522
Kiyoshi Yamaoka and Terumichi Nakagawa
Moment Analysis of Nonlinear Chromatographic Elution Curve Kiyoshi Yamaoka and Terumichl Nakagawa" Faculty of Pharmaceutical Sciences, Kyoto University, Kyoto, Japan (Received June 24, 1974)
A nonlinear (nonlinear distribution isotherm) chromatographic theory is developed using the perturbation theory. A chromatographic system is perturbed by the distribution isotherm with respect to solute concentration, and a material balance describing the nonlinearity is reduced to simultaneous linear partial differential equations according to first-order perturbation theory and solved by the Laplace transform method. From the transformed solution for this equation, the statistical moments are calculated. The equations for the first to the fourth cumulant in the nonlinear ideal chromatographic curve, and for the first ordinary moment in nonlinear nonideal chromatographic curve, are presented. From these equations, quantitative discussions are given describing the effect of the nonlinearity on the shape and position of the elution curve.
Introduction
It is extremely difficult to discuss, analytically, nonlinear chromatographic problems. Ideal chromatography specified by the absence of the diffusion phenomena has been qualitatively discussed by many workers since the beginning of investigations of chromatographic theory.'-& However, to overcome the difficulties in solving nonlinear problems, attempts have been made to linearize the problem by using the perturbation method or by the employment of a computer simulation. Funk and H0ughton,~,7using the finite difference approximation for the computer simulation, Houghton,s appling the perturbation theory to a system which involves a slight deviation of the distribution isotherm from linearity, and Haarhoff and Van der Linde,g taking a sorption effect into account, have advanced nonlinear nonideal chromatographic theory. However, since the mathematical expressions stated in those works are too complicated to apply in practice, it is difficult to obtain the statistical moments of a chromatogram from those preceding results. Oldenkamp and Houghton,lo utilizing the previous work,8 gave qualitative explanations to solute retention in nonlinear chromatography. Recently DeClerk and Buysll applied the moment operator method to nonlinear nonequilibrium system and obtained analytical expressions for the first through the third statistical moments. The moments discussed there, however, were those with respect to axial coordinate z, not in the time domain, so that even retention time is hard to calculate from moment equations. It is suggested from the ample use of statistical moments in linear chromatography that the moment analysis method will also successfully elucidate the profiles of nonlinear chromatographic wrves. Hence, the present paper describes the attempt to solve the nonlinear material balance equation by means of the first-order perturbation theory and to calculate the statistical moments from the Laplace transformed solution in order to give analytical explanations to a nonlinear chromatogram. Nonlinear Ideal Chromatography We first consider the nonlinear chromatographic system with instantaneous equiliblium and no diffusions. For this case, if we represent the nonlinear distribution isotherm by
the mass balance can be described by
(1
+a+
ac + u ;ac iz
EC)G
where tC expresses the perturbation of the distribution ratio due to the nonlinearity of the isotherm, and E represents the shape of the isotherm to be convex toward the Q axis for t < 0, the C axis for t > 0, and linear for t = 0. If the mobile phase contains no solute at t = 0 and solute is , initial and put into the column in the form, f ~ ( t )the boundary conditions become
(3 1 respectively. In the spirit of the first-order perturbation theory, we take C(t,z), expandable for small value of E , in the form C ( 0 , z ) = 0 and C(t,O) = fI(t)
tzo
Substituting eq 4 into eq 2 and neglecting the higher power terms of t, we can reduce eq 2 to the following set of linear partial differential equations.
(1
+
+ u aco/az = o (5) + k ) acl/at + u ac,/az + Y2 aco2/at= o (6) (1
k ) ac,/at
As eq 5 and its solution are well known, further discussions are not developed here. T o solve eq 6 the initial and boundary conditions for C O and C1 are set similarly to eq 3 Co(0,Z) = 0
Ci(0,Z) = 0
Co(t,O) =
fI(t)
C*(t,O) = 0
(7 1 (8)
Combining eq 5 and 7
(9) Substituting eq 9 into eq 6
(1
+
k)
acl/at + u aci/az
t
The Laplace transform of eq 10 becomes The Journal of Physical Chemistry, Vol. 79, No. 5 . 1975
= 0
523
Moment Analysis of Electron Curves c =-l5.0
where s is a complex variable and
dl(s,z) =
1 (s,z
c
=
l5.0
) is defined by
Jme""Cl(t,z) d t
(12)
0
The solution for eq 11with the transformed condition Cl(s, 0 ) = 0 isl2
Hence, eq 4 can be rewritten in Laplace domain as T i m e (sec)
Figure 1. Effect of isotherm nonlinearity on shape and position of elution curve: k = 20, u = 5.0 cm/sec, z = 100 cm, u = 5.0 sec, rn
= 1.0.
To obtain the statistical moments, f ~ ( tis) submitted to the gaussian distribution function (15) where m and u denote solute amount and standard deviation, respectively. Substituting the Laplace transform of eq 15
i ( s ) = m exp(ds2/2)
(16)
into eq 14 and taking natural logarithm results in ",
In C ( s , z ) = In nz --
+
U
kZs
+
Time (sec)
Expanding the last term on the right-hand side of eq 17 for small value of 6 and taking the first term yields
In E ( s , z ) = In m --
+
U
'2s
+
Making use of the following relation between cumulants and the Laplace transformed function
(19) the first through the fourth cumulants can be calculated from eq 18 to give the final results:
Figure 2. Effect of solute amount on shape and position of elution curve: k = 20, u = 5.0 c d s e c , z = 100 cm, u = 5.0 sec, t = -5.0.
tions, p1 and 1 s are linear functions of the solute quantity m, while these cumulants, 112 and 1 4 inclusive, are independent of m for linear chromatography. It is determined from the above equations that a decrease in the retention time and an increase in the peak tailing are proportional to an increase in the solute amount for t < 0, the reverse being true for > 0. It also becomes evident that the nonlinearity of the distribution isotherm exerts no influence over the width and sharpness of a chromatogram. Figures 1 and 2 show such behaviors of the chromatogram as functions of E and m, respectively. These curves were drawn by computer with use of Gram-Charlier series, details of which have been described elsewhere.13
Nonlinear Nonideal Chromatography As the nonideality is termed by taking the diffusion phenomena into account, mass balance for nonlinear nonideal chromatography is represented by12 (1
+
k t EC)
ac/at + u ac/az
=
D a 2 c / a 2 (21)
with the initial and boundary conditions where MI', Mz, M3, and M4 are the first ordinary and the second to the fourth central moments, respectively, and ~1 to 114 are the parameters describing the shape and position of a chromatogram. As readily seen from the above equa-
C(0,Z) = 0 C(t,O) =
?At)
(22)
C(t,m) = 0 The Journal of Physical Chemistry, Vol. 79, No. 5, 1975
Kiyoshi Yamaoka and Terumichi Nakagawa
524
Substituting eq 4 into eq 21 and neglecting the higher power terms of e, eq 21 can be separated to (1
+
k) aco/at
+
u aco/az = D a2co/az2 (23)
and (1
+
k ) aci/at
+
u a c , / a z = D a2cl/az2 - i/,aco2/at (24) Equation 22 is rewritten as Co(0,~) = 0
Co(t,O) =
fI(t)
Co(t,m) = 0 (25)
and Cl(0,z) = 0
Cl(t,O) = 0
C,(t,m) = 0
(26)
for Co and C1, respectively. Equation 23 describing the unperturbed system has been solved by means of the Laplace transform method, and the solution is12
From eq 19 and 31, the first ordinary moment is calculated to be pi = Mi' =
l + k z T--
t
c0(s,2) = where I? denotes the incomplete gamma function which is defined by Then, eq 24 is also solved under the conditions of eq 26 by applying the Laplace transform method (see Appendix), and the solution is obtained as
r,(x) =
JAme-'t'-' d t
(33)
Thus, eq 32 permits analytical discussion about retention of a solute in nonideal and nonlinear chromatography. If c = 0, MI' can be reduced to (1 k ) z / u which agrees with the result in a linear nonideal and instantaneous equilibrium ch~omatography.~~ Nonlinearity of the distribution isotherm gives rise to the last term in the right-hand side of eq 32 which is proportional to the solute amount, m, while the first two terms in the brackets of eq 32 indicate that the diffusibility exerts an effect on solute retention opposite to that of solute amount. Thus, the solute retention will show the positive deviation proportional to the solute amount for > 0, and the reverse being true for E < 0. This agrees with the previous discussion^^^ and also with the experimental trends. The analytical forms of the higher moments for this case are not involved in this work because of the complexity of the mathematical manipulation. However, they may provide more detailed information about the shape of a chromatogram in a nonlinear system. To approach the problem, a computer simulation followed by moment analysis will be discussed elsewhere.16
+
where
and (is a dummy variable for integral. Thus the perturbed solution for eq 21 will be obtained by substituting eq 27 and 28 into the Laplace transform of eq 4. Before that, to ) eq 27, and to avoid the difgive an explicit form to f ~ ( s in ficulty in calculating c02(s, [) in eq 28 which should be obtained by squaring the inverse of eq 27, it is assumed that a solute is introduced into the column in the form approximated to Dirac's 6 lunction and that unperturbed function, C O( t , z ) , can be regarded as a gaussian distribution function. Under these assumptions
where m is a solute amount in a unit of delta function. Squaring the inverse Laplace transform of the above equation, then transforming again the resulting equation to the Laplace domain, one can arrive at
Appendix Laplace transforming of eqn 24 and 26 with respect to t yields
Transforming (Al) again with respect to (s,p ) domain is given as
where
Again, by combining eq 28-30 and the Laplace transform of eq 4,the final solution is shown as follows: Jhe Journal of Physical Chemistry, Vol. 79, No. 5, 1975
and
z,
the solution in
Conductance-Concentration Function
525
Laplace transforms of C andfi with respect to t Laplace transform of C, with respect to t and z Ci distribution ratio between stationary and mobile k phases E perturbation parameter t time Z coordinate along column length linear velocity of the mobile phase 11 solute amount in units of the delta function tu standard deviation of the gaussian distribution U function s ,p complex variables 5 dummy variable for integral D diffusion coefficient Awn statistical moments kl cumulants rh(IC) incomplete gamma function
6,ji
Then, the inverse transform of (A3) leads to the solution in (s,z ) domain
where a =
(5)' +7 1 + ks
Introducing (A2)
e"&(Z-c)}
dc
-
im e,2(s
References and Notes
,L)e(U/2D)(Z-c)
{e G ( Z - 9 ) -
e -Jii(zcCi) dl]
(A5)
and changing the interval of the last integral in the bracket of (A5) and rearranging, one can obtain eq 28.
List of Symbols Q (t,z ) solute concentration in the stationary phase C ( t ,z ) solute concentration i n the mobile phase Co( t ,z ) unperturbed function of solute concentration in the mobile phase C, ( t ,z ) first-order perturbation function of solute concentration i n the mobile phase
(1)J. N. Wilson, J. Amer. Chem. SOC.,62, 1583 (1940). (2) D. DeVault, J. Amer. Chem. SOC.,65,532 (1943). (3)E. Glueckauf, J. Chem. Soc., 1302 (1947). (4)J. Weis, J. Chem. SOC., 297 (1943). ( 5 ) H. C.Thomas, Ann. N.Y. Acad. Sci., 49, 161 (1948). (6)J. E. Funk and G. Houghton, J. Chromatogr., 6, 193 (1961). (7) F. T. Dunckhorst and G. Houghton, hd. fng. Cbem., Fundam., 5, 93 119661. \----,-
(8)G. Houghton, J. Phys. Chem., 67, 84 (1963). (9)P. C.Haarhoff and H. J. Van der Linde, Anal. Cbem., 38, 573 (1966). (IO)R. D. Oldenkamp and G. Houghton. J. Phys. Chem., 67,597(1963). (11) T. S.Buvs and K. DeClerk. Seor. Sci.. 7. 543 11972). i12j L. Lapidk and N. R. Arnundsoi, J. Phys: Chem., 56: 984 (1952). (13)K. Yarnaoka and T. Nakagawa, J. Chromatogr., 92, 213 (1974). (14)J. J. van Deemter, F. J. Zuiderweg, and A. Klinkenberg, Chem. Eng. Sci., 5,271 (1956). (15)A. B. Littlewood, C. S. G. Phillips, and D. J. Price, J. Chem. SOC., 1480 (1955). (16)K . Yamaoka and T. Nakagawa. J. Chromatogr., submitted for publication.
Conductance-Concentration Function for Associated Symmetrical Electrolytes Raymond M. Fuoss Sterling Chemistry Laboratory, Yale University, New Haven, Connecticut 06520 (Received September 20. 1974) Publication costs assisted by the Office of Saline Water
A new model for solutions of symmetrical electrolytes is proposed: Gurney cospheres centered on ions of charge f e , and surrounded by a continuum containing a continuous space charge which integrates to Fe. When two cospheres overlap, the corresponding ions are counted as pairs; the pairs are assumed to contribute nothing to net transport of charge and, as dipoles, to be disregarded in the calculation of screening potentials and activity coefficients. Relaxation field and electrophoretic countercurrent for the model are computed using continuum theory. These are combined to give a three-parameter conductance function A (c; Ao, KA,R ) where A0 is limiting conductance, K A is association constant, and R is the diameter of the cospheres. The terms of the function are given explicitly, and a method of deriving values of the parameter from conductance data is described. The new conductance equation replaces the previous equations, now obsolete, which have been proposed earlier by Fuoss and coworkers.
Most current theories of electrolytic conductance have two features in common: (1) they use the same model (rigid, charged spheres in a continuum); and (2) they start
by calculating the theoretical behavior of the model, assuming full participation of all the ions in the long-range interactions and subsequently postulating a mass action The Journal of Physical Chemistry, Vol. 79, No. 5 , 1975
