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lecture35_umn - Lecture 35 Thermochemistry December 9 2009...

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Lecture 35 December 9, 2009 Thermochemistry
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Microscopic-Macroscopic connection Vast majority of chemical research: not about individual molecules, but macroscopic quantities made up with large numbers of molecules . Behavior of ensembles of molecules is governed by the empirically determined laws of thermodynamics . Most of chemical reactions and many chemical properties are defined in terms of the fundamental variables of thermodynamicsEnthalpy Entropy Free Energy ….. How to convert single-molecule potential energies to ensemble thermodynamic?
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Zero-point Vibrational Energy The Born-Oppenheimer (BO) potential energy surface (PES) is a classical construct. Energies of the various points whose coordinates are defined by the fixed nuclear positions are determined from quantum mechanical (QM) calculations of the electronic energy. When the motion of the nuclei is also computed at the QM level, the energy is tied-up in molecular vibrations . This is true also at T close to absolute zero since the lowest vibrational energy level for any bound vibration is NOT zero.
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Harmonic Oscillator T he energy of the lowest vibrational level ω : vibrational frequency Sum of all of these energies over all molecular vibrations defines the zero-point vibrational energy (ZPVE) Internal energy at 0 K for a molecule (35-1) E ele : energy for the stationary point on the BO PES. ZPVE is isotope dependent since the vibrational frequencies are isotope dependent . 1 2 h " U 0 = E elec + 1 2 h " i i modes #
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Ensemble Properties Collections of molecules in statistical mechanics: one requires that certain macroscopic conditions be held constant by external influence Enumeration of these conditions defines an “ensemble” Canonical ensemble : the constants are: Total number of particles N Volume V Temperature T Ensemble ( N, V, T ). In QM: fundamental function that characterizes the microscopic system is the wave function In SM: fundamental function is the partition function For the canonical ensemble it is written as: (35-2) i over all possible energy states with energy E i , k B is the Boltzmann’s constant Q ( N , V , T ) = e " E i ( N , V )/ k B T i #
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Canonical Ensemble Internal Energy Enthalpy Entropy Gibbs Free Energy We have to find an explicit representation of Q that permits the necessary partial differentiations in eqs. (35-3) to (35-6).
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