2 HPLC THEORY Yuri Kazakevich 2.1 INTRODUCTION The process of analyte retention in high-performance liquid chromatography (HPLC) involves many different aspects of molecular behavior and interac- tions in condensed media in a dynamic interfacial system. Molecular diffusion in the eluent flow with complex flow dynamics in a bimodal porous space is only one of many complex processes responsible for broadening of the chro- matographic zone. Dynamic transfer of the analyte molecules between mobile phase and adsorbent surface in the presence of secondary equilibria effects is also only part of the processes responsible for the analyte retention on the column. These processes just outline a complex picture that chromatographic theory should be able to describe. HPLC theory could be subdivided in two distinct aspects: kinetic and ther- modynamic. Kinetic aspect of chromatographic zone migration is responsible for the band broadening, and the thermodynamic aspect is responsible for the analyte retention in the column. From the analytical point of view, kinetic factors determine the width of chromatographic peak whereas the thermody- namic factors determine peak position on the chromatogram. Both aspects are equally important, and successful separation could be achieved either by opti- mization of band broadening (efficiency) or by variation of the peak positions on the chromatogram (selectivity). From the practical point of view, separa- tion efficiency in HPLC is more related to instrument optimization, column 25 HPLC for Pharmaceutical Scientists, Edited by Yuri Kazakevich and Rosario LoBrutto Copyright © 2007 by John Wiley & Sons, Inc. dimensions, and particle geometry—factors that could not have continuous variation during method development except for the small influence from vari- ation of the mobile phase flow rate. On the other hand, analyte retention or selectivity is mainly dependent on the competitive intermolecular interactions and are influenced by eluent type, composition, temperature, and other vari- ables which allow functional variation. 2.2 BASIC CHROMATOGRAPHIC DESCRIPTORS Four major descriptors are commonly used to report characteristics of the chromatographic column, system, and particular separation: 1. Retention factor (k) 2. Efficiency (N) 3. Selectivity (α) 4. Resolution (R) Retention factor (k) is the unitless measure of the retention of a particular com- pound on a particular chromatographic system at given conditions defined as (2-1) where V R is the analyte retention volume, V 0 is the volume of the liquid phase in the chromatographic system, t R is the analyte retention time, and t 0 is some- times defined as the retention time of nonretained analyte. Retention factor is convenient because it is independent on the column dimensions and mobile- phase flow rate. Note that all other chromatographic conditions significantly affect retention factor. Efficiency is the measure of the degree of peak dispersion in a particular column; as such, it is essentially the characteristic of the column. Efficiency is expressed in the number of theoretical plates (N) calculated as (2-2) where t R is the analyte retention time and w is the peak width at the baseline. Selectivity (α) is the ability of chromatographic system to discriminate two different analytes. It is defined as the ratio of corresponding retention factors: (2-3) α= k k 2 1 N t w R =     16 2 k VV V tt t RR = − = − 0 0 0 0 26 HPLC THEORY Resolution (R) is a combined measure of the separation of two compounds which include peak dispersion and selectivity. Resolution is defined as (2-4) In the following sections the chromatographic descriptors introduced above [equations (2-1)–(2-4)] will be discussed in terms of their functional depen- dencies, specifics, and relationships with different chromatographic and ther- modynamic parameters. 2.3 EFFICIENCY The most rigorous discussion of the formation of chromatographic zone and the mathematical description of zone-broadening is given in reference 1. Here only practically important and useful equations will be discussed. If column properties could be considered isotropic, then we would expect symmetrical peaks of a Gaussian shape (Figure 2-1), and the variance of this peak is proportional to the diffusion coefficient (D) (2-5) At given linear velocity (ν) the component moves through the column with length (L) during the time (t), or (2-6) Lvt= σ 2 2= Dt R tt ww = − + 2 21 21 EFFICIENCY 27 Figure 2-1. Gaussian band broadening . Substituting t from equation (2-6) in equation (2-5), we get (2-7) Expression 2D/ν has units of length and is essentially the measure of band spreading at a given velocity on the distance L of the column. This parameter has essentially the sense of the height equivalent to the theoretical plate and could be denoted as H, so we get (2-8) Several different processes lead to the band-spreading phenomena in the column which include: multipath effect; molecular diffusion; displacement in the porous beds; secondary equilibria; and others. Each of these processes introduces its own degree of variance toward the overall band-spreading process. Usually these processes are assumed to be independent; and based on the fundamental statistical law, overall band-spreading (variance) is equal to the sum of the variances for each independent process: (2-9) In the further discussion we assume the total variance in all cases. In the form of equation (2-8) the definition of H is exactly identical to the plate height as it evolved from the distillation theory and was brought to chro- matography by Martin and Synge [2]. If H is the theoretical plate height, we can determine the total number of the theoretical plates in the column as (2-10) In linear chromatography, each analyte travels through the column with con- stant velocity (u c ). Using this velocity, we can express the analyte retention time as (2-11) Similarly, the time necessary to move analyte zone in the column on the dis- tance of one σ (Figure 2-1) can be defined as t (2-12) τ σ = u c t L u R c = N L H N L =⇒=     σ 2 σσ tot 22 = ∑ i H L = σ 2 σ 2 2 =     D v L 28 HPLC THEORY Substituting both equations (2-11) and (2-12) into (2-10), we get (2-13) Parameter t in equation (2-13) is the standard deviation and expressed in the same units as retention time. Since we considered symmetrical band- broadening of a Gaussian shape, we can use Gaussian function to relate its standard deviation to more easily measurable quantities. The most commonly used points are the so-called peak width at the baseline, which is actually the distance between the points of intersections of the tangents to the peaks inflec- tion points with the baseline (shown in Figure 2-1). This distance is equal to four standard deviations, and the final equation for efficiency will be (2-14) Another convenient determination for N is by using the peak width at the half- height. From the same Gaussian function the peak width on the half-height is 2.355 times longer than the standard deviation of the same peak, and the resulting formula for the number of the theoretical plates will be (2-15) Efficiency is mainly a column-specific parameter. In a gas chromatography column, efficiency is highly dependent on the flow rate. In HPLC, because of much higher viscosity, the applicable flow rate region is not so broad; within this region, variations of the flow rate do not affect column efficiency signifi- cantly. On the other hand, geometry of the packing material and uniformity and density of the column packing are the main factors defining the efficiency of particular column.There is no clear fundamental relationship between the par- ticle diameter and the expected column efficiency, but phenomenologically an increase of the efficiency can be expected with the decrease of the particle diameter, since the difference between the average size of the pores in the par- ticles of the packing material and the effective size of interparticle pores decreases, which leads to the more uniform flow inside and around the parti- cles. From Figure 2-2 it is obvious that the smaller the particles, the lower the theoretical plate height and the higher the efficiency. The general form of the shown dependence is known as Van Deemter function (2-16), which has the following mathematical form: (2-16) HA B v Cv=++ N t w R h =       5 545 1 2 2 . N t w r b =     16 2 N t R =     τ 2 EFFICIENCY 29 where ν is the linear flow velocity, and A, B, and C are constants for given column and mobile phase. Three terms of the above equation essentially represent three different processes that contribute to the overall chromatographic band-broadening. A—represents multipath effect or eddy diffusion B—represents molecular diffusion C—represents mass transfer The multipath effect is a flow-independent term, which defines the ability of different molecules to travel through the porous media with paths of differ- ent length. The molecular diffusion term is inversely proportional to the flow rate, which means that the slower the flow rate, the longer component stays in the column and the molecular diffusion process has more time to broaden the peak. The mass-transfer term is proportional to the flow rate, which means that the faster the flow, the greater the band-broadening. Superposition of all three processes is shown schematically in Figure 2-3. As it could be seen from the comparison of Figure 2-2 and Figure 2-3, all dependencies of the column efficiency on the flow rate follow the theoretical Van Deemter curve.In theory there is an optimum flow rate that allows obtain- ing the highest efficiency (the lower theoretical plate height). 30 HPLC THEORY Figure 2-2. The experimental dependence of the theoretical plate height on the flow velocity for columns packed with same type of particles of different average diameter. As follows from Figure 2-2, the lower the particle diameter, the wider the range of the flow rates where the highest column efficiency is achieved. For columns packed with smaller particles, efficiency is not as adversely affected at faster flow rates, because the mass-transfer term is lower for these columns. Essentially, this means that retention equilibrium is achieved much faster in these columns. Faster flow rates mean higher flow resistance and higher backpressure. It is a modern trend to work with the smaller particles at high linear velocity. However, the overall efficiency of the columns packed with smaller particles (<2µm) is not much higher compared to conventional columns with 3- to 5-µ particles. The comparison of a conventional 15-cm column with 4.6-mm inter- nal diameter packed with 5-µm particles to a column of 15-cm length and 2-mm I.D. packed with 1.7-µm particles shows that the average efficiency of the first column is between 12,000 and 15,000 theoretical plates, and for the second column the efficiency is not much higher: It is on the level of 15,000 to 18,000 theoretical plates.This small increase of the efficiency may only slightly improve the separation; however, the comparison of the run times at the same volumetric flow rates on both columns shows that the separation on the second column can be achieved five times faster. Of course, the ability to increase u depends on the pressure capabilities of the instrument, since pressure is directly proportional to velocity: (2-17) ∆P uL d p = hf 2 EFFICIENCY 31 Figure 2-3. Schematic of the Van Deemter function and its components. where ∆P is the pressure drop across the column, h is viscosity, and f is the flow resistance factor [3]. Therefore, the fastest possible separation requires that the maximum pressure allowed by the instrument be used, assuming that the resolution requirement can still be met. This also means that the speed of analysis is limited by that maximum pressure. As a result, one wants to make the most of the pressure available by reducing the pressure drop across the column as much as possible. This can be achieved by working at higher tem- peratures,using MeCN/water mobile phases instead of methanol/water mobile phases on the same length of column or by using shorter columns. To limit analysis time, the shortest column length possible should be used. Shorter columns have lower pressure requirements, allowing to gain an advan- tage in speed. It must be kept in mind, however, that N will decrease as u increases (for particles ≥3µm), meaning that at faster velocities longer columns are necessary to give the required theoretical plates, thus generating greater operating pressures. 2.4 RESOLUTION In the introduction section we define the term resolution as the ability of the column to resolve two analyte in two separate peaks (or chromatographic zones). In more general form than it was given before, the resolution can be defined as the half of the distance between the centers of gravity of two chro- matographic zones related to the sum of their standard deviations: (2-18) In case of symmetrical peaks, centers of peak gravity could be substituted with the peak maxima; and using the relationship of the peak width with its stan- dard deviation (shown in Figure 2-1), a common expression for the resolution could be obtained: (2-19) The peak width in equation (2-19) could be substituted using expression (2-14), and the resulting equation is (2-20) Expression (2-20) demonstrates that resolution is proportional to the square root of the efficiency. R tt tt N RR RR = − + ⋅ 21 21 2 R tt ww RR = − + () 21 21 1 2 R XX = − + () 21 12 2 σσ 32 HPLC THEORY From the practical point of view, in case of the lack of resolution for some specific separation there are generally two ways to improve it: Increase the efficiency, or increase the selectivity. The efficiency is proportional to the column length: The longer the column, the higher the efficiency, but equation (2-20) shows that the increase of the efficiency increases the resolution only as a square root function (as illustrated in Figure 2-4). At the same time, the increase of the column length leads to the increase of the flow resistance and backpressure, which limits the ability to further increase the column length. If we assume that the peak widths of two adjacent peaks are approximately equal, we can rewrite expression (2-18) in the form (2-21) For symmetrical chromatographic bands, this is the ratio of the distance between peaks maxima to the peak width.The distance between peak maxima is proportional to the distance of the chromatographic zone migration, and the peak width is proportional to the square root of this distance. Figure 2-4 illus- trates this relationship. At low selectivity to achieve the same resolution, one has to use a longer column to increase efficiency and consequently operate under higher-pressure conditions. The relationship between the column length, mobile-phase viscos- ity, and the backpressure is given by equation (2-17), which is the variation of the Kozeny–Carman equation. Expression (2-17) predicts a linear increase of the backpressure with the increase of the flow rate, column length, and mobile phase viscosity.The decrease of the particle diameter, on the other hand, leads to the quadratic increase of the column backpressure. Achievement of good resolution between analytes in complex chro- matogram is the main goal in HPLC method development. Optimal resolu- tion could be achieved by optimization of system efficiency, or selectivity (or R XX = − 21 4σ RESOLUTION 33 Figure 2-4. Relationship between resolution, selectivity, and column length. both). Relationships of the retention, selectivity, and efficiency with the reso- lution has been for long a subject of extensive theoretical studies [4–6], with the goal to express resolution as a function of k, a, and N. Unfortunately, the direct algebraic transformation of expression (2-19) into some form of functional dependence of R on k, α, and N is impossible. Knox and Thijssen were the first to independently propose the transformation based on the assumption of equal peak width (w 2 = w 1 ) and consideration of the retention of the first peak of the pair (k 1 ). The resulting expression is (2-22) For closely eluted peaks and relatively high efficiency of the system, these assumptions do not lead to the significant deviations of equation (2-22) from true resolution given by equation (2-19). Purnell [7] suggested to center attention on the second peak of the pair, thus using the peak width of the second component as a base width (meaning that the width of the first peak is equal to the width of the second peak). This assumption leads to the following equation: (2-23) Both equations do not give a real resolution value; also, the greater the dis- tance between peaks, the higher the error. Said [6] suggests the use of average values instead of selection of the first or second primary peaks, which leads to the following expression for resolu- tion: (2-24) All these expressions give approximate values of resolution; also, the smaller the distance between target peaks in the chromatogram, the closer the values to the true resolution. Detailed analysis of available master resolution equa- tions is given in the B. Karger article [4]. 2.5 HPLC RETENTION In Section 2.1 the main chromatographic descriptors generally used in routine HPLC work were briefly discussed. Retention factor and selectivity are the parameters related to the analyte interaction with the stationary phase and reflect the thermodynamic properties of chromatographic system. Retention factor is calculated using expression (2-1) from the analyte retention time or retention volume and the total volume of the liquid in the column. Retention R k k N = − +     +     a a 1 1 1 2 R k k N = +     −     2 2 1 1 4 a a R k k N = +     − () 1 1 1 1 4 a 34 HPLC THEORY [...]... (2-64) Γ ( c s ) = K H K p ce (2-66) 56 HPLC THEORY where KH is a Henry adsorption constant The final form of the distribution function, accounting for v0 = vm + vs, therefore will be ψ (ce ) = [v0 + ( K p − 1)vs + sK H K p ]ce (2-67) Applying this function into the mass-balance equation (2-33) and performing the same conversions [Eqs (2-34)–(2-39)], the final equation for the analyte retention in binary... protonation is completely suppressed Corresponding capacity factors for neutral and protonated forms of basic analyte could be written in the form kB = kBH = S KB V0 S K + V0 BH ( neutral ) (2-81) ( protonated) (2- 82) Substituting KB and KBH+ from expressions (2-81) and (2- 82) into equation (2-80) and expressing the overall analyte retention in the form of retention factor, we get ... (2- 32) is simplified to ∂y (c )  ∂c −F   =   ∂x  t  ∂t  x (2-33) This equation states that the formation of the analyte concentrational gradient in a fixed moment of time in any place of the column should be equal to the time-dependent variation of the analyte amount The general solution of equation (2-33) could be obtained using two classical relationships for partial derivatives [for simple forms... ) = Vm + Vs df (c ) dc (2- 42) where df(c)/dc is the derivative of the partitioning distribution function For low analyte concentration the distribution function is assumed to be linear and its slope (derivative) is equal to the analyte distribution constant K Equation (2- 42) then could be written in the well-known form VR = Vm + Vs K (2-43) This equation has been derived for the model of the analyte... ratio of Va/S, is the maximum distance of the influence of the surface forces The sum of the excess adsorption value and the product of the equilibrium concentration and adsorbed layer volume represent the total amount of the adsorbate in that layer for any given equilibrium concentration a(ce ) = Γ (ce ) + Va ce (2- 52) Equation (2- 52) is the total adsorption isotherm derived from experimentally measurable... analytical applications of HPLC, all these discrepancies are quietly and conveniently forgotten, and selection of some so-called “nonretained” component as a void volume marker is a common way for void volume measurement In the majority of recent analytical publications, either thiourea or uracil were used as the void volume markers As a disclaimer, we have to say here that for the purposes of analytical... distributed inside this zone In general form, it could be written as ∂y (c )  ∂c − F   dxdt =  dxdt  ∂x  t  ∂t  x (2- 32) where the analyte concentration is a function of both the distance and the time and y(c) is a distribution function that has units of the analyte amount per unit length of the column This distribution function is the key for the solution of equation (2- 32), and the definition of this... solution of the mass-balance equation is the expression for the analyte retention behavior, and it is only valid in the frame of the selected model So far the solution of the mass-balance equation for models with a single dominating process (partitioning or adsorption) was discussed in Sections 2.8 and 2.9 In both cases the solutions have similar form, with the difference in the definition of the parameters... presence of ionic equilibrium for an ionizible basic analyte: B + H + = BH + (2- 72) Above equilibrium is dependent on the mobile-phase pH and the relationship between ionic and nonionic form of the analyte is described by Henderson– Hasselbalch equation c BH+ = 10 pK a − pH cB or c BH+ [H + ] = cB Ki (2-73) where Ki is the analyte ionization constant Both ionic and nonionic forms of the analyte can be... adsorbed in the ionic form, it stays in this form on the surface 59 SECONDARY EQUILIBRIA The left-hand side of equation (2-33) expresses the presence of the gradient of the analyte in the column cross section In the case of an ionizable analyte, there are two forms of the analyte present, and using expression (2-73) the left-hand side of equation (2-33) should be written in the form + ∂c  [H ]   . of view, separa- tion efficiency in HPLC is more related to instrument optimization, column 25 HPLC for Pharmaceutical Scientists, Edited by Yuri Kazakevich. 2 HPLC THEORY Yuri Kazakevich 2.1 INTRODUCTION The process of analyte retention in high-performance liquid chromatography (HPLC) involves