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HPLC Theory and Equations

 

To understand why an HPLC separation occurs it is instructive to have some theoretical explanation.  Chromatographic theory has been developed over a period of 100 years, using models derived from experimental results and then hammered into shape by physical chemists.  For the full monte see the bibliography of HPLC text books.  Otherwise here goes….

 

There are three basic parameters which we can measure from a chromatogram, which affect peak resolution.

Relative Retention Time, k’

 

Relative retention time is a measure of retention time which is independent of flow rate or column length or diameter. It is also known as the Capacity Factor.

 

                                  

Here, tr is the retention time of the peak, and t0 is the retention time of the solvent front (unretained peak).

 

In isocratic chromatography (elution with a constant eluent strength), elution occurs in a given volume of eluent. So if we double the flow rate, the elution volume passes through the column in half the time so we halve the retention time. But k’ remains constant.

 

In isocratic HPLC, we vary k’ by using a stronger or weaker mobile phase. So for example in reversed phase HPLC, using a higher % methanol in the eluent reduces the retention time of all the peaks, but the retention time of the solvent front remains unchanged.

Selectivity, α

Selectivity is a measure of the extent to which two components are separated.

 

 

                           

 

Because k’ is a thermodynamic parameter, and hence constant for a given phase system, so is selectivity.

 

Selectivity is a function of the interaction between the stationary and mobile phases in equilibrium with the sample components. So to change selectivity, we need to change the stationary or mobile phases. By far the greatest changes can be achieved by changing the mobile phase.

 

There are several parameters of the mobile phase which we can change to vary selectivity. It should be noted that selectivity can get worse as well as better!

 

  1. Temperature. Increasing the temperature increases the energy of all the molecules, improving mass transfer in and out of the pores. As a consequence, and as a general rule, peaks are sharper and elute faster at higher temperatures. However this tends to make the peaks closer together, reducing resolution.
  2. pH. Components which are ionisable, such as acidic or basic compounds, give rise to elution times which are very pH dependent. The more charged the species is, the shorter the retention time. So for acidic compounds, retention times are longest at low pH; for basic compounds, retention times are longest at high pH (above 7); and for zwitterionic species (containing both acidic and basic groups such as an amino acid) longest retention time come in the pH region 5-7. Having established the most suitable pH to use, accurate and precise control with a suitable buffer is essential.
  3. Solvent type. Normally we would use three solvents in reversed phase HPLC to achieve the greatest selectivity differences. Acetonitrile has a dipole influence, methanol a slight proton donor effect and tetrahydrofuran a slight proton acceptor effect. Each of these three solvents may be mixed with water to form a reversed phase eluent, and so changing from MeOH:H2O to MeCN:H2O would be expected to give rise to a change in selectivity.
  4. Eluent composition. Changing the eluent composition changes k’, but the rate of change is different for different components. Hence the rate of migration of peaks in the chromatogram as eluent composition is varied  is different, and this can cause peaks to overlap, separate, and even change places.
  5. Buffer type. See the section on buffers for a guide as to which buffer to choose. But changing from one buffer to another can cause selectivity changes.
  6. Buffer concentration. Increasing buffer concentration causes peaks to elute more quickly. But as with %B changes, the effect on some components is greater than others. For a given separation there will be an ideal buffer concentration, and once found, it is important that this is controlled carefully.
  7. Ion Pair Reagent. An ion pair reagent can be used to pair up with polarised sample components, enabling them to elute with less polar species in the chromatogram. It is not a recommended eluent system, because ion pair methods are often not robust. But it does affect selectivity.

Changes to the stationary phase offer less possibilities, but may prove effective.  This can be for example, a change from C18 to C8, or from one type of C18 to another. After each column change, the system must be re-equilibrated thoroughly before evaluating the success or otherwise of the change.

 

Efficiency, N

The above properties define where in the chromatogram the peak maximum occurs and the extent of the separation between them, but they cannot indicate the extent of resolution of the peaks.

 

 

Both the above chromatograms have the same thermodynamic properties but chromatogram (2) is clearly better resolved than chromatogram (1) because the peaks are sharper.

 

Efficiency is calculated from the equations below. Of the two, the second equation where peak width is measured at half the peak height is the most useful, and the easiest to measure.

 

                                   

 

              Or         

 

 

N = efficiency.

tr = Retention time of the peak

wb = the peak width at its base

w1/2 = the peak width at half height

 

To obtain meaningful results, highly accurate measurements of wb or w1/2 is required.  Hence it is normal to use w1/2 because of the difficulty in meaning wb

 

Since tr an w are both measured in either time or volume units, N is essentially dimensionless, and is expressed as a number. It is normal to use to concept of theoretical plates (as in a distillation column), when describing efficiency. Rather than the number of theoretical plates per column, efficiency is sometimes measured in plates per meter (N/m). This is then independent of column length.

 

Efficiency is affected by several parameters:

 

  1. Particle Size. The smaller the particle size of the column packing, the greater the surface area. Since the separation process of adsorption and desorption occurs on the surface of the stationary phase, increasing the surface area increases the efficiency of this process, giving rise to sharper peaks. Note that 90% of the surface area of a packing material is inside the pores, so it is essential that the molecular size of the sample components is not too great to enter the pores. For an 80A pore diameter silica, a maximum molecular weight of about 2000 is recommended, while for a 120A silica, the molecular weight limit is nearer 5000.
  2. Eluent temperature. The higher the eluent temperature, the more efficient the mass transfer process, and hence the sharper the peaks.
  3. Column Length. A longer column gives shaper peaks, until the time in the column is so long that diffusion becomes an issue.
  4. Eluent viscosity. The lower the viscosity of the eluent, the more efficient the transfer process in and out of the pores. This may be achieved either by selecting a non-viscous eluent component such as acetonitrile, or by raising the temperature, or both.
  5. Linear velocity of the eluent. At very low flow rates the diffusion rate of the solute approaches the flow velocity and very poor efficiency is observed.  At higher flow rates the contributions from the other components of band broadening become significant and they increase with flow velocity.  Hence it is anticipated that at high and low flow rates efficiency will be poor and in between an optimum flow corresponding to a maximum efficiency.  This was first plotted by van Deemter

 

 

 

For a given column diameter, eluent linear velocity approximates to flow rate. For a 4.6mm id column, the optimum flow rate for efficiency is about 0.5ml/min, but as can be seen from the above diagram, at 1ml/min the efficiency is not a lot worse, but the analysis time is halved, so as a general rule, 1ml/min is the standard flow rate used in HPLC. Higher flow rates are also possible, with the limiting factor then being the system back pressure.

 

Peak Asymmetry, As

Not all peaks are symmetrical and asymmetry is defined as follows:

 

 

 

 

 

a and b are measured at 50% of peak height

 

The width of a statistical distribution is defined by its standard deviation and this can be applied to a chromatographic peak.  Hence the standard deviation has the same dimensions as retention measurement (usually time, occasionally volume).  The longer a peak is retained the more it is subject to band broadening, and it becomes wider.  Thus the standard deviation of the peak is dependent upon retention time.

 

Peaks should of course be symmetrical, and so the Asymmetry should be as close to 1 as possible. Deviations from this suggest a problem with the method (perhaps the wrong pH has been specified) or a problem with the column.

Resolution, Rs

Resolution between peaks is measured as the ratio of the difference in retention times to the average of their baseline peak widths. So it can be measured directly from the chromatogram using the equation below:

 

 

tr2

=

Retention time of peak 2

tr1

=

Retention time of peak 1

w1

=

Peak width of peak 1

w2

=

Peak width of peak 2

 

                                                                

However it is more usually expressed as shown below, in terms of N, k’ and α:

 

 

                           

 

 

Resolution is expressed as a number, and for baseline resolution of two peaks, a resolution of 1.5 or greater is required.

For a chromatogram containing more than two peaks, the concept of Critical Resolution is useful. This is a measure of the resolution of the least-well-resolved pair of peaks in the chromatogram. By definition therefore, all other peaks in the chromatogram are better resolved.

Solvophobic Theory

Solvophobic theory was developed by Prof C Horvath et al in 1976. It defines six terms which affect retention on a reversed phase column. However of these by far the most important is one which defines retention in terms of energy, showing that the attraction of the sample to the Stationary Phase is because they are both repelled by water. The size of the hydrophobic effect is proportional to the surface area exposed to the water, and by adsorbing onto the column, this surface area is reduced, making adsorption a lower energy state.

Hence the water in the eluent is there to cause retention, and the organic modifier is there to cause elution.

 

The importance of this is that the chain length of the bonded phase has a much smaller effect on selectivity than the mobile phase, and the eluent has the dominant role in governing retention in HPLC.

 

The major driving force for retention comes from the highly ordered structure of the water molecule, which cannot interact with hydrophobic (especially hydrocarbon) molecular surfaces. At the interface to hydrophobic molecules, the water surface is in an elevated energy state, and to minimize this energy, samples retain on the surface of the stationary phase, thereby reducing the exposed surface areas and releasing energy.

The significance of this for method development? Optimise the eluent parameters (%B, temperature, pH, buffer concentration and if relevant, gradient conditions) before optimisation of column choice.

Acid-Base Equilibria

Solvophobic Theory can be used to explain the changes in k’ seen for ionisable species.

 

The equilibria involved are:

 

             

             

             

 

Many compounds and their degradation products contain or basic or acidic groups, and for all of these, their HPLC retention will be very pH-dependent. Charged species will have significantly shorter retention times than the uncharged form.

 

Using solvophobic theory, the charged part of the molecule in an aqueous environment aquires a solvation layer of water molecules, which greatly reduces the hydrophobic surface area exposed to the water. As a consequence, the repulsion forces causing retention are significantly reduced, and hence the species has a lower k’ than the uncharged form.

 

Hence for acidic materials (which form negatively charged ions when ionised) it is preferable to operate at low pH (around pH2-3), the most appropriate pH being determined experimentally. And for basic species (which form positively charged ions when ionised), best results are obtained at high pH (7-10), although care must be taken to select a column which is stable at this pH.

 

For Ion Exchange separations, where separation is based upon the relative charges of the ions in the sample, the principles are exactly the same, but the opposite pH is selected because ionisation is desired (required), whereas in reversed phase, ion-suppression is desired. Cations (positively charged species) are analysed at low pH, and anions (negatively charged atoms or molecules) are separated at high pH. Note that Ion Exchange is generally much less selective than reversed phase and so is not necessarily the first choice when ionisable samples are to be analysed.