Question

How does temprature effect degree of freedom quntum mechanically?????​

Answer 1

Answer:

Explanation:

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:

E=hν=hcλ

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=hcλk

For the 1711 cm−1 mode of water this gives us a temperature of around 2600 K.

Answer 2

$\large\blue{heya mate!!}$

•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.

• For example, oxygen gas (O2) has a rotational temperature of 2.08 K and a vibrational temperature of 2256 K.