Introduction to Modern Liquid Chromatography, Third Edition part 9 ppt

The time spent by the band during its passage through the column varies inversely with the flow rate, so the contribution to band width from longitudinal diffusion decreases for faster flo

(b)

t R (i )

80% B

70% B

3.90 4.00 4.10 4.20 4.30

Time (min)

t R (j )

W

W1/2

h h/2

3.93 min

1.5

Figure2.10 Origin and measurement of peak width (a) Measurement of peak width; (b) peak 3 of Figure 2.6 as a function of %B, shown with the same time scale for each peak.

Because an ideal chromatographic peak has the shape of a Gaussian curve (Section 2.4.2), peak width is sometimes described in terms of the standard deviation

σ of the Gaussian curve, where

and therefore



t R σ

2

(2.9c)

Equation (2.9c) can be expressed in other forms, for example, N = 25(t R /W5σ)2,

where W5σ = 1.25W is the so-called 5σ peak width.

Equation (2.9) can be rearranged to give

or (replacing t Rby Eq 2.5)

W = 4N −0.5 t0(1+ k) (2.10a)

Because values of N are approximately constant for the different peaks in a chromatogram, Equation (2.10) tells us that peak width W will increase in proportion

to retention time A continual increase in peak width from the beginning to the end

of the chromatogram is therefore observed; for example, see the chromatogram of

Figure 2.7c.

The area for a given solute peak normally remains approximately constant when retention time is varied by a change in %B, temperature, or the column—so

peak height h p times peak width W will also be constant For this situation

h p≈ (constant)

W ≈(constant)

t R

(2.11)

That is, as t R increases, peak height decreases An example is shown in Figure 2.10b

for peak 3 of Figure 2.6 as a function of %B A reciprocal change is seen in peak height and width as %B is varied, as predicted by Equation (2.11)

2.4.1 Dependence of N on Separation Conditions

We will begin by summarizing some practical conclusions about how the column

plate number N varies with the column, the sample, and other separation conditions.

In following Section 2.4.1.1, we will examine the theory on which these conclusions

are based N can also be described by

 1

H



where H = L/N is the column plate height H is a measure of column efficiency

per unit length of column; increasing column length (as by replacing a 150-mm long column with a 250-mm column) is therefore a convenient way of increasing

N and improving separation (since H is constant for columns that differ only in

dimensions)

Consider next the log-log plot of Figure 2.11a, which shows how N varies with flow rate F and particle diameter d p(= 2, 5, or 10 μm), while other separation

conditions are held constant As the mobile-phase flow rate F increases from a starting value of 0.1 mL/min, N first increases, then decreases For the present conditions, maximum values of N (indicated by •) are found for flow rates of 0.2

to 1.0 mL/min, depending on particle size d p A 3-fold increase in F, relative to the ‘‘optimum’’ value F opt for maximum N, has only a minor effect on separation (a decrease in N of≈20%) but reduces separation time by 3-fold Therefore flow

rates greater than F opt are usually chosen in practice Flow rates < F opt are highly

undesirable, as this means both lower values of N and longer separation times.

A decrease in particle size generally leads to an increase in N, as seen in Figure 2.11a The occurrence of maximum N is seen to occur at higher flow rates for

smaller particles, which allows faster separations for columns packed with smaller particles (Section 2.4.1.1) The pressure drop across the column (which we will refer

to simply as ‘‘pressure’’ P) increases for smaller particles and higher flow rates (see the dashed lines in Fig 2.11a for P= 2000, 5000, and 15,000 psi) The pressure (in

P (psi)= 2K 5K 15K

10 5

104

103

102

dp= 10 μm

F (mL/min) N

(a)

(b)

150 × 4.6-mm columns

15,000

10,000

5,000

N

F (mL/min)

80 °C,

200 Da

100 × 4.6-mm, 3-μm column

30 °C, 6,000 Da

30 °C,

200 Da

maximum N

5 μm

2 μm

P (MPa) = 14 34 100

maximum N

see Section 2.4.1.2

Figure2.11 Variation of column plate number N with flow rate F, particle diameter d p,

and different conditions Assumes 50% acetonitrile/water mobile phase (a) Conditions:

150× 4.6-mm column, 30◦C, and a sample molecular weight of 200 Da; (- - -) connects points

on curves of N versus F for pressure P = 2000, 5000, and 15,000 psi, respectively (b)

Con-ditions: 100× 4.6-mm column (5-μm particles); other conditions shown in figure All plots based on Equation (2.18a) with A = 1, B = 2, and C = 0.05.

psi) for a packed column can be estimated by

P≈1.25L2η

t0d p

(2.13)

or from Equations (2.7) and (2.13),

P≈ 2500L ηF

Here L is the column length (mm), η is the viscosity of the mobile phase in cP

(see Appendix I for values of η as a function of mobile phase composition and

temperature), t0is in minutes, F is the flow rate (mL/min), d cis the column internal

diameter (mm), and d pis the particle diameter (μm) Other units of pressure (besides psi) are sometimes used in HPLC: bar≡ atmospheres = 14.7 psi; megaPascal (MPa)

= 10 bar = 147 psi

Values of P are also affected somewhat by the nature of the particles within

the column, and how well the column is packed (Section 5.6) Flow restrictions outside the column (tubing between the pump and detector, sample valve, detector flow cell) add to the total pressure measured at the pump outlet, but the sum

of these contributions is usually minor (10–20%) compared to values of P from

Equation (2.13)—the exception to this is HPLC systems designed for operation

>6000 psi with <3-μm particles, which often exhibit a significant pressure in

the absence of a column (e.g., ≥ 1000 psi) Variations in equipment and column permeability can cause Equations (2.13) and (2.13a) to be in error by ±20% or more Despite the ability of most HPLC systems to operate at 5000 to 6000 psi,

it may be desirable to limit the column pressure drop to no more than 3000 psi (Section 3.5) As the pressure typically increases when a column ages, this suggests that the pressure for a routine assay with a new column should not exceed 2000 psi The latter recommendation is conservative, however, and HPLC systems are now commercially available for routine use at pressures of 10,000 to 15,000 psi or higher (Section 3.5.4.3; [21])

Next consider Figure 2.11b, for a 100 × 4.6-mm column of 5-μm particles,

with a mobile phase of 50% acetonitrile/water (note the linear–linear scale and the

narrower, more typical range in values of F) Plots of N versus flow rate are shown for

three different conditions: 30◦C and a 200-Da sample, 30◦C and a 6000-Da sample (e.g., a large peptide), or 80◦C and a 200-Da sample Similar plots of N versus F are

observed for each of these three examples, except that the flow rate for maximum

or optimum N (F opt) is shifted to higher values for higher temperatures, and lower

values for larger sample molecules One conclusion from Figure 2.11b is that higher

flow rates can be used with separations carried out at higher temperatures, which can in turn be used for faster separations (Section 2.5.3.1) Similarly separations of higher molecular-weight samples will generally require lower flow rates (and longer

run times) for comparable values of N (e.g., Table 13.4) The dependence of N

on the column, sample molecular weight, and other conditions is summarized in Table 2.4

2.4.1.1 Band-Broadening Processes That Determine Values of N

The width W and retention time t R of a peak determine the value of N for the

column (Eq 2.9) Various processes within and outside the column contribute to

the final peak volume or width W, as illustrated in Figure 2.12 (note the increases

in band width that result for each process [Fig 2.12a–e], indicated by a bracket

plus arrow alongside the band) Following sample injection, but before the sample enters the column, molecules of a solute will occupy a volume that is usually small

(Section 2.6.1 ) This is illustrated in Figure 2.12a, where individual solute molecules

are represented by • Often the extra-column contribution to band width can be

ignored (Section 3.9), but that depends on the characteristics of the equipment

Table 2.4

Effect of Different Experimental Conditions on Values of the Plate Number N

Effect on F opt, the Value of Effect on N and P of an Increase

Column length L None N increases proportionately

P increases proportionately

Column diameter d c F opt ∝ d c2 None, if flow rate increased in

proportion to d c2(recommended) Column particle size

d p

F opt ∝ 1/d p N decreases (Fig 2.12)

P decreases

Mobile-phase flow

rate F

None N decreases (Fig 2.12)

P increases proportionately

Mobile-phase viscosity

η F optdecreases asη increases N decreases (Eqn 2.19) P increases proportionately

Temperature T(K) F opt increases as T increases b N increases (Fig 2.13c)

P decreases

Sample molecular

weight M

F opt ∝ M −0.6 N decreases (Fig 2.13c) no effect

on P

Note: See discussion of Figures 2.12 and 2.13.

a Assumes flow rates equal or exceed the optimum value for maximum N (F opt), as is often the case.

b Due to an increase in both T and mobile phase viscosity (Eq 2.19).

and the size of the column (small-volume columns packed with small particles are

especially prone to extra-column band broadening) Consider next the longitudinal

diffusion of solute molecules along the column, as illustrated in Figure 2.12b This

process causes band width to increase with time, and it occurs whether or not the mobile phase is flowing The time spent by the band during its passage through the column varies inversely with the flow rate, so the contribution to band width from longitudinal diffusion decreases for faster flow

Eddy diffusion represents another contribution to band broadening

(Fig 2.12c) As molecules of the sample are carried through the column in different

flow streams (arrows) between particles, molecules in slow-moving (constricted

or narrow) streams lag behind, while molecules in fast-moving (wide) streams are carried ahead This contribution to band broadening is approximately independent

of flow rate, and depends only on the arrangement and sizes of particles within the column; band broadening due to eddy diffusion increases for poorly packed

columns Mobile-phase mass transfer (Fig 2.12d) is the result of a faster flow of the

stream center (much like the middle of a river) As flow rate increases, the center of the stream moves relatively faster, and band broadening increases

A final contribution to band broadening within the column is stationary-phase

mass transfer (Fig 2.12e) Some sample molecules will penetrate further into a

particle pore (by diffusion) and spend a longer time before leaving the particle

(e.g., molecule i in Fig 2.12e) During this time other molecules (e.g., j) will have

moved a shorter distance into the particle and spent less time before leaving the

particle Molecules (e.g., j) that spend less time in the particle will move further

( j ) (i )

1

2

3 4

8

7

9

Sample injection

(extra-column band broadening)

Longitudinal diffusion (time dependent)

Eddy diffusion

(flow independent)

Mobile-phase mass transfer (flow dependent)

Stationary-phase mass transfer

(flow dependent)

Flow through detector + connecting tubing (extra-column peak broadening)

1 2

3 4

7

9 8

2

3

1 2

3 4

8

7

9

Detector

particle pore

Figure2.12 Illustration of various contributions to band broadening during HPLC separa-tion Molecules of a solute represented by◦(before migration) and•(after migration); - - ->

indicates movement of a solute molecule

along the column, with a consequent increase in band width This contribution to band broadening increases as the flow rate increases Eventually the band leaves the

column and passes through the detector (Fig 2.12f ), resulting in some additional

extra-column peak broadening—as during introduction of the sample to the column

(Fig 2.12a).

Band-broadening processes as in Figure 2.12 contribute to the final peak width

W as

where W i is the contribution of each (independent) process i to the final peak width.

We can distinguish peak-width contributions that arise either inside or outside of the

column Let W EC represent the sum of extra-column contributions (as in Fig 2.12a plus f ), and let W0indicate the sum of intra-column contributions so that

W2= W0 + W2

The extra-column peak broadening W EC should be relatively minor in a well-designed HPLC system (Section 3.9), so it will be ignored during the following

discussion Because W EC does not depend on values of k, while W0 (defined by

Eq 2.10a) increases with k, extra-column band broadening has its largest effect on

early peaks in the chromatogram

The remainder of this section and Section 2.4.1.2 can be useful for insight into the dependence of N on experimental conditions This discussion also provides

a basis for achieving very fast separations (Section 2.5.3.2) and for otherwise optimizing column efficiency and separation This material is less essential for the everyday use of HPLC separation, however, and can be somewhat challenging The reader may therefore wish to skip to Section 2.4.2, and return to Section 2.4.1.1 and 2.4.1.2 at a later time Nevertheless, the material beginning with Equation (2.17) and especially Section 2.4.1.2 can have great practical value and is very much worth the reader’s attention See also the expanded discussion of band-broadening theory

in [22–25].

The quantity W0 will henceforth be considered equivalent to the peak width

W (Eq 2.10), which can be expressed in terms of Equation (2.14) as

SP

longitudinal eddy mobile-phase stationary-phase diffusion diffusion mass transfer mass transfer

A combination of Equations (2.10) and (2.12) yields

W2=



16

L



t2R



where values of H = L/N for different solutes are approximately independent

of retention time t R for a given column of length L and the same experimental

conditions Therefore

Equation (2.15) can be more directly related to values of N by replacing values of

W2with values of H (Eq 2.15b):

H = H L + H E + H MP + H SP (2.16)

The quantities H L, H E , H MP , and H SP have the same significance as corresponding

values of W in Equation (2.15); H L is the contribution to H by longitudinal diffusion,

and so forth Recalling our discussion above of Figure 2.12, and noting that values

of W2are proportional to values of H (Eq 2.15b), the following expression can be derived from theoretical equations for each of the four contributions to W:

eddy longitudinal mobile-phase plus diffusion mass transfer stationary-phase mass transfer

where the coefficients A, B, and C are each constant for a particular solute, column, and set of experimental conditions If values of F in Equation (2.16a) are replaced

by the mobile-phase velocity u, the so-called van Deemter equation results:

H = A + B

where A, B, and C represent a different set of constants for a particular solute,

column, and set of experimental conditions

Equation (2.16a) is not quite correct, however, because it assumes that all four

contributions to W are independent of each other This is not the case for eddy

diffusion and mobile-phase mass transfer; whenever two inter-particle flow streams combine, remixing occurs, with loss of the velocity profile created by mobile-phase mass transfer (so-called coupling) We must therefore treat these two processes (eddy diffusion and mobile-phase mass transfer) as a single band-broadening event Because eddy diffusion does not vary with flow rate (∝ F0), while mobile-phase mass transfer does (∝ F1), the combination of the two contributions to band width will

vary with some fractional power of F(F n) Experimental studies suggest a dependence

of the combined value of H for eddy diffusion plus mobile-phase mass transfer to

the1/3flow rate power, which leads to an equation of the form

longitudinal eddy diffusion+ mobile- stationary-phase

diffusion phase mass transfer mass transfer

(A, B, and C are still another set of constants) A final, generalized relationship between peak width and experimental conditions can be achieved as follows: A, B, and C of Equation (2.16c) are variously functions of the solute diffusion coefficient

D m and/or particle diameter d p, such that Equation (2.16) can be restated as the so-called Knox equation [25]:

h = Aν0.33+B

Values of the coefficients of Equation (2.17) can be assumed for an ‘‘average’’

separation: A = 1, B = 2, and C = 0.05 (these are very approximate values that

vary somewhat with the nature of the column—and how well it is packed—and

with values of k) Here we define a reduced plate height h,

h= H

d p

(2.18)

and a reduced velocity ν,

ν = u e d p

where u eis the interstitial velocity of the mobile phase, as contrasted with the average

mobile-phase velocity u; cgs units are assumed in Equations (2.18) and (2.18a) The

total porosity of the column (as a fraction of the column volume) is defined asε T, and is composed of the intra-particle porosityε i, plus the inter-particle porosityε e

The quantity u eis then equal to (ε T/ε e )u, where the quantity ( ε T/ε e)≈ 1.6.

The solute diffusion coefficient D m(cm2/sec) can be approximated by a function

of solute molecular volume (V A , in mL), temperature (T, in K), and mobile phase

viscosity (η, in cP) by the Wilke–Chang equation [26]:

D m = 7.4 × 10−8(ψ B M B)0.5 T

Here M B is the molecular weight of the solvent; the association factor ψ B is unity for most solvents, and greater than one for strongly hydrogen-bonding solvents For example,ψ B equals 2.6 for water and 1.9 for methanol (a value ofψ B≈ 2 can be assumed for typical RPC conditions) Equation (2.19) is reasonably accurate for solutes with molecular weights<500 Da, and it can represent a useful approximation

for larger molecules

The Knox equation (Eq 2.17) provides a conceptual basis for optimizing

conditions, so as to provide maximum values of N in the shortest possible time;

it also forms the basis of the predictions of Table 2.4 and the calculated plots of

Figure 2.11 That is, it is possible to predict (approximately) how values of N will

change when any experimental condition is changed The practical application of

Equation (2.17) requires the conversion of values ofν into flow rate Values of F

can be obtained from values ofν as follows: From Equations (2.4a) and (2.7),

where flow rate F is in mL/min, column diameter d c is in mm, and mobile-phase

velocity u is in mm/min Equation (2.18a) for u in mm/min, d p inμm, D m in cm2/

sec, and u e = 1.6u becomes

ν = 2.7 × 10−7ud p

D m

(2.20a)

Combining Equations (2.20) and (2.20a), we have

ν ≈ 5.4 × 10−4Fd p

and

F= 1850d c2νD m

d p

(2.20c)

Figure 2.13a shows a plot of h versus ν(—) based on Equation (2.17) with

values of A = 1, B = 2, and C = 0.05; this plot is independent of the experimental conditions listed in Table 2.4, and therefore applies (approximately) for all condi-tions A minimum value of h ≡ h opt ≈ 2 is found in Figure 2.13a, corresponding

to a value of ν ≡ ν opt ≈ 3 (these ‘‘optimum’’ values of h and ν correspond to maximum values of N) The solid curve of Figure 2.13a is repeated as the solid curve

of Figure 2.13b, except that reduced velocity ν(the x-axis) has now been replaced

by the flow rate F (for a particular set of conditions: 50% acetonitrile-water, 30◦C, 200-Da solute, 4.6-mm diameter column) When any of the conditions of Table 2.4

change for the example of Figure 2.13b, the solid-line plot for 30◦C and a 200-Da solute will be shifted right or left, depending on the effect of the change in condition

on the flow rate for maximum N (F optcorresponding toν opt≈ 3) Table 2.4 predicts

that F opt will increase for an increase in temperature, which is seen (dotted curve)

in Figure 2.13b Similarly an increase in sample molecular weight will decrease F opt

and shift the h– ν plot to the left (also seen in Fig 2.13b as the dashed curve).

Figure 2.11b can be obtained from Figure 2.13b by changing values of h to N (Eqs.

2.12, 2.18)

The reason for changes in F optwith conditions (summarized in Table 2.4) is as

follows: Any change in condition that involves either particle size d por the diffusion

coefficient D mwill change the value ofν, provided that mobile-phase velocity u (and

flow rate F) are held constant To maintain the same value of ν opt≈ 3 for minimum

h and maximum N, the flow rate must then be changed so as to compensate for the

effect of conditions onν For example, an increase in temperature will increase the

diffusion coefficient D m (Eq 2.19) by some factor x, which then results in a decrease

in ν by the same factor (Eq 2.18a) The flow rate corresponding to F opt must

accordingly be increased by the factor x, in order to compensate for the temperature

increase and holdν = ν optconstant

Also shown in Figure 2.13a are plots of each of the three terms that contribute

to Equation (2.17), with the value ofν for minimum h (and maximum N) shown by

the arrow (h ≈ 2, ν ≈ 3) The original Knox equation (Eq 2.17) has subsequently

been refined, leading to modified relationships that are claimed to offer greater accuracy, expanded applicability, and/or greater insight into the basis of column

efficiency [27–32] Additionally values of both B and C depend on the value of

k for the solute [33, 34] when stationary-phase diffusion is taken into account

[35] Consequently Equation (2.17) is mainly useful for practical, semi-quantitative application; it has even been described as a ‘‘merely empirical expression’’ [36] (we do not agree!) Nevertheless, its simplicity, convenience, and fundamental basis continue to recommend it as a conceptual tool for everyday practice