CHAPTER 4 RIGID-ROTOR MODELS AND ANGULAR MOMENTUM EIGENSTATES OUTLINE Homework Questions Attached SECT TOPIC 1. Math Preliminary: Products of Vectors 2. Rotational Motion in Classical Physics 3. Angular Momentum in Quantum Mechanics 4. The 2D Quantum Mechanical Rigid Rotor 5. The 3D Schrödinger Equation: Spherical Polar Coordinates 6. The 3D Quantum Mechanical Rigid rotor 7. Angular Momentum and the Rigid Rotor 8. Rotational Spectroscopy of Linear Molecules 9. Application of QM to Molecular Structure: Pyridine 10. Statistical Thermodynamics: Rotational Contributions to the Thermodynamic Properties of Gases Chapter 4 Homework 1. Calculate the scalar product and cross product of the of the two vectors:       A3i  j2k B  2i 4j3k 2. Consider a rigid rotor in the state characterized by:  Ae2isin2 (a) Verify that  is a solution to the Rigid Rotor Schrödinger Equation (below). What is the eigenvalue (i.e. energy)? Note: You will probably find it useful to use the trigonometric identity, sin2cos21  cos21sin2 (b) Calculate the squared angular momentum, L2, of the rotor. (c) Calculate the z-component of angular momentum, L , of the rotor: z 3. As discussed in class, the rotational motion of a diatomic molecule chemisorbed on a crystalline surface can be modelled as the rotation of a 2D Rigid Rotor. Consider F 2 adsorbed on a platinum surface. The F bond length is 0.142 nm. 2 Calculate the frequency (in cm-1) of the rotational transition of an F molecule from the 2 m = 2 level to the m = 8 level. 4. The first two lines the rotational Raman spectrum of H79Br are found at 50.2 cm-1 and 83.7 cm-1. Calculate the H-Br bond length, in Å. 5. Which of the following molecules will have a rotational microwave absorption spectrum?: H O, H-CC-H, H-CC-Cl, cis-1,2-dichloroethylene, benzene, NH . 2 3 6. The first microwave absorption line in 12C16O occurs at 3.84 cm-1 (a) Calculate the CO bond length. (b) Predict the frequency (in cm-1) of the 7th. line in the microwave spectrum of CO.. (c) Calculate the ratio of the intensities of the 5th. line to the 2nd line in the spectrum at 25 oC (d) Calculate the the initial state (J’’) corresponding to the most intense transition in the microwave absorption spectrum of 12C16O at 25 oC. 7. The CC and C-H bond lengths in the linear molecule, acetylene (H-CC-H) are 1.21 Å and 1.05 Å, respectively (a) What are the frequencies of the first two lines in the rotational Raman spectrum? (b) What are the frequencies of the first two lines in the rotational Mookster absorption spectrum, for which the selection rule is J = +3 ? (c) Calculate the ratio of intensities in the 20th. lowest frequency line to that of the 5th. lowest frequency line in the rotational Raman spectrum at 100 oC. 8. For two (2) moles of the non-linear molecule NO (g) at 150 oC, calculate the rotational 2 contributions to the internal energy, enthalpy, constant pressure heat capacity, entropy, Helmholtz energy and Gibbs energy. The Moments of Inertia are: I = 3.07x10-47 kg-m2 , I = 6.20x10-46 kg-m2 , I = 6.50x10-46 kg-m2. a b c The symmetry number is 2. 9. The molecular rotational partition function of H at 25 oC is qrot = 1.70. 2 (a) What is qrot for D at 25 oC? 2 (b) What is qrot for H at 3000 oC? 2 DATA h = 6.63x10-34 J·s 1 J = 1 kg·m2/s2 ħ = h/2 = 1.05x10-34 J·s 1 Å = 10-10 m c = 3.00x108 m/s = 3.00x1010 cm/s k·N = R A N = 6.02x1023 mol-1 1 amu = 1.66x10-27 kg A k = 1.38x10-23 J/K 1 atm. = 1.013x105 Pa R = 8.31 J/mol-K 1 eV = 1.60x10-19 J R = 8.31 Pa-m3/mol-K m = 9.11x10-31 kg (electron mass) e Rigid Rotor Schrödinger Equation: ^ L2 2  1    1 2  sin()   E     2I 2I sin()   sin2() 2 The L Equation: z  ˆ  L m   z i  Some “Concept Question” Topics Refer to the PowerPoint presentation for explanations on these topics.  Significance of angular momentum operator commutation  Interpretation of |L| and L for rigid rotor in magnetic field z  Amount of required isotopic data to determine structure of linear molecule  HOMO and LUMO electron distributions. Relationship to changes in bond lengths in excited electronic states (see, for example, pyridine)  Equipartition of rotational energy and heat capacity in linear and non-linear molecules Chapter 4 Rigid-Rotor Models and Angular Momentum Eigenstates Slide 1 Outline • Math Preliminary: Products of Vectors • Rotational Motion in Classical Physics • Angular Momentum in Quantum Mechanics • The 2D Quantum Mechanical Rigid Rotor • The 3D Schrödinger Equation: Spherical Polar Coordinates • The 3D Quantum Mechanical Rigid Rotor • Angular Momentum and the Rigid Rotor • Rotational Spectroscopy of Linear Molecules Not Last Topic Slide 2 1 Outline (Cont’d.) •Application of QM to Molecular Structure: Pyridine • Statistical Thermodynamics: Rotational contributions to the thermodynamic properties of gases Slide 3 Mathematical Preliminary: Products of Vectors  Scalar Product (aka Dot Product) Note that the product is a scalar quantity (i.e. a number) Magnitude: Parallel Vectors: Slide 4 2  Cross Product The cross product of two vectors is also a vector. Its direction is perpendicular to both A and B and is given by the “right-hand rule”. Magnitude: Parallel Vectors: Perpendicular Vectors: Slide 5  Expansion by Cofactors Slide 6 3 Outline • Math Preliminary: Products of Vectors • Rotational Motion in Classical Physics • Angular Momentum in Quantum Mechanics • The 2D Quantum Mechanical Rigid Rotor • The 3D Schrödinger Equation: Spherical Polar Coordinates • The 3D Quantum Mechanical Rigid Rotor • Angular Momentum and the Rigid Rotor • Rotational Spectroscopy of Linear Molecules Slide 7 Rotational Motion in Classical Physics Angular Momentum (L)  m Magnitude: Circular Motion: or: where Moment Angular of Inertia Frequency Energy or: Slide 8 4 Comparison of Equations for Linear and Circular Motion Linear Motion Circular Motion Mass m Moment of inertia Velocity v Angular velocity Momentum p=mv Angular momentum Energy Energy or Energy Slide 9 Modification: Rotation of two masses about Center of Mass  m m m 1 r 1 where Slide 10 5 Outline • Math Preliminary: Products of Vectors • Rotational Motion in Classical Physics • Angular Momentum in Quantum Mechanics • The 2D Quantum Mechanical Rigid Rotor • The 3D Schrödinger Equation: Spherical Polar Coordinates • The 3D Quantum Mechanical Rigid Rotor • Angular Momentum and the Rigid Rotor • Rotational Spectroscopy of Linear Molecules Slide 11 Angular Momentum in Quantum Mechanics Classical Angular Momentum Slide 12 6