Physical Chemistry Expert

Chemical Thermodynamics

Fundamental relations:
  dU = TdS - PdV + Σμᵢdnᵢ
  dH = TdS + VdP + Σμᵢdnᵢ
  dA = -SdT - PdV + Σμᵢdnᵢ
  dG = -SdT + VdP + Σμᵢdnᵢ

Maxwell relations:
  (∂T/∂V)S = -(∂P/∂S)V
  (∂T/∂P)S = (∂V/∂S)P
  (∂S/∂V)T = (∂P/∂T)V
  (∂S/∂P)T = -(∂V/∂T)P

Chemical potential:
  μᵢ = (∂G/∂nᵢ)T,P,nⱼ
  μ = μ° + RT·ln(a)  (activity a)
  Ideal gas: μ = μ° + RT·ln(P/P°)
  Solution: μ = μ° + RT·ln(x)  (Raoult)

Gibbs energy of reaction:
  ΔrG = ΔrG° + RT·ln(Q)
  At equilibrium: ΔrG = 0 → ΔrG° = -RT·ln(K)
  K = exp(-ΔrG°/RT)

Temperature dependence:
  Van't Hoff: d(lnK)/dT = ΔrH°/RT²
  Integration: ln(K₂/K₁) = -ΔrH°/R · (1/T₂ - 1/T₁)
  Exothermic: K decreases with T
  Endothermic: K increases with T

Gibbs-Helmholtz:
  (∂(G/T)/∂T)P = -H/T²
  ΔG = ΔH - TΔS
  Spontaneous: ΔG < 0 at constant T, P

Phase Equilibria

Chemical potential equality at equilibrium:
  μ(phase 1) = μ(phase 2)

Clausius-Clapeyron:
  dP/dT = ΔH_trs/(TΔV)
  Liquid-gas (ideal): d(lnP)/dT = ΔvapH/RT²
  → ln(P₂/P₁) = -(ΔvapH/R)(1/T₂ - 1/T₁)

Phase rule (Gibbs):
  F = C - P + 2
  F = degrees of freedom, C = components, P = phases
  Triple point: F = 1 - 3 + 2 = 0  (fixed T,P) ✓
  Two phases: F = 1  (specify T → P fixed)

Activity and fugacity:
  Real gas: μ = μ° + RT·ln(f/f°)
  f = γP  (γ → 1 as P → 0)
  Solution: aᵢ = γᵢxᵢ
  Ideal solution: aᵢ = xᵢ (Raoult's law, γᵢ = 1)

Colligative properties (dilute solutions):
  Boiling point elevation: ΔTb = Kb·m  (Kb = ebullioscopic constant)
  Freezing point depression: ΔTf = Kf·m  (Kf = cryoscopic constant)
  Osmotic pressure: Π = MRT  (M = molarity)
  Vapor pressure lowering: ΔP = xsolute·P*

Raoult's law: P_total = Σ xᵢPᵢ*  (ideal solution)
Henry's law: P = KH·x  (dilute solute, different constant)

Chemical Kinetics

Rate law:
  Rate = k[A]^m[B]^n...
  m, n = reaction orders (must be determined experimentally!)
  Overall order = m + n + ...

Integrated rate laws:
  Zero order: [A] = [A]₀ - kt
    t₁/₂ = [A]₀/2k

  First order: [A] = [A]₀e^(-kt)
    ln[A] = ln[A]₀ - kt
    t₁/₂ = ln2/k = 0.693/k

  Second order: 1/[A] = 1/[A]₀ + kt
    t₁/₂ = 1/(k[A]₀)

Temperature dependence (Arrhenius):
  k = A·exp(-Ea/RT)
  ln(k) = ln(A) - Ea/RT
  ln(k₂/k₁) = -(Ea/R)(1/T₂ - 1/T₁)
  Ea = activation energy (J/mol)
  A = pre-exponential/frequency factor
  Rule of thumb: rate doubles for each 10°C rise (∼50 kJ/mol Ea)

Transition State Theory (Eyring):
  k = (kBT/h)·exp(-ΔG‡/RT) = (kBT/h)·K‡
  ΔG‡ = ΔH‡ - TΔS‡
  ΔH‡ ≈ Ea - RT  (small difference)
  Entropy of activation: negative for bimolecular (more ordered TS)

Steady-state approximation:
  d[intermediate]/dt ≈ 0  (intermediate concentration constant)
  Applies when intermediate is short-lived

Pre-equilibrium approximation:
  Fast equilibrium before slow step
  K = k₁/k₋₁  (rapid equilibrium)
  Rate = k₂·K·[A][B]  (overall rate)

Michaelis-Menten (enzyme kinetics):
  E + S ⇌ ES → E + P
  v = Vmax[S]/(Km + [S])
  Km = (k₋₁ + k₂)/k₁  (Michaelis constant)
  Lineweaver-Burk: 1/v = Km/Vmax · 1/[S] + 1/Vmax

Quantum Chemistry

Born-Oppenheimer approximation:
  Separate nuclear and electronic motion.
  Electronic Schrodinger: Ĥelψel = Eelψel
  Nuclei move on potential energy surface (PES).

Hartree-Fock (HF) theory:
  Each electron moves in average field of all others.
  Ĥ_HF ψᵢ = εᵢψᵢ  (eigenvalue equation)
  HF energy: E_HF = Σεᵢ - Σ(Jᵢⱼ - Kᵢⱼ)/2
  Correlation energy: E_corr = E_exact - E_HF  (missing in HF)

Post-HF methods:
  MP2: Møller-Plesset 2nd order perturbation
  CISD: Configuration interaction singles and doubles
  CCSD(T): gold standard for accuracy
  Cost: HF (N⁴), MP2 (N⁵), CCSD(T) (N⁷)

Density Functional Theory (DFT):
  Hohenberg-Kohn: E is functional of electron density ρ(r)
  E[ρ] = T[ρ] + Vne[ρ] + J[ρ] + Exc[ρ]
  Exc = exchange-correlation functional (approximated)
  Kohn-Sham: solve HF-like equations with effective potential
  Cost: N³ (formally), very popular for large molecules
  Functionals: B3LYP, PBE, M06, ωB97X-D

Basis sets:
  Minimal: STO-3G (one basis function per orbital)
  Split-valence: 6-31G, 6-311G
  Polarization: 6-31G*, 6-31G** (adds d on heavy atoms, p on H)
  Diffuse: 6-31+G* (adds diffuse functions for anions, lone pairs)
  Correlation consistent: cc-pVDZ, cc-pVTZ, cc-pVQZ (for correlated methods)

Statistical Mechanics

Boltzmann distribution:
  Pᵢ = exp(-εᵢ/kBT) / q
  q = molecular partition function = Σᵢ exp(-εᵢ/kBT)

Partition function factorization:
  q = qtrans · qrot · qvib · qelec

Translational:
  qtrans = V(2πmkBT/h²)^(3/2)  (particle in 3D box)
  Thermal de Broglie wavelength: Λ = h/√(2πmkBT)

Rotational (linear molecule):
  qrot = T/σΘrot  (high T limit)
  Θrot = ℏ²/2IkB  (characteristic temperature)
  σ = symmetry number (1 for HCl, 2 for H₂, O₂)

Vibrational:
  qvib = 1/(1-exp(-hν/kBT))  (harmonic oscillator, per mode)
  Θvib = hν/kB  (characteristic temperature)
  High T limit: qvib → kBT/hν

Electronic:
  qelec = g₀ (usually ground state degeneracy only)

Thermodynamic properties from q:
  A = -NkBT(ln q - ln N + 1)  (Helmholtz)
  U = NkBT² (∂lnq/∂T)V
  S = NkB[ln q - ln N + 1 + T(∂lnq/∂T)V]
  Cv = (∂U/∂T)V

Heat capacities:
  Translation: (3/2)R per mole
  Rotation: R (linear), (3/2)R (nonlinear)
  Vibration: R per mode (high T) → 0 (low T, quantum frozen out)
  Equipartition: (1/2)R per quadratic term

Canonical ensemble:
  Q = Σ_states exp(-E_total/kBT)
  For N identical molecules: Q = qN/N! (distinguishable/N!)
  Grand canonical, microcanonical, isothermal-isobaric: other ensembles

Molecular Spectroscopy

def spectroscopy_selection_rules():
    return {
        'Microwave (rotational)': {
            'condition':    'Permanent dipole moment required',
            'selection':    'ΔJ = ±1',
            'spacing':      'B = h/8π²Ic, lines equally spaced at 2B',
            'info':         'Molecular geometry, bond lengths',
            'example':      'HCl: lines at 2B intervals'
        },
        'IR (vibrational)': {
            'condition':    'Change in dipole moment during vibration',
            'selection':    'Δv = ±1 (harmonic), Δv = ±2,±3... (overtones)',
            'fundamentals': '3N-6 modes (nonlinear), 3N-5 (linear)',
            'info':         'Functional groups, force constants',
            'example':      'CO₂: 4 modes, 2 IR active (asymm stretch, bends)'
        },
        'Raman': {
            'condition':    'Change in polarizability during vibration',
            'selection':    'Δv = ±1',
            'complement':   'Often complementary to IR (mutual exclusion for centrosymmetric)',
            'info':         'Symmetric stretches, ring breathing modes'
        },
        'UV-Vis (electronic)': {
            'selection':    'ΔS = 0 (spin allowed), Laporte: Δl = ±1',
            'transitions':  'σ→σ*, n→σ*, π→π*, n→π*, d→d, charge transfer',
            'Beer-Lambert': 'A = εlc  (absorbance = ε × path × concentration)',
            'info':         'Electronic structure, conjugation'
        },
        'NMR': {
            'condition':    'Spin ½ nuclei in magnetic field: ¹H, ¹³C, ¹⁵N, ³¹P',
            'selection':    'ΔmI = ±1',
            'frequency':    'ν = γB₀/2π  (Larmor frequency)',
            'info':         'Chemical environment, connectivity, 3D structure'
        }
    }

def spectroscopic_constants():
    return {
        'Vibrational frequency': 'ν̃ = (1/2πc)√(k/μ)  (k=force constant, μ=reduced mass)',
        'Rotational constant':   'B = ℏ/4πcI  (I=moment of inertia)',
        'Beer-Lambert':          'A = εlc = log(I₀/I)',
        'Einstein coefficients': 'Aᵢⱼ (spontaneous emission), Bᵢⱼ (stimulated)',
        'Franck-Condon':         'Intensity ∝ |⟨v′|v⟩|²  (overlap of vibrational wavefunctions)'
    }

Electrochemistry

Standard electrode potential:
  Half-reaction: Ox + ne⁻ → Red  (E°)
  Standard: 25°C, 1 M concentration, 1 atm
  SHE (standard hydrogen electrode): 2H⁺ + 2e⁻ → H₂, E° = 0.000 V

EMF of cell:
  E_cell = E_cathode - E_anode
  Spontaneous: E_cell > 0 → ΔG < 0
  ΔG° = -nFE°  (n = electrons transferred, F = 96485 C/mol)
  K = exp(nFE°/RT)

Nernst equation:
  E = E° - (RT/nF)ln(Q)
  E = E° - (0.05916/n)log(Q)  at 25°C
  At equilibrium: E = 0, Q = K

Concentration cells:
  Both electrodes same material, different concentrations
  E = (RT/nF)ln(c₂/c₁)

pH measurement:
  Glass electrode: E = const - 0.05916·pH  at 25°C

Butler-Volmer equation:
  i = i₀[exp(αnFη/RT) - exp(-(1-α)nFη/RT)]
  η = E - Eeq (overpotential)
  i₀ = exchange current density
  α = transfer coefficient (~0.5)
  Large anodic η: i ≈ i₀exp(αnFη/RT)  (Tafel equation)

Electrolysis:
  Faraday's laws: m = MIt/nF  (mass deposited)
  Decomposition voltage: E_min = E_cell + overpotentials

Common standard potentials (E°/V):
  Li⁺/Li: -3.04  Na⁺/Na: -2.71  Zn²⁺/Zn: -0.76
  Fe²⁺/Fe: -0.44  H⁺/H₂: 0.00   Cu²⁺/Cu: +0.34
  O₂/H₂O: +1.23   Cl₂/Cl⁻: +1.36  F₂/F⁻: +2.87

Surface Chemistry

Adsorption:
  Physisorption: van der Waals, weak, reversible, multilayer possible
  Chemisorption: chemical bonds, strong, irreversible, monolayer

Langmuir isotherm:
  θ = KP/(1+KP)
  θ = fractional coverage, K = adsorption constant, P = pressure
  Linearized: P/θ = P + 1/K
  Assumptions: monolayer, equivalent sites, no lateral interactions

BET isotherm (Brunauer-Emmett-Teller):
  Extends Langmuir to multilayer
  Used to measure surface area: SA = nmNA·σ
  P/(n(P*-P)) = 1/nm·c + (c-1)/nm·c · P/P*

Freundlich isotherm (heterogeneous surfaces):
  θ = KP^(1/n)  (empirical)
  log(θ) = log(K) + (1/n)log(P)

Catalysis:
  Catalyst lowers activation energy, increases rate
  Not consumed overall
  Langmuir-Hinshelwood: both reactants adsorb, react on surface
  Eley-Rideal: one adsorbed, one from gas phase
  Sabatier principle: optimal catalyst binds neither too weak nor too strong
  Volcano plot: rate vs adsorption energy has maximum

Rates on surfaces:
  Rate = kₛ·θ_A·θ_B  (bimolecular surface reaction)
  TPD (temperature-programmed desorption): measure Ea_des
  LEED: surface structure determination

Transport Properties

Diffusion (Fick's laws):
  First: J = -D·(dc/dx)  (flux proportional to gradient)
  Second: ∂c/∂t = D·∂²c/∂x²
  D = kBT/6πηr  (Stokes-Einstein, sphere radius r)

Viscosity:
  Newton's law: τ = η·(dv/dy)  (shear stress)
  Ideal gas: η = (1/3)ρ⟨v⟩λ  (kinetic theory)
  η increases with T for gases (more collisions)
  η decreases with T for liquids (less intermolecular order)

Thermal conductivity:
  Fourier's law: q = -κ·(dT/dx)
  κ = (1/3)ρCv⟨v⟩λ  (kinetic theory)

Ionic conductivity:
  Λm = κ/c  (molar conductivity)
  Kohlrausch: Λm = Λ°m - K√c  (weak electrolytes)
  Λ°m = Σλ°±  (limiting molar conductivity, additive)
  Mobility: u = v/E (drift velocity per unit field)
  Λ = F(u₊+u₋)

Diffusion coefficient and mobility:
  D = ukBT/ze  (Einstein relation)
  D = RT/NA·ze·friction  (general)

Common Pitfalls

Key Constants

R  = 8.314 J/mol·K
kB = 1.381×10⁻²³ J/K
h  = 6.626×10⁻³⁴ J·s
NA = 6.022×10²³ /mol
F  = 96485 C/mol
c  = 2.998×10⁸ m/s
RT at 25°C = 2.479 kJ/mol
kBT at 25°C = 0.02569 eV

Related Skills

• organic-chemistry-expert: Reaction mechanisms and kinetics
• inorganic-chemistry-expert: Coordination chemistry thermodynamics
• thermodynamics-expert: Classical thermodynamics foundations
• quantum-mechanics-expert: Quantum basis of chemistry
• statistical-mechanics: Microscopic basis of thermodynamics
• electrochemistry-expert: Applied electrochemistry