Review

Review#

1. Checklist of Key Concepts#

Section 2.1: Introduction to Statistical Mechanics#

  1. Microscopic–Macroscopic Connection

    • Macroscopic (thermodynamic) properties can be understood as statistical averages of microscopic properties.

    • “Expected value” (ensemble average) is the central idea:

    \[\langle X \rangle \;=\; \sum_i X_i\,p_i.\]

  2. Arithmetic Average vs. Expected Value

    • Arithmetic average:

    \[\bar{X} \;=\; \frac{1}{M}\sum_{i=1}^M X_i.\]

    • Expected value:

    \[\langle X \rangle \;=\; \sum_i X_i\,p_i, \quad p_i = \text{probability of microstate } i.\]

  3. Microstates and Ensembles

    • Microcanonical ensemble \((N, V, E)\): system is isolated, fixed total energy \(E\).

    • Canonical ensemble \((N, V, T)\): system in thermal contact with a reservoir at temperature \(T\).

    • Grand canonical ensemble \((\mu, V, T)\): system can exchange both energy and particles with a reservoir.

  4. Fundamental Postulate (Microcanonical)

    • For an isolated system, each accessible microstate is equally probable.

Section 2.2: Canonical Ensemble#

  1. Closed System

    • Exchanges energy (heat) with surroundings; no exchange of matter.

  2. Boltzmann Factor and Partition Function

    • Probability of microstate \(i\):

    \[p_i \;=\; \frac{e^{-\beta E_i}}{Q}, \quad \beta \;=\;\frac{1}{k_{\text{B}}\,T}.\]

    • Partition function \(Q\):

    \[Q \;=\; \sum_{i} e^{-\beta E_i},\]

    which normalizes probabilities.

  3. Two-State System

    • A simple example with energies \(E_1\) and \(E_2\).

    • Partition function:

    \[Q \;=\; e^{-\beta E_1} + e^{-\beta E_2}.\]

    • Probabilities (if \(\Delta E = E_2 - E_1\)):

    \[p_1 = \frac{1}{1 + e^{-\beta\,\Delta E}}, \quad p_2 = 1 - p_1.\]

  4. Interpretation of \(Q\)

    • \(Q\) is like an “effective count” of accessible microstates.

    • At low \(T\), only the lowest energy states matter; at high \(T\), many states are accessible.

Section 2.3: Ensemble Averages#

  1. Internal Energy

    • \(U \equiv \langle E \rangle\).

    • In the canonical ensemble:

    \[U \;=\; \frac{1}{Q} \sum_i E_i\, e^{-\beta E_i} \;=\; -\,\bigl(\tfrac{\partial \ln Q}{\partial \beta}\bigr)_{N,V}.\]

  2. Heat Capacity \((C_V)\)

    • Measures how internal energy changes with temperature:

    \[C_V = \bigl(\tfrac{\partial U}{\partial T}\bigr)_{N,V}.\]

    • Also related to energy fluctuations:

    \[\sigma_E^2 \;=\; \langle (E - \langle E\rangle)^2\rangle \;=\; k_{\text{B}}\,T^2\,C_V.\]

  3. Pressure (brief introduction)

    • In the canonical ensemble:

    \[P \;=\; k_{\text{B}}\,T\, \bigl(\tfrac{\partial\ln Q}{\partial V}\bigr)_{N,T}.\]

Section 2.4: Molecular Partition Functions#

  1. Many Identical and Independent Particles

    • For \(N\) distinguishable particles, each in a one‐particle partition function \(q\), the total partition function is

    \[Q \;=\; q^N \quad (\text{if particles are distinct}).\]

    • For indistinguishable particles (e.g., identical atoms/molecules obeying single occupancy),

    \[Q \;=\; \frac{q^N}{N!}.\]

  2. Molecular Partition Function

    • Within Born–Oppenheimer, a molecule’s energy decomposes into translational, rotational, vibrational, and electronic parts:

    \[\varepsilon = \varepsilon_{\text{trans}} + \varepsilon_{\text{rot}} + \varepsilon_{\text{vib}} + \varepsilon_{\text{elec}}.\]

    • Thus,

    \[q \;=\; q_{\text{trans}}\, q_{\text{rot}}\, q_{\text{vib}}\, q_{\text{elec}}.\]

Section 2.5: Particle in a Box#

  1. Quantum Levels

    • For a 1D box \((0 \le x \le L)\),

    \[E_n = \frac{h^2}{8mL^2}\,n^2,\;\; n=1,2,3,\dots\]

    • In 3D (a rectangular box of sides \(L_x, L_y, L_z\)):

    \[E_{n_x,n_y,n_z} \;=\; \frac{h^2}{8m} \left(\frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2}\right).\]

  2. Partition Function (3D Box)

    • For a cubic box of volume \(V = L^3\):

    \[q \;=\; \sum_{n_x=1}^\infty \sum_{n_y=1}^\infty \sum_{n_z=1}^\infty \exp\!\Bigl[ -\beta \frac{h^2}{8mL^2}\,(n_x^2 + n_y^2 + n_z^2) \Bigr].\]

    • At high \(T\) or large \(L\), we approximate the sum by an integral and obtain the classical partition function:

    \[q_{\text{trans}} \;=\; \frac{V}{\Lambda^3}, \quad \Lambda \;=\; \sqrt{\frac{h^2}{2\pi m k_{\text{B}} T}}\,\, \text{(thermal de Broglie wavelength).}\]

  3. Many-Particle System (Ideal Gas)

    • For \(N\) indistinguishable, non-interacting particles:

    \[Q \;=\; \frac{q^N}{N!} \;=\; \frac{1}{N!} \biggl(\frac{V}{\Lambda^3}\biggr)^{\!N}.\]

    • Leads directly to the ideal gas law and to:

    \[U \;=\; \frac{3}{2}\,N\,k_{\text{B}}T, \quad C_V \;=\; \frac{3}{2}\,N\,k_{\text{B}}, \quad P = \frac{N\,k_{\text{B}}\,T}{V}.\]

Section 2.6: Harmonic Oscillator#

  1. Quantum Harmonic Oscillator

    • Energy levels for a 1D harmonic oscillator of frequency \(\omega\):

    \[E_n \;=\; \hbar\omega\,\Bigl(n + \tfrac{1}{2}\Bigr), \quad n=0,1,2,\dots\]

  2. Partition Function

    • Summing over these energy levels:

    \[q \;=\; \sum_{n=0}^{\infty} e^{-\beta E_n} \;=\; \frac{e^{-\frac{1}{2}\,\beta\,\hbar\omega}}{1 - e^{-\beta\,\hbar\omega}}.\]

  3. Ensemble Averages

    • Internal Energy:

    \[U \;=\; \frac{\hbar\omega}{2} \;+\; \frac{\hbar\omega}{\,e^{\beta\,\hbar\omega}-1\,}.\]

    • Heat Capacity:

    \[C_V \;=\; k_{\text{B}} \Bigl(\frac{\hbar\omega}{k_{\text{B}} T}\Bigr)^{2}\, \frac{\,e^{\hbar\omega/(k_{\text{B}}T)}\,} {\bigl(e^{\hbar\omega/(k_{\text{B}}T)} - 1\bigr)^{2}}.\]

    • At high temperature (\(k_{\text{B}}T \gg \hbar\omega\)), each harmonic oscillator recovers the classical limit \(U \to k_{\text{B}}T\) and \(C_V \to k_{\text{B}}\).

Section 2.7: Linear Rigid Rotor#

  1. Energy Levels

    • For a rigid, linear rotor of moment of inertia \(I\):

    \[E_J \;=\; \frac{\hbar^2}{2I}\,J\,(J+1), \quad J=0,1,2,\dots\]

    • Each level \(E_J\) has degeneracy \(g_J = 2J + 1\).

  2. Rotational Partition Function

    • Exact form:

    \[q_{\text{rot}} \;=\; \sum_{J=0}^\infty (2J + 1)\, e^{-\beta \,\frac{\hbar^2}{2I}\,J\,(J+1)}.\]

    • Define the rotational temperature \(\Theta_{\text{rot}} = \frac{\hbar^2}{2k_{\text{B}}I}\).

    • High-temperature limit (\(k_{\text{B}}T \gg \frac{\hbar^2}{2I}\)) gives

    \[q_{\text{rot}} \;\approx\; \frac{T}{\Theta_{\text{rot}}} \quad (\text{for a heteronuclear diatomic}).\]

    • For a homonuclear diatomic, include a symmetry factor \(\sigma=2\), so

    \[q_{\text{rot}} \;\approx\; \frac{T}{\sigma\,\Theta_{\text{rot}}}.\]

  3. Ensemble Averages (High-\(T\) Approximation)

    • Internal Energy (\(U_{\text{rot}}\)):

    \[U_{\text{rot}} \;\approx\; k_{\text{B}}\,T.\]

    (One linear rotor has 2 rotational degrees of freedom \(\rightarrow \frac{2}{2} k_{\text{B}}T\).)

    • Heat Capacity:

    \[C_V^{(\text{rot})} \;\approx\; k_{\text{B}}.\]

Section 2.8: Molecular Statistical Mechanics#

  1. Combining All Degrees of Freedom

    • For a general molecule, the total one‐molecule partition function is

    \[q \;=\; q_{\text{trans}}\, q_{\text{rot}}\, q_{\text{vib}}\, q_{\text{elec}}.\]

    • Extends to polyatomic molecules with more complex rotational constants \(\Theta_{\text{rot,A}}, \Theta_{\text{rot,B}}, \Theta_{\text{rot,C}}\) and multiple vibrational frequencies \(\Theta_{\text{vib},j}\).

  2. Symmetry Considerations

    • Symmetry factor \(\sigma\) must be included for molecules with indistinguishable orientations (e.g., homonuclear diatomics, symmetrical polyatomics).

  3. Summary Table

    • Often, we tabulate \(q_{\text{trans}}, q_{\text{rot}}, q_{\text{vib}}, q_{\text{elec}}\) for different molecule types (linear, nonlinear, spherical top, symmetric top, etc.), applying high-temperature (classical) or more exact quantum results as needed.

2. Checklist of Most Important Equations#

Below is a unified list of the major equations from Sections 2.1–2.8.

A. Expected Value of a General Variable

\[\langle X \rangle \;=\; \sum_i X_i\,p_i \quad\text{or}\quad \int X(\omega)\,p(\omega)\,d\omega.\]

  • Applicability: general definition in statistical mechanics/probability theory.

B. Microcanonical Ensemble (Fundamental Postulate)

\[p_i \;=\; \frac{1}{M} \quad(\text{for all accessible microstates}).\]

  • Applicability: isolated system, \((N, V, E)\).

C. Canonical Ensemble Probability

\[p_i \;=\; \frac{\,e^{-\beta\,E_i}\,}{Q}, \quad \beta = \frac{1}{k_{\text{B}}\,T}.\]

  • Applicability: closed system in thermal contact at \(T\).

D. Canonical Partition Function

\[Q \;=\; \sum_i e^{-\beta\,E_i}.\]

  • Applicability: \((N, V, T)\) ensemble; sum/integral over all microstates.

E. Internal Energy (Canonical)

\[U \;=\; \langle E\rangle \;=\; \frac{1}{Q}\,\sum_i E_i\,e^{-\beta E_i} \;=\; -\bigl(\tfrac{\partial \ln Q}{\partial \beta}\bigr)_{N,V}.\]

F. Heat Capacity at Constant Volume

\[C_V \;=\; \bigl(\tfrac{\partial U}{\partial T}\bigr)_{N,V}.\]

Also,

\[\sigma_E^2 \;=\; k_{\text{B}}\,T^2\,C_V.\]

G. Pressure (Canonical)

\[P \;=\; k_{\text{B}}\,T \,\bigl(\tfrac{\partial \ln Q}{\partial V}\bigr)_{N,T}.\]

H. Molecular Systems: Indistinguishability Factor

\[Q \;=\; \frac{q^N}{N!},\]

  • Applicability: identical, indistinguishable particles (e.g., ideal gases).

G. Translational Partition Function (Particle in a 3D box at high \(T\))

\[q_{\text{trans}} \;=\; \frac{V}{\Lambda^3}, \quad \Lambda = \sqrt{\frac{h^2}{2\pi m k_{\text{B}}T}}.\]

I. Harmonic Oscillator Partition Function (1D)

\[q_{\text{HO}} \;=\; \frac{\,e^{-\tfrac{1}{2}\beta\hbar\omega}\,}{1 - e^{-\beta\,\hbar\omega}}.\]

  • Internal Energy:

\[U_{\text{HO}} \;=\; \frac{\hbar\omega}{2} + \frac{\hbar\omega}{\,e^{\beta\,\hbar\omega}-1\,}.\]

J. Rotational Partition Function (Linear Rotor, High \(T\))

\[q_{\text{rot}} \;\approx\; \frac{T}{\sigma\,\Theta_{\text{rot}}}, \quad \Theta_{\text{rot}} = \frac{\hbar^2}{2k_{\text{B}}\,I}.\]

  • Internal Energy:

\[U_{\text{rot}} \;\approx\; k_{\text{B}}\,T.\]

  • Heat Capacity:

\[C_V^{(\text{rot})} \;\approx\; k_{\text{B}}.\]

K. Polyatomic Molecules

  • General form (neglecting interactions):

\[q \;=\; q_{\text{trans}}\; q_{\text{rot}}\; q_{\text{vib}}\; q_{\text{elec}}.\]

  • For linear vs. nonlinear rotors or multiple vibrational modes, each factor is included appropriately (with possible symmetry factors).