Equipartition, Specific Heats and Mean Free Path: Quick Revision

A brief note on Equipartition, Specific Heats and Mean Free Path. Read it before a test. Try each question at the end before you open its answer.

Degrees of freedom

• Moving: 3 for every molecule.
• Turning: 0 for one atom, 2 for a straight molecule, 3 for a bent one.
• Vibrating: 2 for each active vibration, one kinetic and one potential term.
• Examples: neon $f = 3$, rigid carbon dioxide $f = 5$, rigid methane $f = 6$.

Equipartition

• Each squared term has $\tfrac12 k_BT$ on average.
• One molecule: $E = \tfrac{f}{2}k_BT$.
• A gas: $U = \tfrac{f}{2}nRT$.
• Translation alone: $\tfrac32 k_BT$ for every gas.

Heat capacities

• Gas: $C_v = \tfrac{f}{2}R$, $C_p = C_v + R$, $\gamma = 1 + \tfrac{2}{f}$.
• Polyatomic with $v$ vibrations: $C_v = (3 + v)R$.
• Mixture: $C_v = \dfrac{n_1C_{v1} + n_2C_{v2}}{n_1 + n_2}$. Never average $\gamma$.
• Solid: $C = 3R$ (Dulong and Petit), so $c = 3R/M$.
• Water: about $9R$.
• Low temperatures: turning and vibrating freeze, and $C$ falls. Quantum theory explains it.

Mean free path

• $\lambda = \dfrac{1}{\sqrt{2}\,\pi d^2 n}$, where $d$ is the diameter.
• With $n = p/k_BT$: $\lambda = \dfrac{k_BT}{\sqrt{2}\,\pi d^2 p}$.
• Collisions each second: $\nu = \bar v/\lambda$.
• At a fixed $n$, $\lambda$ does not change with $T$.

A worked example

$2.0\ \text{mol}$ of neon is mixed with $1.0\ \text{mol}$ of rigid carbon dioxide. Find $\gamma$.

1. Each gas. Neon: $C_v = \tfrac32 R$. Carbon dioxide: $C_v = \tfrac52 R$.
2. The mixture. The top line is $2.0\ \text{mol} \times 1.5R + 1.0\ \text{mol} \times 2.5R = 5.5\ \text{mol} \times R$. So $C_v = \dfrac{5.5\ \text{mol} \times R}{3.0\ \text{mol}} = 1.83R$.
3. Gamma. $C_p = 2.83R$, so $\gamma = \dfrac{2.83}{1.83} = 1.55$.

Practice with new numbers

A program to try

The program works through the example above.

Mistakes to avoid

• Giving each term $k_BT$. It is $\tfrac12 k_BT$.
• Counting a vibration once. It counts twice.
• Averaging $\gamma$ for a mixture. Average $C_v$.
• Using the radius in $\lambda$. Use the diameter.

Check yourself

1. How many degrees of freedom does a rigid straight molecule have?

2. What is $C_v$ of a solid, for each mole?

3. The pressure of a gas halves at a fixed temperature. What happens to $\lambda$?

4. Why does $C_v$ of hydrogen fall at low temperatures?