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# Adv Analytical Chemistry II CEM 835

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This 45 page Class Notes was uploaded by Ladarius Rohan on Saturday September 19, 2015. The Class Notes belongs to CEM 835 at Michigan State University taught by Staff in Fall. Since its upload, it has received 39 views. For similar materials see /class/207705/cem-835-michigan-state-university in Chemistry at Michigan State University.

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Date Created: 09/19/15

Spectrochemical Measurements Expressions of Intensity 39 Quantities based on radiometric system not photometric system 0 Basic unit is joule and other SI units sometimes nonSi units for convenience 39 Often de nitions include area volume or solid angle Spectral quantities Bx 7L Partlal quantltles Bk 1 2 k2 Bkdk 1 Total quantities B I Bkdk TABLE 2 1 Radiometric system Quantity Symbols Description Defining equationa Units General Radiant energy Q Energy in the form of radiation J ergs Radiant energy density U Radiant energy per unit U 192 J cm 3 volume 3V Radiant ux or radiant power P Rate of transfer of radiant I 92 W energy at Source Radiant intensity I Radiant power per unit solid 933 W sr 1 angle from a point source an Radiant emittance or radiant M Radiant power per unit area 532 W cm 2 ex1tance 3A Radiant emissivity J Radiant power per unit solid 621 W sr 1 cm 3 angle per unit volume 39 3V Radiance BL Radiant power per unit solid 621 W sr l cmquot2 angle per unit projected area an GAP 621 69 6A cos 0 Receiver Irradiance E Radiant power per unit area E 32 W cm 2 8A I 2 Radlant exposure H Integrated 1rrad1ance H L E dt J cm CEM 835 page 11 Important quantities Radiant flux 1 rate of energy transfer J s391 W Radiant intensity I radiant ux from a point source per unit solid angle QD4n applies to source J s391 sr391 Radiance B radiant intensity I per projected area CD475A applies to source gt depends on angle between detector and radiation propagation direction see Fig 23 Js391sr391cm392 Irradiance E radiant ux CD ontofrom a surface per unit area QDA applies to source or detector Js 1cm392 radiant Exposure H timeintegrated irradiance tAdt J cm392 Fluence often used but meaning is imprecise CEM 835 page 12 Geometric factors Often radiometric quantities include a solid angle or projected area Solid angle l steradian sr is the part of the surface area of a sphere of radius r having an area of r2 2 Asphere 4 7T39 r 4 75 r2 sterad1ans 1n sphere 2 4 71 1257 r 0 x Arc r y r Plane angle One radian 60 Area One steradian Solid angle iIgu b For example intensity is the radiant ux per unit solid angle I 1 4n CEM 835 page 13 How are these quantities related to spectrochemical techniques Emission Spectroscopy Emitted radiation 4 5 E21 hl 21 hcAZI E2 hvz hcKZ E1 hvl hcK1 b 4 5 Thermal electrical A or chemical l I L A energy K2 K1 21 a C 0 Emission refers to thermally excited atoms or molecules ame ICP electrical discharge plasma Interested in number of atoms per unit volume element observed Demission Aji 39hVij 39nj 39V observation volume atoms in excited state j energy per transition Einstein coef cient transition probability j gti CEM 835 page 14 Ilj can be de ned if in thermal equilibrium by Boltzmann expression Demission Aij 39hVij 39nj 39V EjkT gj e oo E kT 39V Z e 1 10 1 fraction of total in state j Z Aij 39hVij 39ntotal The weighting factor gj statistical weight is the number of degenerate states at each energy E g 2 2J 1 J is the total angular momentum LS LSl LS Example g2s2 g2p122 g2p324 Hence radiant ux Qemission can absolutely determine the concentration of atoms in detection volume CEM 835 page 15 Absorption Spectroscopy Incident Transmitted radiation radiation QO P a b E2 hV2 E1 hvl hcAl c Absorbance A given by Beers39 Law related to the measured quantities DO and CD radiant ux by CD A l T l b og ogq 8 c 0 concentration molL39l cell pathlength cm molar absorptiVity L mol391 cm391 lt1 13010 839b39C Luminescence measurements Scattering measurements CEM 835 page 16 Optical Instruments Many spectrochemical instruments share common components 0 a radiation source 0 optics to de ne light paths 0 a sample container 0 a dispersion element 0 detector transducer Speci c names are applied to the various instruments Aperture Photographic film a Spectrochemical or pg fegioimy encoder Exit slit b W Photodetector Dispersion Entrance element Focal slit plane I Q and image transfer system Photodetectors A spectroscope disperses a range of X39s for Visual Viewing A spectrograph disperses a range of X39s onto focal plane for simultaneous measurements by a photographic lm or array detector A monochromator uses entrance slit eXit slit and a dispersion element to separate X39s in space If multiple eXit slits are used the term polychromator is used CEM 835 page 17 A photometer measures intensity but has no provision for 9 scanning X39s can be selected by use of lters A spectrometer includes means of manually or automatically scanning wavelength A spectrophotometer has provision for scanning measurements using two beams of light useful for ratioing incident and transmitted light An interferometer is a nondispersive device that relies on interference to obta1n spectral 1nformatlon A detector is any device whose output is proportional to the intensity of light falling on it A transducer more speci c uses electrical signals CEM 835 page 18 Components of Measurement Instruments Radiation Sources Many radiation sources are based on black body radiation 39 perfect absorber of radiation at all X39s 0 if in thermal equilibrium must also be perfect radiation emitter 2000K E 1750K dWd J m 4 1000 1250K 0 1000 2000 3000 4000 5000 7 visible region 39 391 nm Two obvious points 39 total amount of energy radiated increases rapidly with T U a T4 Stefan39s Law Radiant energy density J cm3 39 position of the maximum spectral radiance kmax blue shifts with increasing T C A 2 max 4965 T where c2 143 8X107 nm K CEM 835 page l9 Energy density U J cm3 is dif cult to measure usually work in radiance B Js1sr1cm2 U c B c B L B V V 47 7 Planck deduced black body equation after consideration of thermodynamics of system with discrete energy levels multiples of hv the beginning of quantum mechanics 2h3 1 B V v C2 ehvkT 1 or in terms of wavelength BKZM 7L5 ehcxkr1 3195 Planck39s radiation law where c12hc2 119x1016 Wnm4cm392sr391 and c2hck same as above 1438x107 nmK CEM 835 page 110 Einstein coef cients Three Einstein coef cients Bij describes probability of absorption from level i gt j Bji describes probability of stimulated emission from level j gti Note These two are simply timereversed processes Aji describes probability of spontaneous emission from level j gti l B A j 8 i i v Absorption Spontaneous Stimulated emission emission The rate of absorption per unit volume s391 cm393 depends on i number of atoms in initial state i ni ii probability of absorption from state i to another state j Bij iii the spectral energy density of incident radiation UV dni B U n dt 1 V 1 absorption removes population from state i so ni decreases CEM 835 page 111 Similarly rate of stimulated emission is The rate of absorption and stimulated emission are the same if there is an equal population in both states Bji 8139 Bij 39gi g is the degeneracy statistical weight Rate of spontaneous emission doesn39t include a UV term E dt If black body is in thermal equilibrium with surroundings rate of absorption and emission must be equal 39Uv ni 39nj Uv nj absorptlon spontaneous stnnulated emission emission U Aji39nj V Bji 39Ili Bji Aji 39nj Alin lt substitutingB g B g ji39 j ij39 i Bjinigigj39nj Aji 39nj CEM 835 page 112 At equilibrium Boltzmann equation can be used to nd nj from ni Aji 39nj Bjini gi gj39nj substituting nj ni eXp hvij kT V Aji 39nj V Bjini gi gj ni eXphVij 1ltT This looks similar in form to Planck39s radiation law B 2hv3 1 v C2 ehvkT1 and gives us the rate of spontaneous emission and absorption UVc 8 h 3 Aji WBij remember1ng BV gjc 4TB Bji39gj Bij39gi CEM 835 page 113 Reflection and Refraction Maxwell39s equations lead to de nition for the velocity of electromagnetic radiation in a vacuum 1 C xSo 39Ho where 80 is the permittivity of freespace 8854x103912 C2N391 m392 uO is the permeability of freespace 4710397 kgmC392 In a medium velocity is reduced VIM The ratio of the velocity in a medium to freespace is refractive indeX gt 100 in a medium 11 varies with wavelength usually increases with frequency called normal dispersion decreases with frequency in region of absorption called anomalous dispersion CEM 835 page 31 7t nm 11 351 1539 458 1525 486 1522 532 1519 644 1515 830 1510 Important Frequency of radiation is xed by source Hence wavelength of radiation in a medium must increase c kz s1ncevot T1 xmedium gt xvacuum When a wave passes from medium with refractive index 111 to medium of refractive index 112 we can write hZ c X nz u 9L1 1121 c 112 CEM 835 page 32 Based on wave representation of electromagnetic radiation and geometry we can quickly deduce the angle of re ection Re ected wavefront Incident wavefront 771 a b 9i 2 93 Law of specular re ectance CEM 835 page 33 The refracted beam does not travel at same velocity as the incident beam v2 v1 411112 first part of the wavefront to strike the interface is retarded preferentially light beam bends towards the interface normal when n2gtm n1 sin61 n2 sin62 Snell39s law of refraction no refraction when 61 00 no transmittance when 61 gt Be critical angle total internal re ection sin 61 sin 62 Snell39s law 111 when sin 62 9O0 61 60 sin 1 111 For airglass 60 z 42 CEM 835 page 34 90 CEM 835 page 35 Fresnel Equations Re ectance losses occur at all at interfaces 0L0 TL pt 1 Conservation Law magnitude increases as the di erence in the refractive indices increases dependent on incidence angle Equation describing the re ectance p00 is the Fresnel equation 1 l sin20i 0 1t2111291 9r 2 Lsin20i 0 1 tan291 9rgti 90 Where 0 i is incidence angle and 01 is refraction angle For the airglass at 589 nm re ectance is about 004 or 4 per interface 10 08 06 Re ectance 04 02 004 CEM 835 page 36 p0 constant for small angles p0 increases rapidly at large angles grazing incidence m Serves several purposes in a spectrometer change the direction of a beam change the polarization of a beam split a beam into two disperse the beam A variety of shapes and materials are available to perform these functions Dispersing prism According to Snell39s Law sinGl sin 62 Snell39s law 111 there will be no dispersion if not is constant dispersion in prism occurs because of the change in refractive index of the prism material as a function of wavelength 0 if prism material exhibits normal dispersion higher frequency shorter wavelength light experiences a higher refractive index than lower frequency longer wavelength light CEM 835 page 37 Light of different wavelengths become divergent and become separated in space angle between incident and refracted beam is called the deviation The variation in deviation with wavelength is called the angular dispersion d6 d6 dn D A dx dn g prism dispersion first term depends on size and shape of the prism and the incidence angle second term prism dispersion depends on the material of the prism and the wavelength 3 glass357 nm 2 194x10 4 nm1 dn 5 1 a glass825 nm 178XlO nm Prisms not often used as dispersion elements because of non constant D A with wavelength produces nonconstant bandwith means range of X39s projected onto eXit slit varies with 9 CEM 835 page 38 Electromagnetic radiation An electromagnetic wave is a transverse wave electric and magnetic elds perpendicular to the propagation direction Plane linearly polarized beam has constant plane containing the electric and magnetic vectors often called unpolarized The timedependent electric eld is E Eosinoa t where E0 is the maximum electric eld strength 0 is the angular frequency 2751 t is time I is the angular phase The angular phase is d0275Xt where X is distance and bo is the phase at x0 275 is number of waves per unit length If two waves maintain the same relative phase difference over i extended period of time ii length they are said to be coherent CEM 835 page 39 Superposition The superposition of two waves states two plane polarized waves can be algebraically summed to produce a resultant wave If waves have same frequency E 2 E1 E2 2 E021 sin03t 11 E022 sin03t 12 Amplitude intensity of wave is E2 E2 2 E1 E22 2 E12 E22 E139E2 E0212 13022 2E0 1 EO Z COS 2 interference term If 1 2 O 27 47 cos0 27 47 l wave amplitude will be reinforced constructive interference If1lt2 7 37 57 cos7 37 57 l wave amplitude will be reduced to zero destructive interference CEM 835 page 310 Interference can result from difference in pathlength If the waves initially start out with same phase the difference in phase 6 due to different paths is 5 1 2 27X1 2nX2 x x 275X1 X2 7 where X1 and X2 are the lengths to the measurement point from source 275 is the number of a complete waves per unit length Thus when 6 O 27 an integral number of wavelengths 275 m 2 n 2 X1 X2 9 5 m 9t 2 construct1ve1nterference 7 when 6 7 37 an integral number of wavelengthsl2 2m 1 5 2 j 2 destruct1ve1nterference Tc CEM 835 page 311 Diffraction Eschellete gratings Parallel grooves etched blazed onto re ective surface asymmetric in profile Groove facet Diffracted ray Incident ray Grating 39 normal N a b Incident light striking long facet is re ected in specular direction With respect to the groove normal light from neighboring grooves travels different distances and so interference occurs in outgoing beam Note angles or and B are de ned With respect to the grating normal not the groove normal Constructive interference occurs When the pathlength difference is an integral number of wavelengths extra pathlength associated With the incident beam is AC AC 2 d sin or extra pathlength associated With the outgoing beam is AD AD 2 d sin B CEM 835 page 312 The total pathlength difference is AC AD AC AD dsinoc sinB m dsin CC sin 3 Grating Formula minimum value of d as M2 because the maximum value of sinoc sinB is 2 The first order m 1 diffraction angle can be calculated for any incidence angle by rearranging the grating formula m s1noc s1nB d m s1nB Y s1n0L where d is found from the groove spacing Important diffraction angle depends on d longer X39s diffracted more than shorter ones 3600 m gt 3500 nm When m0 zero order sinoc sinB or OL B In this case all X39s are diffracted at the same angle If blaze was parallel to the grating plane y 0 the zero order beam would also appear in the specular direction most of the re ected light not dispersed If blaze angle 7 0 specular and zeroorder angles do not correspond and majority of the light is dispersed CEM 835 page 313 Specular re ection Specular re ection Groove normal l3 Incident Incident ray 4 ra Oorder Grating 7 y normal Oorder a b In the special case when incident beam is along the surface normal 0cO and rstorder beam is in specular direction in this case 3 is twice the blaze angle y The wavelength at this angle is called the blaze wavelength m t blaze dsin 0c sin 3 kblaze dSin 3 dsin 2y CEM 835 page 314 Dispersion The angular dispersion D A of the grating can be obtained by differentiating the grating formula with respect to wavelength For constant incidence angle ml dsin 0t sin 3 Grating Formula 1 m DA d E d cosB dsin 0L sin 3 Z d cos 3 sin 0t sin 3 9 cos B sin 0t xed For nearly normal incidence CC is small so 3 is small and so cosB does not change much with k D A does not change much with wavelength much better dispersion element than prism CEM 835 page 315 Monochromators Comprised of 0 dispersive element 0 image transfer system mirrors lenses and adjustable slits an image of the entrance slit is transferred to the eXit slit after dispersion One of the most common arrangements is the CzernyTurner monochromator Entrance Grating SM 81 Collimating mirror M1 Focusing mirror M2 CEM 835 page 316 Wavelength selection Wavelength selection is accomplished by rotating the grating Grating Grating 13 Since angle between the entrance slit grating and exit slit is xed 24 grating formula can be expressed in terms of the grating rotation angle 9 between grating normal and optical axis Sinceoc9and39 mk dsin9 sin9 2dsin 9 coscl the trigonometric identity l2sinABsinAB is sinAcosB Grating formula now in experimental variables 9 the grating rotation angle and I halfangle between the entrance grating and exit and slit CEM 835 page 317 Dispersive characteristics Already mentioned the angular dispersion rate of change of diffraction angle with wavelength for a grating D an ular dis ersion A Cm g P However in monochromator much more interested in dispersion at focal plane eXit slit defined by the linear dispersion D1 gt2 Al K 7 gt 1 x 1 A5 x T Focusing 39 r Ax element 4 TAB gt 2 i l 7 x2 Focal I plane I I r4 f gt dX D1 a lmear d1spers1on units of D1 are mm nm391 or similar For a CzernyTurner arrangement the linear dispersion is D1 f DA where f is the focal length of the focusing exit optic CEM 835 page 318 Sometimes the inverse linear dispersion Rd is used units of 1 nm mm39 or similar d Rd Dfl d 1nverse11near d1spers1on X sin 0t sin 3 A kcos B 1 Rd 2 f D A 9 cosB fsinoc sin 3 Spectral bandpass and the slit function The spectral bandpass nm is the halfwidth of the range of wavelengths passing through the exit slit The geometric spectral bandpass sg Rd W geometric spectral bandpass where Rd is the inverse linear dispersion W is slit width CEM 835 page 319 In a monochromator an image of entrance slit is focused at the eXit slit When input is polychromatic a monochromated version of the image appears at the eXit slit When input is monochromatic image rotating the grating angle 9 Will sweep monochromatic image across the eXit slit W Slit width Fixedpolsition ex1t s it outline I Shtlu ght Moving gt 0 entrance slit image I I i I I l I I I I I I i I I I I I 7 Sg No overlap I I I I I l I l I x I l 39 I I I A 3 sg 25 overlap Z I I I l I I 7 I Direction I I gt of image I 39 travel l A sg 50 overlap I I I l I 7 39 I A III sg 75 overlap l I Z l 7r A 100 overlap I I IA I l i I I I I I I I I l I 39 100 I I I I I I Halfwidth 00 Percentage of image I of ham radiation emerging I Sg from exit slit 39 I 50 l I l l 1 l V o 50 100 200 gto 39 5g 7 0 0 Sg Percentage of image overlap slit function a b CEM 835 page 320 The total intensity t0 measured at the exit slit as image is translated is called the slit function for equal entrance and eXit slits shape is triangular for unequal entrance and eXit slits shape is trapezoidal with a base of s and halfwidth of sg Mathematically the slit function is 9 9 tot 1 x0 sg g x 3 k0 sg s g t0 O elsewhere where 9 is the incident monochromatic wavelength at entrance slit k0 is the wavelength setting of the monochromator the wavelength directed to the center of the eXit slit Resolution Resolution quanti es how well separated two features are at the eXit slit closely related to linear dispersion D1 or angular dispersion DA and physical dimensions of the monochromator through f 0 slit width W CEM 835 page 321 Radiation Sources Continuum sources produce broad featureless range of wavelengths black and gray bodies high pressure arc lamps Line sources produce relatively narrow bands at speci c wavelengths generating structured emission spectrum lasers low pressure arc lamps hollow cathode lamps Line plus continuum sources contain lines superimposed on cont1nuum background medium pressure arc lamps D2 lamp Sources may be continuous or pulsed in time CEM 835 page 21 Continuum sources 0 Continuum sources are preferred for spectroscopy because of their relatively at radiance versus wavelength curves Platinum lead q Glower Re ector H Platinum Platinum wire heater lead L a A 63 U C d Parabolic re ector Anode Window Cathode e a Nernst glower b W lament c D2 lamp d are e are plus re ector CEM 835 page 22 Black body sources Nernst glowers ZrOz YOZ Globars SiC 10001500 K in air 7 lies in IR max 0 relatively fragile 0 low spectral radiance Bi lO394 W39Cm392 nm391 sr391 7500 o Blackbody theoretical at 900 C a Globar 39E 750 P A Nernst glower W a Mantle I 3 739 g 750 3 co I 2 z 75 In 1 4 J 1 l 1 1 1 1 1 1 1 l 1 1 1 A l 1 2 6 10 14 18 22 26 3O 34 38 Wavelength pm a CEM 835 page 23 Heated laments W incandescent lamp QTH 0 20003000 K in evacuated envelope greater radiance UaT4 B7 10392 Wcm39Znm39lsr391 greater UVVis output kmax still in IR QTH heated up to 3600 K wo wg Wg12 gt W12 g WIZ WsIZ Arc sources Hg Xe D2 lamps 0 AC or DC discharge through gas or metal vapor 2070 V 10 mA20 A Ionization necessary for conduction hot cathode thermionic emission cold cathode ignition voltage Nonuniform radiance CEM 835 page 24 50 100 150 150 100 50 Hg are radiance 10 1 I I l lllllll 102 EA uW cm 2 nmquotl ll IIIIIH 10 3 I 39 Irradianee BA from D2 lamp measured by a detector at 25 em CEM 835 page 25 IIHH quotIquot I I Tllllquot I III Bx W cmnzsr391 nlmquot1 9 00 d N d d i y i h h b I I 000I 111111 200300400500600700800 Wavelength nm 39 a high pressure Xe lamp b low pressure Hg lamp Hg arc lamps Hg3P1 gt Hg 1SO hv2537 nm Hg1P1 gt Hg 1SO hv1894 nm 0 if P high gt10 atm pseudocontinuum large current gt5 A many atoms excited I 1 radiance Bx gt10 Wcm39Znm39 sr39 if P low lt1 atm line or line plus continuum small current 1 A 10W radiance Bx ltlO4 WcmZnm1 sr1 selfabsorption at high radiant ux CEM 835 page 26 HIIH W Hg arc lamp selfabsorption CEM 835 page 27 vlam Line sources Generally not much use for molecular spectroscopy useful for luminescence excitation photochemistry eXperiments Where high radiant intensity at one 7 required Arc lamps 0 low pressure lt10 Torr With many different fill vapors Hg Cd Zn Ga In Th and alkali metals 0 excellent wavelength calibration sources Hollow cathode lamps HCL Hollow cathode bk 7 f r Anode Quartz Glass Ne or Ar 0T Pyrex shield at 15 torr window 0 primary line sources in atomic spectroscopy 0 low gas pressure lt10 mtorr linewidths 001 A high currents gtfew mA reduces lifetime and broadens lines 0 single or multielement cathodes 0 moderate radiance Bx 10392 W cm392 nm391sr391 CEM 835 page 28 Electrodeless discharge lamps EDL RF coil Ceramic holder 0 contain a microwave or RFexcited plasma need ignition pulse to start plasma 0 electric eld of RF or microwave drives ions and electrons in plasma no electrodes 0 gas pressures and temperatures relatively low slight pressure broadening linewidths are not as narrow as the HCL lt1A 0 moderate radiance 13 101 W cm392 nm391 sr391 CEM 835 page 29 Lasers intense radiance Bx gt104 Wcm392nm391sr391 nearly monochromatic 00lOl A coherent temporally and spatially directed small divergence pulsed or continuous stable Allow measurements not possible with conventional sources Consider radiation traveling through absorbing medium the change in radiant ux due to absorption is dCD CD ni 6 dz lt change due to absorption length of the medium absorption probability number of molecules in state i radiant ux Similarly for stimulated emission dCD CIgt nj 6 dz lt change due to stimulated emission Total change in ux is amount absorbed minus the amount gained by stimulated emission dd CDcsnj nidz lt total change CEM 835 page 210

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