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KEY EQUATIONS

1. General 4. Equilibrium and electrochemistry Roots of the equation ax2 ϩ bx ϩ c ϭ 0: Defi nition of activity (for ideal systems): 2 6 2 2 2 For an ideal gas: a ϭ P /PЊ, PЊ ϭ 1 bar 5 b b 4ac J J x ϭ Њ Њ ϭ и −1 2a For a solute in an ideal solution: aJ [J]/c , c 1 mol L ϭ Kinetic energy of a particle of mass m and speed v: For a pure liquid or solid: aJ 1 5 1 2 Reaction quotient and equilibrium constant: Ek 2 mv ϩ ¡ ϩ ϭ c d a b Gravitational potential energy of a body of mass m at height h: For the reaction a A b B c C d D, Q aC aD /aA aB 1 Δ 1 E 5 mgh For the equilibrium a A b B c C d D p ϭ c d a b K (aC aD /aA aB )equilibrium Coulomb potential energy of two charges Q 1 and Q 2 at a separation r in a vacuum: Variation of Gibbs free energy of reaction with composition: 5 pe Ep Q 1Q 2/4 0r ¢ 5 ¢ 1 Gr Gr° RT ln Q 2. Structure and spectroscopy Relation between standard reaction Gibbs free energy and equilibrium constant: ␭ ␯ Relation between the wavelength, , and frequency, , of electromagnetic radiation: ¢G ° 52RT ln K ␭␯ 5 c r van ’t Hoff equation: Energy of a photon of electromagnetic radiation of frequency ␯: ¢ 5 ␯ K2 Hr° 1 1 E h ln 5 a 2 b de Broglie relation: K1 R T1 T2 ␭ 5 h/p Relation between K and Kc: ¢ 5 n r 5 Heisenberg uncertainty principle: K (c°RT/P°) Kc P°/Rc° 12.03 K 1 Clausius–Clapeyron equation: ¢p¢x $ U 2 ¢ P Hvap° 2 5 a 1 2 1 b Energy of a particle of mass m in a one-dimensional box of length L: ln P R T T 5 2 2 2 5 1 1 2 En n h /8mL , n 1, 2, . . . Relation between Gibbs free energy and maximum nonexpansion work: Bohr frequency condition: ¢ 5 ␯ 5 2 G we,max at constant temperature and pressure h Eupper Elower Energy levels of a hydrogenlike atom of atomic number Z: Relation between pH and pOH: 1 5 52 2 2 5 pH pOH pKw En Z hR /n , n 1, 2, . . . Formal charge: Relation between acidity and basicity constants of a conjugate acid–base pair: pK 1 pK 5 pK FC 5 V 2 AL 1 1 BB a b w 2 Henderson–Hasselbalch equation: 5 1 3 4 3 4 3. Th ermodynamics pH pKa log( base initial/ acid initial) Ideal gas law: Relation between the Gibbs free energy of reaction and the cell potential: PV 5 nRT ¢ 52 G nFEcell Expansion work against constant external pressure: Relation between the equilibrium constant for a cell reaction and the standard cell 52 ¢ w Pex V potential: Work of reversible, isothermal expansion of an ideal gas from V1 to V2: ln K 5 n FE °/RT 52 r cell w nRT ln(V2 /V1) Nernst equation: First law of thermodynamics: 5 2 Ecell Ecell° (RT/nrF) ln Q ¢U 5 q 1 w Defi nition of entropy change: 5. Kinetics ¢ 5 Average reaction rate: S qrev/T Defi nition of enthalpy: ¢ 3R4 5 1 Rate of consumption of R 52 H U PV ¢t Defi nition of Gibbs free energy: ¢ 3P4 5 2 Rate of formation of P 5 G H TS ¢t Change in Gibbs free energy at constant temperature: Unique average rate for a A ϩ b B ¡ c C ϩ d D: ¢G 5 ¢H 2 T¢S ¢ 3A4 ¢ 3B4 ¢ 3C4 ¢ 3D4 Relation between the constant-pressure and constant-volume molar heat capacities ϭ2 1 521 5 1 5 1 Unique average reaction rate ¢ ¢ ¢ ¢ of an ideal gas: a t b t c t d t 5 1 CP,m CV,m R Integrated rate laws: ϭ Standard reaction enthalpy (X ϭ H) and Gibbs free energy (X ϭ G) from standard For rate of disappearance of A kr[A], 3 4 enthalpies and Gibbs free energies of formation: A t 2 52 3 4 5 3 4 krt ¢X° 5 g n¢X °(products) 2 g n¢X °(reactants), n in moles ln 3 4 krt A t A 0e f f A 0 ¢ 5 ¢ 2 ¢ Xr° g nr Xf°(products) g nr Xf°(reactants), nr a pure number ϭ 2 For rate of disappearance of A kr[A] , Standard reaction entropy: 3A4 ¢S° 5 g nS °(products) 2 g nS °(reactants), n in moles 1 2 1 5 3 4 5 0 1 5 1 1 m m 3 4 3 4 krt A t 1 3 4 3 4 krt 3 4 Kirchhoff ’s law: A t A 0 1 A 0krt A t A 0 ¢ 5 ¢ 1 ¢ 2 H2° H1° CP(T2 T1) Half-life of a reactant in a fi rst-order reaction: 5 Change in entropy when a substance of constant heat capacity, C, is heated from T1 t1/2 (ln 2)/kr to T2: Arrhenius equation: ¢ 5 S C ln(T2/T1) 5 2 ln kr ln A Ea/RT Change in entropy when an ideal gas expands isothermally from V to V : 1 2 Th e rate constant at one temperature in terms of its value at another ¢ 5 S nR ln(V2/V1) temperature: Boltzmann’s formula for the statistical entropy: k2 Ea 1 1 S 5 k ln W ln 5 a 2 b k1 R T1 T2 Entropy change of the surroundings for a process in a system with enthalpy change Th e equilibrium constant in terms of the rate constants: ⌬H: 5 ¢ 52¢ K kforward/kreverse Ssurr H/T