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The Rate Theory of Chromatography
• In the rate theory, a number of different
  peak dispersion processes were
  proposed and expressions were
  developed that described
    • the contribution of each of the
      processes to the total variance of the
      eluted peak
    • the final equation that gave an
      expression for the variance per unit
      length of the column
The processes proposed were

 •Eddy diffusion
 •Longitudinal diffusion
 •Resistance to mass transfer in the
  mobile phase
 •Resistance to mass transfer in the
  stationary phase
This Theory
• Gives more realistic description of the
  processes that work inside a column
• Takes account of the time taken for the
  solute to equilibrate between the
  stationary and mobile phase (unlike the
  plate model, which assumes that
  equilibration is infinitely fast)
• The resulting band shape or a
  chromatographic peak is therefore
  affected by the rate of elution
• It is also affected by the different paths
  available to solute molecules as they travel
  between particles of the stationary phase
• If we consider the various mechanisms which
  contribute to band broadening, we arrive at the
  Van Deemter equation:

                HETP = A + B / u + C u

  where u is the average velocity of the mobile
  phase. A, B, and C are factors which contribute
  to band broadening
The Rate Theory of Chromatography

The rate theory has resulted in a number of different
equations
   All such equations give a type of hyperbolic function
   that predicts a minimum plate height at an optimum
   velocity and, thus, a maximum efficiency. At normal
   operating velocities it has been demonstrated that the
   Van Deemter equation gives the best fit to
   experimental data

           The Van Deemter Equation

        H = A + B/u + u [CM + CS]
The Rate Theory of Chromatography

The rate theory provides another equation that allows
the calculation of the variance per unit length of a
column (the height of the theoretical plate, HETP) in
terms of the mobile phase velocity and other
physicochemical properties of the solute and
distribution system

                    H = σ2/L

 σ = Standard deviation of the band
 H = plate height, which is equal to H/dP
 dP = particle diameter
The Rate Theory of Chromatography

Van Deemter plot
A plot of plate height vs average linear velocity of mobile
phase




Such plot is of considerable use in determining the optimum
mobile phase flow rate
Van Deemter model
            H = A + B/u + u [CM +CS]


A: random movement through stationary phase

B: diffusion in mobile phase

C: interaction with stationary phase

H: plate height

u: average linear velocity u = L/ tM
Van Deemter model
             H = A + B/u + u [CM +CS]




                                       time
Term A
- molecules may travel    Eddy diffusion
unequal distances         MP moves through the column
                          which is packed with stationary
- independent of u        phase. Solute molecules will take
- depends on size of      different paths through the
stationary particles or   stationary phase at random. This
coating (TLC)             will cause broadening of the
                          solute band, because different
                          paths are of different lengths.
Van Deemter model
                  H = A + B/u + u [CM +CS]

Term B
Longitudinal diffusion

      B = 2γ DM                         One of the main causes of
                                        band spreading is
                                        DIFFUSION
 γ:    Impedance factor due to
                                        The diffusion
packing                                 coefficient measures
 DM: molecular diffusion                the ratio at which a
coefficient                             substance moves
B term dominates at low u, and          randomly from a region
                                        of higher concentration
is more important in GC than LC         to a region of lower
since DM(gas) > 104 DM(liquid)          concentration
Van Deemter model
                  H = A + B/u + u [CM +CS]

 Term B

Longitudinal diffusion

      B = 2γ DM
                                   B - Longitudinal diffusion
 γ:    Impedance factor due to     The concentration of analyte is less
packing                            at the edges of the band than at
                                   the centre. Analyte diffuses out
 DM: molecular diffusion           from the centre to the edges. This
coefficient                        causes band broadening. If the
                                   velocity of the mobile phase is high
                                   then the analyte spends less time
B term dominates at low u and is   in the column, which decreases the
more important in GC than LC       effects of longitudinal diffusion.
since DM(gas) > 104 DM(liquid)
Van Deemter model
                         H = A + B/u + u [CM +CS]

  Term C
                                                    Mobile      Elution
Cs: stationary phase-mass transfer                  phase
Cs = [(df)2]/Ds                                Stationary
                                                    phase          Bandwidth
df: stationary phase film thickness                             Slow
                                                                equilibration
Ds: diffusion coefficient of analyte in SP
                                                         Broadened bandwidth
CM: mobile phase–mass transfer

CM = [(dP)2]/DM           packed columns
CM = [(dC)2]/DM           open columns
 dP: particle diameter
 dC: column diameter
Van Deemter model
                      H = A + B/u + u [CM +CS]

                                                    Mobile       Elution
                                                    phase
                                                 Stationary
 Term C (Resistance to mass transfer)                 phase         Bandwidth
                                                                 Slow
                                                                 equilibration



                                                          Broadened bandwidth

The analyte takes a certain amount of time to equilibrate between the
stationary and mobile phase. If the velocity of the mobile phase is high,
and the analyte has a strong affinity for the stationary phase, then the
analyte in the mobile phase will move ahead of the analyte in the
stationary phase. The band of analyte is broadened. The higher the
velocity of mobile phase, the worse the broadening becomes.
• Figure 1 illustrates the effect of these terms,
  both individually and accumulatively. Eddy
  diffusion, the A term, is caused by a turbulence
  in the solute flow path and is mainly unaffected
  by flow rate. Longitudinal diffusion, the B term, is
  the movement of an analyte molecule outward
  from the center to the edges of its band. Higher
  column velocities will limit this outward
  distribution, keeping the band tighter. Mass
  transfer, the C term, is the movement of analyte,
  or transfer of its mass, between the mobile and
  stationary phases. Increased flow has been
  observed to widen analyte bands, or lower peak
  efficiencies.
Van Deemter model Figure 1
    H = A + B/u + u [CM +CS]
Decreasing particle size has been observed to limit
the effect of flow rate on peak efficiency—smaller
particles have shorter diffusion path lengths,
allowing a solute to travel in and out of the particle
faster. Therefore the analyte spends less time
inside the particle where peak diffusion can occur.
Figure 2 illustrates the Van Deemter plots for
various particle sizes. It is clear that as the particle
size decreases, the curve becomes flatter, or less
affected by higher column flow rates. Smaller
particle sizes yield better overall efficiencies, or
less peak dispersion, across a much wider range
of usable flow rates.
Smaller particle sizes yield higher overall peak
efficiencies and a much wider range of usable flow
                  rates (Figure 2)
Resolution
• Ideal chromatogram exhibits a distinct
separate peak for each solute
   in reality: chromatographic peaks often
   overlap
• We call the degree of separation of two
peaks: resolution which is given as
   resolution = peak separation/average
   peak                              width
Resolution

•Resolution =∆ tr / wavg
•let’s take a closer look at the significance of
 the problem:
Resolution

•So, separation of mixtures depends on:
–width of solute peaks (want narrow)
efficiency
–spacing between peaks (want large
  spacing)
selectivity
Example

•What is the resolution of two Gaussian
peaks of identical width (3.27 s) and height
eluting at 67.3 s and 74.9 s, respectively?
•ANS: Resolution = 2.32

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Rate theory

  • 1. The Rate Theory of Chromatography • In the rate theory, a number of different peak dispersion processes were proposed and expressions were developed that described • the contribution of each of the processes to the total variance of the eluted peak • the final equation that gave an expression for the variance per unit length of the column
  • 2. The processes proposed were •Eddy diffusion •Longitudinal diffusion •Resistance to mass transfer in the mobile phase •Resistance to mass transfer in the stationary phase
  • 3. This Theory • Gives more realistic description of the processes that work inside a column • Takes account of the time taken for the solute to equilibrate between the stationary and mobile phase (unlike the plate model, which assumes that equilibration is infinitely fast) • The resulting band shape or a chromatographic peak is therefore affected by the rate of elution
  • 4. • It is also affected by the different paths available to solute molecules as they travel between particles of the stationary phase • If we consider the various mechanisms which contribute to band broadening, we arrive at the Van Deemter equation: HETP = A + B / u + C u where u is the average velocity of the mobile phase. A, B, and C are factors which contribute to band broadening
  • 5. The Rate Theory of Chromatography The rate theory has resulted in a number of different equations All such equations give a type of hyperbolic function that predicts a minimum plate height at an optimum velocity and, thus, a maximum efficiency. At normal operating velocities it has been demonstrated that the Van Deemter equation gives the best fit to experimental data The Van Deemter Equation H = A + B/u + u [CM + CS]
  • 6. The Rate Theory of Chromatography The rate theory provides another equation that allows the calculation of the variance per unit length of a column (the height of the theoretical plate, HETP) in terms of the mobile phase velocity and other physicochemical properties of the solute and distribution system H = σ2/L σ = Standard deviation of the band H = plate height, which is equal to H/dP dP = particle diameter
  • 7. The Rate Theory of Chromatography Van Deemter plot A plot of plate height vs average linear velocity of mobile phase Such plot is of considerable use in determining the optimum mobile phase flow rate
  • 8. Van Deemter model H = A + B/u + u [CM +CS] A: random movement through stationary phase B: diffusion in mobile phase C: interaction with stationary phase H: plate height u: average linear velocity u = L/ tM
  • 9. Van Deemter model H = A + B/u + u [CM +CS] time Term A - molecules may travel Eddy diffusion unequal distances MP moves through the column which is packed with stationary - independent of u phase. Solute molecules will take - depends on size of different paths through the stationary particles or stationary phase at random. This coating (TLC) will cause broadening of the solute band, because different paths are of different lengths.
  • 10. Van Deemter model H = A + B/u + u [CM +CS] Term B Longitudinal diffusion B = 2γ DM One of the main causes of band spreading is DIFFUSION γ: Impedance factor due to The diffusion packing coefficient measures DM: molecular diffusion the ratio at which a coefficient substance moves B term dominates at low u, and randomly from a region of higher concentration is more important in GC than LC to a region of lower since DM(gas) > 104 DM(liquid) concentration
  • 11. Van Deemter model H = A + B/u + u [CM +CS] Term B Longitudinal diffusion B = 2γ DM B - Longitudinal diffusion γ: Impedance factor due to The concentration of analyte is less packing at the edges of the band than at the centre. Analyte diffuses out DM: molecular diffusion from the centre to the edges. This coefficient causes band broadening. If the velocity of the mobile phase is high then the analyte spends less time B term dominates at low u and is in the column, which decreases the more important in GC than LC effects of longitudinal diffusion. since DM(gas) > 104 DM(liquid)
  • 12. Van Deemter model H = A + B/u + u [CM +CS] Term C Mobile Elution Cs: stationary phase-mass transfer phase Cs = [(df)2]/Ds Stationary phase Bandwidth df: stationary phase film thickness Slow equilibration Ds: diffusion coefficient of analyte in SP Broadened bandwidth CM: mobile phase–mass transfer CM = [(dP)2]/DM packed columns CM = [(dC)2]/DM open columns dP: particle diameter dC: column diameter
  • 13. Van Deemter model H = A + B/u + u [CM +CS] Mobile Elution phase Stationary Term C (Resistance to mass transfer) phase Bandwidth Slow equilibration Broadened bandwidth The analyte takes a certain amount of time to equilibrate between the stationary and mobile phase. If the velocity of the mobile phase is high, and the analyte has a strong affinity for the stationary phase, then the analyte in the mobile phase will move ahead of the analyte in the stationary phase. The band of analyte is broadened. The higher the velocity of mobile phase, the worse the broadening becomes.
  • 14. • Figure 1 illustrates the effect of these terms, both individually and accumulatively. Eddy diffusion, the A term, is caused by a turbulence in the solute flow path and is mainly unaffected by flow rate. Longitudinal diffusion, the B term, is the movement of an analyte molecule outward from the center to the edges of its band. Higher column velocities will limit this outward distribution, keeping the band tighter. Mass transfer, the C term, is the movement of analyte, or transfer of its mass, between the mobile and stationary phases. Increased flow has been observed to widen analyte bands, or lower peak efficiencies.
  • 15. Van Deemter model Figure 1 H = A + B/u + u [CM +CS]
  • 16. Decreasing particle size has been observed to limit the effect of flow rate on peak efficiency—smaller particles have shorter diffusion path lengths, allowing a solute to travel in and out of the particle faster. Therefore the analyte spends less time inside the particle where peak diffusion can occur. Figure 2 illustrates the Van Deemter plots for various particle sizes. It is clear that as the particle size decreases, the curve becomes flatter, or less affected by higher column flow rates. Smaller particle sizes yield better overall efficiencies, or less peak dispersion, across a much wider range of usable flow rates.
  • 17. Smaller particle sizes yield higher overall peak efficiencies and a much wider range of usable flow rates (Figure 2)
  • 18. Resolution • Ideal chromatogram exhibits a distinct separate peak for each solute in reality: chromatographic peaks often overlap • We call the degree of separation of two peaks: resolution which is given as resolution = peak separation/average peak width
  • 19. Resolution •Resolution =∆ tr / wavg •let’s take a closer look at the significance of the problem:
  • 20. Resolution •So, separation of mixtures depends on: –width of solute peaks (want narrow) efficiency –spacing between peaks (want large spacing) selectivity
  • 21. Example •What is the resolution of two Gaussian peaks of identical width (3.27 s) and height eluting at 67.3 s and 74.9 s, respectively? •ANS: Resolution = 2.32