Statistical Mechanics | Chemistry

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:

$S = k_B \ln W$

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 $\sum_i n_i \varepsilon_i = E$.

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:

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

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:

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

• 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.

$\blacksquare$

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 $E$.

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).

The total molecular partition function factors into contributions:

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

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).

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

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

$\blacksquare$

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.

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:

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

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), $q_{\text{elec}} > g_0$.

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:

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

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

Theorem 8 (Entropy from Partition Function):

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

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

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

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

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

For an ideal gas:

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

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

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

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:

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

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

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

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 $T = 0$.

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:

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

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

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

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

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): $\Psi(1,2) = -\Psi(2,1)$

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:

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

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:

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

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.

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 $C_V = \frac{3}{2}Nk_B$.

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

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 $T$.

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

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

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.

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 $T \ll \Theta_{\text{vib}}$: $C_{V,\text{vib}} \to 0$.

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 $E$.

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

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.

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.

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.

flowchart TD
    A[Statistical Mechanics] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

• 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.

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).

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%.

Problem: A hypothetical molecule has three energy levels: epsilon_0 = 0, epsilon_1 = 100 cm^-1, epsilon_2 = 300 cm^-1, each non-degenerate. Calculate the molecular partition function at 300 K and the average energy.

Solution: First convert to energy units: k_B T at 300 K = 207 cm^-1 (using k_B = 0.695 cm^-1 K^-1).

q = exp(0) + exp(-100/207) + exp(-300/207) = 1 + 0.617 + 0.236 = 1.853.

Average energy:= (0 x 1 + 100 x 0.617 + 300 x 0.236)/1.853 = (61.7 + 70.8)/1.853 = 71.5 cm^-1 = 0.856 kJ/mol.

The fraction in the ground state is 1/1.853 = 0.540 (54%), meaning roughly half the molecules occupy the ground state at room temperature for these energy spacings.

Common mistake: Forgetting that the partition function is a sum over all states, not just the first two. Also, the average energy is not directly the energy of the most populated state; it is a Boltzmann-weighted average that includes contributions from all accessible states.

$\blacksquare$

Statistical mechanics bridges the gap between individual atoms and the bulk properties we measure in the lab. Imagine a gas with trillions of molecules: you cannot track each one, but you can describe the probability of finding molecules in different energy states. The Boltzmann distribution is the key insight: at higher temperatures, more molecules have enough energy to occupy excited states, but the probability always decreases exponentially with energy. The partition function is the central quantity that encodes all thermodynamic information about a system. It sums up how accessible each energy level is, weighted by the Boltzmann factor. A large partition function means many states are accessible, which corresponds to high entropy. Temperature acts as a scaler: at low T only the ground state matters (low entropy), while at high T many states contribute (high entropy). Quantum statistics reveal that identical particles are fundamentally indistinguishable, leading to Bose-Einstein and Fermi-Dirac distributions that deviate from classical predictions at low temperatures or high densities.

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