Difluoroacetylene - Advanced Physical Chemistry - Exam Paper, Exams of Physical Chemistry

Download Difluoroacetylene - Advanced Physical Chemistry - Exam Paper and more Exams Physical Chemistry in PDF only on Docsity! EXAM INFORMATION Harmonic Oscillator Hamiltonian: Energy Levels: Rigid Rotor Quantum Numbers: J = 0, 1, 2, 3, ... M=0, ±1, ±2, ..., ±J Energy Levels: Moment of Inertia: Linear Polyatomic: Diatomic: Boltzmann Distribution: First Order Perturbation Theory: Spherical Polar Coordinates: STATISTICAL THERMODYNAMICS FORMULAS Rotational Partition Function Non-Linear Molecules: Vibrational Partition Function: 1 2 1 2 1 0,1, 2, 2 2 2n m mk E n n c m m                      2ii rmI 21 212 mm mm rI      dHE )0()1()0( * kTE ii iegN /     Ic h BBhcJJ I h JJEJ 22 2 8 ~~ 1 8 1      20 ).(sin0 sin0 2    RconstRotorRigidfordddV ddrdrdVr 2 2 22 2 1 2 kx dx d H    2/132/1         bba rot Tq   kI h kI h kI h c c b b a a 2 2 2 2 2 2 8 8 8       NVT Q kTU , 2 ln           Nrotrot qQ    2 1 v v T Nvib vib vib v T e hc q Q q k e            NTV Q QkTTSHG ,ln ln ln          INTEGRALS Constants and Conversions: h = 6.63x10-34 J·s ħ = h/2 = 1.05x10-34 J·s k = 1.38x10-23 J/K c = 3.00x108 m/s = 3.00x1010 cm/s NA = 6.02x10 23 mol-1 1 Å = 10-10 m 1 amu = 1.66x10-27 kg 1 J = 1 kg·m2/s2 1 N = 1 kg·m/s2 1 N/m = 1 kg/s2 C2h Character Table Transition Moment: Raman Matrix Elements: u = x, y, or z and v = x, y, or z kMjMiMM zi y i x ii  0000  00 00 00    zeM yeM xeM i z i i y i i x i 0 uvi C2h E C2 i h Ag 1 1 1 1 x 2,y2,z2,xy Bg 1 -1 1 -1 xz,yz Au 1 1 -1 -1 z Bu 1 -1 -1 1 x,y                 0 3 6 0 2 4 0 2 0 16 15 8 3 4 1 2 1 2 2 2 2                dxex dxex dxex dxe x x x x (10) 3. The Rigid Rotor Schrödinger equation is: 2 2 2 2 1 1 sin( ) 2 sin( ) sin ( ) E I                       One of the Rigid Rotor wavefunctions is: siniAe   Verify that this function is an eigenfunction of the Rigid Rotor Hamiltonian and determine the eigenvalue (i.e. the energy, E). HINT: It will prove useful to use the folllowing Trig. Identity at one point in the calculation: 2 2sin cos 1   You MUST show your work or credit. (06) 4. Use the appropriate Statistical Mechanical formulas to show that the rotational contribution to the internal energy of one mole of non-linear molecules is (3/2)RT. You MUST show your work for credit. (12) 5. The vibrational force constant of the methylidyne radical, 12CH, is 450 N/m. The spectroscopic dissociation energy of the CH radical is De = 348 kJ/mol Calculate the thermodynamic dissociation energy, D0, in kJ/mol. (10) 6. Use first-order perturbation theory to determine the energy of the first excited state of the cubic oscillator is: Use the Harmonic Oscillator Hamiltonian (see information sheet) as the unperturbed Hamiltonian and use as the unperturbed first excited state wavefunction and energy. Note: You can leave your answer in terms of , k, , and . 3 2 22 2 x dx d H      2 1/4 1/2(0) /2 (0) 32 2 xAxe where A and E             