Statistical Mechanics

1. Microstates and Macrostates

1.1 Basic Concepts

Definition 1 (Microstate): A complete specification of the state of a system, including the positions and momenta of all particles (or, in quantum mechanics, the quantum numbers of each particle).

Definition 2 (Macrostate): A specification of the system by macroscopic variables (e.g., $N$, $V$, $E$, $T$, $P$).

A single macrostate corresponds to a vast number of microstates. The number of microstates $W$ for a given macrostate is related to entropy:

$S = k_B \ln W$

1.2 Microcanonical Ensemble

Definition 3 (Microcanonical Ensemble): A collection of isolated systems, all with the same $N$, $V$, and $E$. Every accessible microstate is equally probable.

For $N$ distinguishable particles distributed among energy levels $\varepsilon_i$ with occupation numbers $n_i$:

$W = \frac{N!}{n_1!\,n_2!\,\cdots}$

subject to $\sum_i n_i = N$ and $\sum_i n_i \varepsilon_i = E$.

2. The Boltzmann Distribution

2.1 Derivation

Theorem 1 (Boltzmann Distribution): In a system at temperature $T$, the probability of finding a particle in state $i$ with energy $\varepsilon_i$ is:

$p_i = \frac{e^{-\varepsilon_i/k_BT}}{q}$

where $q$ is the molecular partition function:

$q = \sum_i e^{-\varepsilon_i/k_BT}$

2.2 Most Probable Distribution

The most probable distribution maximizes $\ln W$ subject to the constraints $\sum n_i = N$ and $\sum n_i \varepsilon_i = E$. Using Lagrange multipliers:

$n_i^* = N\frac{e^{-\varepsilon_i/k_BT}}{\sum_j e^{-\varepsilon_j/k_BT}}$

2.3 Physical Interpretation

• States with lower energy are more populated.
• The ratio of populations of two states:

$\frac{n_j}{n_i} = e^{-(\varepsilon_j - \varepsilon_i)/k_BT}$

Example 1: At 300 K, the population ratio of the first excited state ($\varepsilon_1$) to the ground state ($\varepsilon_0 = 0$) for an electronic transition of $\varepsilon_1 = 5 \times 10^{-19}$ J:

$\frac{n_1}{n_0} = e^{-\varepsilon_1/k_BT} = e^{-5 \times 10^{-19}/(1.381 \times 10^{-23} \times 300)} = e^{-120.7} \approx 0$

Essentially no population in the excited electronic state at room temperature.

$\blacksquare$

3. The Canonical Ensemble

3.1 Definition

Definition 4 (Canonical Ensemble): A collection of closed systems in thermal contact with a heat bath at temperature $T$. All systems have the same $N$, $V$, $T$ but varying $E$.

3.2 Canonical Partition Function

Theorem 2 (Canonical Partition Function): For $N$ distinguishable particles:

$Q = \sum_j e^{-E_j/k_BT}$

where $E_j$ is the energy of the $j$-th system microstate. For $N$ indistinguishable particles:

$Q = \frac{q^N}{N!}$

where $q$ is the molecular partition function. The $N!$ accounts for indistinguishability (Boltzmann statistics, valid when $n_i \ll g_i$ for all states).

4. Partition Functions

4.1 The Molecular Partition Function

The total molecular partition function factors into contributions:

$q = q_{\text{trans}} \cdot q_{\text{rot}} \cdot q_{\text{vib}} \cdot q_{\text{elec}}$

4.2 Translational Partition Function

Theorem 3 (Translational Partition Function): For a particle of mass $m$ in volume $V$:

$q_{\text{trans}} = \left(\frac{2\pi m k_B T}{h^2}\right)^{3/2}V$

This follows from treating translational motion as a particle in a 3D box and summing over energy levels (or integrating in the classical limit).

Example 2: Calculate $q_{\text{trans}}$ for $\text{N}_2$ ($m = 4.65 \times 10^{-26}$ kg) at 298 K in $V = 0.0248$ m$^3$ (1 mol at 1 atm).

$\Lambda = \frac{h}{\sqrt{2\pi m k_B T}} = \frac{6.626 \times 10^{-34}}{\sqrt{2\pi \times 4.65 \times 10^{-26} \times 1.381 \times 10^{-23} \times 298}} = 1.76 \times 10^{-11} \text{ m}$

$q_{\text{trans}} = \frac{V}{\Lambda^3} = \frac{0.0248}{(1.76 \times 10^{-11})^3} = 4.55 \times 10^{28}$

$\blacksquare$

4.3 Rotational Partition Function

Theorem 4 (Rotational Partition Function): For a linear molecule with moment of inertia $I$:

$q_{\text{rot}} = \frac{T}{\sigma\Theta_{\text{rot}}}$

where $\Theta_{\text{rot}} = \frac{\hbar^2}{2Ik_B}$ is the rotational temperature and $\sigma$ is the symmetry number ($\sigma = 1$ for heteronuclear, $\sigma = 2$ for homonuclear diatomics).

For a nonlinear molecule:

$q_{\text{rot}} = \frac{\sqrt{\pi}}{\sigma}\left(\frac{T^3}{\Theta_A\,\Theta_B\,\Theta_C}\right)^{1/2}$

where $\Theta_A$, $\Theta_B$, $\Theta_C$ are the rotational temperatures about the three principal axes.

4.4 Vibrational Partition Function

Theorem 5 (Vibrational Partition Function): For a harmonic oscillator with frequency $\nu$:

$q_{\text{vib}} = \frac{e^{-h\nu/(2k_BT)}}{1 - e^{-h\nu/(k_BT)}}$

where $\Theta_{\text{vib}} = h\nu/k_B$ is the vibrational temperature. For the zero of energy at the bottom of the potential well (excluding zero-point energy):

$q_{\text{vib}} = \frac{1}{1 - e^{-\Theta_{\text{vib}}/T}}$

For a molecule with $3N - 6$ (nonlinear) or $3N - 5$ (linear) vibrational modes:

$q_{\text{vib}} = \prod_i q_{\text{vib},i}$

4.5 Electronic Partition Function

Theorem 6 (Electronic Partition Function):

$q_{\text{elec}} = g_0\,e^{-\varepsilon_0/k_BT} + g_1\,e^{-\varepsilon_1/k_BT} + \cdots$

where $g_i$ is the degeneracy of level $i$. For most molecules at ordinary temperatures, only the ground state contributes ($q_{\text{elec}} \approx g_0$).

For atoms with accessible excited states (e.g., halogens), $q_{\text{elec}} > g_0$.

5. Thermodynamic Functions from Partition Functions

5.1 Internal Energy

Theorem 7 (Internal Energy): For a system of $N$ molecules:

$U - U_0 = Nk_BT^2\left(\frac{\partial \ln q}{\partial T}\right)_V$

For each contribution:

$U_{\text{trans}} = \frac{3}{2}Nk_BT, \quad U_{\text{rot,linear}} = Nk_BT, \quad U_{\text{rot,nonlinear}} = \frac{3}{2}Nk_BT$

$U_{\text{vib}} = \sum_i \frac{N h\nu_i}{e^{h\nu_i/k_BT} - 1}$

5.2 Entropy

Theorem 8 (Entropy from Partition Function):

$S = \frac{U - U_0}{T} + Nk_B\ln q + Nk_B \quad (\text{distinguishable})$

$S = \frac{U - U_0}{T} + Nk_B\ln\frac{q}{N} + Nk_B \quad (\text{indistinguishable})$

5.3 Helmholtz Free Energy

$A - A_0 = -Nk_BT\ln q \quad (\text{distinguishable})$

$A - A_0 = -Nk_BT\ln\frac{q^N}{N!} \quad (\text{indistinguishable})$

5.4 Gibbs Free Energy

$G - G_0 = -Nk_BT\ln\frac{q}{N} + Nk_BT(V/N)\left(\frac{\partial \ln q}{\partial V}\right)_T$

For an ideal gas:

$G - G_0 = -nRT\ln\frac{q}{N_A} + nRT$

5.5 Chemical Potential

Theorem 9 (Chemical Potential): For an ideal gas:

$\mu = -k_BT\ln\frac{q}{N} = -k_BT\ln\frac{q}{N_A P/k_BT} + k_BT\ln P^\circ = \mu^\circ + k_BT\ln\frac{P}{P^\circ}$

6. The Sackur-Tetrode Equation

6.1 Derivation

Theorem 10 (Sackur-Tetrode Equation): The translational entropy of $N$ indistinguishable ideal gas particles:

$S_{\text{trans}} = Nk_B\left[\frac{5}{2} + \ln\left(\frac{V}{N}\left(\frac{2\pi m k_B T}{h^2}\right)^{3/2}\right)\right]$

For $n$ moles:

$S_{\text{trans}} = nR\left[\frac{5}{2} + \ln\left(\frac{(2\pi m k_B T)^{3/2} k_B T}{P\,h^3}\right)\right]$

6.2 Standard Molar Entropy

At $T = 298.15$ K, $P = 1$ bar:

$S^\circ_m = R\left[\frac{5}{2} + \ln\left(\frac{(2\pi m k_B T)^{3/2} k_B T}{P^\circ\,h^3}\right)\right] + S_{\text{rot}} + S_{\text{vib}} + S_{\text{elec}}$

7. Chemical Equilibrium

7.1 Equilibrium Constant from Partition Functions

Theorem 11 (Statistical Equilibrium Constant): For the reaction $0 = \sum_i \nu_i A_i$:

$K = \prod_i \left(\frac{q_i}{N_A}\right)^{\nu_i}\,e^{-\Delta E_0/RT}$

where $\Delta E_0$ is the energy difference between products and reactants at $T = 0$.

7.2 Activation and Thermodynamics

The equilibrium constant relates to thermodynamic quantities:

$\Delta_r G^\circ = -RT\ln K = \Delta_r H^\circ - T\Delta_r S^\circ$

From statistical mechanics:

$\Delta_r H^\circ = \Delta E_0 + \Delta\left(\sum_i \nu_i k_B T^2 \frac{\partial \ln q_i}{\partial T}\right)$

$\Delta_r S^\circ = R\left[\sum_i \nu_i \ln\frac{q_i e}{N_A} + \sum_i \nu_i T\frac{\partial \ln q_i}{\partial T}\right]$

7.3 Isotope Effects

The equilibrium isotope effect arises from differences in vibrational partition functions (mass dependence of $\Theta_{\text{vib}}$):

$\frac{K_H}{K_D} \approx e^{-(\Theta_{\text{vib},H} - \Theta_{\text{vib},D})/T}$

8. Quantum Statistics

8.1 Identical Particles and Indistinguishability

Theorem 12: Quantum mechanically, identical particles are indistinguishable. The wavefunction must be:

• Symmetric under exchange for bosons (integer spin): $\Psi(1,2) = +\Psi(2,1)$
• Antisymmetric under exchange for fermions (half-integer spin): $\Psi(1,2) = -\Psi(2,1)$

8.2 Bose-Einstein Statistics

Definition 5 (Bose-Einstein Distribution): For bosons:

$\langle n_i \rangle = \frac{1}{e^{(\varepsilon_i - \mu)/k_BT} - 1}$

where $\langle n_i \rangle$ is the mean occupation number of state $i$ and $\mu \leq \varepsilon_0$.

Applications:

• Bose-Einstein condensation: Below a critical temperature, a macroscopic number of particles occupies the ground state.
• Blackbody radiation: Planck distribution (photons are bosons).

Theorem 13 (Planck Distribution): Energy density of blackbody radiation:

$u(\nu)\,d\nu = \frac{8\pi h\nu^3}{c^3}\frac{1}{e^{h\nu/k_BT} - 1}\,d\nu$

8.3 Fermi-Dirac Statistics

Definition 6 (Fermi-Dirac Distribution): For fermions:

$\langle n_i \rangle = \frac{1}{e^{(\varepsilon_i - \mu)/k_BT} + 1}$

At $T = 0$: $\langle n_i \rangle = 1$ for $\varepsilon_i < \mu = \varepsilon_F$ (Fermi energy) and $\langle n_i \rangle = 0$ for $\varepsilon_i > \varepsilon_F$.

The Fermi energy:

$\varepsilon_F = \frac{\hbar^2}{2m}\left(\frac{6\pi^2 N}{V}\right)^{2/3}$

8.4 Classical Limit

When $e^{(\varepsilon - \mu)/k_BT} \gg 1$ (dilute, high-temperature limit), both Bose-Einstein and Fermi-Dirac distributions reduce to the Boltzmann distribution:

$\langle n_i \rangle \approx e^{-(\varepsilon_i - \mu)/k_BT}$

This is the condition $n_i \ll g_i$ (many more states than particles), which holds for most gases at ordinary conditions.

8.5 Electron Gas Model

Definition 7 (Electron Gas): In metals, conduction electrons are treated as a Fermi gas.

The Fermi-Dirac distribution gives:

• At $T = 0$: All states below $\varepsilon_F$ are filled.
• At finite $T$: Electrons near $\varepsilon_F$ are thermally excited; the distribution smears over $\sim k_BT$.

The electronic heat capacity:

$C_{V,\text{elec}} = \frac{\pi^2}{2}Nk_B\frac{T}{T_F}$

where $T_F = \varepsilon_F/k_B \sim 10^4$ K for metals. This explains why electronic contributions to heat capacity are much smaller than the classical prediction $C_V = \frac{3}{2}Nk_B$.

9. The Equipartition Theorem

9.1 Statement

Theorem 14 (Equipartition Theorem): Each quadratic degree of freedom contributes $\frac{1}{2}k_BT$ to the average energy per particle.

9.2 Limitations

The equipartition theorem is classical and fails when $k_BT \ll h\nu$ (quantized energy levels are not approximately continuous). This explains the temperature dependence of heat capacities and the “freezing out” of vibrational modes at low $T$.

10. Heat Capacities

10.1 Translational Heat Capacity

$C_{V,\text{trans}} = \frac{3}{2}Nk_B = \frac{3}{2}nR$

Constant and equal to the equipartition value at all temperatures where the gas behaves ideally.

10.2 Rotational Heat Capacity

For a linear molecule:

$C_{V,\text{rot}} = \begin{cases} 0 & T \ll \Theta_{\text{rot}} \\ \frac{3}{2}nR & T \gg \Theta_{\text{rot}} \end{cases}$

Most diatomics have $\Theta_{\text{rot}} \sim 2$–$10$ K, so rotational heat capacity is fully excited at room temperature. Exception: $\text{H}_2$ has $\Theta_{\text{rot}} = 85$ K.

10.3 Vibrational Heat Capacity

For a single harmonic mode:

$C_{V,\text{vib}} = nR\left(\frac{\Theta_{\text{vib}}}{T}\right)^2 \frac{e^{\Theta_{\text{vib}}/T}}{(e^{\Theta_{\text{vib}}/T} - 1)^2}$

This is the Einstein model. At $T \gg \Theta_{\text{vib}}$: $C_{V,\text{vib}} \to nR$. At $T \ll \Theta_{\text{vib}}$: $C_{V,\text{vib}} \to 0$.

11. The Grand Canonical Ensemble

11.1 Definition

Definition 8 (Grand Canonical Ensemble): Systems in contact with both a heat bath and a particle reservoir. Each system has the same $V$, $T$, $\mu$ but varying $N$ and $E$.

11.2 Grand Partition Function

Theorem 15 (Grand Partition Function):

$\Xi = \sum_{N=0}^{\infty} e^{N\mu/k_BT}Q(N,V,T) = \prod_i \sum_{n_i=0}^{\infty} e^{n_i(\mu - \varepsilon_i)/k_BT}$

For fermions: $\Xi = \prod_i(1 + e^{(\mu - \varepsilon_i)/k_BT})$

For bosons: $\Xi = \prod_i(1 - e^{(\mu - \varepsilon_i)/k_BT})^{-1}$

12. Fluctuations

12.1 Energy Fluctuations

In the canonical ensemble:

$\langle(\Delta E)^2\rangle = \langle E^2\rangle - \langle E\rangle^2 = k_BT^2 C_V$

The relative fluctuation $\langle(\Delta E)^2\rangle/\langle E\rangle^2 \sim 1/N \to 0$ for macroscopic systems.

12.2 Number Fluctuations

In the grand canonical ensemble:

$\langle(\Delta N)^2\rangle = k_BT\left(\frac{\partial N}{\partial \mu}\right)_{T,V} = \kappa_T\,N\,k_BT$

where $\kappa_T$ is the isothermal compressibility. Near the critical point, fluctuations diverge, leading to critical opalescence.

Common Pitfalls

1. Confusing the canonical and microcanonical ensembles. The microcanonical ensemble fixes $E$; the canonical fixes $T$. Fix: Use microcanonical for isolated systems and canonical for systems in contact with a heat bath.
2. Forgetting the $N!$ for indistinguishable particles. $Q = q^N/N!$ (not $Q = q^N$) for indistinguishable particles. Fix: Always include $N!$ for gases; omit for solids (localized particles).
3. Using classical equipartition for vibrational modes at low $T$. Vibrational heat capacity is not constant; it freezes out below $\Theta_{\text{vib}}$. Fix: Use the Einstein model or full quantum partition function.
4. Wrong symmetry number for rotation. $\sigma = 2$ for $\text{H}_2$ but $\sigma = 1$ for HD. Fix: Count the number of indistinguishable orientations of the molecule.
5. Confusing $\varepsilon_0$ and $\Delta E_0$. $\varepsilon_0$ is the ground state energy; $\Delta E_0$ is the energy difference between products and reactants at $T = 0$. Fix: In the equilibrium constant expression, $\Delta E_0$ appears, not individual $\varepsilon_0$ values.
6. Applying Boltzmann statistics when quantum effects matter. The classical limit requires $n_i \ll g_i$. Fix: Use Bose-Einstein or Fermi-Dirac statistics at low $T$ or high density (e.g., electrons in metals, liquid helium).
7. Mixing up energy zero-points. The vibrational partition function depends on where the zero of energy is defined. Fix: Be consistent; if $U_0$ is the zero-point energy, account for it in all thermodynamic functions.

Summary

• Microstate vs macrostate: One macrostate corresponds to $W$ microstates; $S = k_B \ln W$.
• Boltzmann distribution: $p_i = e^{-\varepsilon_i/k_BT}/q$; connects molecular properties to $T$.
• Partition function: $q = \sum e^{-\varepsilon_i/k_BT}$; factors into translational, rotational, vibrational, and electronic contributions.
• Thermodynamic functions from $q$: $U$, $S$, $A$, $G$, $\mu$, and $K$ can all be expressed.
• Sackur-Tetrode: Translational entropy of ideal gases from quantum mechanics.
• Quantum statistics: Bose-Einstein (bosons) and Fermi-Dirac (fermions); reduce to Boltzmann at high $T$ and low density.
• Equipartition: Each quadratic degree of freedom contributes $\frac{1}{2}k_BT$; fails for quantum regime.

Worked Examples

Example 1: Calculating Entropy of an Ideal Gas

Problem: Calculate the molar entropy of neon (Ne, monatomic, M = 20.18 g/mol) at 298 K and 1 atm using the Sackur-Tetrode equation. Solution: S = R[ln((2pi m k_B T/h^2)^(3/2) * (k_B T/P) * e^(5/2))]. With standard values, m = 20.18 x 10^-3 / 6.022e23 = 3.35 x 10^-26 kg. After substitution: S_m = 146.2 J K^-1 mol^-1 (literature value: 146.3 J K^-1 mol^-1).

Example 2: Boltzmann Distribution for a Two-Level System

Problem: A molecule has two energy levels: epsilon_0 = 0 and epsilon_1 = 5.0 x 10^-21 J. At T = 300 K, calculate the fraction of molecules in the excited state. Solution: Boltzmann factor = exp(-epsilon_1/k_B T) = exp(-5.0e-21/(1.38e-23 x 300)) = exp(-1.208) = 0.299. Fraction in excited state = 0.299/(1 + 0.299) = 0.230 or 23.0%.

Cross-References