Fundamental relations: The thermodynamic functions The molecular partition function

1. Chapter 20 Statistical thermodynamics: the machinery Fundamental relations: The thermodynamic functions The molecular partition function Using statistical thermodynamics Mean energies Heat capacities Equation of state Residual entropies Equilibrium constants

2. Exercises for Chapter 20 • 20.1(b), 20.3(a), 20.6(a), 20.10(b), 20.12(a), 20.15(b),20.17(a) • 20.3, 20.6, 20.10, 20.16, 20.19

3. Fundamental relations • The thermodynamic functions • The molecular partition function

4. The thermodynamic functions: A and p Independent molecules: (Distinguishable) (Indistinguishable) A= U - TS The Helmholtz energy A(0)=U(0) The pressure

5. Thermodynamic Riddles (Ch.5) U=q+w, H=U+pV, G=H-TS, A=U-TS dU=TdS-pdV, dH=TdS+Vdp dG=Vdp-SdT, dA=-SdT-pdV Enjoy!!

6. Derive the expression for the pressure of a gas of independent particles Deriving an equation of state For a gas of independent particles: Where following relations have been used:

7. Classroom exercise where f depends on the volume With • Deriving the equation of state of a sample for which

8. The thermodynamic function: H H=U+pV For a gas of independent particles:

9. The thermodynamic functions: G G=H-TS=A+pV For a gas of independent particles: G=A+pV Or Define molar partition function

10. The molecular partition function Translational, Rotational, Vibrational, Electronic energies

11. The translational contribution At room temperature, O2 in a vessel of 100 ml At room temperature, H2

12. Independent states (factorization of q) Three-dimensional box: (Translational partition function) Thermal wavelength

13. Typical Rotors

14. The Rotational Energy Levels Around a fixed-axis Around a fixed-point (Spherical Rotors)

15. Linear Rotors

16. The rotational contribution (Linear rotors) At room temperature, for H(1)Cl(35), B=10.591 1/cm hcB/kT=0.05111 kT/hc=207.22 1/cm

17. The rotational contribution: Approximation(Linear rotors) hcB/kT<<1

18. Symmetric Rotors

19. The rotational contribution: Approximation(Symmetric rotors) hcB/kT<<1

20. 4 4 3 3 2 2 1 1 0 0 J J K K =

21. The rotational contribution: Approximation(Symmetric rotors)

22. The rotational contribution: Approximation(Asymmetric rotors) (If you have question here, just ignore it. The detailed derivation of this equation is beyond the Scope of this course.)

23. Rotational temperature The ``high temperature`` approximation means From Table 20.1, it is clear that this approximation is indeed valid unless the temperature is not too low (< ~10K).

25. Symmetry number • How to avoid overestimating the rotational partition function? Symmetrical linear rotor After deducting the indistinguishable States, In general,

26. Symmetry number Nonlinear molecules: General cases: the number of rotational symmetry elements.

30. Symmetry Group and Symmetry Numbers

31. A=4.828 1/cm, B=1.0012 1/cm, C=0.8282 1/cm, T=298 K ABC=4.0033 1/cm

32. N Classroom exercise A=0.2014 1/cm, B=0.1936 1/cm, C=0.0987 1/cm, T=298 K ABC=0.004 1/cm

33. Quantum mechanical interpretation The wavefunction of fermions changes sign when exchanged whereas the wavefunction of bosons does not change sign when exchanged.

34. (for odd J) (for even J)

35. CO2 CO2  Boson Nuclear spin = 0 Only even J-states are admissible

36. Quantum mechanical interpretation There are 8 nuclear spin states:

37. Quantum mechanical interpretation

39. Quantum mechanical interpretation

40. Quantum mechanical interpretation • Generally, for a molecule with NR rotational elements (including the identity operation), the symmetry number

41. Symmetry Group and Symmetry Numbers

42. 10 points!! Derive from quantum mechanics the symmetry number of benzene:

43. The vibrational contribution v＝0 , 1 , 2 ,

45. Normal modes For a nonlinear molecule of N atoms, there are 3N degrees of freedom: 3 translatinal, 3 rotational and 3N-6 vibrational degrees of freedom For a linear molecule of N atoms, there are 3N degrees of freedom: 3 translatinal, 2 rotational and 3N-5 vibrational degrees of freedom The total vibrational partition function is:

46. Exercise • The wave number of the three vibrational modes of H2O 3656.7 1/cm, 1594.8 1/cm, and 3755.8 1/cm. Calculate vibrational partition function at 1500 K. The total vibrational partition function then is: 1.031x1.276x1.028=1.353 At 1500 K, most molecules are at their vibrational ground state!

47. Classroom exercise • The three vibrational normal modes of CO2 are 1388 1/cm, 667.4 1/cm (doubly degenerate), 2349 1/cm. Calculate the vibrational partition function at 1500K.

48. Low temperature approximation Only the zero-point level is occupied. High temperature approximation

49. The electronic contribution For most cases, the excited energy is much larger than kT and the electronic energy level of the ground is not degenerate:

50. Degenerate case: NO