Year 1 Sem 2/LC Temperature and Matter/Lecture 19
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Average Kinetic Energy
In 3D, we derived last lecture:
Each component contributes equally, so:
Remember from the second module of the course we have:
And the first law of thermodynamics:
At a constant pressure or volume:
We can define an additional value called Enthalpy (free energy):
Hence:
For ideal gases, internal energy comes solely from kinetic energy. As we now have average kinetic energy as a function of $T$, we can now say (for per-atom and per-mole cases respectively):
This works perfectly for monatomic gases, where Mayer’s relation also lets us define $C_P$:
But evidently breaks for diatomic molecules… They seem to have $C_V = 5R/2$? We’ll investigate why later.
Law of Equipartition of Energy
The Law of Equipartition says that any substance in equilibrium at temperature $T$ has an average energy of $k_B T / 2$ per degree of freedom.
A degree of freedom is any term in the expression for the energy of the system which has a squared position or squared velocity term.
In 1D, our expression for kinetic energy is:
And in 3D:
Example
Consider a gas in a harmonic potential, acting as an oscillator in 1D:
Or in a linear potential:
Dulong-Petit Heat Capacity of a Solid
We model the inter-atomic forces between atoms as springs with spring constant $K$. Each spring has vibrational energy given by:
If we assume that the substance forms a perfect cubic lattice where every atom is connected to six other atoms, each spring is shared by two atoms so we say:
This ends up being a surprisingly good fit for pure elements:
Diatomic Molecules
Diatomic molecules can be modelled as being connected by a single spring:
However, they are also free to move around the volume in which they occupy, this gives some extra translational energy:
Finally, they’re also able to rotate around the axis joining them (perpendicularly):
In theory this means that a diatomic molecule has 7 degrees of freedom. We saw earlier however that the values for $C_V$ for a diatomic molecule is $C_V = 5R/2$, only implying 5 DoF? Lets look at some actual data:
Our monatomic gases behave well, with values close to $3R/2$, but our diatomic gases seem to be more varied, with Chlorine having a much higher value between $5R/2$ and very roughly $7R/2$?
We can’t say that our assumption of 7DoF is wrong, because Chlorine sits nicely between them. It means that this cannot, for example, simply lack rotational energy.
The states where a molecule has vibrational energy occurs at larger energies. In order to get a reasonable probability of measuring a state with a certain energy from the Boltzmann factor:
We require $k_BT \not\ll E_i$. This gives a transition, where for some temperatures there is some intermediary probability where there’s a good chance of measuring particles with and without sufficient energy for the final degrees of freedom.
This gives a plot like this:
The translational component is always present, as our gas molecules are always present. The vibrational and rotational modes only appear at certain levels where there is sufficient energy to “unlock” them.
The degrees of freedom can be considered fully excited if:
And if:
There is insufficient energy to expect to encounter any molecules with sufficient energy for the motion associated with the degree of freedom, and it is said to be frozen out.