Statistical Thermodynamics :

1. Statistical Thermodynamics: from Molecule to Ensemble http://www.bodybuilding.com/fun/thermodynamics_training.htm

2. Introductiontothermodynamicstatefunctions • Partitionalfunction (q, [q]=?): Encodeshowthe probabilitiesare partitioned among the different microstates, based on their individual energies (’sum over states’) • (Internal) Energy (E, [E]=kJ/mol): ΔE = w + q • Enthalpy (H, [H]=kJ/mol): H ≡ E + pV = E + RT • Entropy (S, [S]=J/molK): a measure of the number of specific ways in which a thermodynamic system may be arranged (’measureof disorder’) • Gibbs Free energy (G, [G]=kJ/mol): G ≡ H - TS maximum energycan be attained only in a completely reversible process (’chemicalpotential’ ’availableenergy’)

3. StateFunction? Independently: fi

4. Getthe feeling ofEnergyconversion Glucose C6H12O6 + 6 O2 = 6 CO2 + 6 H2O =−2805 kJ/mol =−15.6kJ/g

5. Statisticalthermodynamics • 1 Molecule • 1 Mol

6. Statisticalthermodynamics • 1 Molecule • 1 Mol Ifweknowq and T, then: E(T) and S(T)!!! „Sum over states” Energydistribution of themolecules Energy + Energy E(T) and S(T) H(T) = E(T) + RT G(T) = H(T)– TS(T) Boltzmann distribution microstates Howtogetmicrostates?

7. Microstates • Combintationof Degreeof freedom • External: translation • Internal: rotation, vibration and electronic

8. Translationalstates Whatweneedtoknow: Molecularmass 1 molecule Etrans(T)=3/2NkbT=3/2RT Strans(T)=R[ln(qtrans)+1+3/2] 1 mol Etrans(T=298.15K)=3/2RT=3.7 kJ/mol https://www.youtube.com/watch?v=mXpfO9WhlPA

9. Rotationalstates Whatweneedtoknow: Optimized geometry of the species → Rotationalconstants (Rigid rotor treatment) ≈2x =1x 1 molecule Erot(T)=NkbT=RT Srot(T)=R[ln(qrot)+1] Erot(T=298.15K)=RT=2.5 kJ/mol 1 mol

10. Vibrationalstates Whatweneedtoknow: Optimized geometry of the species Forceconstants (k) → harmonicwavenumber (Harmonicoscillatorapproximation) 1 dimension (diatomicsmolecule) 1 molecule = 1 mol

11. Scalingfactors The vibrational frequencies needadjustment (scale factor)to better match experimental vibrational frequencies. This scaling compensateserrorsfrom: (1) Approximation inthesolution of theelectronic Schrödinger equation. (2) Harmonicoscillatorapproach • Howtogetscaling? • Doityourself • Findit • http://cccbdb.nist.gov/ • literature Experimentalvibrational frequencies (νi), Theoretical vibrational frequencies (ωi). Scalingfactor (c) and its relative uncertainty (): c=0.9608 for B3LYP/6-31G(d)

12. Electronicstates • Usuallyit is notconsidered, butitcan be important: • Spectroscopy • Species havinglowlyingexcitations Energy Units S0/S1/T1 kJ/mol eV nm S1 303.9 3.15 393.6 1.223 Å/1.239 Å/1.230Å 1.223 Å/1.238 Å/1.230Å T1 287.9 2.98 415.5 1.478 Å/1.434 Å/1.453Å 1.393 Å/1.408 Å/1.400Å 1.390 Å/1.409 Å/1.401Å 1.386 Å/1.371 Å/1.376Å 1.389 Å/1.370 Å/1.375Å 1.397 Å/1.430 Å/1.425Å 1.401 Å/1.433 Å/1.426Å 1.486 Å/1.393 Å/1.406Å 1.112 Å/1.111 Å/1.096Å 1.207 Å/1.261 Å/1.299Å S0 0

13. Jablonski diagram Energy Jablonski diagram Potentialenergy diagram Energy Electronicexcitedstate (e.g. S1) Rotationally, vibrationally and electronicallyexcitedstate Rotationalgroundstate, vibrationallyandelectronicallyexcitedstate Rotationallyexcited, vivrationalgroundstate and electronicallyexcitedstate Rotational and vibrationalgroundstate, electronicallyexcitedstate microstate Elecronicgroundstate (e.g. S0) Rotationallyvibrationallyexcitedstateand elecronicgroundstate Rotationalgroundstate, vibrationallyexcitedstateand elecronicgroundstate Rotationallyexcited, vibrational and elecronicgroundstate Rotational, vibrational and elecronicgroundstate(firstmolecularstate!) Interatomicdistance

14. Thermodynamicsterminology E0≡Etot+ZPVE Zero-pointcorrectedenergy E°(T)≡Etot+EthermalThermal-correctedenergy H°(T)=E°(T)+RT Standard enthalpy (pV=nRT!) G°(T)≡H°(T)-TS°(T) Standard Gibbs free energy Macroscopicproperties: Microstates P(T) X inenergydimension Etot (firstmolecularstate!) ZPVE P(0K) H°(rmin)(T) E°(rmin)(T) E0(rmin) G°(rmin)(T) Ethermal(T=0K)=ZPVE Ethermal(T)=Etrans(T)+Erot(T)+Evib(T) (+Eelec(T)) Interatomicdistance Minimum of potentialenergysurface (Etot(rmin))

15. Thermodynamicsterminology ZPVE Zero-point correction= 0.799551 Thermal correction to Energy= 0.855985 Thermal correction to Enthalpy= 0.856929 Thermal correction to Gibbs Free Energy= 0.699092 Sum of electronic and zero-point Energies= -2392.526502 Sum of electronic and thermal Energies= -2392.470067 Sum of electronic and thermal Enthalpies= -2392.469123 Sum of electronic and thermal Free Energies= -2392.626960 E (Thermal) CV S KCal/Mol Cal/Mol-Kelvin Cal/Mol-Kelvin Total 537.139 206.443 332.196 Ecorr Hcorr Gcorr E0=Etot+ZPVE E°(T)=Etot+Ecorr H°(T)=Etot+Hcorr G°(T)=Etot+Gcorr Etot+ZPVE Etot+Ecorr Etot+Hcorr Etot+Gcorr X inenergydimension Etot Hcorr Ecorr Gcorr H°(rmin)(T) E°(rmin)(T) G°(rmin)(T) Interatomicdistance Minimum of potentialenergysurface (Etot(rmin))

16. Referencestate? Differentdefinition of referencestateinexperimentand theory, conversionneeded Experimental Theory e.g. B3LYP/6-31G(d) E(kJ/mol) E(Hartree) Ref (Theory): 0Hartree 9C6+(g)+8H+(g)+2Cl17+(g) +3O8+ (g)+120e- (g) 9184.246kJ/mol =9·716.68+ 8·217.998+ 2·121.301+ 3·249.18 -1490.021129 Hartree =9·-37.843920+ 8·-0.497912+ 2·-460.133882+ 3·-75.058263 93C(g)+82H(g)+22Cl(g) +33O(g) Ref (Experiment): 0 kJ/mol 9Cgrafit+4H2(g)+Cl2 (g) +1.5O2 (g) fH° -404.5 kJ/mol -1493.673273 Hartree C9H8Cl2O3