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2. Molecular stucture/Basic spectroscopy The electromagnetic spectrum Spectral region fooatocadr atomic and molecular spectroscopy E. Hecht (2nd Ed.) Optics, Addison-Wesley Publishing Company,1987 Per-Erik Bengtsson Spectral regions Mo lecu lar spec troscopy o ften dea ls w ith ra dia tion in the ultraviolet (UV), visible, and infrared (IR) spectltral reg ions. • The visible region is from 400 nm – 700 nm • The ultraviolet region is below 400 nm • The infrared region is above 700 nm. 400 nm 500 nm 600 nm 700 nm Spectroscopy: That part of science which uses emission and/or absorption of radiation to deduce atomic/molecular properties Per-Erik Bengtsson Some basics about spectroscopy E = Energy difference = c /c h = Planck's constant, 6.63 10-34 Js ergy nn = Frequency E hn = h/hc /l E = h = hc / c = Velocity of light, 3.0 108 m/s = Wavelength 0 Often the wave number, , is used to express energy. The unit is cm-1. = E / hc = 1/ Example The energy difference between two states in the OH-molecule is 35714 cm-1. Which wavelength is needed to excite the molecule? Answer = 1/ =35714 cm -1 = 1/ = 280 nm. Other ways of expressing this energy: E = hc/ = 656.5 10-19 J E / h = c/ = 9.7 1014 Hz Per-Erik Bengtsson Species in combustion Combustion involves a large number of species Atoms oxygen (O), hydrogen (H), etc. formed by dissociation at high temperatures Diatomic molecules nitrogen (N2), oxygen (O2) carbon monoxide (CO), hydrogen (H2) nitr icoxide (NO), hy droxy l (OH), CH, e tc. Tri-atomic molecules water (H2O), carbon dioxide (CO2)etc), etc. Molecules with more fuel molecules such as methane,,, ethene, etc. than three atoms also molecules formed in rich flames Per-Erik Bengtsson Quantum mechanics According to quantum mechanics all energy levels in atoms and molecules are discrete and are given by the solution to the Schrödinger equation: 2 2 V r E = wavefunction 2 h = h/2 (h = Plancks constant) m m 1 2 reduced mass m1 m2 E = energy levels VilV = potential energy Since the selections rules only permit certain transitions a spectrum can be very precisely calculated Diatomic molecules: Structure A diatomic molecule consists of two - - - nuclilei an d an e lec tron c lou d. - + - + - - The positive nuclei repel each other. - - The negative electrons attract the positive nuclei and prevent them from separation. rgy ee When all electrons have their lowest En possible energies, the molecule is in its ground electronic state. Ground electronic 0 state Per-Erik Bengtsson Diatomic molecules: electronic states We observe an outer loosely bound electron. By the addition of energy, it can be moved to another ”orbit” at higher energy, and the molecule end up in an excited electronic state. + + rgy ee En Different electronically excited states Ground 0 electronic state Per-Erik Bengtsson Absorption For many molecules of interest in combustion, absorption of radiation in the ultraviolet and visible spectral region can lead to electronic excitation of a + + molecule. This absorption process can be represented in an energy level diagram: The energy levels are discrete, and the rgy ee energy difference can be used to En calculate the exact wavelength of the E radiation needed for absorption: E = hc/ 0 Per-Erik Bengtsson Diatomic molecules: Structure The electronic states of molecules are in analogy with electronic states of atoms. In addition molecules can vibrate and rotate . In - - - the Born-Oppenheimer approximation these - motions are initially treated separately: + - + - - - Etot= Eel + Evib + Erot - To more easily describe vibrational and rotational energies, a simpler model for the molecule will be used. Consider the diatomic molecule as two atoms joined by a string. Time scales: Electronic interaction ~10-15s Vibrations ~10-14s Rotations ~10-13-10-12 s Per-Erik Bengtsson Diatomic molecules: vibrations Molecules can also vibrate at different frequencies. BtBut onl y spec ifidiific discre te vibrational frequencies occur, corresponding to specific energies. Per-Erik Bengtsson Vibrational energy levels The vibrational energy states give v=2 a fine structure to the electronic yy Excited states. v=1 Electronic v=0 }state Energ v=0 is the lowest vibrational frequency, then higher v numbers correspond to higher vibrational frequencies. While th e separati on b et ween electronic energy states is around 20000 cm-1,,p the separation between vibrational energy states is on the order of 2000 cm-1. v=2 Ground v=1 Electronic }state 0 v=0 Per-Erik Bengtsson Vibrational energy levels Solutions to Schrödinger equation for harmonic oscillator with 2 give V(r) = ½ k (re-r) 1 Ev/hc = Gv ω(v1/2) [cm ] G0= 1/2 ,G1= 3/2 , G2= 5/2 v = 1 = G(v') - G(v") = (v + 1 + 1/2) - (v + 1/2) = V'- corresponds to levels in the excited state V" – corresponds to levels in the ground state Vibrational energy levels A better description of the energy is given by the Morse function: 2 V Deq.[1exp{a(req. r)}] Dissociation energy, D . where a is a constant for a particular e molecule. Energy corrections can now be introduced. 2 1 Gv e (v1/ 2)exe (v1/ 2) [cm ] v = 1, 2, Per-Erik Bengtsson Vibrational population The transition from v=0 to v=1 then occurs at; 2 2 1 Gv 1 G v 0 e 11/ 2e xe 11/ 2 e 1/ 2 e xe 1/ 2 e 1 2xe cm Whereas the transition from v=1 to v=2 (hot band) occurs at G(v=2) - G(v=l) = (1-4 ) cm-1 Population distributiondistribution of of N N2 e e 2 1 0,9 n oo 0,8 v=0 0,7 v=1 Population Nv; 0,6 v=2 l populati 0,5 aa vv3=3 0,4 Nv ~ exp[-G(v)/kT] 0,3 0,2 Fraction 0,1 0 200 700 1200 1700 2200 2700 3200 Temperature (K) Per-Erik Bengtsson Diatomic molecules: rotations Molecules can also rotate at different frequencies. But only specific discrete rotational frequencies occur, corresponding to specific energies. Per-Erik Bengtsson Rotational energy levels J = 6 The rotational energy states give a v=2 fine structure to the vibrational yy states. v=1 v=0 Energ J=0 is the lowest rotational energy state, then higher J numbers correspond to higher rotational J = 5 states. While th e separati on b et ween vibrational energy states is around J = 4 2000 cm-1,,p the separation between rotational energy states is on the order of a few to hundreds of cm-1. v=2 J = 3 v=1 J =2= 2 J = 1 0 v=0 J = 0 Per-Erik Bengtsson Energy levels for rigid rotator Solutions to Schrödinger equation with V(r) = 0, gives m1 m C 2 2 h r r EJ J(J 1) 1 2 8 2I r0 2 m1m2 I r0 m1 m2 E h F r J(J 1) BJ(J 1) [cm1] J hc 82cI Rotational transitions F = F (J') - F (J") = B J'(J'+l) - B J" (J"+l) J=J = ±l F=2BJF = 2BJ J' - corresponds to levels in the excited state J" – corresponds to levels in the ground state Rotational population distribution The rotational population distribution can be N (2J 1)eBJ(J 1)hc/kT calculated as If f (J) (2J 1)eBJ(J1)hc/kT 0,09 0,08 Nitrogen f (J) 0,07 then on 0 0,06 T=300 K T 0,05 0,04 populati kT 1 ee gives J 0,03 max 0,02 T=1700 K 2Bhc 2 0,01Relativ 0 0 3 6 9 12 15 18 21 24 27 30 33 36 39 Rotational quantum number, J Per-Erik Bengtsson A coupling rotation-vibration Total energy Tv,j given by: 2 Tv,J Gv FJ e v 1/ 2 e x e v 1/ 2 BJJ 1 A transition from v’ to v’’ occurs at: BJ'J'1 BJ"J"1 Consider two cases: A coupling rotation-vibration: Examp le HCl ab sorpti on HITRAN HITRAN is an acronym for high- resolution transmission molecular ABSORPTION, which is a database using spectroscopic parameters to predict and simulate the absorption spectra of different molecules Example: The CO vibrational band (0-1) P Branch R Branch Different broadening mechanisms: Collisional broadeninggppg and Doppler broadening Lorentz profile: 1 bL gL () 22 () 0 bL Doppler profile: exp( (ln 2)(0 )2 ) ln 2 bD gD () bD Voigt profile—convolution of Doppler andLd Loren tz ian line s hape func tion: ''' gggdVDL() ( ) ( ) Absorption simulation H2O+CO2 H2O+CO2 CO2 H2O Isotope abundance: 1% Simulated concentration: 500 ppm Detect limit: 5 ppm Transitions between electronic states Absorption and emission spectra are A spppecies specific The energy difference (and thereby v=1 the wavelength) can be calculated v=0 using molecular parameters for the involved electronic states. nn bsorptio mission X AA EE Rotational energy levels in the first excited vibrational state. v=1 } v=0 } Rotational energy levels in the ground vibrational state. Equilibrium distance Per-Erik Bengtsson Molecular structure - OH Q P R Per-Erik Bengtsson Summaryyp 1: Temperature effects At low tempp(erature (room-T) Molecules generally • rotate at lower frequencies. • vibrate at lower frequencies At high temperature (flame-T) Molecules generally • rotate at higher frequencies. • vibrate at higgqher frequencies Per-Erik Bengtsson Summaryyp 2: Temperature effects Population distribution over energy states at flame temperature, ability bb (also high rotational frequencies, not only lowest vibrational frequency) Pro yy Population distribution over energy states at room temperature, obabilit (Low rotational frequencies, lowest vibrational frequency) rr P 0 Energy The population distribution is often what is probed when temperatures are measured Per-Erik Bengtsson Summary: Molecular physics Each molecule is at a certain time in a specific rotational state, J, in a specific vibrational energy level, v, and in a specific electronic energy level.
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