - nb) = nRT$. $a$ is the attraction, $b$ the excluded volume.
• $T_c = \dfrac{8a}{27Rb}$, $P_c = \dfrac{a}{27b^2}$, $V_c = 3b$, $Z_c = \tfrac{3}{8}$, and $T_B = \dfrac{a}{Rb}$.
• A gas below $T_c$ can be liquefied by pressure. Above $T_c$ it cannot.

A worked example

Find the rms speed of argon, $M = 39.95\ \text{g/mol}$, at $350\ \text{K}$. Then find the temperature at which helium, $M = 4.003\ \text{g/mol}$, has the same rms speed.

1. Argon. $u_{\text{rms}} = \sqrt{\dfrac{3 \times 8.314 \times 350}{0.03995}} = 467\ \text{m/s}$.
2. Same speed. The speed depends on $T/M$, so $T_{\text{He}} = 350 \times \dfrac{4.003}{39.95}$.
3. The temperature. $T_{\text{He}} = 35.1\ \text{K}$.

Practice with new numbers

A program to try

The program finds the critical temperature, the critical pressure and the Boyle temperature from the constants $a$ and $b$.

Mistakes to avoid

• The molar mass in g/mol in $u = \sqrt{3RT/M}$. Use kg/mol.
• One speed for all molecules. The molecules have a spread of speeds.
• A wrong sign in $Z$. $Z < 1$ means attraction, and $Z > 1$ means that the size of the molecules wins.
• Forgetting $nb$ in the volume. The free volume is $V - nb$.
• Trying to liquefy a gas above $T_c$ by pressure.

Check yourself

1. How does the rms speed change when the temperature rises from $200\ \text{K}$ to $800\ \text{K}$?

2. Which is smaller, the most probable speed or the average speed?

3. Can nitrogen, $T_c = 126.2\ \text{K}$, be liquefied at room temperature by pressure alone?

Sources

• OpenStax, Chemistry 2e, Chapter 9, sections 9.5 and 9.6.
• CRC Handbook of Chemistry and Physics, for the van der Waals and critical constants.