Year 1 Sem 2/LC Temperature and Matter/Lecture 17
← Previous lecture · Module · Next lecture →
Recap from last lecture:
- We can predict the microscopic behaviour of a system using a probability density function defined using macroscopic terms.
- We can predict the likelihood of a given energy state begin populated for a system (discrete or continuous) with the Boltzmann factor.
-
For a two level system with $E_1 < E_0$ we have: $$\Pr\left(E_i\right) = \frac{\exp\left(\frac{-E_i}{k_B T}\right)}{\exp\left(\frac{-E_0}{k_B T}\right) + \exp\left(\frac{-E_1}{k_B T}\right)}$$
-
While not explicitly covered last lecture, this trivially generalises to a system with $N$ levels: $$\Pr\left(E_i\right) = \frac{\exp\left(\frac{-E_i}{k_B T}\right)}{\sum_{j=0}^{N-1} \exp\left(\frac{-E_j}{k_B T}\right)}$$
Degeneracies
In an atom, electrons can have the same energy in multiple different combinations (multiple ways of having the same result). For example, considering only one energy level, electrons can be spin up or spin down. This gives two possible options, hence a degeneracy of two.
With electrons in shells, the degeneracy gives the number of electrons which can fit in a shell. The Pauli exclusion principle says that no two atoms can occupy a shell if they share a quantum number, so the number of possible combinations of distinct states is equal to the number of possible electrons in the shell.
Given the degeneracy $g(E)$, we can modify the Boltzmann Factor accordingly:
For $n = 1$, $g(E) = 2$, for $n = 2$, $g(E) = 4$, and so on.
Boltzmann Factors and Ideal Gases
For an ideal gas, we have some assumptions (in terms of energy) that have been discussed previously. The most important of these in terms of energy is:
- Internal energy all comes from the average kinetic energy of the molecules (temperature) and not from potential energy.
-
The Boltzmann Factor for an ideal gas hence takes the form: $$\Pr(E_i) \propto \exp\left(\frac{-E_K}{k_B T}\right)$$
Our other assumptions are:
- The gas is in thermal equilibrium, hence $\langle E_K\rangle$ is constant.
- No gravity (as molecules are extremely light).
- Assume that the average separation of each molecule is much larger than the molecule size, hence collisions are infrequent.
- There is a very large number of molecules $\sim 10^{23}$ to allow us to treat the gas statistically without high statistical uncertainties.
- All molecules have the same mass and size. This is not a good assumption for gases made up of multiple component gases mixed.
- We assume that the direction of molecular motion is effectively random, i.e. is unbiased to any particular direction.
- Molecules can collide, but do so elastically (and as previously mentioned, infrequently).
- Molecules are not relativistic, so $E_K = 0.5 mv^2$ holds.
Model of Ideal Gas
We can split the velocity of a gas into three cartesian coordinates by adding in quadrature:
By assumption 6, we assume that all of these components are effectively equal, so we can consider the x-direction alone initially (and extrapolate out).
We want to consider the probability of having a velocity somewhere between $v_x$ and $v_x + dx$.
And using the standard integral: $\int_{-\infty}^{\infty} e^{-ax^2} \, dx = \sqrt{\pi / a}$, we have:
And as $a = m/2k_BT$:
And as the Boltzmann factor for this x-directional motion is $\exp \left(mv_x^2 / 2k_B T\right)$:
We finally have:
As we’re experiencing random gas motion, the distribution is centred on 0, as velocity can be positive or negative.
As we increase the temperature, we increase the chance of occupying higher energy states (i.e. higher $v_x$).
This distribution has no degeneracies, as there’s only one way for a particle to have a specific $v_x$, which is by travelling at that value. It either has a specified velocity, or it simply doesn’t.
3D Model
Now we have a single component:
We can extend this to 3D:
Density of States
While we had no degeneracies in each component direction, we do have degeneracies for $v$ itself. For example, $v^2 = 100$ could mean that x, y are zero and $z = 10$, or any other number of combinations.
We therefore want to define the degeneracies as a function of $v$. As a model, consider a grid of infinitesimally small dots in 2D (but for the actual application we would want a 3D model).
Our final velocity in 2D is $v^2 = v_x^2 + v^2_y$, so if each axis represents possible values for $v_x$ and $v_y$, $v$ is given as the circumference of a circle with radius $v$. The density of states $g(v) dv$ is given as the ring formed between $v$ and $v + dv$.
In 2D, the degeneracy is therefore given by:
In 3D, rather than a ring this would be a spherical shell surface:
Applying this to the Maxwell-Boltzmann distribution, we obtain:
And as $v^2 = v_x^2 + v_y^2 + v_z^2$:
This finally gives us the Maxwell-Boltzmann distribution we encountered before (although we have derived it for molecules vs moles):
