# Variation of degrees of freedom with temperature

I read in my friend's thermodynamics notes that polyatomic molecules have a different degree of freedom at high temperatures (viz. non-linear triatomic molecules have 9 dof but only 7 are accessible at low temperatures). I was wondering at what temperature the switch takes place. Say i am told that there exists a non-linear tri-atomic gas whose temperature is say 500K. Is this temperature high enough for the change in dof? Is the given information not enough?

And what happens at high temperatures that causes the change in dof?

• Search term that may help "quantum freeze out". – dmckee --- ex-moderator kitten Jun 28 '18 at 16:42
• "polyatomic atoms" can't be right. – Javatasse Jun 28 '18 at 17:20
• i'm sorry. i meant molecules. i'll edit it right away. – Ujjwal Barman Jun 29 '18 at 21:05

The reason that new degrees of freedom open up at higher temperatures is because, with the possible exception of translational kinetic energy, degrees of freedom are quantized. Due to quantum mechanics, the molecules can only vibrate/rotate/get excited with certain discrete energies, and there is a lowest energy at which this happens. The particular energy of this "lowest excited state" determines the temperature at which the degree of freedom "turns on," by which we mean that it is accessible to a large number of particles in the ensemble (in reality, you'll nearly always have a few highly excited particles at any temperature simply due to the nature of the statistical distribution of the particles, but in most situations this tiny fraction is irrelevant). As such, the temperature at which new degrees of freedom turn on is highly dependent on the specific material that is being examined.

Water, for example, has rotational energy levels at very low energies (low enough to be excited by microwaves) due to its asymmetry, while its vibrational energy levels are somewhat higher (can only be excited by higher-energy infrared radiation). Incidentally, this is why both infrared radiation and microwaves are perceived as heat - they put energy directly into one of these degrees of freedom, and this energy redistributes to the translational degrees of freedom to give a higher temperature.

Water is a bit of a complicated case, as it's a nonlinear asymmetric molecule, so for more concrete predictions, the numbers to look for are the rotational temperature and the vibrational temperature, which are material-specific, and tend to only be calculated for linear molecules. The rotational temperature is typically much lower than the vibrational temperature, as rotational motion typically has a much lower first excited state than vibrational motion. Some typical values for each are found on Wikipedia (https://en.wikipedia.org/wiki/Rotational_temperature, https://en.wikipedia.org/wiki/Vibrational_temperature). For example, oxygen gas (O$_2$) has a rotational temperature of 2.08 K and a vibrational temperature of 2256 K. This means, at room temperature, the typical energy of the oxygen gas molecules is more than large enough to excite rotational modes, but vibrational modes will be essentially out of reach.

As such, if you were to heat oxygen gas to above 2256 K, then you would see a jump in heat capacity corresponding to the vibrational modes now being accessible places to store energy; likewise, cooling oxygen to below 2 K will cause a decrease in heat capacity, as the rotational modes are no longer easily accessible.

Let's consider the water molecule as it's a nice simple triatomic. There is a nice page illustrating the normal vibrational modes on the University of Liverpool web site.

The lowest energy vibration has a wavenumber of $1711$ cm$^{-1}$. For vibrational excitations the spacing between energy levels is:

\begin{align} E &= h\nu \\ &= \frac{hc}{\lambda} \end{align}

which gives an energy of about $0.21$ eV if we plug in the wavenumber above.

If we consider a gas at some temperature $T$ then the kinetic energy of the gas molecules is of order $kT$, so that means when the molecules collide we have about $kT$ worth of energy that could go into exciting vibrational transitions. But those vibrational transitions can only be excited if their energy spacing is less than $kT$, and actually it needs to be significantly less than $kT$. If the energy spacing is more than $kT$ then there simply isn't enough energy around to cause vibrational transitions so the degree of freedom represented by the vibrational transitions is inaccessible and in effect doesn't exist. That's why the number of degrees of freedom is reduced.

We can get an idea of the temperature at which the vibrational degrees of freedom become accessible simply by setting the vibrational energy equal to $kT$. This gives us:

$$T = \frac{hc}{\lambda\,k}$$

For the $1711$ cm$^{-1}$ mode of water this gives us a temperature of around $2600$ K.

All this "degrees of freedom" counting is a dangerous business. The behaviour around the threshold energies is complex (and around it the heat capacity will have intermediate values).

The point is, that degrees of freedom "freeze out", when $T \to 0$. If the temperature gets lower than the energy gap to the lowest lying excitation, the contribution to the heat capacity will be exponentially suppressed, because the probability of finding the system in the exited states will go like $e^{-E/T}$. The excitation energies for rotational and vibrational degrees of freedom show such limits due to quantum mechanics.

This can be easily shown by considering a harmonic oscillator coupled to a heat bath, as demonstrated in this post.

And even the translational degrees of freedom, whose excitation spectrum is continuous, will freeze out at sufficiently low temperatures due to the fact that the particles are either Bosons, so the classical ideal gas description will break down (and has to be replaced by the ideal quantum gas description). For Fermions the heat capacity gets linear for $T \ll T_F$. For Bosons the calculation get more complicated due to Bose-Einstein condensation. The result is an even stronger suppression of the heat capacity.