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Contribution to the theory of chromatography
JOURNAL
CQNTRIBUTION
TO THE THEORY
LINEAR
NON-EQUILIBRIUM
EUGENE
KTJCERA
Institute of Plcysical (Received
237
OB CHROMATOGRAFNY
Chemistry,
Novcmbcr
OF CHROMATOGRAPHY
ELUTION
Czechoslovai~
Academ,y
CHROMATOGRAPHY
of Seienccs,
Prague
(CzecJ~oslovakia)
zGt11, 1964)
Recent theories of gas chromatography and similar separation methods only satisfactorily solve the linear type of equilibrium chromatography’-5 and some simple types of non-equilibrium chromatography o--O.The present communication describes a theory which simultaneously takes .into account the longitudinal diffusion in the mobile phase, the radial diffusion inside the porous grains of the packing material, the finite rate of mass transfer through the boundary (which may comprise diffusion in the thin layer of the immobile phase on the surface of the grains) and the sorption on the internal surface of grains. In spite of the fact that no analytical solution can be given even for the simplest model of the column with a granular filling, which will be described below, it is possible to obtain valuable informatlon based on the known Laplace transform of the solution. Let an infinitely long column with section $Jbe uniformly filled with the packing consisting of small porous grains of uniform size having the shape of spheres (index v = 3) or “infinite” cylinders (V = 2) of radius 22 or “infinite” plates (v = I) with thickness d = 2R. Among the internal co-ordinates only the co-ordinate Y, which is measured in the direction from centers (or axes or planes) of symmetry perpendicularly to the surface of the grains, is significant. The area of the geometric surface of grains per unit volume of the column is given by the formula SV = v(x - a)/ri where txis the external porosity, A homogeneous distribution of pores in the grains is supposed. The average area of boundary surface between pores and the external free volume per unit volume of the column is PV = /3S, where f3 is the internal porosity. “Incompressible” carrier flows through the free external volume along axis z of the column, with velocity ZG.The rate of flow through the whole cross section J!Jof the column is then given by $ = +zc. It is assumed that the concentration C of the compound being separated in the free volume is the same in the whole cross section of the column at any moment, thus depending only on the longitudinal co-ordinate z and time f. The flow of the given compound is defined hy the velocity of the carrier and by the diffusion and therefore it may be written as follows:
ac
ac
z+~~az-“~g=Q~
82C
The grain is presumed to be small enough so that the concentration C of the given compound may be assumed to be the same along the whole grain at any moment, only being dependent on the location of the grain center. Inside the gmins, J. Clwomalog.,
19 (1965)
237-243
E. ltTJ6ERA
238
concentrations c (Y, a, i) and 9t(L/,z, t) do not depend on angular co-ordinates. Further, sorption on the internal surface of the grains (boundary between pores and the solid part of grains) may occur. Then it is valid for concentration in the pores:
ac -A
at
D?
@G
U-I
&e-~&z
ac >
=
Qn=--8
where c is the concentration in the pores and 31is the concentration of the sorbed compound on the internal surface; both concentrations are given per unit volume of the pores. For the flow of the compound in the direction of the normal to the external surface of the pores ,
and since Pv is the external pore area in unit volume of the column, the rate of decay of the substance from the pores in unit volume is equal to jR& and also to cxQC,as the rate of influx of the compound into the carrier is the same as the rate of decay from the pores. Hence
Further, due to the symmetry aG
of the grains the following condition is valid
I
ar
It is assumed that the transfer and sorption equations’ : and
Qc = ---Wc(Ic&
-
c/r=R)
Qn =
-
n)
-H,(K,G
rates are defined by the linear
where K, and K, are equilibrium constants and Hc and Hn are mass transfer coefficients. The system of equations is solved for these initial and boundary conditions:
c (2, t)
=
c (2,$1= c (I/, x, i!) =
c (Y, z, t) =
CI (4 C$ (2)
9.2(Y, 2, 15)= IZ$(2)
ti (Y, z, t) =
I
for f = 0, -
o for t < o and z =
rt: 00
co
where Ct (x), cs (2) and mt(a) describe the initial distribution pound. J. Clwomatog., 19 (19%)
237-243
of the introduced
com-
240
E., KUthRA
For chromatography it is not necessary to know solutions c, B (Y, z, t) or c’, % (Y, x, s), respectively. For other boundary conditions and for initial relations (z) and ct (z) = K,Ct (z) ; the expression for C (a, s) was given in ref. 33. %I(2) = .I’;?2c~ In general, it is impossible to carry out the inverse transformation and to find an analytical expression for C (a, t) even for the simplest forms of the feed band. However, the expansion
can be found using Hermite
polynomialsl2:
El Lx%(t) = zl
(- I)lCfi! (2t)n-21c k--o Iz!(n 212) !
,where n
-
[ 2
1
I
for
f
even
The expansion
c1 f-
32;
=
coefficients
n -
2
I
for
odd 32.
a7&are given by the formula
based on the orthogonality of Hermite polynomials. Let p’k be the I&th moment of function C (t) defined by equation13:
where
sco
mk =
tw(t)dl!
0
and let ,#_Jk: be the I+th central moment Pk =
Goj-r(’ -pl’)kC(t)dt
Then the coefficients
al = a2 =
defined by the equation:
(+) (-,,&‘)t
= j.
can be expressed through
/-lb_2
the moments
as follows:
0
I
a,
m0
=
dzrclc2
Further,
El
0
=9 (w’%)
I)k,‘h-2k,U2k
zVz!(rc -2h)!(2,u2)?
the following
swdkC(W’ = J. Clrvornalog.,
(_
c k=" (-
I)*jLy
237-248
property dkC( s) dSk-
of the Laplace
transformation
is known:
LINEAR
NON-EQUILIBRIUM
This
ELUTION
is the key
24=
CHROMATOGRAPHY
for the calculation
of the moments
and coefficients
of the
FIermite expansion of the function C (t) from the known transform 2: (s), The number of necessary terms in the expansion series will depend on the required accuracy. The above mentioned method was used in ref. g for a more simple model, and for other initial and boundary conditions. In practice the following forms of the feed bands are of importance: For liquid-solid chromatography the given compound is initially sorbed on a definite part of the column (0, L). If a macroporous sorbent is considered in which sorption on the internal surface occurs, the sampling may be described by the conditions : =
G (4
Cf (z) =
vsytg (z) =
IZ$ =
vzt(z) =
0
0,
const. + 0 for0
for z < o and z > L.
For a micro-porous sorbent, the sorption on the internal surface cannot be distinguished from the sorption in the pores of the grains and, therefore, it is possible to describe the sampling by the conditions : IiT,
=
0, uti(Z)
cg (2) =
cc =
C$(2) =
0
=
0, C*(z)
const , + 0
=
0
for 0 < 2 < L for a c
o and z c
L.
For these cases it would be more convenient to solve the model in which only the section (0, L) of the column,is filled by the sorbent and, therefore, they will not be analysed here. In the case of gas chromatography the given compound is injected into the carrier in an “infinitesimally” short section of the column at the start of the experiment, and thus the feed band may be characterised by conditions:
where M is the total mass of injected compound and 6 (z) is the Dirac function. The Dirac function S (a - z’) has the following property :
s b
f (@(z
-
z’) dz = f(d)
In this case the transform
m = C(z,S)/Er
&
Mexp =
if a e 2’ c: b if 2’ -z a or 2’ * b.)
=o
a
(Note:
of the solution
(2)
is simplified
to the expression
[$-&-,”
,sL] . _.-.-___ J.
CkYO~atOg.,
(2’2)
19
(1965)
237-248
E.
242
ItUc!%RA
The resulting moments up to the fifth central moments are given by the fdlowing formulas :
pot =
PO =
(34
I ;
(94
PI
+
cc2
P3
I 2Dp2L
--
=
12D,L -_
+
+
ZLfj
SD,2 ---&-
>
[I +
EKc(l
64DP3 -[I + %e >
+
+
Kn)12 +
EK,(I
+
Kn)13
(34
+
+
U3
6L A_ 26
(
+
cc4 =
2 1GDp3L
I 2D,VY -__d
___-u7
+
+
5
+
NC
E(I +
+
-_--
Kn)2
2(x +
Kn +
F,
.@(I + K,)3 ___-Ii!,2
+
48DpL
+
+
R2 2(1
KTZF
___-A--V2(Y + 2) (Y +
+
243 ---264 EKC I (-_--7
4) + Dr
1g6Dp2
+
[I +
zsKn<( 1 + --Kn) v-w--HCH,,!
+
+(I
+
+ J.GhYomatO~.,
3$(1 ---.
Dr
Y(V +
2)
+
zz_
1
+
H 782
2)2 (Y +
+
6rKn( _-_-_
H$
~$K,(I E3(1 _-__- + JT?d4+ WC”
19 (I&)
K,#
237-248
+
JX,,)l
V(V +
4) (v +
( R2 I _ -_--
+
K,z)
6)
(5lJ + 12)E(I ___A_--Hc
+
EICc(I
K,)2
Hc2H n
+
I + I;TcNn
Kn)2
+
&Kn(2 -I- 3Kd -__-I_-H,Hn2
+
Km)2
(2 + 3Kn)K’n -___H.-2
+
(ti ----
+
12)Kn HP,
+
+ )
LINEAR
NON-EQUILIBRIUM
240DpsL2 --u--
+
E(1 + Hc
IC*)3 -
ELUTLON
3920Dp4L @
+
17408D,G 810
+
1 (
K, z;
+
CWROMAZOGRAPWY
12oD,L3 zt”-+--
+
243
c1 +
)
EK(l
1440Dp2L
2(1 v3(v +
+ Ic7d3 2) (v +
‘f
Kn)lG
+
(31)
48000~~ -Zb3
+
4)
Kg .(----WC+ En+--~-2-+---yEj~g.y--+IJ,G 1+ E(1 + Xn)2
2EKn(l
E2(1 +
Ii’,
+
Kn)
K,
>
+
(E$ + 4q
+
6Sn( __---
I +
+ 280;;p2L + 9$92K~3p + sKc(l
Icn)2
e3(1 --
N&7,.
+
1 (
EKn(2 + 3-T(n) I<71 ---+IJ,,a HcN.n3
E21i’c2
12oL3 213
+
+
+
Kn)4
+
3e21t’n(1 + ---HCW,
H,3
72oD& -_ u3
+
Kn)]
K7d2
+
1440DP2 --264
+
Ii2 (I + Km)2 + 41 + Kd2 + Kn ----[ Dr
v(v
+
2)
13,
1+ !I + Y3(Y +
2>2+$-(v
_--- (I 3 Kn) v’c(v + 2) (v + (6~ + ----
(14~ +
Kd3
12).K,(1
+
+
(9
+
+ I<,,) __-_--
24)e2(I ----I$,3
4)
2Kd
+
R2
H,3 6&Kn(1
6)
+
2-T(,)
+
+
dcn(1 -----
HcHn2 4E3Kn(l
+ JCn)3 + --
H,3H,
--HC
+
Kn)
H,zH,2
+
2ov
Icfl.14
4E3(I ---
+
2x7s)
(1 -k 2Kn)
--
48)Kn.
48)&7t(1 + -v--e LTJcH7&
+
Kn)’
+
12E21c7,(I
-I-
K7J3
+
+-TC7d3
+
HtpH,
Ii,3
2) +
(201) +
Hn
_t- &(I
+
Ic7d5
+
I-E,4
H?h3 3E2K7a( I + -
Kd2
+(
1
o,v(v (I
+
48).$1
+
2EKn(I
+
Hczr,3
2K7l)
+
xn
g4
J. Ciwornalog., 19 (1965)
1 -
237-248
E.
244 In the case of equilibrium chromatography K ?t = o) there is a well known solution4* 6 I4:
CP! = C(ZJ)/E=&
- %mt>2 exp --- CL > @,rt
-M-• -I- EJG wnll),pt I
=
(
=
where ucf = ---,
u
anclD,f
I -j- EICc
=
-I +
(R2/Dy =
o, E-EC=
KUh3Rk
W, =
CO,
(4)
DP E&-
It therefore follows from equations
(3) that
PO = PO’ = r ; pl
=
0; L =---I_--; 2def
(I +EKc)
EKc)2 =
(If 12D,3L Us
Pug=
ND,3 +-yi3-
)
2De.f Ucf2
2De.+ -Uef3
8Def2 %.f4
-I- -;
(I
(5)
gGoD# +
+ 240Dp3L3 --u-
pi; =
24oD,fL3 --A %ft”
=
+
+
3920Dp4L T-3920Db.f4L %f3
218,. )
(I +
&cc)4
=
9GoDc.t4,
uep ’ +
r’4;l”a”G)
(I +
E.Ti'# =
17408Def” -I- ---y-pj--*
DLSCUSSLOM
The first moment avtd tlze retention time of a chrow8atogra$hic peak
Let a detector measuring the concentration as a function of the time be placed at a given place L. in the column. The following elution times may be determined from an ideal chromatogram : The time i?Rof the peak maximum is defined by BC(L, 8) --
at
I t+=
OS
i.e. the time when the maximum concentration
is registered at the point of detections, The time 8,sof the peak median is defined by
sp
C(L, t)dt
=
& l: c(L, t>dt,
i.e. the time when half area of the peak is registered. J. Clwomatog.,
19 (WW
237-248
(6)
LINEAR
NON-EQULLIBRLUM
245
CWROMATOGRAPHY
ELUTION
The time to of the center (of gravity) or the mean of the peak is defined by
slxB
1c=pll=
o
&qL,d)
clt
Great significance has been accorded to the time t,,, ac(t,,z) az I
Z==L
= 0,
the time when the-maximum of the immediate concentration distribution in the column passes through the point of detection?. Our task is to determine the retention time defined as tn* = L/zcef by measuring some of the times defined above. Assuming the theory of equilibrium chromatography can be employed, it is possible to determine a relation among all these times based on formula (4). It is generally known that i!R* = t,,. The equation
i.e.,
erfc
tJ
%f2(tR*)2
4~cfk
_JE)
--exp
(2.2.2)
crfc (Jyp;”
+ $zJ
=
I
the relation between tR* and ts can be derived from the condition (6) and approximating,
Ior
The relation tin’ = J
tR2
+
29A
iR
f
w?f2
was derived in a previous papeP.
;f c2
Finally, the relation
is derived in this study and it can be found by rearranging the expression for the first moment (3b). In contrast to other relations, this relation is valid even for nonequilibrium chromatography. For this reason it is suggested that in accurate measurements it is necessary to measure the center (of gravity) of the chromatographic peak and not its maximum. Only this value can give us reliable data for the determination of the retention time which it is necessary to know for the determination of the partition coefficient K = .sKc(~ + ICn). The difference between to and tn* depending on the coefficient of longitudinal diffusion D,, can be neglected for large retention times, A notable finding is that the motion of the center (of gravity) of the peak does not depend on the following quantities : the shape of the grains (factor Y), the coefficient of radial diffusion Dr and the mass transfer coefficients HC and Hn, but depends only on the equilibrium constants K C and I’,), the coefficient of longitudinal diffusion Dp and the velocity of the carrier zc. J.
Chromatog.,
19 (Wd
237-248
E.
246
IW&3RA
0 %Aw secoazd ce9ztral Inon5ent and tlae widtlz of the chvomatogra$lzic peak The second central moment - variance - depends on all factors characterising the column, its filling and transport of the given compound through the column. It increases for the given place L of the co1um.n together with the increasing coefficient of the longitudinal diffusion, the equilibrium constants and the size of grains R. It decreases with the increasing coefficient of radial diffusion, the mass transfer coeffkients and with the increasing symmetry of grains characterised by coefficient V. The significance of the shape factor may be assigned to this coefficient, supposing that its value ranges from 2.5 to 3 for the usual shape of the chromatographic grains. For a given ratio s, the chromatographic peak will be narrowest for the column filled by spherically-shaped grains. The standard deviation x = dz defines the width of the pealclGglO. The value 2 x ,&i is considered as the width of the curve. The ratio o = 2 3c ,/2/pjs.defining the relative width of the peak has a large significance for the determination of the separation efficiency of the columrP. The optimal velocity zto for which the relative width of the peak has a minimal value can be found from the condition dw/dzb = o. Let DP depend on the velocity zc and let this dependence be given by an approximate relationship =P = Do + Am + Bu2 (e.g. ref. 16) where Do is the coefficient of diffusion, A and B are empirical (eventually derived theoretically) constants being dependent on the properties of the column, its filling and carrier. Substitution of the expressions (3b, c) into the expression for o gives :
where .A? (I + K,)2 + E(I + K,)2 %=
After
+
Ir'n
D, - ?J(Y + 2) --TT:-+Nn -_--_-_--_--[I + EKc(l + Kn)]2
differentiation,
(F
+
the following
12;fox)u2
-
(Do
+
‘9
algebraic
_
t$?$~
equation
_
,5D02,L
of the fourth
=
degree
o
is obtained for zco, which is practically impossible to solve. For a large distance L, the terms containing L in the denominator may be neglected and thus the equation is simplified to the following form: (13 + ~)zta The optimum -210 =
Dou =
o
velocity
JBf
J. Clwontatog.,
Do
19 (1963) 237-218
z&O is then given by the formula
LINEAR
NON-EQUILIBRIUM
ELUTION
247
CHROMATOGRAPHY
The tlaird central rraoment am? the asynwaetry of the peal3 The third central moment characterizes the asymmetry of the peak, If this moment is positive, and this is always true for the given moment (sd), it means that the front boundary of the peak is sharp and the rear boundary of the peak is diffuse and the maximum of the peak is shifted in front of the center (of gravity) of the peak. The dependence of the third moment on the factors considered is similar to the case of the second moment. Central moments of higher orders define in more detail the character of broadening and asymmetry of peaks. Preliminary calculation shows that moments up to the fifth order are probably insufficient for theoretical description of the chromatographic peak by m.&ans of the Wermite expansion. More cletails on the solution of the problem are given in ref. 17. ACKNOWLEDGEMENT I
should like to thank to Dr. OTTO GRUBNER
for valuable
comments,
SUAIMARY
The model of elution chromatography is solved using a Wermite polynomials expansion, The column is infinitely long and filled along the whole length, the detector being placecl at a distance L from the feeding point. The following factors were considered: longitudinal diffusion in the mobile phase, finite rate of mass transfer through the surface of porous grains, radial diffusion inside the grains and finite rate of sorption on the internal surface of the grains. Uniform shape of the grains and uniform distribution of the filling in the column are assumed. The effect of the eddy diffusion may be included in the longitudinal diffusion coefficient. The effect of the pressure gradient and non-uniform concentration in the cross-section of the column and along the boundary of individual grains is neglected. A method of describing chromatographic peaks by means of statistical moments was used and the central moments up to the fifth order were calculated. The first moment is of basic significance for the determination of retention time. It is shown that it depends only on the partition coefficients and on the longitudinal diffusion, while it is not affected by transport phenomena in the grain and across the surface film or by the shape and size of the grain, The effect of the longitudinal diffusion. on the determination of retention time from the first moment (i.e. the center of gravity) of the peak is not significant. The second central moment has a significance for the determination of the peak width and like all higher moments it depends on all factors characterizing transport of the given compound through the column. It is possible to determine theoretically the optimum velocity of the carrier for which the relative width of the peak reaches a minimum value. lxEFEl‘CENCES I A. J, I?. MARTIN AND 33. L. M, SYNGE, Biociwm. J., 35 (1941) 2 D. IXC VAULT, J, Am. Cltem. Sot., 65 (1943) 532. . 3 E. GLUECKAUF, J. Cl&em. Sot., (1949) 32So.
1355.
J. Clwomalog.,
19 (1965)
237-24s
E.
248
KUbD!A
4 H. R&cK, Chm.
5 6 7 8 g
Ingv. Tech., 28 (1956) 489. J. J. CARBERRY, Nature, 189 (1961) 39s. L. LAPXDUS AND N. R. AWJNDSON, J. Pitys. Clccm., 56 (1952) 984. I?. R. KASTEN, L. LAP~DUS AND N. R. AMUNDSON, J. Phys. Chem.. 56 (1952) 683. S. KAZUO AND A. TAKASFII, J. C/tern. Sot. Jap., Ind. Clcem. Sect., 60 (1957) 1.42. M. KunfN, Colleclion Czech. Chem. Commacn., in press.
IO G. DOYZTSCW, Anlcitung zum pvaidisc?$en Gebvaacclc dcv Laplace Transformation, Oldenbourg, Mtinchen, xg5f.5. II J. MAL+, persons1 communicat;ion, 12 G. SZEGLS, OrthogonaE PoZyvwmiaEs, American Mathemat;ical Society (Colloquiuti Publications, Vol. XXIII), New York, 1939, p. IOI. 13 I% L. VAN DER WAERDEN, Mathematiscke Statistik, SpriIIger, BdiII, 1957. 14 E. ICUEERA AND 0. GRUBNER, Collection Czech. Chem. Commurt., zg (1964) 1782. r.5 A. A. ZNUKHOVITSKII AND N. b& TURKELTAUB, Gazovaia chvomatogvafia (Gas Clzromatogvaplzy). GosLoptechizdat, Moscow, 1962, p.’ I 2 .. IG H. PURNELL, Gas Chromatography, J. Wiley and Sons, New York, IgGz, p. 104. 17 E. KuC~xzn, Tit&s, 1nst;itut;oof l?hysical Chemistry, CSAV, Prague, 1964. J. GIwomatog.,
19 (1965) 237-248
CQNTRIBUTION
TO THE THEORY
LINEAR
NON-EQUILIBRIUM
EUGENE
KTJCERA
Institute of Plcysical (Received
237
OB CHROMATOGRAFNY
Chemistry,
Novcmbcr
OF CHROMATOGRAPHY
ELUTION
Czechoslovai~
Academ,y
CHROMATOGRAPHY
of Seienccs,
Prague
(CzecJ~oslovakia)
zGt11, 1964)
Recent theories of gas chromatography and similar separation methods only satisfactorily solve the linear type of equilibrium chromatography’-5 and some simple types of non-equilibrium chromatography o--O.The present communication describes a theory which simultaneously takes .into account the longitudinal diffusion in the mobile phase, the radial diffusion inside the porous grains of the packing material, the finite rate of mass transfer through the boundary (which may comprise diffusion in the thin layer of the immobile phase on the surface of the grains) and the sorption on the internal surface of grains. In spite of the fact that no analytical solution can be given even for the simplest model of the column with a granular filling, which will be described below, it is possible to obtain valuable informatlon based on the known Laplace transform of the solution. Let an infinitely long column with section $Jbe uniformly filled with the packing consisting of small porous grains of uniform size having the shape of spheres (index v = 3) or “infinite” cylinders (V = 2) of radius 22 or “infinite” plates (v = I) with thickness d = 2R. Among the internal co-ordinates only the co-ordinate Y, which is measured in the direction from centers (or axes or planes) of symmetry perpendicularly to the surface of the grains, is significant. The area of the geometric surface of grains per unit volume of the column is given by the formula SV = v(x - a)/ri where txis the external porosity, A homogeneous distribution of pores in the grains is supposed. The average area of boundary surface between pores and the external free volume per unit volume of the column is PV = /3S, where f3 is the internal porosity. “Incompressible” carrier flows through the free external volume along axis z of the column, with velocity ZG.The rate of flow through the whole cross section J!Jof the column is then given by $ = +zc. It is assumed that the concentration C of the compound being separated in the free volume is the same in the whole cross section of the column at any moment, thus depending only on the longitudinal co-ordinate z and time f. The flow of the given compound is defined hy the velocity of the carrier and by the diffusion and therefore it may be written as follows:
ac
ac
z+~~az-“~g=Q~
82C
The grain is presumed to be small enough so that the concentration C of the given compound may be assumed to be the same along the whole grain at any moment, only being dependent on the location of the grain center. Inside the gmins, J. Clwomalog.,
19 (1965)
237-243
E. ltTJ6ERA
238
concentrations c (Y, a, i) and 9t(L/,z, t) do not depend on angular co-ordinates. Further, sorption on the internal surface of the grains (boundary between pores and the solid part of grains) may occur. Then it is valid for concentration in the pores:
ac -A
at
D?
@G
U-I
&e-~&z
ac >
=
Qn=--8
where c is the concentration in the pores and 31is the concentration of the sorbed compound on the internal surface; both concentrations are given per unit volume of the pores. For the flow of the compound in the direction of the normal to the external surface of the pores ,
and since Pv is the external pore area in unit volume of the column, the rate of decay of the substance from the pores in unit volume is equal to jR& and also to cxQC,as the rate of influx of the compound into the carrier is the same as the rate of decay from the pores. Hence
Further, due to the symmetry aG
of the grains the following condition is valid
I
ar
It is assumed that the transfer and sorption equations’ : and
Qc = ---Wc(Ic&
-
c/r=R)
Qn =
-
n)
-H,(K,G
rates are defined by the linear
where K, and K, are equilibrium constants and Hc and Hn are mass transfer coefficients. The system of equations is solved for these initial and boundary conditions:
c (2, t)
=
c (2,$1= c (I/, x, i!) =
c (Y, z, t) =
CI (4 C$ (2)
9.2(Y, 2, 15)= IZ$(2)
ti (Y, z, t) =
I
for f = 0, -
o for t < o and z =
rt: 00
co
where Ct (x), cs (2) and mt(a) describe the initial distribution pound. J. Clwomatog., 19 (19%)
237-243
of the introduced
com-
240
E., KUthRA
For chromatography it is not necessary to know solutions c, B (Y, z, t) or c’, % (Y, x, s), respectively. For other boundary conditions and for initial relations (z) and ct (z) = K,Ct (z) ; the expression for C (a, s) was given in ref. 33. %I(2) = .I’;?2c~ In general, it is impossible to carry out the inverse transformation and to find an analytical expression for C (a, t) even for the simplest forms of the feed band. However, the expansion
can be found using Hermite
polynomialsl2:
El Lx%(t) = zl
(- I)lCfi! (2t)n-21c k--o Iz!(n 212) !
,where n
-
[ 2
1
I
for
f
even
The expansion
c1 f-
32;
=
coefficients
n -
2
I
for
odd 32.
a7&are given by the formula
based on the orthogonality of Hermite polynomials. Let p’k be the I&th moment of function C (t) defined by equation13:
where
sco
mk =
tw(t)dl!
0
and let ,#_Jk: be the I+th central moment Pk =
Goj-r(’ -pl’)kC(t)dt
Then the coefficients
al = a2 =
defined by the equation:
(+) (-,,&‘)t
= j.
can be expressed through
/-lb_2
the moments
as follows:
0
I
a,
m0
=
dzrclc2
Further,
El
0
=9 (w’%)
I)k,‘h-2k,U2k
zVz!(rc -2h)!(2,u2)?
the following
swdkC(W’ = J. Clrvornalog.,
(_
c k=" (-
I)*jLy
237-248
property dkC( s) dSk-
of the Laplace
transformation
is known:
LINEAR
NON-EQUILIBRIUM
This
ELUTION
is the key
24=
CHROMATOGRAPHY
for the calculation
of the moments
and coefficients
of the
FIermite expansion of the function C (t) from the known transform 2: (s), The number of necessary terms in the expansion series will depend on the required accuracy. The above mentioned method was used in ref. g for a more simple model, and for other initial and boundary conditions. In practice the following forms of the feed bands are of importance: For liquid-solid chromatography the given compound is initially sorbed on a definite part of the column (0, L). If a macroporous sorbent is considered in which sorption on the internal surface occurs, the sampling may be described by the conditions : =
G (4
Cf (z) =
vsytg (z) =
IZ$ =
vzt(z) =
0
0,
const. + 0 for0
for z < o and z > L.
For a micro-porous sorbent, the sorption on the internal surface cannot be distinguished from the sorption in the pores of the grains and, therefore, it is possible to describe the sampling by the conditions : IiT,
=
0, uti(Z)
cg (2) =
cc =
C$(2) =
0
=
0, C*(z)
const , + 0
=
0
for 0 < 2 < L for a c
o and z c
L.
For these cases it would be more convenient to solve the model in which only the section (0, L) of the column,is filled by the sorbent and, therefore, they will not be analysed here. In the case of gas chromatography the given compound is injected into the carrier in an “infinitesimally” short section of the column at the start of the experiment, and thus the feed band may be characterised by conditions:
where M is the total mass of injected compound and 6 (z) is the Dirac function. The Dirac function S (a - z’) has the following property :
s b
f (@(z
-
z’) dz = f(d)
In this case the transform
m = C(z,S)/Er
&
Mexp =
if a e 2’ c: b if 2’ -z a or 2’ * b.)
=o
a
(Note:
of the solution
(2)
is simplified
to the expression
[$-&-,”
,sL] . _.-.-___ J.
CkYO~atOg.,
(2’2)
19
(1965)
237-248
E.
242
ItUc!%RA
The resulting moments up to the fifth central moments are given by the fdlowing formulas :
pot =
PO =
(34
I ;
(94
PI
+
cc2
P3
I 2Dp2L
--
=
12D,L -_
+
+
ZLfj
SD,2 ---&-
>
[I +
EKc(l
64DP3 -[I + %e >
+
+
Kn)12 +
EK,(I
+
Kn)13
(34
+
+
U3
6L A_ 26
(
+
cc4 =
2 1GDp3L
I 2D,VY -__d
___-u7
+
+
5
+
NC
E(I +
+
-_--
Kn)2
2(x +
Kn +
F,
.@(I + K,)3 ___-Ii!,2
+
48DpL
+
+
R2 2(1
KTZF
___-A--V2(Y + 2) (Y +
+
243 ---264 EKC I (-_--7
4) + Dr
1g6Dp2
+
[I +
zsKn<( 1 + --Kn) v-w--HCH,,!
+
+(I
+
+ J.GhYomatO~.,
3$(1 ---.
Dr
Y(V +
2)
+
zz_
1
+
H 782
2)2 (Y +
+
6rKn( _-_-_
H$
~$K,(I E3(1 _-__- + JT?d4+ WC”
19 (I&)
K,#
237-248
+
JX,,)l
V(V +
4) (v +
( R2 I _ -_--
+
K,z)
6)
(5lJ + 12)E(I ___A_--Hc
+
EICc(I
K,)2
Hc2H n
+
I + I;TcNn
Kn)2
+
&Kn(2 -I- 3Kd -__-I_-H,Hn2
+
Km)2
(2 + 3Kn)K’n -___H.-2
+
(ti ----
+
12)Kn HP,
+
+ )
LINEAR
NON-EQUILIBRIUM
240DpsL2 --u--
+
E(1 + Hc
IC*)3 -
ELUTLON
3920Dp4L @
+
17408D,G 810
+
1 (
K, z;
+
CWROMAZOGRAPWY
12oD,L3 zt”-+--
+
243
c1 +
)
EK(l
1440Dp2L
2(1 v3(v +
+ Ic7d3 2) (v +
‘f
Kn)lG
+
(31)
48000~~ -Zb3
+
4)
Kg .(----WC+ En+--~-2-+---yEj~g.y--+IJ,G 1+ E(1 + Xn)2
2EKn(l
E2(1 +
Ii’,
+
Kn)
K,
>
+
(E$ + 4q
+
6Sn( __---
I +
+ 280;;p2L + 9$92K~3p + sKc(l
Icn)2
e3(1 --
N&7,.
+
1 (
EKn(2 + 3-T(n) I<71 ---+IJ,,a HcN.n3
E21i’c2
12oL3 213
+
+
+
Kn)4
+
3e21t’n(1 + ---HCW,
H,3
72oD& -_ u3
+
Kn)]
K7d2
+
1440DP2 --264
+
Ii2 (I + Km)2 + 41 + Kd2 + Kn ----[ Dr
v(v
+
2)
13,
1+ !I + Y3(Y +
2>2+$-(v
_--- (I 3 Kn) v’c(v + 2) (v + (6~ + ----
(14~ +
Kd3
12).K,(1
+
+
(9
+
+ I<,,) __-_--
24)e2(I ----I$,3
4)
2Kd
+
R2
H,3 6&Kn(1
6)
+
2-T(,)
+
+
dcn(1 -----
HcHn2 4E3Kn(l
+ JCn)3 + --
H,3H,
--HC
+
Kn)
H,zH,2
+
2ov
Icfl.14
4E3(I ---
+
2x7s)
(1 -k 2Kn)
--
48)Kn.
48)&7t(1 + -v--e LTJcH7&
+
Kn)’
+
12E21c7,(I
-I-
K7J3
+
+-TC7d3
+
HtpH,
Ii,3
2) +
(201) +
Hn
_t- &(I
+
Ic7d5
+
I-E,4
H?h3 3E2K7a( I + -
Kd2
+(
1
o,v(v (I
+
48).$1
+
2EKn(I
+
Hczr,3
2K7l)
+
xn
g4
J. Ciwornalog., 19 (1965)
1 -
237-248
E.
244 In the case of equilibrium chromatography K ?t = o) there is a well known solution4* 6 I4:
CP! = C(ZJ)/E=&
- %mt>2 exp --- CL > @,rt
-M-• -I- EJG wnll),pt I
=
(
=
where ucf = ---,
u
anclD,f
I -j- EICc
=
-I +
(R2/Dy =
o, E-EC=
KUh3Rk
W, =
CO,
(4)
DP E&-
It therefore follows from equations
(3) that
PO = PO’ = r ; pl
=
0; L =---I_--; 2def
(I +EKc)
EKc)2 =
(If 12D,3L Us
Pug=
ND,3 +-yi3-
)
2De.f Ucf2
2De.+ -Uef3
8Def2 %.f4
-I- -;
(I
(5)
gGoD# +
+ 240Dp3L3 --u-
pi; =
24oD,fL3 --A %ft”
=
+
+
3920Dp4L T-3920Db.f4L %f3
218,. )
(I +
&cc)4
=
9GoDc.t4,
uep ’ +
r’4;l”a”G)
(I +
E.Ti'# =
17408Def” -I- ---y-pj--*
DLSCUSSLOM
The first moment avtd tlze retention time of a chrow8atogra$hic peak
Let a detector measuring the concentration as a function of the time be placed at a given place L. in the column. The following elution times may be determined from an ideal chromatogram : The time i?Rof the peak maximum is defined by BC(L, 8) --
at
I t+=
OS
i.e. the time when the maximum concentration
is registered at the point of detections, The time 8,sof the peak median is defined by
sp
C(L, t)dt
=
& l: c(L, t>dt,
i.e. the time when half area of the peak is registered. J. Clwomatog.,
19 (WW
237-248
(6)
LINEAR
NON-EQULLIBRLUM
245
CWROMATOGRAPHY
ELUTION
The time to of the center (of gravity) or the mean of the peak is defined by
slxB
1c=pll=
o
&qL,d)
clt
Great significance has been accorded to the time t,,, ac(t,,z) az I
Z==L
= 0,
the time when the-maximum of the immediate concentration distribution in the column passes through the point of detection?. Our task is to determine the retention time defined as tn* = L/zcef by measuring some of the times defined above. Assuming the theory of equilibrium chromatography can be employed, it is possible to determine a relation among all these times based on formula (4). It is generally known that i!R* = t,,. The equation
i.e.,
erfc
tJ
%f2(tR*)2
4~cfk
_JE)
--exp
(2.2.2)
crfc (Jyp;”
+ $zJ
=
I
the relation between tR* and ts can be derived from the condition (6) and approximating,
Ior
The relation tin’ = J
tR2
+
29A
iR
f
w?f2
was derived in a previous papeP.
;f c2
Finally, the relation
is derived in this study and it can be found by rearranging the expression for the first moment (3b). In contrast to other relations, this relation is valid even for nonequilibrium chromatography. For this reason it is suggested that in accurate measurements it is necessary to measure the center (of gravity) of the chromatographic peak and not its maximum. Only this value can give us reliable data for the determination of the retention time which it is necessary to know for the determination of the partition coefficient K = .sKc(~ + ICn). The difference between to and tn* depending on the coefficient of longitudinal diffusion D,, can be neglected for large retention times, A notable finding is that the motion of the center (of gravity) of the peak does not depend on the following quantities : the shape of the grains (factor Y), the coefficient of radial diffusion Dr and the mass transfer coefficients HC and Hn, but depends only on the equilibrium constants K C and I’,), the coefficient of longitudinal diffusion Dp and the velocity of the carrier zc. J.
Chromatog.,
19 (Wd
237-248
E.
246
IW&3RA
0 %Aw secoazd ce9ztral Inon5ent and tlae widtlz of the chvomatogra$lzic peak The second central moment - variance - depends on all factors characterising the column, its filling and transport of the given compound through the column. It increases for the given place L of the co1um.n together with the increasing coefficient of the longitudinal diffusion, the equilibrium constants and the size of grains R. It decreases with the increasing coefficient of radial diffusion, the mass transfer coeffkients and with the increasing symmetry of grains characterised by coefficient V. The significance of the shape factor may be assigned to this coefficient, supposing that its value ranges from 2.5 to 3 for the usual shape of the chromatographic grains. For a given ratio s, the chromatographic peak will be narrowest for the column filled by spherically-shaped grains. The standard deviation x = dz defines the width of the pealclGglO. The value 2 x ,&i is considered as the width of the curve. The ratio o = 2 3c ,/2/pjs.defining the relative width of the peak has a large significance for the determination of the separation efficiency of the columrP. The optimal velocity zto for which the relative width of the peak has a minimal value can be found from the condition dw/dzb = o. Let DP depend on the velocity zc and let this dependence be given by an approximate relationship =P = Do + Am + Bu2 (e.g. ref. 16) where Do is the coefficient of diffusion, A and B are empirical (eventually derived theoretically) constants being dependent on the properties of the column, its filling and carrier. Substitution of the expressions (3b, c) into the expression for o gives :
where .A? (I + K,)2 + E(I + K,)2 %=
After
+
Ir'n
D, - ?J(Y + 2) --TT:-+Nn -_--_-_--_--[I + EKc(l + Kn)]2
differentiation,
(F
+
the following
12;fox)u2
-
(Do
+
‘9
algebraic
_
t$?$~
equation
_
,5D02,L
of the fourth
=
degree
o
is obtained for zco, which is practically impossible to solve. For a large distance L, the terms containing L in the denominator may be neglected and thus the equation is simplified to the following form: (13 + ~)zta The optimum -210 =
Dou =
o
velocity
JBf
J. Clwontatog.,
Do
19 (1963) 237-218
z&O is then given by the formula
LINEAR
NON-EQUILIBRIUM
ELUTION
247
CHROMATOGRAPHY
The tlaird central rraoment am? the asynwaetry of the peal3 The third central moment characterizes the asymmetry of the peak, If this moment is positive, and this is always true for the given moment (sd), it means that the front boundary of the peak is sharp and the rear boundary of the peak is diffuse and the maximum of the peak is shifted in front of the center (of gravity) of the peak. The dependence of the third moment on the factors considered is similar to the case of the second moment. Central moments of higher orders define in more detail the character of broadening and asymmetry of peaks. Preliminary calculation shows that moments up to the fifth order are probably insufficient for theoretical description of the chromatographic peak by m.&ans of the Wermite expansion. More cletails on the solution of the problem are given in ref. 17. ACKNOWLEDGEMENT I
should like to thank to Dr. OTTO GRUBNER
for valuable
comments,
SUAIMARY
The model of elution chromatography is solved using a Wermite polynomials expansion, The column is infinitely long and filled along the whole length, the detector being placecl at a distance L from the feeding point. The following factors were considered: longitudinal diffusion in the mobile phase, finite rate of mass transfer through the surface of porous grains, radial diffusion inside the grains and finite rate of sorption on the internal surface of the grains. Uniform shape of the grains and uniform distribution of the filling in the column are assumed. The effect of the eddy diffusion may be included in the longitudinal diffusion coefficient. The effect of the pressure gradient and non-uniform concentration in the cross-section of the column and along the boundary of individual grains is neglected. A method of describing chromatographic peaks by means of statistical moments was used and the central moments up to the fifth order were calculated. The first moment is of basic significance for the determination of retention time. It is shown that it depends only on the partition coefficients and on the longitudinal diffusion, while it is not affected by transport phenomena in the grain and across the surface film or by the shape and size of the grain, The effect of the longitudinal diffusion. on the determination of retention time from the first moment (i.e. the center of gravity) of the peak is not significant. The second central moment has a significance for the determination of the peak width and like all higher moments it depends on all factors characterizing transport of the given compound through the column. It is possible to determine theoretically the optimum velocity of the carrier for which the relative width of the peak reaches a minimum value. lxEFEl‘CENCES I A. J, I?. MARTIN AND 33. L. M, SYNGE, Biociwm. J., 35 (1941) 2 D. IXC VAULT, J, Am. Cltem. Sot., 65 (1943) 532. . 3 E. GLUECKAUF, J. Cl&em. Sot., (1949) 32So.
1355.
J. Clwomalog.,
19 (1965)
237-24s
E.
248
KUbD!A
4 H. R&cK, Chm.
5 6 7 8 g
Ingv. Tech., 28 (1956) 489. J. J. CARBERRY, Nature, 189 (1961) 39s. L. LAPXDUS AND N. R. AWJNDSON, J. Pitys. Clccm., 56 (1952) 984. I?. R. KASTEN, L. LAP~DUS AND N. R. AMUNDSON, J. Phys. Chem.. 56 (1952) 683. S. KAZUO AND A. TAKASFII, J. C/tern. Sot. Jap., Ind. Clcem. Sect., 60 (1957) 1.42. M. KunfN, Colleclion Czech. Chem. Commacn., in press.
IO G. DOYZTSCW, Anlcitung zum pvaidisc?$en Gebvaacclc dcv Laplace Transformation, Oldenbourg, Mtinchen, xg5f.5. II J. MAL+, persons1 communicat;ion, 12 G. SZEGLS, OrthogonaE PoZyvwmiaEs, American Mathemat;ical Society (Colloquiuti Publications, Vol. XXIII), New York, 1939, p. IOI. 13 I% L. VAN DER WAERDEN, Mathematiscke Statistik, SpriIIger, BdiII, 1957. 14 E. ICUEERA AND 0. GRUBNER, Collection Czech. Chem. Commurt., zg (1964) 1782. r.5 A. A. ZNUKHOVITSKII AND N. b& TURKELTAUB, Gazovaia chvomatogvafia (Gas Clzromatogvaplzy). GosLoptechizdat, Moscow, 1962, p.’ I 2 .. IG H. PURNELL, Gas Chromatography, J. Wiley and Sons, New York, IgGz, p. 104. 17 E. KuC~xzn, Tit&s, 1nst;itut;oof l?hysical Chemistry, CSAV, Prague, 1964. J. GIwomatog.,
19 (1965) 237-248