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
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.=
p>
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
(de -=
¥
à +¥=
span>)
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>
=
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 / s=
span>L
=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=
span>
largeur à la "base" du pic à 13,5 % de sa hauteur :=
s =3D w /
4
- par mesure de d=
span> 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 :=
b>
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.: =
span>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.
=
o:p>
=
o:p>
=
o:p>
=
o:p>
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=
p>
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=
sup> :
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.
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=
|
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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=
span>
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=
span>
vmort/debit=
àtmort=
span>
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=
span>
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=
o:p>
ZoomData
EndPrgm=
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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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