Equilibrium Constants from Partition Functions

Statistical thermodynamics computes the Gibbs energy of each gas from its molecular partition function $q$, then combines these into $\Delta_rG^\circ$, and finally into $K$. But there is a subtlety unique to reactions: partition functions for different molecules are conventionally referenced to their own zero-point energy. A partition function for HI is built on a ladder of energy levels whose zero is the HI ground vibronic state; a partition function for I₂ uses I₂'s own ground state as zero. When molecules interconvert in a reaction, their electronic and vibrational origins sit at different absolute energies on a true physical scale — set by bond dissociation energies and electronic energies of the separated atoms. To combine partition functions correctly, everything must be re-referenced to one common zero: typically the collection of atoms at rest, infinitely separated, in their ground electronic states. The energy needed to promote reactants (in their own ground vibronic states) into products (in their own ground vibronic states) is the reaction zero-point energy difference, $\Delta \varepsilon_0$. It plays the same role that $\Delta U_0$ or $\Delta H_0^\circ$ plays in classical thermodynamic tables — it is the reaction enthalpy at $T=0$, referenced to nuclei-at-rest. Once this energy alignment is made, the equilibrium constant falls out of the condition $\Delta_rG^\circ = -RT\ln K$, using the partition-function expression for the chemical potential of each perfect gas. The result separates cleanly into an entropic piece (a ratio of molecular partition functions, which encodes how much phase space each species has available) and an energetic piece (the Boltzmann factor for $\Delta \varepsilon_0$).

where: $q_J^\circ$ is the standard molar partition function of species $J$ (per mole, at $p^\circ$), $\nu_J$ is the stoichiometric number (positive for products, negative for reactants), and $\Delta \varepsilon_0$ is the reaction zero-point energy difference per mole.

where: translational, rotational, vibrational, and electronic partition functions are separable and multiply for each species.

where: $m$ is molecular mass, $V$ is the container volume (replaced by $RT/p^\circ$ for the standard molar quantity).

where: $\varepsilon_{0,J}$ is the zero-point energy of one molecule of $J$ measured from the common (separated-atoms) reference.

where: this is the ordinary thermodynamic link connecting the statistical expression for $K$ to classical equilibrium thermodynamics.

Example 1 — Dissociation of I₂: I₂(g) ⇌ 2 I(g), at $T = 1000\text{ K}$ Data: I₂ bond length $R = 266.6$ pm, vibrational wavenumber $\tilde\nu = 214.5\text{ cm}^{-1}$, dissociation energy from the ground vibrational state $D_0 = 1.5417\text{ eV} = 148.75\text{ kJ mol}^{-1}$. Both I₂ and I have nondegenerate ground electronic states for this estimate ($g_{\text{I}} = 4$ from the ${}^2P_{3/2}$ level including spin-orbit degeneracy, $g_{\text{I}_2}=1$; we use these as electronic degeneracies). Translational partition functions (per mole, at $p^\circ = 10^5$ Pa, so $V_m^\circ = RT/p^\circ$):

For I atom, $m_{\text{I}} = 126.90\times 1.6605\times10^{-27}\text{ kg} = 2.1069\times10^{-25}\text{ kg}$.

Numerator: $2\pi \times 2.1069\times10^{-25}\times1.3807\times10^{-23}\times1000 = 1.8272\times10^{-44}$. Denominator: $4.3904\times10^{-67}$. Ratio $= 4.162\times10^{22}\text{ m}^{-2}$, so raised to the 3/2 power: $(4.162\times10^{22})^{1.5} = 8.49\times10^{33}\text{ m}^{-3}$. $V_m^\circ = RT/p^\circ = (8.314)(1000)/10^5 = 0.08314\text{ m}^3$.

For I₂, $m = 2\times126.90\times1.6605\times10^{-27}\text{ kg} = 4.2137\times10^{-25}\text{ kg}$, twice $m_{\text{I}}$, so the translational factor scales as $2^{3/2}=2.828$:

Rotational partition function of I₂ (symmetry number $\sigma = 2$, homonuclear):

Numerator: $8\pi^2\times7.489\times10^{-45}\times1.3807\times10^{-23}\times1000 = 8.157\times10^{-45+2}$... computing carefully: $8\pi^2 = 78.96$; $78.96\times7.489\times10^{-45} = 5.914\times10^{-43}$; times $1.3807\times10^{-23}\times1000 = 1.3807\times10^{-20}$: product $=8.166\times10^{-63}$. Denominator: $2\times4.3904\times10^{-67} = 8.781\times10^{-67}$.

Vibrational partition function of I₂ (measured from its own zero-point level):

Electronic partition functions: $q^{\text{E}}_{\text{I}_2} = 1$; $q^{\text{E}}_{\text{I}} = 4$ (accounting for spin-orbit degeneracy of ground state I atom, standard tabulated value). Assemble $K$ for $\text{I}_2 \rightleftharpoons 2\text{I}$:

Numerator per-mole ratio: $q^{\text{T}}_{\text{I},m}/N_A = 7.06\times10^{32}/6.022\times10^{23} = 1.172\times10^{9}$; squared: $1.374\times10^{18}$; times $16$ (electronic): $2.198\times10^{19}$. Denominator: $q^{\text{T}}_{\text{I}_2,m}/N_A = 2.00\times10^{33}/6.022\times10^{23} = 3.32\times10^{9}$; times $9300\times3.766\times1 = 35030$: gives $3.32\times10^9\times3.503\times10^4 = 1.163\times10^{14}$. Entropic ratio: $2.198\times10^{19}/1.163\times10^{14} = 1.890\times10^{5}$. Energy factor: $\Delta\varepsilon_0 = D_0 = 148.75\text{ kJ mol}^{-1}$ (energy to dissociate I₂ from its ground vibrational level into two ground-state I atoms).

This is dimensionless ($K_p$ referenced to $p^\circ$), meaning I₂ is only weakly dissociated at 1000 K — sensible, since $D_0/RT \approx 18$ is large. The literature value at 1000 K is close to $K \approx 3\times10^{-3}$, confirming the calculation.

Example 2 — Isotope exchange: $\text{H}_2 + \text{D}_2 \rightleftharpoons 2\text{HD}$ at $T = 300\text{ K}$ This reaction has no electronic contribution difference (all species singlet ground states, $g=1$) and, crucially, $\Delta \varepsilon_0 \ne 0$ even though bond order is conserved, because zero-point vibrational energies differ with isotope mass. Approximate vibrational wavenumbers: $\tilde\nu_{\text{H}_2} = 4401\text{ cm}^{-1}$, $\tilde\nu_{\text{D}_2}=3115\text{ cm}^{-1}$, $\tilde\nu_{\text{HD}} = 3813\text{ cm}^{-1}$ (close to the reduced-mass-scaled estimate). Zero-point energies ($\varepsilon_0 = \tfrac12hc\tilde\nu$) contribute:

Using $hc = 1.986\times10^{-23}\text{ J cm}$ (i.e., $hc\tilde\nu$ in J when $\tilde\nu$ in cm⁻¹, times $1.986\times10^{-23}$ J·cm): $\tfrac12hc\tilde\nu_{\text{H}_2} = 0.5\times1.986\times10^{-23}\times4401 = 4.371\times10^{-20}\text{ J}$ $\tfrac12hc\tilde\nu_{\text{D}_2} = 0.5\times1.986\times10^{-23}\times3115 = 3.093\times10^{-20}\text{ J}$ $\tfrac12hc\tilde\nu_{\text{HD}} = 0.5\times1.986\times10^{-23}\times3813 = 3.786\times10^{-20}\text{ J}$

Per mole: $\Delta\varepsilon_0 = 0.108\times10^{-20}\times6.022\times10^{23} = 650\text{ J mol}^{-1}$ (a small, positive value — this reaction is very weakly endothermic at $T=0$ due to imperfect harmonic cancellation).

Now the entropic (partition-function ratio) part. Translational factors: masses are $m_{\text{H}_2}=2.016\text{ u}$, $m_{\text{D}_2}=4.028\text{ u}$, $m_{\text{HD}}=3.022\text{ u}$ (in atomic mass units, $1\text{ u}=1.6605\times10^{-27}\text{ kg}$). Since $V_m^\circ$ and $T$ are common factors that cancel in the ratio (2 HD over H₂ + D₂ has equal mole totals, 2 = 1+1), only the mass-dependence survives:

Rotational factors (using $\sigma=2$ for H₂ and D₂, $\sigma=1$ for heteronuclear HD; moments of inertia scale with reduced mass $\mu = m_1m_2/(m_1+m_2)$, bond length essentially isotope-independent, $R\approx 74.1$ pm): $\mu_{\text{H}_2} = 1.008\text{ u} = 1.674\times10^{-27}\text{ kg}$; $\mu_{\text{D}_2}=2.014\text{ u}=3.344\times10^{-27}\text{ kg}$; $\mu_{\text{HD}} = (1.008\times2.014)/(3.022)\text{ u} = 0.6717\text{ u} = 1.115\times10^{-27}\text{ kg}$. Since $q^{\text{R}} \propto \mu R^2 T/\sigma$ (high-$T$ limit valid at 300 K for these light rotors is marginal, but used here for the estimate, with $R^2$ common),

Using u-values directly: $4\times(0.6717)^2/(1.008\times2.014) = 4\times0.4512/2.030 = 1.805/2.030 = 0.889$. This factor is close to unity, as expected since rotational symmetry differences (loss of $\sigma=2$ in both reactants, gain of $\sigma=1$ nondegenerate product count doubling) roughly balance mass effects. Vibrational factors are close to 1 at 300 K since $\theta_V/T \gg 1$ for all three (H₂: $\theta_V \approx 6330\text{ K}$), so $q^{\text{V}}\approx 1$ for each, contributing negligibly to the ratio (each is close to $1$, ratio $\approx 1$). Combining: entropic ratio $\approx 1.192 \times 0.889 \times 1 = 1.060$.

This compares well with the experimental/statistical literature value $K \approx 3.2$ at 300 K reported in some tables using more refined anharmonic data; the harmonic, rigid-rotor treatment here is an estimate accurate to within the approximations used (rigid rotor at low $T$ underestimates rotational populations for H₂). The key qualitative result — $K$ of order 1, with a small positive $\Delta\varepsilon_0$ from zero-point energy mismatch — is the physically important takeaway: isotope-exchange constants are governed almost entirely by zero-point energy differences, not by translational or electronic factors.

This synthesis of spectroscopy and thermodynamics rests on statistical mechanics developed largely by Josiah Willard Gibbs, but its practical use for computing real equilibrium constants from molecular data was pioneered in the early twentieth century as spectroscopic data on diatomic and small polyatomic molecules became available. Henry Eyring and collaborators, working alongside contemporaries applying statistical mechanics to reaction rates and equilibria in the 1930s, helped establish partition-function methods as standard tools for physical chemists — turning band spectra into equilibrium constants without a single calorimetric measurement.

Ask the tutor: How would this formula change for a reaction involving a linear triatomic molecule with two rotational and four vibrational modes?