12.3 Equipartition of Energy, Specific Heats and Mean Free Path

Heat one mole of helium in a sealed container by one kelvin and it takes $12.5\ \text{J}$. Do the same with one mole of nitrogen and it takes $20.8\ \text{J}$ — two-thirds as much again — although at the same temperature the molecules of both gases have exactly the same mean translational kinetic energy, $\tfrac32k_BT$. Where does the extra energy go? A nitrogen molecule is a dumb-bell of two atoms: besides flying about, it can tumble end over end. Heating it means speeding up the tumbling as well as the flying.

This section counts the independent ways — the degrees of freedom — in which a molecule can hold energy, and states the law of equipartition of energy: in thermal equilibrium each one carries, on average, the same energy $\tfrac12k_BT$. That single rule predicts the molar heat capacities $C_v$ and $C_p$ and their ratio $\gamma$ for monatomic, diatomic and polyatomic gases, and the heat capacity $3R$ of solids — and it shows, where it fails, the first hint of quantum physics. Finally you will see why a molecule moving at $500\ \text{m/s}$ takes so long to cross a room: it travels only about a ten-thousandth of a millimetre — its mean free path — before it hits another molecule and changes direction.

Degrees of freedom

To describe the position of a single atom you need three coordinates, $x$, $y$ and $z$, and its kinetic energy has three independent parts: $\tfrac12mv_x^2+\tfrac12mv_y^2+\tfrac12mv_z^2$. A rigid diatomic molecule can also rotate. Take the line joining its two atoms as the $z$-axis: it can spin about the $x$- and $y$-axes, with rotational kinetic energies $\tfrac12I_x\omega_x^2$ and $\tfrac12I_y\omega_y^2$ (Section 6.3). Spin about its own axis does not count — the atoms are so nearly point-like that the moment of inertia about that axis is negligible. If the bond is not rigid, the two atoms can also vibrate along it, like two masses joined by a spring, adding a vibrational kinetic energy and an elastic potential energy, $\tfrac12\mu\left(\dfrac{dr}{dt}\right)^2+\tfrac12k(r-r_0)^2$.

For translation you have already proved it: $\tfrac12m\overline{v^2}=\tfrac32k_BT$, so each of $\tfrac12m\overline{v_x^2}$, $\tfrac12m\overline{v_y^2}$, $\tfrac12m\overline{v_z^2}$ is $\tfrac12k_BT$ (Result 12.6). The law says that collisions do the same to every other kind of motion: if the molecules tumbled more slowly than their share, collisions would on average speed up the tumbling at the expense of translation, until the shares were equal. The law was found by Maxwell and Boltzmann; it is a result of classical (Newtonian) statistical physics, and it holds when every mode can take up energy freely — we shall see when that fails.

Specific heat capacities of gases

The molar heat capacity at constant volume, $C_v$, is the heat needed per mole per kelvin when no work is done, so all of it goes into internal energy: $C_v=\dfrac{1}{n}\dfrac{\Delta U}{\Delta T}$. At constant pressure the gas also expands and does work, and for an ideal gas $C_p=C_v+R$ (Mayer's relation, from Chapter 11, Thermodynamics). Equipartition now predicts both.

How good are the predictions? Measured values at room temperature (in $\text{J mol}^{-1}\text{K}^{-1}$, with $\tfrac32R=12.5$, $\tfrac52R=20.8$, $3R=24.9$):

The agreement for monatomic and simple diatomic gases is excellent — a triumph of the molecular picture, since these numbers were measured with calorimeters long before anyone could see a molecule. For $\mathrm{CO_2}$ and $\mathrm{CH_4}$ the measured values are higher than the rigid prediction because some of their vibrations already take up energy at room temperature.

Specific heat capacity of solids — and where equipartition fails

In a solid each atom is held near a fixed position by its neighbours, as if by springs, and vibrates about it in three independent directions. Each direction is an oscillator with a kinetic and a potential term, so each atom has $f=6$ squared terms and mean energy $6\times\tfrac12k_BT=3k_BT$. A solid hardly expands when heated, so the work done is negligible and $C_p\approx C_v$.

Where classical theory fails. Equipartition predicts heat capacities that do not depend on temperature. Experiment disagrees at low temperatures. The $C_v$ of hydrogen gas falls from $\tfrac52R$ to $\tfrac32R$ below about $100\ \text{K}$, as though its molecules forgot how to rotate, and rises towards $\tfrac72R$ above about $2000\ \text{K}$, when the vibration finally joins in. The heat capacity of every solid falls towards zero as $T\to0$, and for diamond it is well below $3R$ even at room temperature. The explanation came from quantum theory: energy can be taken up by rotation and vibration only in steps of a definite size, and when $k_BT$ is much smaller than the step, those modes are "frozen out" and receive no share. Classical equipartition is the high-temperature limit.

Mean free path

If air molecules move at $500\ \text{m/s}$, why does the smell of perfume take minutes to cross a room? Because a molecule does not travel in a straight line for long. It collides with another molecule, is knocked in a new direction, collides again, and so wanders along a zigzag path. Its net progress is very slow.

Model each molecule as a sphere of diameter $d$. Two molecules collide if their centres pass within $d$ of each other. So, as a molecule moves a distance $L$, it hits every molecule whose centre lies inside a cylinder of radius $d$ — cross-sectional area $\pi d^2$ — around its path. That cylinder has volume $\pi d^2L$ and contains $n_V\pi d^2L$ molecules, so the molecule makes that many collisions in the distance $L$, and the distance per collision is $\dfrac{1}{n_V\pi d^2}$. Allowing for the motion of the other molecules (what matters is the relative speed, which on average is $\sqrt2$ times the speed of one molecule) reduces this by a factor $\sqrt2$.

Exercises

Take $R=8.31\ \text{J mol}^{-1}\text{K}^{-1}$, $k_B=1.38\times10^{-23}\ \text{J/K}$, $N_A=6.02\times10^{23}\ \text{mol}^{-1}$ and $1\ \text{atm}=1.013\times10^5\ \text{Pa}$. Treat diatomic gases as rigid and all gases as ideal unless told otherwise.