Review

Review#

1. Checklist of Key Concepts#

  1. Gibbs Free Energy & Reaction Extent

    • Define the extent of reaction \(x\) so that for
      \(\ce{\nu_A A(g) + \nu_B B <=> \nu_Y Y + \nu_Z Z}\), \(n_{\ce{A}} = n_{\ce{A},0}-\nu_{\ce{A}} x\), etc.

    • The driving force is

      \[\left. \frac{\partial G}{\partial x} \right|_{T,P} = \sum_i \nu_i\,\mu_i = \Delta G_{\text{rxn}}.\]

      • If \(\Delta G_{\text{rxn}}<0\), reaction proceeds forward.

      • If \(\Delta G_{\text{rxn}}=0\), equilibrium.

      • If \(\Delta G_{\text{rxn}}>0\), reaction moves backward.

  2. Reaction Quotient & Equilibrium Constant

    • Reaction quotient \(Q\) compares partial pressures to standard state:

      \[Q = \prod_i \left( \frac{P_i}{P^{\circ}} \right)^{\nu_i},\]

      so \(\Delta G_{\text{rxn}} = \Delta G_{\text{rxn}}^\circ + RT\ln Q\).

    • At equilibrium, \(\Delta G_{\text{rxn}}=0\implies Q=K\), the equilibrium constant:

      \[K = \exp\left(-\frac{\Delta G_{\text{rxn}}^\circ}{RT}\right).\]

    • Interpretation: \(Q<K\) drives products; \(Q>K\) drives reactants.

  3. Gas‐Phase Example & ICE Analysis

    • Dimerization \(\ce{2NO2(g) <=> N2O4(g)}\): use an ICE table with extent \(x\) to express mole numbers and mole fractions; define

      \[K_p = \frac{P_{\ce{N2O4}}/P^\circ}{(P_{\ce{NO2}}/P^\circ)^2}.\]

    • Solve for \(x_\text{eq}\) at various temperatures (e.g., 250 K, 298 K, 350 K) and pressures.

    • Observe that increasing \(T\) (endothermic dimerization) shifts equilibrium toward reactants.

  4. Temperature Dependence: van’t Hoff Equation

    • From \(\Delta G_{\text{rxn}}^\circ = -RT\ln K\) and Gibbs–Helmholtz,

      \[\begin{split}\left. \frac{d\ln K}{dT}\right|_P \;=\;\frac{\\Delta H_{\text{rxn}}^\circ}{R\,T^2}.\end{split}\]

    • Enables prediction of how \(K\) changes with \(T\), and rationalizes the data trend in the \(\ce{NO2}\)–\(\ce{N2O4}\) system.

  5. Microscopic Basis via Partition Functions

    • Helmholtz energy \(A=-RT\ln Q\) for a mixture of ideal gases with \(\;Q(T,V,\{N_i\})=\prod_i Q_i(T,V,N_i)\).

    • Chemical potentials \(\mu_i=-RT\,(\partial\ln Q/\partial N_i)\), leading to the same equilibrium condition \(\sum\nu_i\,\mu_i=0\).

    • From these, one can compute \(K\) purely from molecular partition functions, e.g., for \(\ce{H2(g) + I2(g) <=> 2HI(g)}\).

2. Checklist of Most Important Equations#

Equation

Variables

Meaning & Use Cases

Caveats/Approximations

\(\Delta G_{\text{rxn}} = \sum_i \nu_i\,\mu_i \;=\;\left(\frac{\partial G}{\partial x}\right)_{T,P}\)

\(\nu_i\): stoichiometric coefficients (products positive)
\(\mu_i\): chemical potential of species \(i\)

Reaction driving force. Use this to assess spontaneity and locate equilibrium \(\left( \Delta G_{\text{rxn}} = 0 \right)\).

Assumes constant \(T,P\) and only \(PV\) work; applies generally to closed, simple systems.

\(\mu_i = \mu_i^\circ + RT\ln\left( P_i / P^{\circ} \right)\)

—

Defines chemical potential in terms of measurable quantities. Controls mass exchange; at equilibrium between phases or compartments, \(\mu_i\) equal in each.

Requires proper choice of standard state and correction for nonideal behavior.

\(\Delta G_{\text{rxn}} = \Delta G_{\text{rxn}}^\circ + RT\ln Q\)

\(\Delta G_{\text{rxn}}^\circ\): standard‐state free energy change
\(Q=\prod \left( P_i / P^{\circ} \right)^{\nu_i}\): reaction quotient

Relates instantaneous driving force to composition. Use to predict direction of reaction under non‐standard conditions.

—

\(K = \exp\left(-\frac{\Delta G_{\text{rxn}}^\circ}{RT}\right)\)

\(K\): equilibrium constant (can be \(K_P\), \(K_c\), etc.)

Defines equilibrium composition. Use standard thermodynamic data (\(\Delta G_{\text{rxn}}^\circ\)) to compute \(K\).

Assumes temperature‐independent \(\Delta G_{\text{rxn}}^\circ\) unless corrected; different \(K\) for different standard states (pressure vs concentration).

\(Q = \frac{\left( y_{\ce{Y}} P / P^{\circ} \right)^{\nu_{\ce{Y}}} \, \left( y_{\ce{Z}} P / P^{\circ} \right)^{\nu_{\ce{Z}}}}{\left( y_{\ce{A}} P / P^{\circ} \right)^{\nu_{\ce{A}}} \, \left( y_{\ce{B}} P / P^{\circ} \right)^{\nu_{\ce{B}}}}\)

\(y_i\): mole fraction of gas \(i\)
\(P\): total pressure

Specific form of \(Q\) for ideal‐gas mixtures. Use in ICE analyses to solve for equilibrium extent \(x\).

Ideal‐gas assumption; neglects nonidealities.

\(\frac{d\ln K}{dT} = \frac{\Delta H_{\text{rxn}}^\circ}{R\,T^2}\)

\(\Delta H_{\text{rxn}}^\circ\): standard enthalpy change

van’t Hoff: predicts how \(K\) shifts with \(T\).

Assumes \(\Delta H_{\text{rxn}}^\circ\) constant over the temperature range; in reality, \(C_p\) corrections may be needed.

\(A = -RT\ln Q,\quad \mu_i = -RT\left(\frac{\partial\ln Q}{\partial N_i}\right)_{T,V}\)

\(A\): Helmholtz free energy of the mixture
\(Q\): total partition function

Connects molecular partition functions to macroscopic free energies and chemical potentials. Enables microscopic calculation of \(K\).

Exact only if full partition functions (translational, rotational, vibrational, electronic) are known.