A molecule's shape decides how many independent ways it can store thermal energy. Every molecule gets 3
translational DOF; rotation and vibration depend on geometry. Pick a molecule, drag the temperature, and
watch which motions "wake up" — and how that changes Cv.
A linear molecule has only 2 rotational DOF, not 3: spinning point masses about the bond axis itself
changes nothing, since their moment of inertia about that axis is essentially zero. Each vibrational
mode carries two quadratic energy terms (kinetic + potential), so equipartition gives it a
full R, not R/2, once it's classically active:
The vibrational term is the Einstein heat-capacity function — the same math used for the
two-solid thermal-contact simulation, applied per normal mode instead of per oscillator. Each mode
has its own characteristic temperature θi (real spectroscopic values, in Kelvin);
the term smoothly turns on from 0 to 1 as T sweeps through θi.
For the three linear molecules, ½DOFrot is only the T≫θrot
limit — this sim actually evaluates rotation from the real quantum rigid-rotor partition
function, $z(T)=\sum_J (2J{+}1)e^{-\theta_{\text{rot}} J(J+1)/T}$, and differentiates it numerically
for Cv,rot. N₂ and CO₂ have θrot of a few kelvin or less, so
they stay deeply classical everywhere shown; H₂'s θrot≈85K sits right in
the visible range, which is why it's included here.
Why do vibrations (and sometimes rotations) "freeze out"?
Classical equipartition says every quadratic energy term should get ½kBT on
average, with no dependence on temperature at all. That's wrong once level spacing matters: a
quantum harmonic oscillator's energy levels are spaced by ħω, and when kBT is
much smaller than that spacing, there isn't enough thermal energy to reach the first excited level.
The mode sits frozen in its ground state and simply can't absorb heat — it contributes nothing
to Cv.
Once kBT ≫ ħω (T ≫ θi), the level spacing becomes
negligible compared to the thermal energy available, the mode behaves classically again, and it
contributes its full R. This is exactly why diatomic gases show Cv=5/2R at room
temperature (vibration frozen) but climb toward 7/2R only at very high T.
Rotation is quantized the same way (rigid-rotor levels EJ∝J(J+1)), with its own
θrot=ħ²/(2IkB). But θrot depends on the moment
of inertia I, and I grows fast with atomic mass and bond length — so for most real molecules
θrot is a few kelvin, far below any temperature where they're still a gas, and
rotation looks "always classical." H₂ is the exception: it's light enough (small I) that
θrot≈85K, so cooling it below roughly 50–100K visibly drops
Cv,rot/R below 1 — a real, historically important clue that pushed physicists toward
quantum statistics in the first place. Translational spacing, by contrast, is so tiny for any
macroscopic box that its freeze-out is never relevant to a real gas.
Argon (Ar) — 3D Model
Translation (always 3 axes shown, corner)
Rotation axis/axes
Vibration amplitude ≈ how "unfrozen" each mode is at this T. For multi-mode molecules, check just one mode below (and raise T) to see its shape in isolation. Drag to orbit, scroll to zoom.
