Statistical Thermodynamics: from Molecule to Ensemble 1

Introduction to thermodynamic state functions Partitional function (q, [q]=?): Encodes how the probabilities are 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 (’measure of disorder’) Gibbs Free energy (G, [G]=kJ/mol): G ≡ H - TS maximum energy can be attained only in a completely reversible process (’chemical potential’ ’available energy’) 2

State Function? Independently: fi 3

SedentaryEstimated Energy Requirements (kJ per day) YearFemaleMale Get the feeling of Energy conversion Glucose C 6 H 12 O O 2 = 6 CO H 2 O 4

Statistical thermodynamics 1 Molecule 1 Mol 5

+ Statistical thermodynamics 1 Molecule 1 Mol How to get microstates? If we know q and T, then: E(T) and S(T) !!! „Sum over states”Energy distribution of the molecules Energy microstates Boltzmann distribution E(T) and S(T) H(T) = E(T) + RT G(T) = H(T) – TS(T) 6

Microstates Combintation of Degree of freedom – External: translation – Internal: rotation, vibration and electronic 7

Translational states What we need to know: Molecular mass E trans (T)=3/2Nk b T=3/2RT S trans (T)=R[ln(q trans )+1+3/2] 1 molecule 1 mol E trans (T=298.15K)=3/2RT=3.7 kJ/mol 8

Rotational states Energy level diagram ≈2x =1x What we need to know: Optimized geometry of the species → Rotational constants (Rigid rotor treatment) E rot (T)=Nk b T=RT S rot (T)=R[ln(q rot )+1] 1 molecule 1 mol E rot (T=298.15K)=RT=2.5 kJ/mol 9

Vibrational states What we need to know: Optimized geometry of the species Force constants (k) → harmonic wavenumber (Harmonic oscillator approximation) 1 molecule 1 mol 1 dimension (diatomics molecule) 10

Scaling factors The vibrational frequencies need adjustment (scale factor) to better match experimental vibrational frequencies. This scaling compensates errors from: (1) Approximation in the solution of the electronic Schrödinger equation. (2) Harmonic oscillator approach How to get scaling? (1)Do it yourself (2)Find it (a) (b)literature c= for B3LYP/6-31G(d) 11

Electronic states Usually it is not considered, but it can be important: -Spectroscopy -Species having low lying excitations Å/1.239 Å/1.230Å Å/1.434 Å/1.453Å Å/1.409 Å/1.401Å1.393 Å/1.408 Å/1.400Å Å/1.370 Å/1.375Å Å/1.430 Å/1.425Å Å/1.238 Å/1.230Å Å/1.371 Å/1.376Å Å/1.393 Å/1.406Å Å/1.433 Å/1.426Å Å/1.261 Å/1.299Å Å/1.111 Å/1.096Å S0/S1/T1S0/S1/T1 Energy S0S0 T1T1 S1S kJ/mol eV nm Units 12

13 Jablonski diagram Energy Interatomic distance Rotational, vibrational and elecronic ground state (first molecular state!) Rotationally excited, vibrational and elecronic ground state Rotational ground state, vibrationally excited state and elecronic ground state Rotationally vibrationally excited state and elecronic ground state Rotational and vibrational ground state, electronically excited state Rotationally excited, vivrational ground state and electronically excited state Rotational ground state, vibrationally and electronically excited state Rotationally, vibrationally and electronically excited state Electronic excited state (e.g. S 1 ) Elecronic ground state (e.g. S 0 ) Jablonski diagram Potential energy diagram Energy microstate

Thermodynamics terminology E 0 ≡E tot +ZPVEZero-point corrected energy E°(T)≡E tot +E thermal Thermal-corrected energy H°(T)=E°(T)+RT Standard enthalpy (pV=nRT!) G°(T)≡H°(T)-TS°(T)Standard Gibbs free energy X in energy dimension Minimum of potential energy surface (E tot (r min )) E 0 (r min ) Interatomic distance E tot G°(r min )(T) H°(r min )(T) E°(r min )(T) 14 P(T) ZPVE E thermal (T=0K)=ZPVE E thermal (T)=E trans (T)+E rot (T)+E vib (T) (+E elec (T)) Microstates Macroscopic properties: (first molecular state!) P(0K)

Thermodynamics terminology E 0 =E tot +ZPVE E°(T)=E tot +E corr H°(T)=E tot +H corr G°(T)= E tot +G corr X in energy dimension Minimum of potential energy surface (E tot (r min )) Interatomic distance E tot G°(r min )(T) H°(r min )(T) E°(r min )(T) 15 G corr Zero-point correction= Thermal correction to Energy= Thermal correction to Enthalpy= Thermal correction to Gibbs Free Energy= Sum of electronic and zero-point Energies= Sum of electronic and thermal Energies= Sum of electronic and thermal Enthalpies= Sum of electronic and thermal Free Energies= E (Thermal) CV S KCal/Mol Cal/Mol-Kelvin Cal/Mol-Kelvin Total G corr H corr E corr ZPVE E corr H corr E tot +G corr E tot +H corr E tot +E corr E tot +ZPVE

Reference state? E(kJ/mol) 16 E(Hartree) 9 3 C(g)+8 2 H(g)+2 2 Cl(g) +3 3 O(g) kJ/mol =9· · · · Hartree =9· · · · Theory e.g. B3LYP/6-31G(d) Experimental Hartree Different definition of reference state in experiment and theory, conversion needed 9C grafit +4H 2 (g)+Cl 2 (g) +1.5O 2 (g) 0 kJ/mol kJ/mol fH°fH° 9C 6+ (g)+8H + (g)+2Cl 17+ (g) +3O 8+ (g)+120e - (g) 0 Hartree C 9 H 8 Cl 2 O 3 Ref (Theory): Ref (Experiment):

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