EQUIPARTITION AND DEGREES OF FREEDOM

By 1860 had the distribution of molecular speeds; now he asked how energy shares itself out among the ways a molecule can move. His answer, sharpened by Boltzmann over the next decade, is one of the most powerful counting arguments in physics. In thermal equilibrium energy distributes itself with perfect democracy: every independent quadratic term in the energy — every degree of freedom that enters as a square — carries, on average, exactly the same amount, $\tfrac{1}{2}k_B T$.

A monatomic atom can only fly through space, so it has three quadratic terms, one for each component of velocity: $\tfrac{1}{2}mv_x^2 + \tfrac{1}{2}mv_y^2 + \tfrac{1}{2}mv_z^2$. A diatomic molecule can also tumble end over end, adding two rotational terms. A bound atom in a crystal can store energy in the stretch of its bonds as well as in its motion. Count the squares and equipartition hands you the energy.

If a molecule's energy contains $f$ quadratic degrees of freedom, the Equipartition theorem says its average energy is $f$ helpings of $\tfrac{1}{2}k_B T$:

$\langle E \rangle = \frac{f}{2}\,N k_B T$

In words: the internal energy of $N$ molecules is just the number of quadratic degrees of freedom, times $\tfrac{1}{2}k_B T$, times the number of molecules. Nothing about the chemistry survives — only the count $f$ matters. Helium with $f = 3$, nitrogen with $f = 5$, a crystal atom with $f = 6$: each is a different value of one small integer.

Differentiate the energy with respect to temperature and you have the heat capacity — the energy it takes to warm the substance by one degree:

$C_v = \frac{f}{2} N k_B \quad\Longrightarrow\quad C_v^{\text{molar}} = \frac{f}{2} R$

In words: each degree of freedom contributes $\tfrac{1}{2}R$ — about $4.16$ J/(mol·K) — to the molar heat capacity. Monatomic gases should sit at $\tfrac{3}{2}R \approx 12.5$, diatomic gases at $\tfrac{5}{2}R \approx 20.8$, and simple solids at $3R \approx 24.9$. That last number is the Dulong–Petit law, measured forty years before anyone could explain it — and here it falls out of pure counting, $f = 6$ for an atom held by springs in three directions. A single theorem reproduces a table of heat capacities that took a century of calorimetry to assemble.

Equipartition also delivers a number promised earlier in the branch. The ratio of heat capacities at constant pressure and constant volume,

$\gamma = \frac{C_p}{C_v} = \frac{f+2}{f}$

In words: $\gamma$ is fixed entirely by the degrees of freedom, because heating at constant pressure always costs one extra $R$ of expansion work. For a monatomic gas $f = 3$ gives $\gamma = \tfrac{5}{3}$; for a diatomic gas at room temperature $f = 5$ gives $\gamma = \tfrac{7}{5}$. Those are exactly the values that set the speed of sound in air and the steepness of an adiabat — the empirical exponents from the study of isothermal and adiabatic processes, now explained rather than measured.

The theorem is too good. If every degree of freedom always carried $\tfrac{1}{2}k_B T$, heat capacities would be constants of nature. They are not. Cool hydrogen gas below about $100$ K and its heat capacity slides from $\tfrac{5}{2}R$ down to $\tfrac{3}{2}R$ — the molecule stops rotating, behaving like a monatomic atom. Classically this is impossible: a dumbbell can always be set spinning, however gently.

The resolution is quantum. Rotational energy comes in discrete steps of size $\sim\hbar^2/I$, and when $k_B T$ falls below that gap there is simply not enough thermal energy to excite even the first rotation. The mode goes silent — it freezes out. Equipartition assumed energy was continuous; it is not.

The same failure strikes solids from the other direction. Dulong–Petit holds at room temperature, but cool a crystal and its heat capacity collapses toward zero — far below the classical $3R$. took the first quantum shot at it in 1907, treating every atom as an identical oscillator with quantised energy; the curve bent the right way but fell too fast. In 1912 did it properly, treating the vibrations as collective sound-like waves of the whole lattice — quantised, and now called phonons.

The temperature scale of the collapse is the Debye temperature: above it a solid behaves classically and obeys Dulong–Petit; below it the high-frequency lattice modes freeze out one by one and $C_v$ falls as $T^3$. Diamond's Debye temperature is over $2000$ K, which is why diamond's heat capacity is anomalously low at room temperature — most of its vibrational modes are still frozen. A second warning bell that classical physics was incomplete.

The third failure was fatal. Apply equipartition to the electromagnetic field in a hot cavity and you must give $\tfrac{1}{2}k_B T$ to each of its standing-wave modes — and there are infinitely many of them, packed ever more densely toward short wavelengths. The predicted energy diverges: every warm object should blaze with infinite ultraviolet light. This is the ultraviolet catastrophe, and it is equipartition pushed to the point of absurdity.

In 1900 Max Planck stopped it the only way it could be stopped — by declaring that the modes could not take energy continuously, only in quanta $E = h\nu$. High-frequency modes need a quantum too large for the available $k_B T$, so they stay dark, and the energy is finite. Planck thought he was patching thermodynamics; he had started quantum mechanics. Equipartition, the triumph of the classical kinetic theory, turned out to be the precise place where that theory broke — three cracks, each opening onto the quantum world told in full in the QM branch.

Behind all three cracks stands the same molecular picture that gives the Maxwell–Boltzmann distribution its shape and makes Brownian motion visible — a picture that, by 1910, no one could any longer deny.