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Equation for evolution of temporal width of a solute band migrating in chromatographic column
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Equation for evolution of temporal width of a solute band migrating in chromatographic column Jan Leppert a,∗, Leonid M. Blumberg b, Peter Boeker a,c a
Institute of Agricultural Engineering, University of Bonn, Nussallee 5, D-53115 Bonn, Germany Advachrom, P.O. Box 1243, Wilmington, DE 19801, USA c HyperChrom SA, 21 Avenue Monterey, L-2163 Luxembourg, Luxembourg b
a r t i c l e
i n f o
Article history: Received 2 August 2019 Revised 25 September 2019 Accepted 20 October 2019 Available online xxx Keywords: Peak width Temporal variance of a solute band Residency General elution equation
a b s t r a c t An alternative differential equation to model the development of the temporal width of a solute band during its migration in a chromatographic column is developed. This model uses the solute residency, the inverse of the solute velocity, as parameter, which has the advantage to be an always finite quantity even at the column outlet in GC–MS where carrier gas velocity approaches infinity. The new differential equation is derived from a known equation describing the evolution of the spatial variance of migrating band. Supplemental material illustrating solutions of the newly developed equation in several special cases is provided. © 2019 Elsevier B.V. All rights reserved.
1. Introduction
plate height for the solute located at z. In smooth medium, σ z can be found from differential equation [6]:
Several parameters can be used for describing the widths of the peaks in chromatograms and the widths of the solute bands within columns. Among them are [1] peak/band width at half height, at base, variance, standard deviation (square root of variance). However, only the latter two can be theoretically predicted from operational conditions for non-Gaussian peaks. Only these two parameters are used throughout this report making its results applicable to any peak/band shapes. The medium of chromatographic analysis (the column, the mobile phase, their conditions) can be uniform (the same along the column at any given time), or non-uniform otherwise [2,3]. The analysis itself is static (like isothermal GC, isocratic LC, etc.) if its conditions do not change with time, or dynamic otherwise (temperature-programmed GC, gradient LC, etc.) [4,5]. The degree of non-uniformity of the medium affects the complexity of the band width equations [6]. In the majority of practical applications, the medium is smooth [6] (within the solute band, the plate height is nearly constant and the solute velocity is nearly linear function of distance). Let x be an arbitrary longitudinal distance from the column inlet, z be that distance to a solute center of mass, σ z = σ z (z) and u be, respectively, spatial standard deviation (in units of length) and velocity of a solute located at z, and H == H(z) be the column
dσz2 2σz2 =H+ dz u
∗
Corresponding author. E-mail address: [email protected] (J. Leppert).
∂ u ∂ x x=z
(1)
All quantities in Eq. (1), including the (spatial) gradient, ∂ u/∂ x, of u are measured at the time t = t (z ) of the solute migration from the inlet to z. Eq. (1) can be used to study evolution of the width of a solute band during its migration within a column [7], effects of non-uniform conditions on column performance [8,9], etc. Eq. (1) can also be used for theoretical [8,9] and/or numerical [10] prediction of peak widths in chromatograms. Let σ L and uo be the solute spatial standard deviation and velocity at the column outlet. Quantity:
σ=
σL uo
(2)
is typically and in this report interpreted as the peak temporal standard deviation (in time units)1 [11]. Eq. (2) has a shortcoming – it has a singularity in GC–MS where the column outlet is at vacuum, i.e. po = 0. The product, pu, is the same at any location along a column. Therefore, ideally, when the outlet pressure (po ) approaches zero, uo in Eq. (2) approach infinity. Similarly, infinite gas decompression at the column outlet 1 Strictly speaking, quantity σ defined in Eq. (2) is different from true (temporal) standard deviation of a peak. However, relative difference, on the order of 1/N (N is plate number) [7], is practically insignificant and typically ignored. The same approach is used throughout this report.
https://doi.org/10.1016/j.chroma.2019.460645 0021-9673/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: J. Leppert, L.M. Blumberg and P. Boeker, Equation for evolution of temporal width of a solute band migrating in chromatographic column, Journal of Chromatography A, https://doi.org/10.1016/j.chroma.2019.460645
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causes infinite band expansion so that σ L becomes infinity. The singularity due to infinity in the numerator (σ L ) and denominator (uo ) in Eq. (2) causes computational problems in numerical evaluation of GC–MS chromatograms in spite of the fact that the ratio σ L /uo in Eq. (2) (the standard deviation, σ , of a peak in chromatogram) is finite. The latter fact suggests that the singularity in GC–MS at po = 0 can be avoided if quantity
σt =
σz u
(3)
is considered instead of σ z . Quantity σ t represents the time that it takes for spatial standard deviation (σ z ) of a solute band to travel through coordinate z. Similarly to σ in Eq. (2), quantity σ t can be interpreted as the temporal standard deviation of the band. It has been shown elsewhere [8] that transformation of Eq. (1) for σ z into equation for σ t yields under static conditions:
dσt2 = Hr 2 , (static conditions ) dz
(4)
where
r=
1 u
also be viewed as a known one. If rm and k are known functions, rm = rm (x,y) and k == k(x,y), of x and y then a solute retention time (tR ) can be found from general elution equation [7,17]: tR
∫ 0
L dt = ∫ rm (z )dz 1 + k(t ) 0
where k(t ) = k(z(t ), t ), rm (z ) = rm (z, t (z ) ). Generally, only the functions rm (x,y) and k = k(x,y) could be known for a particular solute, but not functions z = z(t) and t = t(z). Finding the latter two functions can be a part of solving Eq. (8). Moreover, functions z = z(t) and/or t = t(z) can be complex even if they are known a priori. All this means that, generally, Eq. (8) can only be solved numerically by iterations starting from z = 0 and t = 0 although, there are operational conditions leading to substantial simplifications of Eq. (8) and to its analytical solutions [7,17]. All variables in Eq. (8) are finite and, therefore, it is suitable for numerical retention time evaluations. 2.2. Peak width
(5) A more detailed form of Eq. (1) can be expressed as:
It has been pointed out [8] that the Giddings compressibility factor [11,12] follows directly from Eq. (4) (see Appendix A for details). Eq. (4) was also used [8,13] for evaluation of optimal operational conditions and optimal performance of a chromatographic column (GC and SFC) in non-uniform static conditions. Quantity r defined in Eq. (5) can be called as the residency of a solute (or, more generally, of a traveling object) [13] – the larger is an objects residency, (r), at some coordinate z the longer it resides in vicinity of z. The importance of quantity (1/u) in studies of properties of solute migration in chromatography was recognized in several publications [3,13–15]. The purpose of this report is to extend Eq. (4) to dynamic (uniform and non-uniform) conditions. 2. Theory
dσz2 (z ) 2σz2 (z ) = H (z ) + dz u (z )
(6)
where
1 um
(7)
can be interpreted as the mobile phase residency. Let, in addition to the earlier introduced ones, variable y be an arbitrary time and t == t(z) be the time of a solute migration to location z. Both y and t are measured since the sample introductions. Quantity k is a never negative (k ≥ 0). If the mobile phase flows in one direction and never stops (um > 0) then z(t) is a monotonically increasing function and its inversion, t(z), always exists and is a monotonically increasing function as well. This implies that, if one function z(t) or t(z) is known then the other one can 2
In packed column LC, um and k are typically denoted as u0 and k , respectively.
(9)
Eq. (9) yields after differentiation and further transformations:
σt2 (z ) ∂ 1 (10) 2 r (z ) ∂ x r (x, y ) x = z y = t (z ) 2 2 2 2σt dr (z ) 2σt (z ) ∂ r (x, y ) 1 dσt (z ) − 3 = H (z ) − 3 ∂x x = z r 2 (z ) dz r (z ) dz r (z ) y = t (z ) d dz
σt2 (z ) r 2 (z )
= H ( z ) + 2r ( z )
⎛
To predict a chromatogram, one needs to predict retention time (tR ) and width of a peak corresponding to each solute. Following is a brief review of a known equation expressed in a form suitable for numerical retention time prediction. Transport velocity (u) [16,17] of a solute migrating in a chromatographic column can be expressed as [7,17] u = um /(1 + k ) where um is the mobile phase transport velocity [16] and k is the solute retention factor.2 Eq. (5) for the solute residency (r) can be expressed as:
r = (1 + k )rm
∂ u(x, y ) ∂x x = z y = t (z )
Eqs. (3) and (5) allow one to express σ z as σz (z ) = σt (z )/r (z ). Substitution of this relationship together with Eq. (5) in
2.1. Retention time
rm =
(8)
⎞
dσt2 (z ) 2σt2 ⎜ dr (z ) = H (z )r 2 (z ) + ⎝ dz − dz r (z )
∂ r (x, y ) ⎟ ∂x x = z ⎠ y = t (z )
(11) (12)
The last equation contains the difference between full and partial derivatives of r by z and x, respectively. The difference can be simplified recognizing that the full derivative can be expressed as [7]:
dr (z ) = dz
∂ r (x, y ) ∂ r (x, y ) + r (z ) ∂x ∂y
Eq. (12) becomes:
(13)
dσt2 (z ) ∂ r (x, y ) = H (z )r 2 (z ) + 2σt2 dz ∂y x = z y = t (z )
(14)
All variables in this equation (H, r and σ t ) are finite under all known and feasible operational conditions including conditions at the column outlet in GC-MS. Also finite under these conditions is the temporal rate, ∂ r/∂ y, of change in r. As a result, Eq. (14) is suitable for numerical integration in all known and feasible practical conditions. Equation similar to Eq. (14) is known from Lan and Jorgenson [15]. In Lan-Jorgenson equation (LJE), σ t is a true temporal standard deviation of a solute band while σ t in Eq. (14) is only a close approximation to that standard deviation (footnote 1 ). However, the LJE has its own disadvantages. Among them is a requirement
Please cite this article as: J. Leppert, L.M. Blumberg and P. Boeker, Equation for evolution of temporal width of a solute band migrating in chromatographic column, Journal of Chromatography A, https://doi.org/10.1016/j.chroma.2019.460645
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that the third temporal central moment of a solute band has negligible effect on σ t . This limits LJE to the cases where it is a priory known that temporal profiles of the solute bands are symmetric. No such requirement is necessary for validity of Eq. (14). Examples of analytical solutions for special cases of Eq. (14) are provided in the Appendix A. 3. Conclusions A Differential equation for evolution of temporal variance of a solute band migrating in a chromatographic column has been developed. Unlike previously known equations for spatial variance of the band, the newly developed equation is free of singularities and, therefore, is more suitable for numerical integration. It has been also demonstrated that a minor modification in known equations for retention time removes their possible singularities and makes them suitable for numerical integration.
3
with the Giddings compressibility factor jG
9( p2i + p2o )( pi + po )
2
jG =
(A.9)
2
8( p2i + p2o + pi po )
The Giddings compressibility factor ranges from 1 for weak decompression ( pi = po) to 1.125 for strong decompression ( po = 0). Due to Eq. (A.8), the plate number N = (tR /σ )2 [1] can be expressed as:
N=
tR2
σ2
=
L jG H
(A.10)
indicating that N can be up to 1.125 times lower than the ratio L/H. Appendix B. Constant solute migration velocity in a non-uniform dynamic velocity field with fixed H
This work was supported by the German Research Foundation (DFG) [grant number 290980359] and HyperChrom SA, Luxembourg.
This is a model of a velocity gradient, where the solutes migrate with their velocity equal to the velocity of advancement of the temperature gradient. As a result, each solute “experiences” during its migration a fixed temperature and, therefore, migrates with a fixed velocity. A velocity field with a positive gradient is:
Appendix A. Isothermal gas chromatography
u+ (x, y ) =
Acknowledgements
Under isothermal conditions the temperature of a column is constant over time and uniform over the length of the column. As a result ∂ r/dy in Eq. (14) becomes zero and we obtain Eq. (4).
dσ = Hr 2 , (static conditions ) dz 2 t
(A.1)
The solute residency r at point z is given by
r (z ) = rm,i (1 + k ) 1 −
p2 1 − 2o pi
z L
(A.2)
with the retention factor k, inlet mobile phase residency rm, i and pi the inlet and po outlet pressure. The solution for the temporal variance σ 2 at the end of the column is the integration of the right-hand side of eq. (A.1) for z over the length of the column L. L
σ 2 = H ∫ r 2 dz
(A.3)
0
σ =
2 HL(1 − k )rm,i
2
p2 1 + 2o pi
u− (x, y ) =
(A.4)
The inlet mobile phase residency rm, i can be expressed with m
pi tm = j po L
(A.5)
(A.6)
r+ (x, y ) =
(A.7)
Eqs. (A.5), (A.6) and (A.7) substituted into Eq. (A.4) results in
σ
2
H = jG tR2 L
y tR 1 + tR L 1 + xL
(B.3)
and for the negative solute velocity gradient the residency field is
r− (x, y ) =
tR 1 + xL L 1 + ty R
(B.4)
2σt,2± t2 dσ±2 = H R2 ± dz L+z L
(A.8)
(B.5)
with the ‘+’ sign for the positive and the ‘-‘ sign for the negative solute velocity gradient. The solution for an ideal injection, σ 2 (z = 0 ) = 0, in case of the positive solute velocity gradient is 2 +
t2 (z ) = H R2 L
z2 z+ L
(B.6)
At the end of the column, at z = L, Eq. (B.6) becomes
σ+2 (L ) = 2tR2
and the void time tm .
tR tm = 1+k
(B.2)
The solute velocity in the apex of the solute zone is constant, u+ (z, t (z ) ) = u− (z, t (z ) ) = L/tR . The spatial derivative of the positive gradient velocity u+ is positive and the spatial derivative of the negative gradient velocity u− is negative for 0 ≤ x ≤ L and 0 ≤ y ≤ tR . The corresponding residency field for the positive solute velocity gradient is
σ
with the James-Martin compressibility factor j
3 po ( pi + po ) j= 2( p2i + po pi + p2o )
y L 1 + tR tR 1 + xL
The differential equation (15) is for these gradients
the average mobile phase velocity um = L/t :
rm,i
(B.1)
A velocity field with a negative gradient is
The solution of this integral is 2
L 1 + xL tR 1 + ty R
H L
(B.7)
In the case of the negative solute velocity gradient the solution for ideal injection is
σ−2 (z ) = H
tR2 L2 z + Lz2 + L2 (L + z )2
z3 3
(B.8)
Please cite this article as: J. Leppert, L.M. Blumberg and P. Boeker, Equation for evolution of temporal width of a solute band migrating in chromatographic column, Journal of Chromatography A, https://doi.org/10.1016/j.chroma.2019.460645
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At the end of the column, at z= L, Eq. (B.8) becomes
σ−2 (L ) =
7 2H t 12 R L
(B.9)
The temporal variance in the case of the moving gradient with positive solute velocity gradient is 3.4 times larger than the temporal variance with a negative solute velocity gradient. The peak width is consequently 1.8 times bigger for the positive gradient compared to the peak width with the negative gradient. References [1] IUPAC, Nomenclature for chromatography, Pure Appl. Chem. 65 (1993) 819–872. [2] J.J. van Deemter, F.J. Zuiderweg, A. Klinkenberg, Longitudinal diffusion and resistance to mass transfer as causes of nonideality in chromatography, Chem. Eng. Sci. 5 (1956) 271–289. [3] J.C. Giddings, Plate height of nonuniform chromatographic columns. gas compression effects, coupled columns, and analogous systems, Anal. Chem. 35 (1963) 353–356, doi:10.1021/ac60196a026. [4] L.M. Blumberg, Limits of resolution and speed of analysis in linear chromatography with and without focusing, Chromatographia 39 (1994) 719–728, doi:10.1007/BF02274589. [5] L.M. Blumberg, Erratum. limits of resolution and speed of analysis in linear chromatography with and without focusing, Chromatographia 40 (1995) 218, doi:10.1007/BF02272175. [6] L.M. Blumberg, Variance of a zone migrating in a linear medium II. timevarying non-uniform medium, J. Chromatogr. 637 (1993) 119–128, doi:10.1016/ 0021- 9673(93)83204- 6.
[7] L.M. Blumberg, Temperature-Programmed Gas Chromatography, Wiley-VCH, Weinheim, 2010. [8] L.M. Blumberg, T.A. Berger, Variance of a zone migrating in a non-uniform time invariant linear medium, J. Chromatogr. 596 (1992) 1–13, doi:10.1016/ 0021- 9673(92)80197- 3. [9] L.M. Blumberg, Theory of gradient elution liquid chromatography with linear solvent strength: Part 2. peak width formation, Chromatographia 77 (2014) 189–197, doi:10.1007/s10337- 013- 2556- x. [10] F. Aldaeus, Y. Thewalim, A. Colmsjö, Prediction of retention times and peak widths in temperature-programmed gas chromatography using the finite element method, J. Chromatogr. A 1216 (2009) 134–139, doi:10.1016/j.chroma. 2008.11.038. [11] J.C. Giddings, S.L. Seager, L.R. Stucki, G.H. Stewart, Plate height in gas chromatography, Anal. Chem. 32 (1960) 867–870, doi:10.1021/ac60163a043. [12] G.H. Stewart, S.L. Seager, J.C. Giddings, Influence of pressure gradients on resolution in gas chromatography, Anal. Chem. 31 (1959) 1738, doi:10.1021/ ac60154a001. [13] L.M. Blumberg, Erosion of efficiency in non-uniform linear chromatography, J. High. Resolut. Chromatogr. 16 (1993) 31–38, doi:10.1002/jhrc.1240160106. [14] P.D. Schettler, M. Eikelberger, J.C. Giddings, Nonideal gas corrections for retention and plate height in gas chromatography, Anal. Chem. 39 (1967) 146–157, doi:10.1021/ac60246a004. [15] K. Lan, J.W. Jorgenson, Theoretical investigation of the spatial progression of temporal statistical moments in linear chromatography, Anal. Chem. 72 (20 0 0) 1555–1563, doi:10.1021/ac990533u. [16] L.M. Blumberg, G. Desmet, Kinetic performance factor – a measurable metric of separation-time-pressure tradeoff in liquid and gas chromatography, J. Chromatogr. A 1567 (2018) 26–36, doi:10.1016/j.chroma.2018.06.048. [17] L.M. Blumberg, Migration and elution equations in gradient liquid chromatography, J. Chromatogr. A 1599 (2019) 35–45, doi:10.1016/j.chroma.2019.03.057.
Please cite this article as: J. Leppert, L.M. Blumberg and P. Boeker, Equation for evolution of temporal width of a solute band migrating in chromatographic column, Journal of Chromatography A, https://doi.org/10.1016/j.chroma.2019.460645
JID: CHROMA
[m5G;October 31, 2019;17:25]
Journal of Chromatography A xxx (xxxx) xxx
Contents lists available at ScienceDirect
Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma
Short communication
Equation for evolution of temporal width of a solute band migrating in chromatographic column Jan Leppert a,∗, Leonid M. Blumberg b, Peter Boeker a,c a
Institute of Agricultural Engineering, University of Bonn, Nussallee 5, D-53115 Bonn, Germany Advachrom, P.O. Box 1243, Wilmington, DE 19801, USA c HyperChrom SA, 21 Avenue Monterey, L-2163 Luxembourg, Luxembourg b
a r t i c l e
i n f o
Article history: Received 2 August 2019 Revised 25 September 2019 Accepted 20 October 2019 Available online xxx Keywords: Peak width Temporal variance of a solute band Residency General elution equation
a b s t r a c t An alternative differential equation to model the development of the temporal width of a solute band during its migration in a chromatographic column is developed. This model uses the solute residency, the inverse of the solute velocity, as parameter, which has the advantage to be an always finite quantity even at the column outlet in GC–MS where carrier gas velocity approaches infinity. The new differential equation is derived from a known equation describing the evolution of the spatial variance of migrating band. Supplemental material illustrating solutions of the newly developed equation in several special cases is provided. © 2019 Elsevier B.V. All rights reserved.
1. Introduction
plate height for the solute located at z. In smooth medium, σ z can be found from differential equation [6]:
Several parameters can be used for describing the widths of the peaks in chromatograms and the widths of the solute bands within columns. Among them are [1] peak/band width at half height, at base, variance, standard deviation (square root of variance). However, only the latter two can be theoretically predicted from operational conditions for non-Gaussian peaks. Only these two parameters are used throughout this report making its results applicable to any peak/band shapes. The medium of chromatographic analysis (the column, the mobile phase, their conditions) can be uniform (the same along the column at any given time), or non-uniform otherwise [2,3]. The analysis itself is static (like isothermal GC, isocratic LC, etc.) if its conditions do not change with time, or dynamic otherwise (temperature-programmed GC, gradient LC, etc.) [4,5]. The degree of non-uniformity of the medium affects the complexity of the band width equations [6]. In the majority of practical applications, the medium is smooth [6] (within the solute band, the plate height is nearly constant and the solute velocity is nearly linear function of distance). Let x be an arbitrary longitudinal distance from the column inlet, z be that distance to a solute center of mass, σ z = σ z (z) and u be, respectively, spatial standard deviation (in units of length) and velocity of a solute located at z, and H == H(z) be the column
dσz2 2σz2 =H+ dz u
∗
Corresponding author. E-mail address: [email protected] (J. Leppert).
∂ u ∂ x x=z
(1)
All quantities in Eq. (1), including the (spatial) gradient, ∂ u/∂ x, of u are measured at the time t = t (z ) of the solute migration from the inlet to z. Eq. (1) can be used to study evolution of the width of a solute band during its migration within a column [7], effects of non-uniform conditions on column performance [8,9], etc. Eq. (1) can also be used for theoretical [8,9] and/or numerical [10] prediction of peak widths in chromatograms. Let σ L and uo be the solute spatial standard deviation and velocity at the column outlet. Quantity:
σ=
σL uo
(2)
is typically and in this report interpreted as the peak temporal standard deviation (in time units)1 [11]. Eq. (2) has a shortcoming – it has a singularity in GC–MS where the column outlet is at vacuum, i.e. po = 0. The product, pu, is the same at any location along a column. Therefore, ideally, when the outlet pressure (po ) approaches zero, uo in Eq. (2) approach infinity. Similarly, infinite gas decompression at the column outlet 1 Strictly speaking, quantity σ defined in Eq. (2) is different from true (temporal) standard deviation of a peak. However, relative difference, on the order of 1/N (N is plate number) [7], is practically insignificant and typically ignored. The same approach is used throughout this report.
https://doi.org/10.1016/j.chroma.2019.460645 0021-9673/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: J. Leppert, L.M. Blumberg and P. Boeker, Equation for evolution of temporal width of a solute band migrating in chromatographic column, Journal of Chromatography A, https://doi.org/10.1016/j.chroma.2019.460645
JID: CHROMA 2
ARTICLE IN PRESS
[m5G;October 31, 2019;17:25]
J. Leppert, L.M. Blumberg and P. Boeker / Journal of Chromatography A xxx (xxxx) xxx
causes infinite band expansion so that σ L becomes infinity. The singularity due to infinity in the numerator (σ L ) and denominator (uo ) in Eq. (2) causes computational problems in numerical evaluation of GC–MS chromatograms in spite of the fact that the ratio σ L /uo in Eq. (2) (the standard deviation, σ , of a peak in chromatogram) is finite. The latter fact suggests that the singularity in GC–MS at po = 0 can be avoided if quantity
σt =
σz u
(3)
is considered instead of σ z . Quantity σ t represents the time that it takes for spatial standard deviation (σ z ) of a solute band to travel through coordinate z. Similarly to σ in Eq. (2), quantity σ t can be interpreted as the temporal standard deviation of the band. It has been shown elsewhere [8] that transformation of Eq. (1) for σ z into equation for σ t yields under static conditions:
dσt2 = Hr 2 , (static conditions ) dz
(4)
where
r=
1 u
also be viewed as a known one. If rm and k are known functions, rm = rm (x,y) and k == k(x,y), of x and y then a solute retention time (tR ) can be found from general elution equation [7,17]: tR
∫ 0
L dt = ∫ rm (z )dz 1 + k(t ) 0
where k(t ) = k(z(t ), t ), rm (z ) = rm (z, t (z ) ). Generally, only the functions rm (x,y) and k = k(x,y) could be known for a particular solute, but not functions z = z(t) and t = t(z). Finding the latter two functions can be a part of solving Eq. (8). Moreover, functions z = z(t) and/or t = t(z) can be complex even if they are known a priori. All this means that, generally, Eq. (8) can only be solved numerically by iterations starting from z = 0 and t = 0 although, there are operational conditions leading to substantial simplifications of Eq. (8) and to its analytical solutions [7,17]. All variables in Eq. (8) are finite and, therefore, it is suitable for numerical retention time evaluations. 2.2. Peak width
(5) A more detailed form of Eq. (1) can be expressed as:
It has been pointed out [8] that the Giddings compressibility factor [11,12] follows directly from Eq. (4) (see Appendix A for details). Eq. (4) was also used [8,13] for evaluation of optimal operational conditions and optimal performance of a chromatographic column (GC and SFC) in non-uniform static conditions. Quantity r defined in Eq. (5) can be called as the residency of a solute (or, more generally, of a traveling object) [13] – the larger is an objects residency, (r), at some coordinate z the longer it resides in vicinity of z. The importance of quantity (1/u) in studies of properties of solute migration in chromatography was recognized in several publications [3,13–15]. The purpose of this report is to extend Eq. (4) to dynamic (uniform and non-uniform) conditions. 2. Theory
dσz2 (z ) 2σz2 (z ) = H (z ) + dz u (z )
(6)
where
1 um
(7)
can be interpreted as the mobile phase residency. Let, in addition to the earlier introduced ones, variable y be an arbitrary time and t == t(z) be the time of a solute migration to location z. Both y and t are measured since the sample introductions. Quantity k is a never negative (k ≥ 0). If the mobile phase flows in one direction and never stops (um > 0) then z(t) is a monotonically increasing function and its inversion, t(z), always exists and is a monotonically increasing function as well. This implies that, if one function z(t) or t(z) is known then the other one can 2
In packed column LC, um and k are typically denoted as u0 and k , respectively.
(9)
Eq. (9) yields after differentiation and further transformations:
σt2 (z ) ∂ 1 (10) 2 r (z ) ∂ x r (x, y ) x = z y = t (z ) 2 2 2 2σt dr (z ) 2σt (z ) ∂ r (x, y ) 1 dσt (z ) − 3 = H (z ) − 3 ∂x x = z r 2 (z ) dz r (z ) dz r (z ) y = t (z ) d dz
σt2 (z ) r 2 (z )
= H ( z ) + 2r ( z )
⎛
To predict a chromatogram, one needs to predict retention time (tR ) and width of a peak corresponding to each solute. Following is a brief review of a known equation expressed in a form suitable for numerical retention time prediction. Transport velocity (u) [16,17] of a solute migrating in a chromatographic column can be expressed as [7,17] u = um /(1 + k ) where um is the mobile phase transport velocity [16] and k is the solute retention factor.2 Eq. (5) for the solute residency (r) can be expressed as:
r = (1 + k )rm
∂ u(x, y ) ∂x x = z y = t (z )
Eqs. (3) and (5) allow one to express σ z as σz (z ) = σt (z )/r (z ). Substitution of this relationship together with Eq. (5) in
2.1. Retention time
rm =
(8)
⎞
dσt2 (z ) 2σt2 ⎜ dr (z ) = H (z )r 2 (z ) + ⎝ dz − dz r (z )
∂ r (x, y ) ⎟ ∂x x = z ⎠ y = t (z )
(11) (12)
The last equation contains the difference between full and partial derivatives of r by z and x, respectively. The difference can be simplified recognizing that the full derivative can be expressed as [7]:
dr (z ) = dz
∂ r (x, y ) ∂ r (x, y ) + r (z ) ∂x ∂y
Eq. (12) becomes:
(13)
dσt2 (z ) ∂ r (x, y ) = H (z )r 2 (z ) + 2σt2 dz ∂y x = z y = t (z )
(14)
All variables in this equation (H, r and σ t ) are finite under all known and feasible operational conditions including conditions at the column outlet in GC-MS. Also finite under these conditions is the temporal rate, ∂ r/∂ y, of change in r. As a result, Eq. (14) is suitable for numerical integration in all known and feasible practical conditions. Equation similar to Eq. (14) is known from Lan and Jorgenson [15]. In Lan-Jorgenson equation (LJE), σ t is a true temporal standard deviation of a solute band while σ t in Eq. (14) is only a close approximation to that standard deviation (footnote 1 ). However, the LJE has its own disadvantages. Among them is a requirement
Please cite this article as: J. Leppert, L.M. Blumberg and P. Boeker, Equation for evolution of temporal width of a solute band migrating in chromatographic column, Journal of Chromatography A, https://doi.org/10.1016/j.chroma.2019.460645
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that the third temporal central moment of a solute band has negligible effect on σ t . This limits LJE to the cases where it is a priory known that temporal profiles of the solute bands are symmetric. No such requirement is necessary for validity of Eq. (14). Examples of analytical solutions for special cases of Eq. (14) are provided in the Appendix A. 3. Conclusions A Differential equation for evolution of temporal variance of a solute band migrating in a chromatographic column has been developed. Unlike previously known equations for spatial variance of the band, the newly developed equation is free of singularities and, therefore, is more suitable for numerical integration. It has been also demonstrated that a minor modification in known equations for retention time removes their possible singularities and makes them suitable for numerical integration.
3
with the Giddings compressibility factor jG
9( p2i + p2o )( pi + po )
2
jG =
(A.9)
2
8( p2i + p2o + pi po )
The Giddings compressibility factor ranges from 1 for weak decompression ( pi = po) to 1.125 for strong decompression ( po = 0). Due to Eq. (A.8), the plate number N = (tR /σ )2 [1] can be expressed as:
N=
tR2
σ2
=
L jG H
(A.10)
indicating that N can be up to 1.125 times lower than the ratio L/H. Appendix B. Constant solute migration velocity in a non-uniform dynamic velocity field with fixed H
This work was supported by the German Research Foundation (DFG) [grant number 290980359] and HyperChrom SA, Luxembourg.
This is a model of a velocity gradient, where the solutes migrate with their velocity equal to the velocity of advancement of the temperature gradient. As a result, each solute “experiences” during its migration a fixed temperature and, therefore, migrates with a fixed velocity. A velocity field with a positive gradient is:
Appendix A. Isothermal gas chromatography
u+ (x, y ) =
Acknowledgements
Under isothermal conditions the temperature of a column is constant over time and uniform over the length of the column. As a result ∂ r/dy in Eq. (14) becomes zero and we obtain Eq. (4).
dσ = Hr 2 , (static conditions ) dz 2 t
(A.1)
The solute residency r at point z is given by
r (z ) = rm,i (1 + k ) 1 −
p2 1 − 2o pi
z L
(A.2)
with the retention factor k, inlet mobile phase residency rm, i and pi the inlet and po outlet pressure. The solution for the temporal variance σ 2 at the end of the column is the integration of the right-hand side of eq. (A.1) for z over the length of the column L. L
σ 2 = H ∫ r 2 dz
(A.3)
0
σ =
2 HL(1 − k )rm,i
2
p2 1 + 2o pi
u− (x, y ) =
(A.4)
The inlet mobile phase residency rm, i can be expressed with m
pi tm = j po L
(A.5)
(A.6)
r+ (x, y ) =
(A.7)
Eqs. (A.5), (A.6) and (A.7) substituted into Eq. (A.4) results in
σ
2
H = jG tR2 L
y tR 1 + tR L 1 + xL
(B.3)
and for the negative solute velocity gradient the residency field is
r− (x, y ) =
tR 1 + xL L 1 + ty R
(B.4)
2σt,2± t2 dσ±2 = H R2 ± dz L+z L
(A.8)
(B.5)
with the ‘+’ sign for the positive and the ‘-‘ sign for the negative solute velocity gradient. The solution for an ideal injection, σ 2 (z = 0 ) = 0, in case of the positive solute velocity gradient is 2 +
t2 (z ) = H R2 L
z2 z+ L
(B.6)
At the end of the column, at z = L, Eq. (B.6) becomes
σ+2 (L ) = 2tR2
and the void time tm .
tR tm = 1+k
(B.2)
The solute velocity in the apex of the solute zone is constant, u+ (z, t (z ) ) = u− (z, t (z ) ) = L/tR . The spatial derivative of the positive gradient velocity u+ is positive and the spatial derivative of the negative gradient velocity u− is negative for 0 ≤ x ≤ L and 0 ≤ y ≤ tR . The corresponding residency field for the positive solute velocity gradient is
σ
with the James-Martin compressibility factor j
3 po ( pi + po ) j= 2( p2i + po pi + p2o )
y L 1 + tR tR 1 + xL
The differential equation (15) is for these gradients
the average mobile phase velocity um = L/t :
rm,i
(B.1)
A velocity field with a negative gradient is
The solution of this integral is 2
L 1 + xL tR 1 + ty R
H L
(B.7)
In the case of the negative solute velocity gradient the solution for ideal injection is
σ−2 (z ) = H
tR2 L2 z + Lz2 + L2 (L + z )2
z3 3
(B.8)
Please cite this article as: J. Leppert, L.M. Blumberg and P. Boeker, Equation for evolution of temporal width of a solute band migrating in chromatographic column, Journal of Chromatography A, https://doi.org/10.1016/j.chroma.2019.460645
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At the end of the column, at z= L, Eq. (B.8) becomes
σ−2 (L ) =
7 2H t 12 R L
(B.9)
The temporal variance in the case of the moving gradient with positive solute velocity gradient is 3.4 times larger than the temporal variance with a negative solute velocity gradient. The peak width is consequently 1.8 times bigger for the positive gradient compared to the peak width with the negative gradient. References [1] IUPAC, Nomenclature for chromatography, Pure Appl. Chem. 65 (1993) 819–872. [2] J.J. van Deemter, F.J. Zuiderweg, A. Klinkenberg, Longitudinal diffusion and resistance to mass transfer as causes of nonideality in chromatography, Chem. Eng. Sci. 5 (1956) 271–289. [3] J.C. Giddings, Plate height of nonuniform chromatographic columns. gas compression effects, coupled columns, and analogous systems, Anal. Chem. 35 (1963) 353–356, doi:10.1021/ac60196a026. [4] L.M. Blumberg, Limits of resolution and speed of analysis in linear chromatography with and without focusing, Chromatographia 39 (1994) 719–728, doi:10.1007/BF02274589. [5] L.M. Blumberg, Erratum. limits of resolution and speed of analysis in linear chromatography with and without focusing, Chromatographia 40 (1995) 218, doi:10.1007/BF02272175. [6] L.M. Blumberg, Variance of a zone migrating in a linear medium II. timevarying non-uniform medium, J. Chromatogr. 637 (1993) 119–128, doi:10.1016/ 0021- 9673(93)83204- 6.
[7] L.M. Blumberg, Temperature-Programmed Gas Chromatography, Wiley-VCH, Weinheim, 2010. [8] L.M. Blumberg, T.A. Berger, Variance of a zone migrating in a non-uniform time invariant linear medium, J. Chromatogr. 596 (1992) 1–13, doi:10.1016/ 0021- 9673(92)80197- 3. [9] L.M. Blumberg, Theory of gradient elution liquid chromatography with linear solvent strength: Part 2. peak width formation, Chromatographia 77 (2014) 189–197, doi:10.1007/s10337- 013- 2556- x. [10] F. Aldaeus, Y. Thewalim, A. Colmsjö, Prediction of retention times and peak widths in temperature-programmed gas chromatography using the finite element method, J. Chromatogr. A 1216 (2009) 134–139, doi:10.1016/j.chroma. 2008.11.038. [11] J.C. Giddings, S.L. Seager, L.R. Stucki, G.H. Stewart, Plate height in gas chromatography, Anal. Chem. 32 (1960) 867–870, doi:10.1021/ac60163a043. [12] G.H. Stewart, S.L. Seager, J.C. Giddings, Influence of pressure gradients on resolution in gas chromatography, Anal. Chem. 31 (1959) 1738, doi:10.1021/ ac60154a001. [13] L.M. Blumberg, Erosion of efficiency in non-uniform linear chromatography, J. High. Resolut. Chromatogr. 16 (1993) 31–38, doi:10.1002/jhrc.1240160106. [14] P.D. Schettler, M. Eikelberger, J.C. Giddings, Nonideal gas corrections for retention and plate height in gas chromatography, Anal. Chem. 39 (1967) 146–157, doi:10.1021/ac60246a004. [15] K. Lan, J.W. Jorgenson, Theoretical investigation of the spatial progression of temporal statistical moments in linear chromatography, Anal. Chem. 72 (20 0 0) 1555–1563, doi:10.1021/ac990533u. [16] L.M. Blumberg, G. Desmet, Kinetic performance factor – a measurable metric of separation-time-pressure tradeoff in liquid and gas chromatography, J. Chromatogr. A 1567 (2018) 26–36, doi:10.1016/j.chroma.2018.06.048. [17] L.M. Blumberg, Migration and elution equations in gradient liquid chromatography, J. Chromatogr. A 1599 (2019) 35–45, doi:10.1016/j.chroma.2019.03.057.
Please cite this article as: J. Leppert, L.M. Blumberg and P. Boeker, Equation for evolution of temporal width of a solute band migrating in chromatographic column, Journal of Chromatography A, https://doi.org/10.1016/j.chroma.2019.460645