8.2 Energy level and its degeneracy

1. 0 1 2 3 4 5 8.2 Energy level and its degeneracy Energy levels are said to be degenerate, if the same energy level is obtained by more than one quantum mechanical state. They are then called degenerate energy levels. The number of quantum states at the same energy level is called the degree of degeneracy.

2. A molecular energy state is the sum of an electronic (e), nuclear (n), vibrational (v), rotational (r) and translational (t) component, such that:

3. The degree of freedom of movement • Translation: x,y,zF=3

4. Rotation • For linear molecules, F=2 • For non-linear molecules, F=3

5. Vibration • A polyatomic molecule containing n atoms has 3n degrees of freedom totally. Three of these degrees of freedom can be assigned to translational motion of the center of mass, two or three to rotational motion. • 3n-5 for a linear molecule; • 3n-6 for a nonlinear molecule

6. CO2 has 3×3-5 = 4 degrees of freedom of vibration; nonlinear molecule of H2O has 3×3-6 = 3 degrees of freedom of vibration.

7. 8.2.1 Translational particle The expression for the allowed translational energy levels of a particle of mass m confined within a 3-dimensional box with sides of length a, b, c is Where h＝6.626×10-34J·s，nx, ny, nz are integrals called quantum numbers. The number of them is 1,2,…∞ . If a=b=c, equation becomes

8. all energy levels except ground energy level are degenerate. Example At 300K, 101.325 kPa, 1 mol of H2 was added into a cubic box. Calculate the energy level εt,0 at ground state, and the energy difference between the first excited state and ground state.

9. Solution Take the H2 at the condition as an ideal gas, then the volume of it is The mass of hydrogen molecule is

10. the energy difference is so small that the translational particles are excited easily to populate on different excited states, and that the energy changes of different energy levels can be think of as a continuous change approximately.

11. 8.2.2 Rigid rotator (diatomic) The equation for rotational energy level of diatomic molecules is ： where J is rotational quantum number, I is the moment of inertia (转动惯量) μis the reduced mass (折合质量), The degree of degeneracy is

12. 8.2.3 One-dimensional harmonic oscillator Where v quantum number，when v=0，the energy is called zero point energy. One dimensional harmonic vibration is non-degenerate.

13. 8.2.4 Electron and atomic nucleus The differences between energy levels of electron motion and nucleus motion are big enough to keep the electrons and nuclei stay at their ground states. Both degree of degeneracy, ge,0, for electron motion at ground state and degree of degeneracy, gn,0, for nucleus motion at ground state are different for different substances, but they are constant for a given substance.

14. 8.5 Computations of the partition function • 8.5.1 Some features of partition functions • (1) at T=0, the partition function is equal to the degeneracy of the ground state. • (2) When T is so high that for each term εi/kT=0, • (3) factorization property If the energy is a sum of those from independent modes of motion, then

15. The partition functions for 5 mode motions are expressed as

17. 8.5.2 Zero-point energy • zero-point energy is the energy at ground state or the energy as the temperature is lowered to absolute zero. • Suppose some energy level of ground state is ε0, and the value of energy at level i is εi, the energy value of level i relative to ground state is • Taking the energy value at ground state as zero, we can denote the partition function as q0.

18. Since εt,0≈0, εr,0=0, at ordinary temperatures.

19. The vibrational energy at ground state is • therefore • the number of distribution in any levels does not depend on the selection of zero-point energy.

20. 8.5.3 Translational partition function Energy level for translation The partition function

22. For a gas at ordinary temperature α2<<1, the summation converts into an integral. Take qt,x as an example

23. In like manner, From mathematic relations in Appendix

24. Example Calculate the molecular partition function q for He in a cubical box with sides 10cm at 298K. • Solution The volume of the box is V=0.001m3. The mass of the He molecule is 0.004/(6.022×1023)=6.6466×10-27kg. Substituting these numbers and the proper natural constants, we have

25. For ideal gas,

26. 8.5.4 Rotational partition function The rotational energy of a linear molecule is given by εr = J(J+1)h2/8π2I and each J level is 2J+1 degenerate. define the characteristic rotational temperature

27. Θr<<T at ordinary temperature, The summation can be approximated by an integral Let J(J+1)=x, hence J(2J+1)dJ=dx, then

28. For a homonuclear diatomic molecule, such as O2, it comes back to the same state after only 180o rotation. where σ is called the symmetry number. σ is the number of indistinguishable orientations that a molecule can exhibit by being rotated around symmetry axis. It is equal to unity for heteronuclear diatomic molecules and is equal to 2 for mononuclear diatomic molecules. For HCl, σ = 1; and for Cl2, σ = 2.

29. 8.5.5 Vibrational partition function Vibrational energies for one dimensional oscillator are Vibration is non-degenerate, g=1. The partition function is

30. Define the characteristic vibrational temperature

31. Characteristic vibrational temperatures are usually several thousands of Kelvins except for very low frequency vibrational modes. we cannot use integral instead of summation in the calculation of vibrational partition function.

32. At low T， ， ，according to mathematics

33. take the ground energy level as zero, For NO, the characteristic vibrational temperature is 2690K. At room temperature Θv/T is about 9; the , indicating that the vibration is almost in the ground state.

34. 8.5.6 Electronic and nuclear partition function Energy difference is large, so electrons are generally at ground state, all terms except first one in the summation expression is negligible.

35. If the quantum number of total angular momentum for electronic motion is j, the degeneracy is (2j+1). Then the electronic partition function can be written as • A rare exception is halide atoms and NO molecule. The difference between the ground state and the first excited state of them are not so large, the second term in the summation has to be considered.

36. Nuclear motion Nuclear motion is always in the ground state at ordinary chemical and physical process because of large energy difference between ground and first excited state. Its partition function has the form of where I is a quantum number of nuclear spin.

37. 8.6 Thermodynamic energy and partition function

38. 8.6 Thermodynamic energy and partition function Substitute this equation into equation (8.48), we have

39. Substitute the factorization of partition function for q Only qt is the function of volume, therefore

40. If the ground energy is specified to be zero, then

41. It tells us that the thermodynamic energy depends on the zero point energy. Nε0is the total energy of system when all particles are localized in ground state. It (denoted as U0) can also be thought of as the energy of system at 0K. Then,

42. U0 can be expressed as the sum of different energies

43. The calculation of • (1) The calculation of

44. The calculation of The degree of freedom of rotation for diatomic or linear molecules is 2, the contribution to the energy of every degree is also ½ RT for a mole substance.

45. The calculation of Usually, Θv is far greater than T, the quantum effect of vibration is very obvious. When Θv/T>>1, Showing that the vibration does not have contribution to thermodynamic energy relative to ground state.

46. If the temperature is very high or theΘv is very small, thenΘv/T<<1, the exponential function can be expressed as

47. For monatomic gaseous molecules we do not need to consider the rotation and vibration, and the electronic and nuclear motions are supposed to be in their ground states. The molar thermodynamic energy is

48. For diatomic gaseous molecules vibration and rotation must be considered. If only lowest vibrational levels are occupied, the molar thermodynamic energy is

49. If all vibrational energies are equally accessible, the molar thermodynamic energy for vibration is • The molar thermodynamic energy for diatomic molecules is then

50. 8.7 Heat capacity and partition function • The molar heat capacity, CV,m, can be derived from the partition function. Replace q with We can see from above equations that heat capacity does not depends on the selection of zero point of energy.