Statistical thermodynamics — partition functions, Q_trans, Q_rot, Q_vib, Q_elec; computing thermodynamic properties

1. What is a partition function?

WHY this form? The probability of occupying level $i$ is the Boltzmann factor $p_i = g_i e^{-\varepsilon_i/k_BT}/q$. For probabilities to sum to 1 we need a normalizing constant — that constant is $q$. So $q$ literally counts the effectively accessible states at temperature $T$.

• At $T\to 0$: only ground state matters, $q\to g_0$ (just its degeneracy).
• At $T\to\infty$: $e^{-\varepsilon_i/k_BT}\to 1$ for all levels, $q\to$ number of states (huge).

From molecular $q$ to system $Q$

For $N$ independent, indistinguishable molecules (an ideal gas): $Q = \frac{q^N}{N!}$ The $N!$ removes overcounting of states that differ only by swapping identical molecules.

WHY factorize? If total energy is a sum of independent contributions, the Boltzmann exponential of a sum becomes a product — so partition functions multiply: $\varepsilon = \varepsilon_{trans}+\varepsilon_{rot}+\varepsilon_{vib}+\varepsilon_{elec}\;\Rightarrow\; q = q_{trans}\,q_{rot}\,q_{vib}\,q_{elec}$

2. The master equations (derive once, use forever)

HOW we get $U$ (derivation from scratch): Mean energy is the probability-weighted sum $U-U_0 = \langle\varepsilon\rangle N = N\frac{\sum_i \varepsilon_i e^{-\beta\varepsilon_i}}{q}.$ Notice $\sum_i \varepsilon_i e^{-\beta\varepsilon_i} = -\dfrac{\partial}{\partial\beta}\sum_i e^{-\beta\varepsilon_i} = -\dfrac{\partial q}{\partial\beta}$. Therefore $U-U_0 = -\frac{N}{q}\frac{\partial q}{\partial\beta} = -N\frac{\partial \ln q}{\partial\beta} = -\frac{\partial \ln Q}{\partial\beta}.$ Converting $\partial\beta \to \partial T$ via $\beta=1/k_BT$, $d\beta = -1/(k_BT^2)\,dT$ gives the $k_BT^2(\partial\ln Q/\partial T)$ form. That's the whole engine.

3. The four contributions

(a) Translational — $q_{trans}$

Particle in a 3-D box, energy spacings tiny → treat as continuum: $q_{trans} = \frac{V}{\Lambda^3},\qquad \Lambda = \frac{h}{\sqrt{2\pi m k_BT}}$ $\Lambda$ is the thermal de Broglie wavelength — the "quantum size" of the molecule.

WHY $V/\Lambda^3$? $q_{trans}$ counts how many "quantum boxes" of size $\Lambda^3$ fit in volume $V$. More room or higher $T$ (smaller $\Lambda$) → more accessible states.

Contribution: $U_{trans}=\tfrac32 Nk_BT$, $C_{V,trans}=\tfrac32 Nk_B$ (equipartition: 3 translational DOF).

(b) Rotational — $q_{rot}$ (linear molecule)

Levels $\varepsilon_J = hcB\,J(J+1)$, degeneracy $g_J = 2J+1$, $B$ = rotational constant. When $k_BT \gg hcB$ (usual at room $T$), replace the sum by an integral: $q_{rot} = \frac{k_BT}{\sigma\, hcB}$ $\sigma$ = symmetry number (=1 for heteronuclear like HCl, =2 for homonuclear like $N_2$) — it removes indistinguishable rotated orientations.

Contribution: $U_{rot}=Nk_BT$, $C_{V,rot}=Nk_B$ (2 rotational DOF for linear).

(c) Vibrational — $q_{vib}$ (harmonic oscillator)

Levels $\varepsilon_v = (v+\tfrac12)h\nu$. Measuring from ground state, $\varepsilon_v = v h\nu$, a geometric series: $q_{vib}=\sum_{v=0}^\infty e^{-v\,h\nu/k_BT} = \frac{1}{1-e^{-h\nu/k_BT}}$ WHY geometric? $\sum_{v=0}^\infty x^v = 1/(1-x)$ with $x=e^{-h\nu/k_BT}$.

Define vibrational temperature $\theta_v = h\nu/k_B$. Then $U_{vib}=Nk_B\frac{\theta_v}{e^{\theta_v/T}-1},\qquad C_{V,vib}=Nk_B\left(\frac{\theta_v}{T}\right)^2\frac{e^{\theta_v/T}}{(e^{\theta_v/T}-1)^2}.$ At high $T$: $q_{vib}\to T/\theta_v$, $C_{V,vib}\to Nk_B$ (full equipartition). At low $T$: vibrations "freeze out", $C_{V,vib}\to 0$.

(d) Electronic — $q_{elec}$

Electronic gaps are usually huge ($\gg k_BT$), so only the ground state contributes: $q_{elec}=g_0\quad(\text{ground-state degeneracy})$ For most molecules $g_0=1$. Exceptions: O atom, NO (low-lying excited states), radicals.

4. Worked examples

5. Common mistakes (steel-manned)

6. Forecast-then-Verify

Flashcards

Connections

• Boltzmann distribution — origin of the weighting $e^{-\varepsilon/k_BT}$
• Equipartition theorem — high-$T$ limits of each contribution
• Ideal gas law — derived from $p=k_BT(\partial\ln Q/\partial V)_T$
• Sackur–Tetrode equation — absolute entropy of monatomic gases
• Rotational and vibrational spectroscopy — provides $B$ and $\nu$ that feed $q_{rot},q_{vib}$
• Heat capacity of gases — temperature dependence explained by freezing-out of modes

Concept Map

Hinglish (regional understanding)