2302
Anal. Chem. 1983, 55, 2302-2309
milians-Universitat Munchen, G.F.R.
LITERATURE CITED ( I ) Ettre, L. S. J . Chromatogr. Scl. 1978, 76, 396-417. (2) Julln, B. G.; Vandenborn, H. W.; Kirkland, J. J. J . Chromatcgr. 1975, 7 72, 443-453. (3) Locke, D. C.; Dhlngra. B. S.; Baker. A. D. Anal. Chem. 1982, 54, 447-450. (4) Willmott, F. W.; Doiphln, R. J. J . Chromatogr. Scl. 1974, 72, r- -i ~ .~- -. 7nn (5) Krejci, M.; Tesarik, K; Rusek, M.; Pajurek, J. J . Chromatogr. 1981, 278, 167-178. (6) McGuffln, V. L.; Novotny, M. unpublished research, Indlana Unlversb, Bloomington, IN, 1982.(7) McGuffln, V. L.; Novotny, M. Anal. Chem. 1981, 53, 946-951. (8) Karmen, A.; Gluffrlda, L. Nature (London) 1984, 207, 1204-1205. (9) Sevcik, J. Chromatographla 1973, 6, 139-148. (10) Kolb, 6.; Blschoff, J. J . Chromatogr. Scl. 1974, 72, 625-629. (11) Brazhnikov, V. V.; Gur'ev, M. V.; Sakodynsky, K. I.Chromatogr. Rev. 1970, 72, 1-41. (12) Slais, K.; Krejci, M. J . Chromatogr. 1974, 97, 181-186. (13) Compton, B. J.; Purdy, W. C. J . Chromatogr. 1979, 769, 39-50. (14) Lubkowitz, J. A.; Semonlan, B. P.; Galobardes, J.; Rogers, L. B. Anal. Chem. 1978, 50, 672-676. (15) Yang. F. J. J . Chromatogr. 1982, 236,265-277. (16) Gluckrnan, J. C.; Hlrose, A.; McGuffln, V. L.; Novotny, M. Chromatographla 1983, 77, 303-309. (17) Tsuda, T.; Novotny, M. Anal. Chem. 1978, 50, 271-275. (18) McGuffin, V. L.; Novotny, M. J . Chromatogr. 1983. 255, 381-393. (19) Hirata, Y.; Novotny, M.; Tsuda, T Ishii, D. Anal. Cbem. 1979, 57, 1807-1809. (20) Hirata, Y.; Novotny, M. J . Chromatogr. 1979, 786, 521-528. (21) McGuffln, V. L.; Novotny, M. Anal. Chem. 1983, 55, 560-583. (22) Jacob, K.; Vogt, W.; Knedel, M. Lleb/gs Ann. Chem. 1979, 878-865. (23) Jacob, K.; Falkner, C.; Vogt, W. J . Chromatogr. 1978, 767, 67-75. (24) Jacob, K.; Maler, E.; Schwertfeger, G.; Vogt, W.; Knedel, M. Blomed. Mass Spectrom. 1978, 5 , 302-311.
-
(25) Deo, P. G.; Howard, P. H. J . Assoc. Off. Anal. Chem. 1978, 67, 2 10-2 13. (26) Semonian, B. P.; Lubkowitz, J. A,; Rogers, L. B. J . Chromatogr. 1978, 757, 1-10, (27) Patterson, P. L. J . Chromafogr. 1978, 767, 381-397. (28) Gore, R. C.; Hannah, R. W.; Pattacini, S. C.; Porro, T. J. J . Assoc. Off. Anal. Chem. 1971, 54, 1040-1082. (29) Hartmann, C. H. Anal. Chem. 1971, 4 3 , 113A-125A. (30) Lubkowitz, J. A.; Glajch, J. L.; Semonlan, B. P.; Rogers, L. B. J . Chromatogr. 1977, 733, 37-47. (31) Sternberg, J. C. "Advances In Chromatography"; Giddings, J. C., Keiler, R. A., Eds.; Marcel Dekker: New York, 1966; Vol. 2; pp 205-270. (32) Kirkland, J. J.; Yau, W. W.; Stoklosa, H. J.; Diiks, C. H. J . Chromatogr. Sci. 1977, 15, 303-316. (33) Chesier, S. N.; Cram, S. P. Anal. Cbem. 1971, 4 3 , 1922-1933. (34) Taylor, G. R o c . R . SOC.London, Ser. A 1953, 279, 186-203. (35) Bayer, F. L. J . Chromatogr. Scl. 1977, 15, 580-581. (36) McGuffln, V. L. Ph.D. Dlssertatlon, Indiana Universlty, Bloomlngton, IN, 1983. (37) Bowman, M. C.; Beroza, M. J . Assoc. Off. Anal. Cbem. 1967, 50, 926-933. (38) McCallum, N. K. J . Chromatogr. Sci. 1973, 7 7 , 509-514. (39) Heenan, M. P.; McCallum, N. K. J . Chromatogr. Scl. 1974, 72, 89-90.
RECEIVED for review April 4,1983. Accepted August 25,1983. This research was supported by the Department of Health and Human Services, Grant No. GM 24349. V.L.M. was the recipient of a full-year graduate fellowship bestowed by the American Chemical Society, Division of Analytical Chemistry, and sponsored by the Upjohn Co. Preliminary results of this research were presented at the Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy,Atlantic City, NJ, 1982.
Pressure Dependence of Diffusion Coefficient and Effect on Plate Height in Liquid Chromatography Michel Martin and Georges Guiochon* Ecole Polytechnique, Laboratoire de Chimie Analytique Physique,' 91128 Palaiseau, France
The dlffuslon coefflclent of solutes In solvents Is a function of pressure, the relative varlatlon being of the order of lo4 bar-'. Thus, the reduced linear velocity varies along the column and the average plate height Is not equal to the local plate helght at column outlet. The dtfference between these plate helgMs Increases markedly wlth Increasing velocltles. Accordingly, losses In plate number would appear at large velocities when coupling several columns, but they would be lnslgnlflcant around the velocity correspondingto the mlnlmum plate height because of a compensation effect. Because of this pressure effect on dlffuslon coefficient, the values of the coefflclents of the plate height equation derived from a least-squares flt of the data differ slgnlflcantly from their local values. Errors exceedlng 30 and 100% can be observed on the values of A and C, respectively. This effect of the longitudinal pressure gradlent Is parallel to the thermal effect. I t may be somewhat reduced by the axial temperature gradient but Interacts wlth the radlal temperature effect, 80 as to Increase their common contrlbutlon to band broadening.
Pressure can influence in many ways the peak dispersion in a chromatographiccolumn. The most obvious effect of inlet pressure comes from the fact that it determines the mobile phase velocity in the column and, hence, changes the plate
height for the solute. Such an effect has been extensively studied and is described in almost every text on chromatography (1,2). If however, the column is operated with a given mobile phase at a fixed average velocity, but with different average pressures, one will observe a change in the plate height value of the solute due mainly to the pressure dependence of its diffusion coefficient. Such an effect is accounted for in gas chromatography through pressure correction factors. Pressure effects on mobile phase parameters are quite often considered as negligible in liquid chromatography (LC), because the compressibility and the pressure dependence of viscosity of liquids are relatively weak. Typical values are 104/bar for the former and 104/bar for the latter. Thus, when operating a chromatographic column under a pressure drop of 100 bars (approximately 1500 psi), the change in the viscosity of the mobile phase reaches 10% between the inlet and the outlet of the column. Such a variation must be taken into account for accurate calculations of the chromatographic behavior, as indicated in a previous study about the effect of pressure on the retention time and the retention volume of an inert compound in LC (3). Pressure dependences of density and viscosity tend to increase, respectively, the elution volume and the elution time of an unretained solute above the values expected for a noncompressible solvent of constant viscosity. Beside these effects on an inert compound, pressure can thermodynamicallyaffect the distribution coefficients and the capacity ratios of retained solutes, offering the possibility of
0003-2700/83/0355-2302$01.50/00 1983 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 55, NO. 14, DECEMBER 1983
-
_ _ _ I l _ 1 1 l - 1
~
1 1 1 1 1 -
_
_
_
_
1
_
1
_
_
_
2303 _
Table I. Linear Regression Analysis of the Viscosity and Diffusion Coefficient vs. Pressure Data on Cyclohexane ( 1 2 ) According to the Following Equations 7) = n0[l t a”(P - Po)] 1/D = (l/Do)[l t a’(P - Po)] _ l l l l _ l l l l
a
temp, K
rlol cp
313 333 358 3 83
0.686 0.424 0.265 0.190
viscosity data a”, bar-’ 1.60 x 1 0 - ~ 2.38 x 1 0 - ~ 3.18 X 3.55 x 1 0 - ~
r2 = coefficient of determination.
rzu
where DABis the diffusion coefficient, Tis the absolute temperature, VB is the viscosity of the solvent, r A is the radius of the spherical solute, and k is the Boltzmann constant. In the following, subscripts A and B refer to the solute afid solvent, respectively. Hence, most of the modifications to eq 1 can be written, for dilute solutions, as = constant --f(V) qB
1.44 X l o u 3 1.65 x 10-3 2.13 X 2.95 x 10-3
10-5 10-5 10-5 10-5
r2a
0.995 0.987 0.996 0.986
l__-l_.llll_-_-_l_.
PRESSURE DEPENDENCE OF THE DIFFUSION COEFFICIENT Little is known about the basic relationships between diffusion coefficients in liquids and the properties of the molecules involved (7) and even less about their pressure dependence. It is symptomatic there is not any mention about this dependence in either Bridgman’s authoritative work on high pressure (8) or Reid, Prausnitz, and Shenvood’s book on the properties of gases and liquids (9). Several semiempirical equations have been proposed, however, for the prediction of the diffusion coefficients (9). Most of them are empirical modifications of the fundamental Stokes-Einstein equation established for rigid spherical molecules in dilute solutions from the hydrodynamic theory (7, 9,10)
T
1.97 x 2.81 x 4.61 x 7.89 x
0.994 0.981 0.981 0.944
a pressure-induced selectivity (4-6). There are other pressure effects which have not been considered, those on mass-transfer properties, which have direct bearing on band broadening. The diffusion coefficient of a solute in a solvent is a function of the pressure. This phenomenon and its consequences are discussed in the following.
DAB
diffusion coefficient data D o ,cmz/s a’,bar-’
___
(2)
where f ( V)is a function of the molar volume of solute or of some combination of the molar volumes of the solute and solvent, which in either case has the dimension of Va, the value of a being in the range 1/3 to 2/3 (9). For liquids, the widely used Wilke-Chang expression for the prediction of D A B has the € o m of eq 2 (12)
reciprocal of the solvent viscosity. This inverse dependence, which originates from the Stokes-Einstein relation as well as from the Eyring rate theory (7), has been questioned for solvents with viscosities higher than about 1 CPbut can be considered as valid for low-viscosity solvents, such as the typical LC mobile phases used in normal as well as reversed phases (9). In these conditions, it comes from eq 2, when noting that f ( V) is proportional to V-O
with a ranging from 0.33 to 0.67. Equation 4 states that the relative variation of the diffusion coefficient with pressure, P, can be calculated from the relative variation of the solvent viscosity with pressure, (dq,/qB)/dP, and the solute compressibility, -(aVA/ VA)/aP. Since, as noted at the beginning of this paper, the viscosity term in the right-hand side (RHS) of eq 4 is 1 order of magnitude larger, in absolute value, than the compressibility and a is smaller than 1,the pressure dependence of the diffusion coefficient is approximately equal, but reversed in sign, to the pressure dependence of the solvent viscosity, that is, about W3/bar. This is in agreement with the recent experimental data obtained by Jonas et al. in a study on cyclohexane, although such a molecule is clearly nonspherical and hence does not fill the requirements for the application of the Stokes-Einstein relationship (12). The relationship between solvent viscosity and pressure in the pressure range of chromatographic experiments can be satisfactorily expressed by TB
‘IBo[l
+ a”(p - Po)]
(5)
where q b is the viscosity at the reference pressure Po,usually the atmospheric pressure, a” is the coefficient of variation of the viscosity with pressure, corresponding to the first term in the RHS of eq 4 (3). Since D A B is inversely proportional to VB and since the solute compressibility X A = --(9VA/VA)/aP is small in comparison with the viscosity term of eq 4, the variation of the diffusion coefficient with the pressure can be represented as
DAB = (l/Dmo)[l + (a”- ~ x A ) (-PPo)]
(6)
or where M B is the solvent molecular weight and D A ~VA, , and VB are expressed in cm2/s, cm3/mOl, and P. respectively. C$B is a dimensionless association factor equal to 2.6 for water, 1.9 for methanol, 1.5 for ethanol, and 1.0 for nunassociated solvents. The Wilke-Chang correlation is usually found satisfying for diffusion of organic solutes in either aqueous or organic solvents, although it can lead to errors as large as 30% in the prediction of diffusion coefficients of typical LC solute-solvent systems (9). The most significant fact, for our purpose, in eq 1to 3 is the proportionality between the diffusion coefficient and the
DAB
( ~ / D A B J+ [ ~a‘(P - Poll
(7)
where D e is the diffusion coefficient at the reference pressure Poand a‘ the coefficient of variation of Dm with P, assumed to be constant in the pressure range Poto P. a’ should be similar to a”. This can be checked a8 shown in Table I by linear regression analysis of the data of Jonas et al. on cyclohexane, although they actually measured the self-diffusion coefficient D A A of cyclohexane. However, for our purpose, self-diffusion can be considered as a special case of mutual diffusion where solute A is diffusing in solvent A. It is shown from Table I that a’ and a” are of the same order of magnitude
2304
ANALYTICAL CHEMISTRY, VOL. 55, NO. 14, DECEMBER 1983
and that they both increase with increasing temperature. A linear relationship between 1/D and P, leading to d equal to about 2.5 X bar-’, has also been observed by McCool and Woolf from self-diffusion measurements on cyclohexane (13). Similar relationships are found for the self-diffusion coefficients of methanol (14, pyridine (15), and methylcyclohexane (16) leading to a’ values of 3 X bar-’, 7.5 X 10“. bar-’, and 2.5 X bar-’, respectively. Similarly, a linear regression analysis, according to eq 7 , of the data of Koeller and Drickamer on diffusion of carbon disulfide in various 50/50 organic mixtures, gives a’ values of the order of magnitude of bar-’ when the number of data is larger than 6 and the correlation coefficient larger than 0.99 (17). Work on the high-pressure dependence of transport coefficients has been reviewed by Le Neindre (18). The pressure dependence of mutual diffusion coefficients has been recently investigated, for the first time to our knowledge, and leads to a’ values ranging from 0.9 to 1.1 X bar-’ for eight different solute-solvent systems (19). In the following, eq 6 and 7 are used to represent the pressure dependence of the infinite dilution mutual diffusion coefficient of solute A in solvent B, which is the one of chromatographic interest.
EFFECTOFTHEPRESSUREDEPENDENCEOF D A B ON THE PLATE HEIGHT In liquid chromatography,the variations of the plate height, H, of solute A with the mobile phase velocity, uB, are conveniently represented in reduced parameters by the semiempirical Knox equation (20)
B h =-+ Y
+ CV
(8)
v = vO[l
+ (a’’ - XB - axA)(P - P o ) ]
(14)
+ a(P - Po)]
(15)
or v = vo[l
where u0 is the reduced velocity at a reference pressure, conveniently the column outlet, usually atmospheric, pressure; a is the coefficient of variation of v with P, which is typically bar. One is now seeking for an expression of the apparent plate height in terms of the local plate height parameters and the hydrodynamic characteristicsof the column. The combination of eq 8 and 15 gives the expression of the local plate height
The apparent plate height is then given according to eq 11 by a sum of three integrals, hA,hB, and hc, the subscripts referring to the local plate height parameter entering each integral with
1
hA = Avol/’$o [1 4- a(P -
h = H/d,
v = UB~P/DAB
(9)
(10)
d, being the particle diameter. A , B, and C are constant for a given solute-solvent stationary phase system, packing configuration, and temperature. Typical values of A, B , and C are, respectively 1, 1.8, and 0.05. Since, as noted in the previous section, the diffusion coefficient varies with the pressure, the reduced velocity will also change with pressure; hence, because of the pressure gradient in the chromatographic column, the plate height will vary with the abscissa z in the column, scanning some range of the plate height curve, h(v), between the inlet and the outlet of the column of length L. The apparent reduced plate height, happ,observed on the chromatogram is then an average given by
h, = CvoJ
0
[l
+ a(P - Po)]d(z/L)
which gives by combining with eq 4 and noting that the U B term in eq 12 is the negative of the compressibility of solvent
B, XB = -(avB/vB)/aP -dv/v =-
aP
+-~ VaPB / +~ -aB~ Va pA / ~ A(13)
~VB/VB
aP
Since the compressibility terms in this equation are one order of magnitude sm$ler than the viscosity term, it appears that the reduced velocity will vary with the pressure according to
(20)
In order to integrate eq 18 to 20, it is necessary to know the pressure profile P(z) in the column, which is given after the Darcy law by
where k is the column permeability. qB is related to the pressure P after eq 5 and U B through its compressibility XB by UB
=
uBo[l
uBoe-xs(~po)
- X B ( P - Po)]
(22)
where uB0 is the solvent velocity at the outlet pressure Po. From eq 21, 22, and 5 , one obtains
dP
UB~VB~
k
dz = -
N
[ I +a”(p-P0)1[1 - X B ( P - P o ) ]
dP
[I
The derivation of this equation, which assumes a constant velocity all along the column is given in the Appendix. From eq 10, one gets
(19)
and 1
where h and v are the reduced plate height and reduced velocity given by
d(z/L)
+ (a”- X B ) ( P - Po)] (23)
It is more convenient for the simplicity of the following calculations to equate (a” - xB)with a in this equation, which amounts to neglect axAin front of (01” - xB),a very reasonable approximation. Then, eq 23 can be integrated as
which gives [ l + a(P - Po)]= exp[aAPo(l - 2/15)]
(25)
where AP, is the column pressure which would be obtained if the solvent B were incompressible,of constant viscosity q ~ ~ , and flowing in the column at the velocity U B ~
ANALYTICAL CHEMISTRY, VOL. 55, NO. 14, DECEMBER 1983
2305
Table 11. Range of Values of the Coefficient P for Columns Used in LC P,a range of ~ , b dp,w L, cm bar bar 11 2.7-18.9 3 3 37 8.9-63.0 3 10 4 0.96-6.8 5 5 8 1.9-13.6 5 10 16 3.8-27.2 5 20 40 9.6-68.0 5 50 1 0.24-1.7 10 10 3 0.72-5.1 30 10 5 1.2-8.5 10 50 cgsu; 12, = 1x lo+. a D A B , ~ B=, 1 x between 0.24 X lom7and 1.7 X lo-' cgsu.
DAB,VB,
I
I
10
With the help of eq 25 and 26, the integration of eq 18 to 20 becomes straightforward and gives for hap,
1
3 A ~ ~ ~ / ~ [ e x p ( a A P-, /131)+ Cvo[exp(aaPo)- 11 (27)
I
I
I
20
I
30
I
I
I
40
% Figure 1. Variations of the apparent reduced plate height with the reduced velocity at the column outlet pressure as a function of the a0 value. Local plate height parameters are as follows: A = 1, 6 = 1.8, and C = 0.05. The a@ values are indicated on each curve.
by a serial development of the exponentials limited to the first three terms, which gives
It is convenient to express h, in terms of the outlet reduced velocity, yo, since the plate height curves are generally reported in terms of hap,vs. v,-,. vo is given by yo
= UB~~P/DAB~
(28)
or
and is related to Do through eq 26 by elimination of U B , or P
o
= Pvo
(29)
with
P=
DABor]BoL
- DABoqBoL
(30)
kOdd
kdP
where ko is the dimensionless permeability coefficient, equal to k/d,2. P represents the limiting value, at low pressures, of the pressure drop leading to a unit increase of the reduced velocity. The range of values of the product D ~ Vhas B been shown to be between 0.5 and 1.7 X lo-' (in cgs units) in normal phase chromatography and between 0.24 and 1.3 X lo-' in reversed-phase chromatography (22). Values of @ computed for different values of the product DABqB are given in Table I1 for a number of typical columns. In practice, @ will be between 1 and 10 bars, being large for columns packed with small particles, especially for long columns. The combination of eq 27 and 29 gives
- exp(-apvo)] 3AvO1I3[eip( : u o )
- l]
+
I
+ Cvo[exp(apuo) - 11
(31)
This equation relates the experimentally measured apparent (average) reduced plate height to the outlet reduced velocity calculated by using eq 28 where D A B o is the outlet, usually atmospheric, mutual diffusion coefficient of the solute-solvent system generally determined from eq 3. This relationship between hap,and vo depends on one parameter which is the product of the two constants a and ,8. Typical values of a and ,8 are, re&ectively, 10-3/bar and 10 bars, which corresponds to DAB,= 10" cm2/s, VB, = lo-' P,L = 12.5 cm, KO = and d, = 5 pm, so the product a,8 is typically Since the chromatographic columns are, most often, operated in the reduced velocity range from 3 to 30, eq 31 can be simplified
It appears that in the reduced velocity range noted above, where the first term in eq 32 and 33 is negligible, the apparent reduced plate height will be higher than the expected value obtained for replacing u by uo in the classical Knox equation (8)which is the limiting equation obtained when a0 = 0 in eq 31. The gap between the two values will increase with increasing reduced velocities. For instance, for typical values of A and C , 1.0 and 0.05, respectively, when CUP= 2 X lom2, this gap reaches 0.9%, 3.9%, 6.9%, and 10.8% for uo values of 5,10,15, and 20, respectively. The apparent reduced plate height curves are shown in Figure 1for a given set of parameters, A, B , and C, and different values of the product ap. Accordingly, the plate height may have an upward curvature at large velocities, while at velocities around the optimum one, the effect is rather small or even negligible. This is due to the relatively small pressure drop in these conditions, in addition to the fact there is a partial compensation of the pressure effect on the diffusion coefficient between the B term, on one hand, and the A and C terms on the other hand. Indeed, if one was working at velocities well below the optimum one, the apparent plate height will be slightly smaller than the one at the column outlet, in agreement with eq 31-33, since, in this case, the B term, which reflects the contribution of axial molecular diffusion, is predominant. However, one should never work at velocities smaller than the optimum one, for which one has long analysis times as well as low plate numbers. Eventually, for a certain velocity, this compensation will be complete. A consequence of the dependence of hap,on AP, is that the total plate number is not conserved when coupling two or several chromatographic columns together at constant v,,, independently of losses associated with the design of the
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ANALYTICAL CHEMISTRY, VOL. 55, NO. 14, DECEMBER 1983
fittings and, possibly, with the disappearance of the infinite diameter effect. This is because the pressure drop AP, and consequently /3 increase, during the coupling, approximately by a factor equal to the number of coupled columns. For example, with the typical values given above for A, B, and C, if the value of a/3for the assembly of columns is 8 X and vo = 10, the plate number loss reaches 8.7%, 13.0%, or 14.3% when, respectively, 2,5, or 10 columns, supposed identical, are coupled (cf. eq 31). The loss for 10 identical columns in the same conditions is only 4.4% a t vo = 5, however. It should be emphasized, however, that this applies only to the design and construction of long columns operated with large pressure drops. As a is of the order of 1to 3 X (cf. Table I, third column) a value of a/3 of 8 X is achieved with a column of, at least, 50 cm when 5-pm particles are used and several meters with 10-pm particles. Most experiments made so far involve the combination of too few columns to permit the observation of this effect, which can be obscured by the influence of coupling devices. Furthermore, the plate number loss decreases markedly with decreasing velocity. In the case of very long combinations, relatively low reduced velocities have to be used for pressure drop considerations. The effect could become significant with 3-pm particles, however.
DETERMINATION OF THE MASS-TRANSFER COEFFICIENTS A AND C In liquid chromatography, it has been found useful to determine the A and C coefficientsof eq 8 for a better evaluation of the column performance. Small values of these Coefficients are desirable. The value of coefficient A gives an indication on how well the column is packed while the coefficient C is related to the kinetics of mass transfer in the stationary phase. The determination of A and C is usually done by least mean squares regression methods. Generally, the B coefficient, which influences h mostly at low reduced velocities, is not determined by such methods since plate height measurements are essentially carried out a t velocities larger than the one corresponding to the minimum of the plate height curve and B is supposed equal to 1.8, for packed columns (22). Since the apparent plate height differs from the local plate height at the outlet of the column because of the pressure dependence of the reduced velocity, essentially due to the pressure dependence of the diffusion coefficient, it is likely that the apparent parameters A’ and C’ determined from regression analysis of the apparent plate height curve, happvs. vo, according to eq 8 will differ from the true local plate height parameters A and C, respectively. If we assume that the local parameters A and C are equal to 1.00 and 0.05, respectively, then the apparent coefficients A’and C’determined by least mean squares regression from the observed plate height curve, happvs. vo, according the following equation hap, = (B/v,) Arvo1/3 C’vo (34)
+
+
differ from the true parameters A and C, respectively, by 3.5% and 27.3% if a@ equals 0.01 and 7.5% and 57.5% if a@ equals 0.02. In this example the regression is made with YO ranging from 2 to 20, the interval between each point being 1. These calculations are made with a computer; hap, is calculated for different values of v0 from eq 31 using given values of B , A, C , and cup; A‘ and C‘ are then determined from eq 34. It is noticeable that C’is always larger than C and A’always smaller than A. These errors on A and C are influenced by the measurement conditions, that is, range of vo and interval between successive values of v0. This latter factor is rather insignificant. Indeed, when changing the vo interval from 0.5 to 2 in the range 2-20, the error made on A changes from 3.53 to 3.51% if a@ = 0.01
Table 111. Influence of the Range of Variations ofy o on the Determination of A ’ and C’a % re1 crp
uo
0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.05 0.05 0.05
A‘
error on A
C’
% re1 error on C
0.96 5 0.973 0.980 0.925 0.945 0.960 0.908 0.8 7.9 0.844 0.772 0.842 0.892
3.5 2.7 2.0 7.5 5.5 4.0 .9.0 12.1 15.6 22.8 15.8 10.8
0.0636 0.0615 0.0593 0.0787 0.0738 0.0692 0.0813 0.0861 0.0912 0.135 0.117 0.102
27.3 22.9 18.7 57.5 47.6 38.3 62.5 72.1 82.4 169 134 103
range
2-20 2-15 2-10 2-20 2-15 2-10 5- 20 10-20 15-20 2- 20 2-15 2-10
a B = 1.8; A increment, 1.
= 1;C =
0.05; v o range, 2-20;
vo
-
Table IV. Influence of the Parameter ap on the Determination of A‘ and C’a aP
A’
error on A
C‘
% re1 error on C
0.01 0.02 0.05
0.965 0.925 0.772
3.5 7.5 22.8
0.0636 0.0787 0.135
27.3 57.5 169
% re1
a B = 1.8; A increment, 1.
= 1;C =
0.05; v o range, 2-20; v o
and from 7.50 to 7.48% if a0 = 0.02, while the error on C changes from 27.26 to 27.38% if a/3 = 0.01 and from 57.38% to 57.70% if (UP = 0.02. However, the range of variations of vo may significantly influence the relative difference between A and A’and C and C’, respectively. In Table 111, the values of A’and C’as well as the relative differences, (A - A?/A and (C’- C)/C, are indicated as a function of the vo range and the value of a@ for typical values of B, A, and C (B = 1.8, A = 1, C = 0.05) with an increment between successive values of vo equal to 1, as in the following. As might be expected from Figure 1, the effect of the vo range is quite significant. Indeed, when measurements (or calculations of h,,,) are done at increasing large values of vo, the difference between happ and h(vo) is increasingly large; therefore C’, which is mostly determining the shape of the plate height curve at large values of vo, appears increasingly larger than C. That A’ is smaller than A, and increasingly smaller than A when vo increases, may be explained by the fact that, as seen in Figure 1, the apparent curve, happ vs. vo becomes less concave downward at intermediate values of vo than the h vs. vo curve, this downward concavity being due to the A and A’terms. From Table 111, it can be concluded that the errors on A and C decrease when the upper limit of the vo range decreases and increase when the lower limit of the vo range increases. Therefore, determination of A‘and C’ should be made in a small v0 range near the optimum reduced velocity, but in this cme, the level of precision in the determination of C’ becomes poor. One may expect that the differences between A ’and A and C’and C will increase with increasing values of a@since this parameter is responsible for the deviation of hap, from h. Table IV shows that these relative differences increase slightly faster with ap than a@. While the error on A is relatively moderate, the one on C is important and can exceed 100%. It is interesting to see how A’and C’differ from A and C, respectively, when A or C are varied. These relative differences are indicated in Table V for several values of A ranging from 0.7 to 2. When A increases, at constant C, the absolute
ANALYTICAL CHEMISTRY, VOL. 55, NO. 14, DECEMBER 1983
Table V. Influence of A on the Determination of A’ and C’ a % re1 % re1 A A’ error on A C’ error on C 0.7 1 1.5 2
0.630 0.925 1.416 1.907
9.9 7.5 5.6 4.6
0.0753 0.0787 0.0845 0.0903
50.6 57.5 69.1 80.6
a B = 1.8;C = 0.05;ap = 0.02;uo range, 2-20;vo increment, 1.
Table VI. Influence of C on the Determination of A’ and C’ % re1 % re1 C A’ error on A C‘ error on,C 0.01 0.025 0.05 0.1 0.25 0.5
0.960 0.947 0.925 0.881 0.750 0.532
4.0 5.3 7.5 11.9 25.0 46.8
0.0259 0.0456 0.0787 0.145 0.343 0.674
159 82.8 57.5 44.8 37.2 34.7
a B = 1.8;A = 1;ap = 0.02;v o range, 2-20;uo increment, 1.
difference between A’ and A increases while the relative difference (error) decreases. In the same time the difference between C and C’and the relative bias increases. In Table VI are shown the relative differences for several values of C ranging from 0.01 to 0.5 a t constant A equal to 1. As C increases, A’decreases, so the relative error on A increases, while the relative difference between C and C’decreases. C’ appears to overestimate the true value of C by as much as 100% in some cases, especially when C is small. Therefore it will be difficult to determine accurately very small values of c.
DISCUSSION Tables 111-VI show that the plate height parameters A’and C’ determined by least mean squares regression of apparent data according to the Knox equation (34)may differ significantly from the true local parameters A and C, especially for C’, because of the influence of pressure on diffusion coefficients. Since A‘ is always smaller than A, which measures the efficiency of the packing technique (23),the column will appear better packed than it really is. Conversely, as C’is always larger than C, which is related to the kinetics of mass transfer in the stationary phase, this stationary phase will appear worse than it actually is. Moreover, these appreciations of the stationary phase and packing qualities depend on the conditions of measurement of the plate height curve (range of uo, value of ab). It is not easy to give experimental evidence of the effect of the pressure dependence of diffusion coefficient on plate height. One could, however, make plate height rneasurementa with given column, solute, solvent, and temperature by changing the outlet pressure from atmospheric to elevated pressure, Pa,while keeping the flow rate constant. The difference in reduced plate heights between two measurements at different outlet pressures may then be approximated by using eq 3, 28, 29,30,and 32,when the A and B terms are small compared to the C plate height term, as
where Pi is the inlet pressure when the outlet pressure is Pa. This gap can be maximized when working at the highest possible inlet pressure with PJPi = 0.5, using a solvent giving
2307
a maximum ( Y / ( + B M B ) O . ~value at room temperature with solute and stationary phase leading to a large VA0% value. The column configuration should maximize the d:/L ratio but, in the same time, the injection time and detection response time contributions to the plate height should be negligible. Besides, the precision on the plate height should be sufficiently large so that the experimental fluctuations in the determination of h should be much smaller than the difference Ah. It is obvious that some of these requirements are contradictory. For example, high values of the inlet pressure and of the ratio d:/L will lead to very fast analyses, but in the same time, the external contributions to the plate height are likely to be significant. Furthermore, it is now well-known that in a chromatographic column, radial and longitudinal temperature gradients are developed, brought about by viscous heat dissipation (24-30). These gradients undoubtly have an effect on the plate height as they influence the diffusion coefficient and the capacity factor of the solutes. Changes in column configuration induce changes in the magnitude of this effect. However, it is not yet clear whether the temperature effect is larger than the effect of the pressure-induced variation of the diffusion coefficient discussed above. Anyway, both effects have their origin in the longitudinal pressure gradient existing in any chromatographic column. Accordingly, we feel, under the light of the results discussed in the previous section, that the effect of the temperature gradients cannot be discussed separately from those of the pressure gradient. It is clear, for example, that the upward curvature of the plate height curves in Figure 6 of ref 28 plotted at large reduced velocities ( u = 20-70) can be explained at least partly by the pressure induced variation of the diffusion coefficient, in agreement with Figure 1 of the present paper. The pressure effect on prate height via the diffusion coefficient results from the fact that the reduced velocity decreases from the inlet to the outlet of the column, so that the mean reduced velocity is larger than its value at column outlet, leading to a larger plate height than expected. If now, the thermal effect is present, the range of variation of the reduced velocity from the inlet to the outlet of the column will be larger than without this effect, since the viscous heat dissipation results in another source of progressive increase in the diffusion coefficient. However, the mean reduced velocity may be somewhat lower than in the absence of the thermal effect, and, consequently, the plate height may also be lower, as recently indicated by Katz et al. in a study of pressure effects in LC (31). However, the radial temperature gradient constitutes the essential contribution of the thermal effect to the plate height. It cannot be compensated by the longitudinal pressure gradient. It is expected to be larger for thermostated than for unthermostated columns, for which the rate of heat exchange at the column wall is lower (32). This is in agreement with the results of Katz et al. who found a twice larger effect for thermostated columns. This may indicate that, in their experimental conditions, the radial temperature effect is larger than the combined effect of axial pressure and thermal gradients. As the temperature effects decrease rapidly with decreasing column diameter, the pressure effect described in the present study will be predominant with narrow bore columns. Because of the significant influence on the plate height of, a t least, the diffusion coefficient effect, which may be reinforced, in some instances, by the temperature effect, it is doubtful that experimental plate height measurements will decisively prove the correctness of a theoretical plate height equation over another slightly different one. This is true, particularly, for the value of the exponent of the “eddy”
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ANALYTICAL CHEMISTRY, VOL. 55, NO. 14, DECEMBER 1983
dispersion coupling term, which has been stated to be equal to 1 (I), (33),or (34). For this reason, we feel that a simple semiempirical equation, like the Knox equation (8), is best appropriate for primary description of the velocity effect on plate height.
ACKNOWLEDGMENT Discussions with Henri Colin about the experimental evidence of the pressure-induced variation of the diffusion coefficients and assistance of Alain Jaulmes for the computer calculations are greatly acknowledged. APPENDIX The apparent plate height, Happ,is given in terms of the plate number measured by the retention time of the peak, tR, and its time variance, ut:, a t the outlet of the column by
The retention time is given by
where R is the retention factor of the solute, that is, its migration velocity in the column relative to the one of the solvent. The time standard deviation, of the peak uto,at the column outlet, is related to the standard deviation in distance units along the column uz at z = L, assuming that the solvent velocity is constant all along the column, that is, neglecting its very small variations due to the solvent compressibility (3), by
(-43) a,,, in turn, is obtained from the basic definition of the local plate height, H, by (I)
Combining eq A1 to A4, one obtains
which gives, using the definition of reduced plate height (eq
9)
local reduced plate height (eq 8) contribution of the term to the apparent plate height (eq 17) contribution of the l / v o term to the apparent plate height (eq 17) contribution of the vo term to the apparent plate height (eq 17) apparent reduced plate height local plate height apparent plate height column permeability (eq 21 and following equations) or Boltzmann constant (eq l ) dimensionless column permeability coefficient column length solvent molecular weight (eq 3) apparent plate number (eq A l ) pressure column inlet pressure (eq 35) reference or column outlet pressure radius of the spherical solute (eq 1) solute retention factor (eq A2) retention time (eq Al) temperature mobile phase velocity (eq 10) mobile phase velocity at column outlet pressure (eq 22) solute molar volume solvent molar volume distance from the column inlet coefficient of relative variation of the reduced velocity with pressure (eq 15) coefficient of relative variation of the reciprocal of the diffusion coefficient with pressure (eq 7) coefficient of relative variation of the viscosity with pressure (eq 5) limiting value, at low pressure, of the pressure drop leading to a unit increase of the reduced velocity (eq 30) difference between the apparent reduced plate height and the reduced plate height at column outlet (eq 35) limiting column pressure drop given by equation 26 eluent viscosity eluent viscosity at pressure Po reduced velocity reduced velocity at pressure Po time standard deviation at column outlet standard deviation in distance unit at column outlet dimensionless association factor (eq 3) solute compressibility solvent compressibility
LITERATURE CITED GLOSSARY a
A A’
B
c C’ d $AB
DAB^
exponent of the molar volume in the diffusion coefficient expression (eq 2) coefficient of the v1t3 term in the local plate height equation (eq 8) coefficient of the v ~ term ~ in / the ~ plate height equation derived by least-squares fit of the experimental data (eq 34) coefficient of the 1/v term in the local plate height equation (eq 8) coefficient of the v term in the local plate height equation (eq 8) coefficient of the vo term in the plate height equation, derived by least-squares fit of the experimental data (eq 34) average particle diameter mutual diffusion coefficient of solute A in solvent B mutual diffusion coefficient of solute A in solvent B a t reference pressure Po
Giddings, J. C. “Dynamics of Chromatography. Part I. Principles and Theory”; Marcel Dekker: New York, 1965; Chapter 2. Snyder, L. R.; Klrkland, J. J. “Introduction to Modern Liquid Chromatography”, 2nd ed.; Wliey-Interscience: New York, 1979; Chapter 2. Martin, M.; Biu, G.; Gulochon, G. J. Chromatogr. Sci. 1973, 1 1 , 641. Glddings, J. C. Sep. Scl. 1966, 1 , 73. Bidiingmeyer, B. A.; Hooker, R. P.; Lochmuiler, C. H.; Rogers, L. B. Sep. Sci. 1989, 4 , 439. Bidlingmeyer, B. A.; Rogers, L. B. Sep. Scl. 1972, 7 , 131. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. “Transport Phenomena”; Wiley: New York, 1960; p 515. Brldaman, P. W. “The Physics of High Pressure”; G. Bell and Sons, Ltd.: London, 1931. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. “The Properties of Gases and Liquids”, 3rd ed.; McGraw-Hill: New York, 1977; Chapter 11. Tvrrell. H. J. V. I n “Diffusion Processes”: Sherwood, J. N., et ai. Eds.; Gbrdon and Breach Science Publishers: London, 1971; Vol. 1, Chapter 2.1. Wilke, C. R.; Chang, P. AIChE J. 1955, 1 , 264. Jonas, J.; Hasha, D.; Huang, S. G. J. Phys. Chem. 1980, 8 4 , 109. McCool, M. A.; Woolf, L. A. High Temp.-High Pressures 1972, 4 . 65. Jonas, J.; Akai, J. A. J. Chem, fhys. 1977, 6 6 , 4946. Fury, M.; Munie, G.; Jonas, J. J . Chem. fhys. 1979, 70, 1260. Jonas, J.; Hasha, D.; Huang, S, G. J. Chem. fhys. 1979, 7 1 , 3996. Koeller, R. C.; Drlckamer, H. G. J. Chem. fhys. 1853, 21, 575. Le Nelndre, B. Rev. fhys. Chem. Jpn. 1981, 5 0 , 36. Atwood, J. G.; Goldstein, J. J. fhys. Chem., in press.
Anal. Chem. 1983, 55,2309-2312 (20) Done, J. N.; Kennedy, G. J.; Knox, J. H. “Gas Chromatography 1972”; Perry, S. G., Ed.; Applied Science Publishers Ltd: London, 1973; p 145. (21) Guiochon, G. J. Chromatogr. 1979, 785, 3. (22) Kennedy, G. J.; Knox, J. H. J. Chromatogr. Scl. 1972, 70, 549. (23) Martin, M.; Guiochon, G. Chromatographla 1977, 70, 194. (24) Glddlngs. J. C. “Dynamics of Chromatography. Part I . Principles and Theory”; Marcel Dekker: New York, 1965; Chapter 5. (25) Asshauer, J.; Halasz, I.J. Chromatogr. Scl. 1974, 72, 139. (26) Martin, M.; Eon. C.; Guiochon, G. J. Chromatogr. 1974, 99, 357. (27) Martin, M. Th6se de Doctorat d’Etat, UniversLB Paris VI, 1975, Chapter 2. (28) Lln. H.J.; Horvath, C. Chem. Eng. Sci. 1981, 36, 47.
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(29) Poppe, H.; Kraak, J. C.; Huber, J. F. K.; Van den Berg, J. H. M. Chromatographla 1981, 74, 515. (30) Abbott, S.; Achener, P.; Slmpson, R.; Klink, F. J. Chromatogr. 1981, 278, 123. (31) Katz, E.; Ogan, K.; Scott, R. P. W. J . Chromatogr. 1983, 260, 277. (32) Poppe, H.; Kraak, J. C. Lecture presented at the VIIth International Symposium on Column LC, Baden-Baden, May 1983. (33) Huber, J. F. K. Ber. Bunsenges. Phys. Chem. 1973, 77, 179. (34) Horvath, C.; Lin, H.-Y. J. Chromatogr. 1976, 726, 401.
RECEIVED for review May 23,1983. Accepted August 18,1983.
Coulostatic Pulse Amperometry for Liquid Chromatography/Electrochemistry Detection Anthony C. Barnes’ and Timothy A. Nieman*
School of Chemical Sciences, University of Illinois, 1209 West California Street, Urbana, Illinois 61801
Small charge pulses are repetltively injected Into the cell to maintain the the worklng electrode at the deslred potential. Because the method Is not interfered with by charging current, the potential can be changed rapidly; 10-point voltammograms are run in 1-2 e. Caffeic acid and o-chlorophenol are quantitated at fixed potential In a flow Injection system over the range of 1-100 pM with 4 % preclslon. Hydrodynamic voltammograms were run for caffeic acid solutions as dilute as 0.5 pM. For a llquid chromatographic separation of 0 - , m-, and p-amlnophenol, the detection llmlt Is 40 pmol Injected; voltammograms can be run rapidly enough to obtain E,,2 values for all eluting substances from a single Injectlon.
In recent years liquid chromatography with electrochemical detection (LCEC) has become an established technique for the quantitation of electroactive species. General articles have covered the basics of the technique (1-4) and the large volume of recent literature is well covered in recent reviews (5,6) and a bibliography (7).LCEC has proven exceptional sensitivity and ease of use but has as a limitation, the general ineffectiveness of conventional LCEC techniques in applying potential scanning methods for qualitative characterization of eluting species within the time constraint of a chromatographic peak. This limitation arises from the large background and charging currents at solid electrodes and problems due to large and/or variable uncompensated resistances in flow cell configurations. Thus, LCEC detectors usually involve dc amperometry or coulometry at f i e d potential. To achieve greater selectivity, investigators have also used normal pulse (8, 9), differential pulse (8-11),ac (11,12),and square wave (13) techniques all at fixed potential. Some work has been done with square wave voltammetry as a scanning LCEC detector at a static mercury drop, and scan rates of 500 mV in 2 s were obtained (15). Because coulostatic methods are not interfered with by iR drop or charging current (16,17),a coulostatic LCEC detector should offer advantages over existing LCEC detectors. The coulostatic technique may be used to mimic controlled potential amperometry while maintaining the above mentioned ‘Present
MI 49001.
address: U p j o h n Co., 7000 S. Portage Rd., Kalamazoo,
coulostatic characteristics. For this method we propose the name coulostatic pulse amperometry (CPA). The CPA method should provide the ability to perform potential scanning on a time scale short enough to be compatible with the time restrictions of eluting LC peaks and thus obtain qualitative information (hydrodynamic Ellz values) concerning eluting species. Furthermore, CPA should allow electrochemical detection for LC systems in which the mobile phase is of low conductivity. In earlier reports we have demonstrated use of coulostatics as a rapid scanning voltammetric technique (18) and preliminary results for application of coulostatic detection in flowing streams (19)and liquid chromatography (20). This paper details the development and application of an LCEC detector using coulostatic pulse amperometry. THEORY The development of computer-controlled electrochemical instrumentation has prompted a rebirth of interest in coulostatic analysis (18,21,22). In coulostatics, a charge pulse of known coulombic content is injected into a cell on a time scale much more rapid than the time scale for electron transfer between the electrode and electroactive species in solution. Therefore, essentially all of the injected charge goes toward charging the double layer capacitance with a resulting change in the potential of the working electrode. After pulse application, external circuit connections to the working electrode are broken. The only mechanism for charge to leave the double layer capacitance is then via faradaic reactions at the electrode-solution interface. The higher the concentration of electroactive species, the faster charge leaks off the double layer capacitance and the faster the electrode potential decays back toward its equilibrium value. Coulostatic techniques are unique in that the stimulation of the cell (via the charge pulse) and the measurement of electrode potential are separated in time. Because there is no current passing through the cell between successive charge injections, there is no iR drop in solution. Also, because double layer charging occurs only during the time of the pulse application, there is no interference due ‘to charging current during the time between successive pulses. In single-potential coulostatic pulse amperometry (CPA), small charge pulses are repetitively applied to the cell to maintain the working electrode at the desired potential within an acceptably narrow tolerance. Every time the electrode potential decays outside this tolerance range, a small current
0003-2700/83/0355-2309$01.50/00 1983 American Chemical Society
