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Indroduction to Statistical Mechanics: Building Up to the Bulk

Shaun Williams, PhD

Atomic Structure

Atomic Structure

Quantization of Properties

Quantizer Energy

A graph of energy versus time in a classical picture is a straight, continuous line. The same graph for a quantum system is a staircase in which only certain values are possible.
  • In a classical system, energy is a continuous function
  • In a quantum system, energy comes only in discrete values

Problem with the quantum nature

Particle in a Three-Dimensional Box

Energy Levels of a Three-Dimensional Box

A plot of the energy equation for a particle in a 3D box shows few energies that are low and the number of individual levels increase at higher energy.

States of Such a Particle

Use of Quantum Mechanics

Quantum States of Atoms

Energy Levels of the Hydrogen Atom

The energy levels of the hydrogen atom as far apart energetically at low energy and get closer together at higher energies.

Quantum State of Electrons

Term Symbols

Electron Spin

The Pauli Exclusion Principle

Degrees of Freedom

Translational Energy

Electronic State Transitions (UV-vis)

Vibrational State Transitions (IR)

Vibrational Modes

Vibrational Energies

Rotational States Transions (microwave)

Rotational State Degeneracies

Bulk Properties

Some Definitions

Balloon Example


Difficulty with QM

Macroscopic Properties


Statistical Mechanics

Common Assumptions in Statistical Mechanics

  1. Chemically identical molecules share the same physics
  2. Macroscopic variables are continuous variables
  3. Measured properties reflect the ensemble average
    • If we fix some values (e.g. volume and moles)
    • Ensemble - Set of all quantum states, same values
    • Microstate – a unique quantum state
    • \(\Omega\) – number of microstates in the ensemble

Ergodic Hypothesis

12-Member Ensemble

  • Here are 12 distinct microstates
  • All have the same properties
  • Let the particle exert a pressure of p0 on the walls next to it
  • Half the top walls have two particles next to it and half have one
  • \(\expect{p_{top}} = 1.5p_0\)
Consider a square. In that square there are three circles, two of one color and one of a third color. The circles must be on a corner of the square. There are 12 possible ways of arranging the three circles in the square that are all unique.

The Microcanonical Ensemble

  • Approaching a problem, we must decide which parameters are allowed to vary
  • More flexible is more realistic but more difficult to solve
  • If we fix all the extensive variables, \(E\), \(V\), and \(N\) we have our first ensemble – the microcanonical ensemble
Four particles are in boxes with the same volume. Those boxes that have the same total energy of the system are a microcanonical ensemble.

Properties of Microcanonical Ensemble

Advantages of Microcanonical Ensemble


Boltzmann Entropy

Entropy and Microstates

As the number of microstates increases, the entropy of the system increases. This increase is very fast at first, but slows down quickly.

\(\Omega\) versus \(g\)

Size of \(\Omega\)

Two Non-Interacting Subsystems

Gibbs Definition of Entropy

Example of Calculating \(\Omega\)

With six total particles, three in one energy state, two in the next higher energy state, and one in the next higher energy state, there are 60 distinct ways of identifying which particle (by number) is in what state.

The Ensemble Size

Using Stirling's Approximation

$$ S = k_\mathrm{B} \left[ \ln N! - \ln \prod_{i=1}^k N_i! \right] $$

Rewriting Entropy in Terms of Probability

$$ S = k_\mathrm{B} \sum_{i=1}^k N_i \left[ \ln N - \ln N_i \right] $$

Temperature and the Partition Function

The Ideal Gas

System and Reservoir

  • The box has:
    • Fixed volume, \(V\)
    • Fixed number of molecules, \(N\)
  • Energy will no longer be held constant
  • Reservoir strongly interacts with system
A part of a container of gas is identified as the system. The system has an energy and an ensemble size. The rest of the gas, the reservoir, has its own energy and ensemble size.

The Reservoir

  • The reservoir has:
    • \(E_r \gg E\)
    • \(E+E_r = E_T \gg E\)
  • \(\Omega_r\left(E_r\right) \gg \Omega\left(E\right)\)
  • The ensemble size of the universe is $$ \Omega_T\left(E_T\right) = \sum_E \Omega\left(E\right) \Omega_r \left(E_r\right) $$
A part of a container of gas is identified as the system. The system has an energy and an ensemble size. The rest of the gas, the reservoir, has its own energy and ensemble size.

\(\mathcal{P}(i)\) and Number of Microstates

Ensemble Average

A set of nine possible arrangements of the particles in a reservoir all with the same arrangement of particles in the system.

A Problem

Continuing to Solve the Problem

Rewriting the Entropy Equation

Solving the Issues


In a plot of energy as a function of entropy, the slope of the curve is the temperature.

A New Ensemble

More Equations, Now Within the Canonical Ensemble

Solving the Equation

$$ \sum_{i=1}^\infty \mathrm{P}(i) = \frac{\Omega_r\left(E_T\right)}{\Omega_T\left(E_T\right)} \sum_{i=1}^\infty e^{-\bfrac{E_i}{k_\mathrm{B}T}} = 1 $$

Canonical Partition Function and Distribution


$$ \mathcal{P}(i) = \frac{e^{-\bfrac{E_i}{k_\mathrm{B}T}}}{Q(T)} $$

Distribution in a 3D Box

As the value of the temperature increase as compared to the lowest energy of a system, the system has more and more particles in higher and higher energy state.

Finding Another Likelihood


Rotational Partition Functions

Comparison is rotational energy level population between a system at 77K and one at 298K. The system at 298K shows a lot more particles exist in quite high excited state rotational energy levels compared to the same system at 77K.

Check of Zero Energy

Example 2.1

At 298 K, calculate the ratio of the number of \(\chem{NH_3}\) molecules in the excited state to the number in the ground state, where the excited state is (a) the \(0^-\) state of the inversion, which lies \(0.79\,\mathrm{cm}^{-1}\) above the \(0^+\) ground state; and (b) the \(1^+\) state, which lies \(932.43\,\mathrm{cm}^{-1}\) above the \(0^+\). (The wavenumber unit, \(\mathrm{cm}^{-1}\), is conventionally used by spectroscopist as an energy unit, based on the relation between the transition energy in the experiment and the reciprocal wavelength of the photon that induces the transition: \(E_{photon}=\frac{hc}{\lambda}\). Because the energy is inversely proportional to the wavelength, energy is given in units of \(\frac{1}{\text{distance}}\).)

Example 2.2

At a temperature of 1000 K, how many vibrational states of \(\chem{H_2}\) are populated by at least 1% of the molecules, given the vibrational constant \(\omega_e=4395\,\mathrm{cm}^{-1}\) and the vibrational energy (relative to the ground state)of approximately \(E_{vib}=\omega_e\nu\)?

Example 2.3

You discover a molecular system having the energy levels and degeneracies $$ \varepsilon = c\left(n-1\right)^6;\;g=n;\;\text{for }n=1,2,3,\dots\text{ and }k_\mathrm{B}T=400c $$ Evaluate the partition function and calculate \(\mathcal{P}(\varepsilon)\) for each of the four lowest energy levels.

The Ideal Gas Law

The Original Problem - Pressure of an Ideal Gas

Gas particle bouncing off the walls inside a box. The walls of the box have an area A and the length of each side of the box is S.

Moving A Wall

  • We momentarily free one wall
  • It reduces its forces therefore its pressure on the gas, \(P_{min}\)
  • The gas pushes the wall out infinitesimally, \(ds\)
Gas particle bouncing off the walls inside a box. The walls of the box have an area A and the length of each side of the box is S.

The Pressure Equation

Cyclic Rule for Partial Derivatives

Determining Entropy Change with Volume

More On Our Gas

Even More on Our Gas

The Entropy

So What?