Introductory Physical Chemistry
Introductory Physical Chemistry CH 331
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This 6 page Class Notes was uploaded by Sienna Shields on Thursday October 15, 2015. The Class Notes belongs to CH 331 at North Carolina State University taught by Stefan Franzen in Fall. Since its upload, it has received 25 views. For similar materials see /class/223995/ch-331-north-carolina-state-university in Chemistry at North Carolina State University.
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Date Created: 10/15/15
Chemistry 331 Lecture 5 Rotational Spectroscopy NC State University The Dipole Moment Expansion The permanent dipole moment of a molecule oscillates about an equilibrium value as the molecule vibrates Thus the dipole moment depends on the nuclear coordinate Q 5H o H an where p is the dipole operator 000 00 0 Rotational Transitions Rotational transitions arise from the rotation of the permanent dipole moment that can interact with an electromagnetic field in the microwave region of the spectrum MQ Mu bowOf The total wave function The total wave function can be factored into an electronic a vibrational and a rotational wave function P We va Mm 1 1 if mm mm 1 1de I nmu xymsinededw 1 xllGMunvalMSinededt Interaction with radiation An oscillating electromagnetic field enters as Eocosmt such that the angular frequency hm is equal to a vibrational energy level difference and the transition moment is Mm Bf Yj Mcos9Ystin9ded bow9i Interaction with radiation The choice of cose means that we consider zpolarized microwave light In general we could considerx orypolarized as well X S39neoos um W MN 216 y S39nesln u p sinecostbz sinesin jcosek EEK H3 3 Rotation in two dimensions The angular momentum is JZ pr J z P m Using the deBroglie relation p hAwe also have a condition for quantization of angular motion JZ hrA Classical Rotation In a circular trajectoryJZ pr and E JZ2ZI l is the moment of inertia Mass in a circle I mr2 Diatomic l pr2 m r 1 A 4 m2 mimz Reduced mass it m1 m2 The 2D rotational hamiltonian The wavelength must be a whole number fraction of the circumference for the ends to match after each circuit The condition 2mm Acombined with the deBroglie relation l3ads to a quantized egtltpressionJZ mh Z Z The hamiltonian is m 545 2 13 The 2D rotational hamiltonian Solutions of the 2D rotational hamiltonian are sine and cosine functions just like the particle in ox Here the boundary condition is imposed by the circle and the fact that the wavefunction must not interfere with itself The 2D model is similar to condition in the Bohr model of the atom The 3D rotational hamiltonian There are two quantum numbers J is the total angular momentum quantum number M is the zcomponent of the angular momentum The spherical harmonics called YJM are functions whose probability YJM2 has the well known shape of the s p and d orbitals etc J 0 is s M 0 J1isp M101 J2isd M21012 Space quantization in 3D A Solutions ofthe rotational Schrodinger equation have energies E h2 JJ 121 Specification ofthe azimuthal uantum number rrt implies that the angular momentum about lhleaXlSlSJZ Z This implies a xed orientation between the total angular component i 7 Thex andycomponents cannot be known due to the Uncertainty principle Sphenca harmomc for PE 0056 mmmnmm mm mm w w 5mm mum quotmum Wm number innnmwwamwvnz mm numhu w myquot Wm Sphenca harmomc for P1 0056 A wmmpnmnnmm Manama quantum mm 1 1 Pn wnmxa x mum mumquot mavmmmn Sphenca harmomc for moose pmmmmm 1 quotmm M when wmuamw m i i pumman x mhxd 1m i 722 quantum mummy m mmm mavmmmn Sphenca harmomc for P2 0056 Sphenca harmomc for P2 0056 wmmpnmnnmm m N where Sphenca harmomc for P2cose pmmmmm mW39Pvmm r Msmr39lamxxe mumquot mavmmmn Rotational Wavefunctions O8 J0 J1 J2 These are the spherical harmonics Y which are solutions of the angular Schrodinger equation The degeneracy of the solutions The solutions form a set of 2J 1 functions at each energy the energies are E hJJ 1Z A set of levels that are equal in energy is called a degenerate set Rotational Transitions Electromagnetic radiation can interact with a molecule to change the rotational state Typical rotational transitions occurin the microwave region of the electromagnetic spectrum There is a selection rule that states that the quantum ange only by or 1 for an allowed rotational transition AJ 1 Jl V Orthogonality of wavefunctions The rotational wavefunctions can be represented as the product of sines an cosines ignoring normalization we have s 1 p cose sinecoso sinesino d 123cos79 r 1 cos1 ecos2 coszesianp cosesinecoso cosesinesino The differential angular element is SinededqzAn over the limits 9 Oton and st 0t027r I The angularwavefunctions are orthogonal Orthogonality of wavefunctions For the theta integrals we can use the substitution x cose and dlt sinede For example for s and ptype rotational wave functions we have ltsipgtmrcosesmederxdx5 427120 Question Which of the following statements is true A The number of z projection of the quantum numbers is 2J1 B The spacing between rotational energy levels increases as 2J1 C Rotational energy levels have a degeneracy of 2J1 D All of the above Q Li esti Oi i Wm m the Miniwig 321mm i m a Yhenumhevm zpmmnmmuammmmm 2M 5 We waninghmenMannnaienemviewimnmase a 2w 1 Wimai enemviemixham a degenemcymzm mi a the am m mm mm u m i Q Li esti OH We manner Marnnai imam mm are n hngnnai man that a mm mm may D mmmam Question We manner Marnnai imam mm are n hngnnai man that The moment of inertia m M mm mm mm W MWMWW W m m WWW we in a a g 2 Nme Divaie mmem mam isvmm MW mg unequai mamsn Pure rotational spectra Apure mtatianai medmm isamained bv micmvrave absum ian Iiherange in v avenumbers ism 2 mama seie iarr miesdidatethat the change in quantum numbermus be m 1 and AM 39Amaiemie must passess a gmund sate dipaie mamem in arderta have s Pure mum medium Energy level spacing Energy ieveis quot1 1 i 7 Energvaevenues m i 2M 5 Wan The rotationai constant A pure rotational spectrum 2 5 msgmwwuW320 mg iine waning isms pmpamanai 39 tame mum mnstam i 1 7 7 X h h K m nm a i 7 77 i w mm x 3quot CW Apuvevmmmnaispe mm isabsevvedintne micmv ave va ge m eiedmmagnetic spe mm Question Question Pave Matianai mediunw Pave Matianai mediunw i maxvgen i ammuxygen a Dialamicnmagen a Dialamicnmagen c Waiev c Waiev 0 Datum mam 0 Datum mam A typical rovibrational spectrum Nmetna ne Maiianai medmm is Demeved a vihvatmnai quotequan
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