I read a statement about participation theory.

The equipartition theorem is generally valid only at high temper- ature, so that the thermal energy is larger than the energy gap between quantized energy levels. Results based on the equipar- tition theorem should emerge as the high-temperature limit of more detailed theories.

Does it mean that, we can't apply the theorem at low temperature?

Let assume room temperature of diatomic atom has 5 degrees of freedoms and therefore can't we write the mean energy be $\frac{5}{2} kT$.

  • 2
    $\begingroup$ Just as a side note, room temperature is not really considered to be cold. Now, deduce away... $\endgroup$ – ClassicStyle Aug 31 '15 at 16:29
  • $\begingroup$ May I know the energy range where the quipartition theory would be valid? $\endgroup$ – Numerical Person Aug 31 '15 at 16:34

Here is the heat capacity of nitrogen gas (a diatomic molecule) as a function of temperature:

heat capacity of nitrogen verses temperature

$C_V$ is the heat capacity at constant volume; $C_p$ is the heat capacity at constant pressure. Below 500 kelvin, $C_V$ is close to $(5/2) N_A k_B$ (about 20.8 joules per kelvin per mole), where $N_A$ is Avagadro's number and $k_B$ is the Boltmann constant. This is an example of equipartition: there are 5 degrees of freedom per molecule, consisting of 3 translational and 2 rotational. But note that at higher temperature the heat capacity increases. Eventually it tends to $(7/2) N_A k_B$ (about 29 J/K per mole). This is because the molecule has two more degrees of freedom, associated with vibration, so it is the equipartition theorem once again.

But this raises the question, why aren't all 7 degrees of freedom contributing at lower temperatures? It is owing to the fact that the equipartition theorem is indeed a high-temperature result, which only applies to those degrees of freedom such that the energy level gaps, especially the one from the ground to first excited state, are small compared to $k_B T$. In this example the vibrational energy levels have separations of order $k_B T_v$ where $T_v \simeq 1000$ kelvin, therefore the vibrational motion is not excited by thermal effects unless the temperature is of this order, and the full equipartition result is only achieved at temperatures large compared to $T_v$.

  • $\begingroup$ And at lower temperatures, the rotational energy of diatomic gases can also start to show quantization effects. In nitrogen it does not matter, but it is clearly measurable in H2. $\endgroup$ – Pieter Feb 28 at 0:07

According to quantum theory, expected average energy of harmonic oscillator when in contact with thermal reservoir of temperature $T$ is

$$ \epsilon_{av} = \frac{\hbar \omega}{2}+ \frac{\hbar \omega}{e^{\frac{\hbar\omega}{k_B T}}-1}. $$

If $k_B T \gg \hbar \omega$, this is approximately equal to

$$ k_B T, $$

else the average energy is lower.

For other models (not harmonic oscillator), things may be similar.

  • $\begingroup$ How will I correlate your equation with the expected heat capacity for diatomic molecule. Can you please do a bit math to connect the equations? $\endgroup$ – Numerical Person Aug 31 '15 at 16:46

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