Equipartition of Energy Tool

Physics Thermodynamics • Kinetic Theory of Ideal Gases

Written by STEM Calculators Team Published January 3, 2026 Updated February 24, 2026

4. Equipartition of Energy Tool

Equipartition (classical) assigns $\tfrac{1}{2}kT$ per quadratic degree of freedom. For $n$ moles: $U=\tfrac{f}{2}\,nRT$ and $C_v=\tfrac{f}{2}R$ (if $f$ is constant). With “freeze-out”, $f=f(T)$ and $C_v(T)=\dfrac{dU}{dT}$ changes with temperature.

Inputs support: pi, e, sqrt(), sin, cos, exp, log. Use * for multiplication.

Inputs

Molecule type (preset $f$): Atoms $N$ (polyatomic only): Amount $n$ (mol): Temperature $T$: $T$ unit: Include vibrations (high-T limit): Temperature model: Graph y-axis: Energy blocks per DOF:

For a diatomic at room temperature (typically vibrations “off”): $f\approx 3+2=5$ so $U=\tfrac{5}{2}nRT$ and $C_v=\tfrac{5}{2}R$.

Advanced: customize degrees + freeze-out temperatures

Translational DOF (usually 3): Rotational DOF (0/2/3): Vibrational modes $m$: DOF per mode (equipartition): Rotation activation $T_{\mathrm{rot}}$ (K): Vibration activation $T_{\mathrm{vib}}$ (K): Activation sharpness (bigger = steeper):

Freeze-out is a smooth toy model: rotational and vibrational contributions “turn on” around $T_{\mathrm{rot}}$, $T_{\mathrm{vib}}$. It’s meant to explain why measured $C_v$ changes with temperature (quantum effects).

Graph + animation controls

Graph $T_{\min}$ (K): Graph $T_{\max}$ (K): Progress $\tau\in[0,1]$: Play speed: Animation mode: Shade “high-T levels” band:

Graph supports drag-to-pan, wheel-to-zoom, and double-click/Reset view.

Ready

Steps

Enter values and click Solve.

Energy partition (visual)

Stacked energy bar + optional DOF blocks

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$\,C_v(T)\,$ graph (“why $C_v$ changes”)

Heat capacity at constant volume vs temperature

Hover: — Marker: —

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Theory — Equipartition of Energy

1) The equipartition theorem (classical)

In classical statistical mechanics, each independent quadratic term in the energy contributes an average energy $\tfrac{1}{2}kT$ per molecule (or $\tfrac{1}{2}RT$ per mole).

\[ \langle E \rangle = \frac{f}{2}kT \qquad\Rightarrow\qquad U = \frac{f}{2}\,nRT \]

Here $f$ is the number of quadratic degrees of freedom (DOF): translation, rotation, and (at high enough $T$) vibration.

2) Degrees of freedom $f$: translation, rotation, vibration

Translation: always $f_{\text{trans}}=3$ in 3D.
Rotation: monatomic: $0$; diatomic/linear: $2$; nonlinear polyatomic: $3$.
Vibration: each vibrational mode contributes $2$ quadratic terms (kinetic + potential), so $f_{\text{vib}}=2m$ where $m$ is the number of vibrational modes.

For a molecule with $N$ atoms:

\[ m = \begin{cases} 3N-5 & \text{(linear)}\\[2pt] 3N-6 & \text{(nonlinear)} \end{cases} \qquad\Rightarrow\qquad f_{\text{vib}} = 2m \]

Example (diatomic): $f \approx 3 + 2 = 5$ at moderate $T$ (translation + rotation). At high $T$, vibration activates and $f \to 7$.

3) Heat capacity at constant volume

If $f$ is constant (pure classical equipartition):

\[ U = \frac{f}{2}nRT \quad\Rightarrow\quad C_v = \left(\frac{\partial U}{\partial T}\right)_V = \frac{f}{2}nR \]

Per mole: $C_{v,\mathrm{mol}}=\tfrac{f}{2}R$. This predicts plateaus like $\tfrac{3}{2}R$, $\tfrac{5}{2}R$, $\tfrac{7}{2}R$, …

4) Why $C_v$ changes with temperature (quantum freeze-out)

In reality, not all modes are accessible at low temperature because energy levels are quantized. A “frozen” mode contributes little to $U$ and $C_v$.

A simple explanatory model is to let the effective degrees of freedom depend on $T$: $f=f(T)$. Then $U(T)=\tfrac{f(T)}{2}nRT$ and

\[ C_v(T) = \frac{dU}{dT} = \frac{f(T)}{2}nR + \frac{nRT}{2}\frac{df}{dT}. \]

When a new set of modes “turns on”, $df/dT>0$ and $C_v(T)$ can rise above its plateau temporarily (a bump). This tool uses a smooth activation curve to illustrate that behavior.

5) Worked example (diatomic at room temperature)

At room temperature, vibrations are often effectively frozen for many diatomic gases, so take $f=5$:

\[ U = \frac{5}{2}nRT,\qquad C_v = \frac{5}{2}R\ \ (\text{per mole}). \]

Interpretation: translation (3) + rotation (2) active; vibration not yet contributing significantly.

Steps

Energy partition (visual)

\(\,C_v(T)\,\) graph (“why \(C_v\) changes”)

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