1. Mathematical models of the chromatographic process

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1 1. athematcal models of the chromatographc process - What determnes retenton tme n L? - What causes pea broadenng n L? - Why are the L peas often asymmetrc? - Why s partton chromatography much more popular than orpton chromatography? 1 athematcal modelng of chromatography We wll loo at three aspects: ompound s retenton tme Pea wdth Pea shape Partton and orpton chromatography Partton chromatography: Lqud-lqud chromatography Partton between two lquds The most common s reversed phase (RP) L Gas-lqud chromatography Partton between lqud and gas Partton and orpton chromatography Adsorpton chromatography: Lqud-sold chromatography Adsorpton on a sold from a lqud Gas-sold chromatography Adsorpton on a sold from the gas phase Partton s sometmes also called dstrbuton 3 4 Partton between lqud and gas Lqud-gas partton: Henry s law H, P P H, Henry s constant of compound Depends on compound, lqud, temperature molar concentraton of compound n the lqud Partal pressure of compound n the gas phase Partton between two lquds Lqud-lqud: Dstrbuton law v v v1 d, v1 d, v1 molar concentraton of compound n lqud 1 v molar concentraton of compound n lqud d, Dstrbuton coeffcent of between lquds 1 and d, depends on compound and lquds, less on temperature 5 6 1

2 Lnear relatonsp Both relatonshps gve the same lnear relatonshp lnear sorpton sotherm (Henry sotherm): n the statonary phase Lnear chromatography hromatographc model whereby lnear sorpton sotherm descrbes partton of the analyte molecules between statonary and le phases s called lnear chromatography If possble then t s useful to wor under the lnear chromatography condtons: Ths s a lot easer to acheve n partton chromatography than n orpton chromatography n the le phase 7 8 When does the lnear chroamtography model hold? Both statonary phase (SP) and le phase (P) are homogenous There s no possblty that the molecules of the same compound can be retaned wth dfferent strengths n ether phase The volumes of the statonary and le phases are much larger than the amount of compound There s no nteracton between the molecules of compound In realty these condtons only partally hold d, Dstrbuton law n L d, concentraton of n SP concentraton of n P Dstrbuton coeffcent of between S and P d, depends on the compound and the phases, less on temperature 9 10 Dstrbuton law n L apacty factor After rearrangng: n V d, n V n number of moles of n SP n number of moles of n P V volume of SP V volume of P After rearrangng: V d, ' V n capacty factor of Depends on, d,, and volumes of the phases n 11 1

3 Snce: then: t R, t d, apacty factor ' V V t R, t retenton tme of dead tme of the system t t R, t t 13 Retenton tme of compound After rearrangng: V t + 1 R, t d, V Ths s the an equaton of eluton ompound s retaned the stronger, The lower s the P flow rate (.e. hgher t ) The longer s the column The more What s factors the partton causeequlbrum strong retenton? shfted to the SP The larger s the SP volume The lower s the P volume 14 Pea wdth Pea wdth ncreases on eluton.e. separaton effcency decreases Ths s descrbed by the van Deemter equaton 15 van Deemter equaton H A B u S u u H heght of the theoretcal plate (HETP) u lnear flow rate of P A Eddy dffuson term (several flow paths) B longtudnal dffuson term S, mass transfer coeffcents n SP and P H u H S u u B/u A HETP (μm) lnear velocty (mm/s) 16 Average dameter of SP partcles γd fs( ') d f H λdp + + u + u D Resstance of sorbent to dffuson oeffcent of unformty of SP partcles Dffuson coeffcent of the analyte n the P S apacty factor functons Thcness of SP layer on on the sorbent partcle f ( ') d p D u Dffuson coeffcent of analyte n the SP 17 How wll effcency change f we ae partcles more unform?... ae partcles smaller?... Increase column temperature?... Increase the vscosty of the le phase? 18 3

4 Pea shape If The sorpton sotherm s lnear (.e. Henry sotherm) There are many factors causng pea broadenng The factors act n both drectons The factors nfluence all analyte molecules wth the same probablty,.e. wthout dfferentatng molecules n dfferent parts of the pea Then the pea has the shape of the Gauss dstrbuton (Normal dstrbuton) 19 Detetor Detector sgnaal sgnal Pea wth Gauss shape 4.5 t R 3.5 Analüüd Analyte p pea Pg Pea õrgus: heght Pg Pea pndala: area.0 h w S y h e ( t tr ) w Retenton Retentsoonaeg, tme, mn 0 In real lfe peas usually have tals Asymmetry factor 4.5 Detetor Detector sgnaal sgnal Pg Asymmetry asümmeetrafator: factor: w A s w w 1 w 10% Retenton Retentsoonaeg, tme, mn Optmal values of the parameters Talng s a problem: Resoluton deterorates Quanttatve accuracy becomes worse Lmt of detecton gets hgher An acceptable value: A s < Nonlnear sorpton sotherm Also n partton L the sorpton sotherm s often nonlnear The most common reason n reversed phase L: Dual retenton: partton + orpton Adsorpton occurs on Resdual slanol groups etal catons, present n slca as mpurtes Why are peas often asymmetrc? 3 4 4

5 RP statonary phases The majorty of statonary phases are based on slca On slca surface: Slanol groups These are dervatzed durng producton: H 3 H 3 S + l S R S O S R + Hl H 3 H 3 The fuller s the dervatzaton, the better Some always reman undervatzed: resdual slanol groups How to mnmze the number of resdual slanols? End-cappng Besdes l-s(h 3 ) -R some smaller-molecule dervatzaton reagent s added Sheldng Instead of l-s(h 3 ) -R the l-s(t-bu) -R reagent s used 5 6 Slanol groups Slanol groups are of three types (smplfed): The orpton ablty s the stronger, the hgher s the acdty Thep a values range from 3 to 15 Usually S HO S S a) b) c) S Isolated gemnal vcnal 7 etal catons etal catons greatly enhance the orpton ablty of slca They can be Free (d) or Embedded n the lattce (e) Because of ther valence propertes the metal catons are postvely charged n the lattce + + d) e) S 8 Adsorpton depends on the compound Such orpton nfluences frst of all: Polar compounds Especally strongly: basc compounds Often a base s added to the P ompounds that gve strong metal complexes Physcal bacground? Two sorpton processes run n parallel: 1. Partton Large volumes of phases All analyte moleculs have space n the SP Lnear sotherm. Adsorpton Small number of orpton centres Only a small part of the analyte molecules can be orbed Retenton by orpton s stronger than by partton

6 odelng orpton Assumptons (1): Adsorpton only occurs on orpton centres onomolecular orpton Analyte molecules do not nteract wth each other The number of centres s lmted All centres are energetcally equvalent odelng orpton Assumptons (): There are two processes runnng smultaneously: orpton and desorpton Ther relatve rates determne the orpton equlbrum The rate of orpton s proportonal to the number of free centres and the number of non-orbed analyte molecules n the lqud phase The rate of desorpton s proportonal to the number of occuped centres 31 3 v Adsorpton rate ( _max Adsorpton rate Rate constant of orpton oncentraton of analyte n the P axmum surface concentraton of analyte molecules Surface concentraton of analyte molecules Te extent of surface occupaton: θ ) _max 33 Desorpton rate v des des Desorpton rate Desorpton rate constant Surface concentraton of analyte molecules 34 Equlbrum There s equlbrum f Therefore: v v des _max des ( ) 0 Analyte concentraton on surface Reorgansng: _max + des Brngng n orpton equlbrum constant: des

7 Langmur s sotherm... We get the Langmur s sotherm: _max + 1 Shape of Langmur s sotherm ontsentratsoon the surface pnnal _max + 1 _max 10 _max ontsentratsoon n the le phase faass 38 Shape of Langmur s sotherm If the number of centres s large compared to the number of analyte molecules then the sotherm s can be approxmated by the Henry sotherm If the number of centres s small then almost all of them are occuped Drawbacs of Langmur s sotherm Adsorpton centres are not energetcally equvalent Analyte molecules nteract among themselves. Adsorpton can occur on molecules already orbed. So the assumpton of monomolecularty does not hold The overall sorpton sotherm xed nonlnear sotherm ontsentratsoon the surface faass Adsorpton partton ontsentratsoon n the le phase faass xed sotherm 41 Pea shape wth nonlnear sotherm on Detetor the surface sgnaal orptsoondesorptsoonprotsessdesorpton processes orpton orptsoondesorptsoonprotsessdesorpton processes orpton Gauss Gaussan õvera 0.5 pea ujulne p Retentsoonaeg, mn n the le phase 4 7

8 onclusons (1) It s good f the chromatographc process s based on partton, wth lnear sorpton sotherm Symmetrc peas For ths: ether no orpton centres or they must be occuped oble phase addtve can be used, whch orbs strongly and does not let analyte molecules to orb Statonary phase must be of hgh qualty: low number of resdual slanols and metal catons and ther actvty low: - endcappng, sheldng - acd washng 43 onclusons () In orpton chromatography the wor should be done n the Henry regon : Large specfc surface Lots of centres on the surface entres energetcally smlar Low amounts of analytes should be njected In realty t s almost never deal Talng s frequent n orpton chromatography Why? Ths s one of the reasons why partton chromatography s more popular than orpton chromatography 44 onclusons (3) Talng s more pronounced wth analytes that orb stronger Frst of all polar and basc compounds Why? Also compounds that gve strong metal complexes any other orpton models Freundlch s sotherm onomolecular orpton entres are not energetcally equvalent The actvty of the centres decreases logarthmcally Tjomn s sotherm Analogous, but the decrease s lnear BET soterm Polymolecular orpton any more emprcal models a 90 emprcal models have been proposed for descrbng pea shape All descrbe unsymmetrcal peas Some even descrbe doubled peas Applcatons of the models Deeper understandng Pea deconvoluton L smulatons V.B. D arco, G.G. Bomb J. hrom. A 001, 931,