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Article paru dans le Bull= etin de l’Union des Physiciens – Vol 99 – Décembre 2005=

 « Réalisation d’un simulateur de chromatogrammes »=

Dr Thierry BRIERE - Professeur Agrégé =

Département de Chimie - Université de La Réunion

Faculté des sciences et des technologies

15 avenue René Cassin - 97715 -St Denis messagerie- cedex 9<= o:p>

briere@univ-reunion.fr

 www.chimie-briere.com

 

Rés= umé : Cet article présente de façon détaillée la réalisation d’un simulateur de chromatogrammes. Ce simulateur prend en compte tous les paramètres qui influent sur la séparation : caractéristiques physiques de la colonne, diamètre des particules de phase stationnaire, constantes de partage= et sélectivité, débit de l’éluant. On pourra réaliser simplement ce simulateur, soit sur un micro ordinateur avec= un tableur, soit à l’aide d’une calculatrice alpha numérique graphique. Un éditeur de chromatogramme avec rapport d’analyse est aussi proposé, il permettra d’éditer des chromatogrames proches de la réalité pouvant être utilisés pour réaliser facilement des exercices. Toutes les notions théoriques nécessaires sont rappelées et démontrées dans une première partie, puis on passe à la réalisation pratique. Les programmes pour calculatrices = TI 89-TI92-Ti92 plus sont donnés, on pourra les adapter à d’autres calculatrices. Les fichiers Excell ou TI sont téléchargeables à l’adresse : http://www2.univ-reunion/~briere<= /a>

 

I) Notions théoriques nécessaires :

I-1) La co= urbe de Gauss :

Les chromatogrammes présentent un profil gaussien, dans l’idéal ils peuvent donc être décrits par la cour= be de Gauss. On va ici utiliser une formule simplifiée dont l’équation est la suivante :

y =3D 1 / s<= /span>   exp [- ( x – m )2= / (2 s2)<= /span> ]

m =3D valeur moyenne =3D abscisse du sommet du pic gau= ssien, en chromatographie il s’agira du temps de rétention du compos&eac= ute; tR.

s =3D &eacut= e;cart type du pic : il définit la largeur du pic chromatographique.

Propriétés remarquables:

Deux points d'inflexion à x =3D m - s et x =3D m + s

G(m) =3D 1 / s =3D h (hauteur du pic)

G( m - = s ) =3D G( m + s ) =3D 1/s exp (-1/2) =3D 0,6065 / s  =3D 60,5 % de h<= /p>

G( m - = 2s ) =3D G( m + 2s ) =3D 1/s exp (-2) =3D 0,1353 / s =3D 13,5 % de h

G( m - 1,1774 s ) =3D G( m + 1,1774 s ) =3D 0,5000 / <= span style=3D'font-family:Symbol'>s =3D 50 % de h

Intégration =3D surface du pic

(de -= ¥ à +¥) ò G(x) dx =3D 2,507 =3D S (surface du pic)

(de m - 2s &= agrave; m + 2s) ò G(x) dx =3D 2,393 =3D 95,4 % de S

I-2) Grand= eurs chromatographiques

I-2-1) Efficacité théorique d'une colonne :

Lors de sa progression dans la colonne, le solut&eacut= e; subit une diffusion longitudinale. Les pics sont donc de plus en plus large= s au fur et à mesure que les temps d'élution augmentent. Simultanément, et "toute choses égales par ailleurs"= ; la hauteur des pics proportionnelle à 1/s diminue. L'allure d'un chromatogramme idéal est alors la suivante :<= /p>

 

=

On peut imaginer, &agr= ave; la limite que l'écart type d'un pic devienne égal à la longueur de la colonne entière. Le composé occuperait alors t= oute la colonne à lui seul. Il serait alors impossible de séparer = deux composés sur une telle colonne dont l'efficacité serait minim= ale. Inversement, plus les pics sont étroits et plus la colonne sera efficace.  L'efficacité= de la colonne est par analogie avec la théorie de la distillation exprimée en nombre de plateaux théoriques. Plus le nombre de plateau sera important et plus la colonne sera efficace. La colonne la moins efficace aura par définition un seul plateau théorique et une colonne très efficace possédera au contraire un très g= rand nombre de plateaux théoriques.

 

 

3D"Zone3D"Zone
3D"Zone
 

 

 

 

 

 

 

 

 

 

 


Soit L la longueur de la colonne et N son nombre de pl= ateaux théoriques, la longueur équivalente à un plateau théorique (H.E.P.T) est : h =3D L / N.

Pour une colonne de 1 plateau théorique on pose= :

L =3D h =3D sL soit sL2 =3D L *= h

sL2 =3D L * L / N =3D L2 / N soit : N =3D L2 / = sL2

Le temps de rétention d'un composé est p= ar définition le temps pour lequel sa concentration est maximale &agrav= e; la sortie de la colonne, c'est à dire dans le détecteur. A cet instant, l'écart type de la courbe de Gauss est sL pour le profil de concentration et st =3D s pour le chromatogramme en fonction du temps.

Pour parcourir la distance L, le composé a mis = le temps tR.

C'est le déplacement de l'éluant qui &ag= rave; provoqué ce déplacement du composé. La vitesse de ce "déplacement" est propportionnelle au débit de l'éluant que nous allons supposer constant. Il y a donc proportionnalité entre la distance parcourue et le temps.

On a donc : L / sL =3D tR / s  soit : N =3D L2 / sL2 =3D tR2 / s= 2

On pourra donc déterminer le nombre de plateaux théorique de la colonne à partir du chromatogramme en fonctio= n du temps.

Il faudra pour cela déterminer s, cette détermination pourra se= faire soit :

- en prenant la demi largeur du pic à 60,6 % de= sa hauteur : mesure directe de s

- par mesure de w largeur à la "base" du pic à 13,5 % de sa hauteur := s =3D w / 4

- par mesure de d largeur à mi hauteur du pic :  s =3D d /  2,354

N =3D tR= 2 / s2

s =3D d /&nb= sp; 2,354  =3D=3D> 1 / <= span style=3D'font-family:Symbol'>s2 =3D ( 2,354 / d )2 =3D 5,54 / d2 =3D=3D> N =3D 5,54 = tR2 / d2

s =3D w / 4 =3D=3D> 1 / s2 =3D 16 / w<= /span>2 =3D=3D> N =3D 16 tR<= sup>2 / w2

Dans la pratique, en raison de la déformation d= es pics à la base on utilise généralement, la largeur à mi hauteur d pour la détermination de N.

I-2-2) Efficacité réelle d’une colonne :

L’efficacité théorique d’une colonne est une grandeur relative, on préfère utiliser une grandeur plus réaliste pour pouvoir comparer des colonnes de concept= ions différentes ; On définit alors le nombre effectif de plateaux Neff.

Neff =3D t’R2 / s<= sup>2 ou Neff =3D 16 t= 217;R2 / w2 ou Neff =3D 5,54 t’R2 / d2

t’R est le temps de rétention réduit du composé.:  t’R =3D tR - tm

tm e= st le temps mort de la colonne il correspond au temps que met le solvant ou un soluté non retenu pour traverser la colonne.

L’utili= sation de l’efficacité réelle permet ainsi de comparer entre elles des colonnes de géométrie très différentes. En effet le temps mis par le solvant pour traverser la colonne dépend étroitement de la géométrie de celle-ci et en particul= ier de son volume interstitiel.

 

 

 

 

I-2-3) G= randeurs de rétention

I-2-3-1)= Volume mort : Vm

Il correspond au volume occupé par la phase = mobile dans la colonne, c’est à dire au volume interstitiel de celle-= ci. Si on suppose un débit D constant :

VM =3D tM D avec tM =3D temps mort


On voit qu’il est inutile d’augmenter trop fortement les valeur= s de a ou k car on tend rapidement ver= s un pallier et la résolution gagnée est faible devant l’allongement de la durée d’analyse.

I-3) Influ= ence du débit – Equation de Van Deemter ou de Knox<= /p>

Dans tout ce qui précède on n’a pa= s tenu compte de l’influence du débit de la phase mobile qu’on a supposé constant. Or il est évident que ce débit va modifier le chromatogramme et influencer la séparation des constitua= nt. A priori on s’attend à ce que la séparation soit meille= ure avec un débit plus faible.

Il existe une équation dite équation d= e Van Deemter qui relie la hauteur équivalente à un plateau théorique (HEPT) à la vitesse linéaire moyenne de la p= hase mobile u :

= H.E.P.T =3D h(u) =3D A + B/u + C u

Historiquement l’équation de Van Deemter s’appliquait uniquement à la chromatographie en phase gazeuse (C.P.G) mais elle a été généralisée &agr= ave; la chromatographie liquide (équation de Knox) La courbe obtenue est = une branche d’hyperbole qui passe par un minimum dont on peut classiqueme= nt trouver les coordonnées par annulation de la dériv&eacut= e;e.:

h’= ;(u) =3D C – B / u2

C ̵= 1; B / u2 =3D 0

u opt =3D (B/C)1/2    et   h(uopt) =3D hmin =3D A + (B*C)1/2 + B / (B/C)1/2=

Dans la pratique on cherche à déterminer= le minimum de la courbe qui correspond à la HEPT minimale soit au nombre maximal de pla= teau et donc à la séparation optimale. Pour cela on procède à plusieurs chromatographie d’un même composé ave= c le même éluant mais avec plusieurs débit différents= . On mesure l’efficacité de la colonne pour ces divers débit= et on trace la courbe correspondante.

On peut éventuellement déterminer également les trois paramètres A, B et C, on peut procé= ;der par régressions linéaires multiples ou déterminer graphiquement A, B et  C en sa= chant que la courbe tend vers la droite h =3D A + C u quand u devient grand. On p= eut ensuite trouver B par l’ordonnée du minimum de la courbe.

Remarque : Le terme A est appelé fa= cteur de remplissage, il est lié a la qualité et à la régularité du remplissage de la colonne, avec les colonnes actuelles ce terme est le plus souvent très faible voire nul.

I-4) Evalu= ation à priori du débit optimal en chromatographie liquide = ;:

Une formule empirique permet d’estimer simplemen= t le débit optimal d’utilisation d’une colonne à parti= r du diamètre dp des particules de la colonne, de sa porosité et de ses caractéristiques géométrique= s. On peut aussi déterminer son efficacité attendue.

uopt (cm.s-1) =3D 0,5 / dp (mm) et hopt (mm) =3D 3 dp (m<= /span>m)

Dopt =3D uopt * VM / = L

VM =3D e= VI =3D ¼ e p dint2 * L

Dopt =3D uopt * ¼ e p<= /span> dint2

Nattendu= e =3D L / hopt (avec L et hopt exprimées ave= c la même unité)

Cette détermination a priori de N n’est qu’approximative mais permet de connaître l’ordre de gran= deur de N.

On voit que diminuer la taille des particules augmente l’efficacité de la colonne mais parallèlement entraîne une diminution du débit optimal et donc un allongement des temps de rétention et du temps d’analyse.

Si on fait l’hypothèse simplificatrice qu= e le terme A est nul on peut déterminer facilement B et C et donc déterminer N pour un débit quelconque. En effet :

uopt =3D (B/C)0.5

hopt =3D A + (B*C)0.5 + B / (B/C= )0.5 =3D (B*C)0.5 + B / (B/C)0.5

uopt2 =3D B / C

C =3D B / uopt2

B * C =3D B2 / uopt2<= /p>

(B*C)0.5 =3D B / uopt

B= / (B/C)0.5 =3D B / uopt

h= opt =3D A + (B*C)0.5 + B / (B/C)0.5

hopt = =3D 0 + B / uopt + B / uopt

hopt = =3D 2 B / uopt

B =3D h= *uopt / 2

C =3D B= / uopt2

C =3D h= opt*uopt / 2 / uopt2

C =3D h= opt / 2 / uopt

avec hopt (cm) =3D 3 dp /10000 et uopt= (cm.s-1) =3D 0,5 / dp

Soit finalement :

B =3D 3 * dp = /10000 * 0,5 / dp / 2  =3D 0,75 10= -4

C =3D 3 dp /1= 0000 / 2 / ( 0,5 / dp ) =3D 3 10-4 dp2


Hauteur équivallente réelle : hreel (en cm) =3D 0,75 10-4 / u + 3 10= -4 * dp2 *u

II) Concep= tion du simulteur de chromatogrammes :

Nous allons maintenant décrire la réalisation pratique du simulateur de chromatogrammes qui va nous permettre de récapituler et de bien assimiler toutes les grandeurs fondamentales de la chromatographie et de voir leurs diverses interdépendances. Nous nous limitons a deux composés seulemen= t, on suppose que les deux composés ont des concentrations et des facte= urs de réponse identiques. On pourra facilement faire intervenir ces deux facteurs si on le désire comme dans l’éditeur de chromatogrammes proposé. L’utilisation d’un tableur (ici EXCEL) et de macros permettra de visualiser très rapidement l’effet des divers facteurs sur la séparation. Le programme po= ur calculatrice est moins spectaculaire a cet égard .

II-1) Les données préalables :

Les paramètres fondamentaux qu’il faut fi= xer au préalable sont les suivants :

Param&egra= ve;tres physiques de la colonne et de son garnissage :

Longueur en cm :=   long – Diamètre interne en cm : dint – Porosité : poro – Diamètre des particules de phase stationnaire en mm : dpart

Param&egra= ve;tres de séparation :

Constante de partage du composé  le plus retenu : KeqB

Sélectivité : a

Débit en mL.min-1 : debit

II-1) Données déduites des paramètres fondamentaux :

Volume interne de la colonne en mL : Vint =3D lon= g * p * dint2 / 4

Volume mort de la colonne en mL : Vmort =3D poro = * Vint

Temps mort en min : tmort =3D Vmort / débi= t

Vitesse linéaire moyenne en cm.s-1 : umoy =3D long / tmort/60

Vitesse linéaire optimale en cm.s-1 : uopt =3D 0,5 / dpart

Débit optimal en ml/min : dopt =3D Vmort *= uopt * 60 / long

hauteur équivalente optimale en cm : hopt = =3D 3 dpart /10000

Param&egra= ve;tres de Knox : h =3D A + B / u + C * u

A est supposé nul

h =3D B / u + C * u

Attention aux unités : u en cm.s-1 ; h en cm ; B en cm2.s-1; C en s-1

B =3D 3 * dpart /10000 * 0,5 / dpart / 2  =3D 0,75 10-4

C =3D 3 dpart /10000 / 2 / ( 0,5 / dpart ) =3D 3 10-4 dpart2

Hauteur équivallente réelle (en cm)= : heel =3D 0,75 10-4 / u + 3 10-4 * dpart2 *= u

Efficacité de la colonne : Nreel = =3D long / hreel * 1000

On suppose que l’efficacité est la m&ec= irc;me pour les deux composés.

 

 

 

 

3D"Zone3D"Zone
Caractéristiques des pics chromatographiques :

 

Rés= olution : R

On peut la déterminer de 4 facons différentes :

R « vrai » : 1/2 (tRb – tRa) / (sigmaA + sigmaB)

R approximation 1 : Rapp1 =3D ½ (tRb ̵= 1; tRa) / (tRa + tRb)

R approximation 2 : Rapp2 =3D ¼ * NeffB0.5 * (a – 1) / a

R Purnel : Rpurnel =3D ¼ Nreel0.5 * (a – 1) / a * kret2 / (1+ kret2)

 

On a tous les éléments pour simuler le chromatogramme. On peut réali= ser un programme sur calculatrice (voir page suivante) ou utiliser une feuille = de calcul EXCEL pour réaliser cette simulation. Voir : Feuille Excel

On pourra utiliser cette feui= lle Excel pour bien comprendre l’effet des divers paramètres et le= urs influences sur la séparation.

 

Remarque : Il peut arriver parfois de pe= tits problèmes à cause du pas utilisé pour le traçage des courbes. Les pics peuvent être mal définis et donc peu jol= is il faudrait augmenter le nombre de points pour le traçage en diminua= nt le pas, mais cela augmente énormément le temps de calcul et d’affichage des courbes. On a donc choisit un compromis raisonnable, = mais rien n’empêche de le modifier si on le désire.

Ainsi,  certains pics peuvent parfois disparaître, il s’agit d’un artefact du aussi à la “résolution graphique” si les pics sont trop fins leur maximum peut ne pas être dans la liste des valeurs de temps choisies,= les pics seront alors déformés et peu jolis voire dans les cas extrêmes disparaîtrent totalement du graphique. Ces problèmes sont néanmoins très peu fréquents. No= us proposons également une feuille de calcul EXCEL permettant l’édition de chromatogrammes avec rapport d’analyse, cet éditeur permettra   d’obtenir facilement des chromatogrammes proches de l’as= pect des chromatogrammes réels, on pourra les utiliser pour des exercices= par exemple.<= o:p>

 

 

 

 

 

PROGRAMME POUR TI 89 - TI 92 - TI 92 PLUS=

 

Simulchrom

()

Prgm

NewProb

setMode("Exact/Approx","APPROXIMATE")<= /span>

Dialog<= /span>

Title  "simul chro"

Text  "simulateur de chromatogrammes"

Text  "    par Thierry Briere= "

EndDlog=

Text  "donnees fondamentales&qu= ot;

Input  "porosité",po= ro

Input  "longueur en cm",lon= g

Input  "diametre interne en cm",dint

Input  "diametre particules en <= /span>mm",dpart=

Input  "K équilibre B",keqb

Input  " selectivite a",alp= ha

0.5/dpartàuopt

p*dint^2/4*longàvint

poro*vintàvmort

vmort*uopt*60/long= àdopt

Disp  "uopt "&string(u= opt)&"cm/s"

Disp  "debit optimal "&string(dopt)&"mL/min"

 Input  "debit en ml.min-1",= debit

vint-vmort<= span style=3D'font-size:11.0pt;mso-bidi-font-size:10.0pt;font-family:Wingdin= gs; mso-ascii-font-family:Arial;mso-hansi-font-family:Arial;mso-char-type:s= ymbol; mso-symbol-font-family:Wingdings'>àvstat

vmort/debit= àtmort

long/tmort/60àumoy

 3*dpart/10000àhopt

7.5 10-5/umoy+0.0003*dpart^2*umoyàhreel

long/hreel<= span style=3D'font-size:11.0pt;mso-bidi-font-size:10.0pt;font-family:Wingdin= gs; mso-ascii-font-family:Arial;mso-hansi-font-family:Arial;mso-char-type:s= ymbol; mso-symbol-font-family:Wingdings'>ànreel

Dialog<= /span>

Title  "Données déduites"

Text  "volume interne "&string(vint)&" mL"

Text  "volume mort "&string(vmort)&" mL"

Text  "volume stationnaire "&string(vstat)&" mL"

Text  "temps mort "&string(tmort)&" min"

EndDlog=

Dialog<= /span>

Title  "Données déduites"

Text  "u optimal "&string(uopt)&" cm/s"

Text  "h optimal "&string(hopt)&" cm"

Text  "u moy reelle "&string(umoy)&" cm/s"

Text  "h reelle "&string(hreel)&" cm"

Text  "efficacite "&string(nreel)&" pltx"

EndDlog

keqb/alphaàkeqa

keqa*vstat/vmortàkreta

vmort+keqa*vstatàvra

vra/debitàtra

tra/nreel^(0.5) à= ;sigmaa

tra-tmortàtreda

(treda/sigmaa)^2àneffa

 

 

 

 

 

 

 

 

 

 

 

Dialog

Title  "Compose A"

Text  "kret A "&string(kretb/alpha)

Text  "Keq A "&string(= keqa)

Text  "VR A "&string(vra)&" mL"

Text  "tR A "&string(tra)&" min"

Text  "tred A "&string(treda)&" min"

Text  "sigma A "&string(sigmaa)&" min"

Text  "N eff A "&string(neffa)&" pltx"

EndDlog

 

keqb*vstat/vmortàkretb

vmort+keqb*vstatàvrb

 vrb/debitàtrb

trb/nreel^(0.5) à= ;sigmab

trb-tmortàtredb

 (tredb/sigmab)^2àneffb

 

Dialog

Title  "Compose B"

Text  "kret B "&string= (kretb)

Text  "Keq B "&string(= keqb)

Text  "VR B "&string(vrb)&" mL"

Text  "tR B "&string(trb)&" min"

Text  "tred B "&string(tredb)&" min"

Text  "sigma B "&string(sigmab)&" min"

Text  "N eff B "&string(neffb)&" pltx"

EndDlog

 

0.5*(trb-tra)/(sigmaa+sigmab) à= ;rvrai

0.5*nreel^(0.5)*(t= rb-tra)/(trb+tra) à= ;rap1

1/4*neffb^(0.5)*(a= lpha-1)/alphaàrap2

1/4*nreel^(0.5)*(a= lpha-1)/alpha*kretb/(1+kretb) à= ;rpurnel=

 <= /span>

Dialog<= /span>

Title  "Résolution"<= o:p>

Text  "R vrai "&string(rvrai)

Text  "R app 1 "&string(rap1)

Text  "R app 2 "&strin= g(rap2)

Text  "R Purnel "&string(rpurnel)

EndDlog

seq(t,t,0,1.3*trb,trb/50) à= ;listt

1/sigmab*exp(-(listt-trb)^2/2/sigmab^2) à= ;picb

1/sigmab*exp(-(listt-tra)^2/2/sigmaa^2) à= ;pica

pica+picbàlistchro

NewPlot  1,2,listt,listchro,,,,5

ZoomData

EndPrgm=

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Aspect de la feuille de calcul EXCELL :

=

 

Exemple de chromatogramme obtenu avec l’éditeur :

=

 

 

Conclusion = :

La réalisation de ce simulateur de chromatogrammes permet de bien comprendre les relations entre= les diverses grandeurs chromatographiques et leurs influences sur la qualit&eac= ute; de la séparation. En s’amusant avec lui on assimilera peu &agr= ave; peu le jeu des interactions croisées entre tous ces paramètre= s. En particulier on pourra appréhender concrètement l’optimisation d’une séparation en tenant compte du temps d’analyse ce qui constitue le problème quotidien du chromatographiste. Il ne s’agit là que d’une premi&egrav= e;re approche mais pédagogiquement intéressante par son coté ludique.

 

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