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# Rate theory

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### Rate theory

1. 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. 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. 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. 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. 5. The Rate Theory of ChromatographyThe rate theory has resulted in a number of differentequations 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. 6. The Rate Theory of ChromatographyThe rate theory provides another equation that allowsthe calculation of the variance per unit length of acolumn (the height of the theoretical plate, HETP) interms of the mobile phase velocity and otherphysicochemical properties of the solute anddistribution system H = σ2/L σ = Standard deviation of the band H = plate height, which is equal to H/dP dP = particle diameter
7. 7. The Rate Theory of ChromatographyVan Deemter plotA plot of plate height vs average linear velocity of mobilephaseSuch plot is of considerable use in determining the optimummobile phase flow rate
8. 8. Van Deemter model H = A + B/u + u [CM +CS]A: random movement through stationary phaseB: diffusion in mobile phaseC: interaction with stationary phaseH: plate heightu: average linear velocity u = L/ tM
9. 9. Van Deemter model H = A + B/u + u [CM +CS] timeTerm A- molecules may travel Eddy diffusionunequal 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 thestationary particles or stationary phase at random. Thiscoating (TLC) will cause broadening of the solute band, because different paths are of different lengths.
10. 10. Van Deemter model H = A + B/u + u [CM +CS]Term BLongitudinal diffusion B = 2γ DM One of the main causes of band spreading is DIFFUSION γ: Impedance factor due to The diffusionpacking coefficient measures DM: molecular diffusion the ratio at which acoefficient substance movesB term dominates at low u, and randomly from a region of higher concentrationis more important in GC than LC to a region of lowersince DM(gas) > 104 DM(liquid) concentration
11. 11. Van Deemter model H = A + B/u + u [CM +CS] Term BLongitudinal diffusion B = 2γ DM B - Longitudinal diffusion γ: Impedance factor due to The concentration of analyte is lesspacking at the edges of the band than at the centre. Analyte diffuses out DM: molecular diffusion from the centre to the edges. Thiscoefficient causes band broadening. If the velocity of the mobile phase is high then the analyte spends less timeB term dominates at low u and is in the column, which decreases themore important in GC than LC effects of longitudinal diffusion.since DM(gas) > 104 DM(liquid)
12. 12. Van Deemter model H = A + B/u + u [CM +CS] Term C Mobile ElutionCs: stationary phase-mass transfer phaseCs = [(df)2]/Ds Stationary phase Bandwidthdf: stationary phase film thickness Slow equilibrationDs: diffusion coefficient of analyte in SP Broadened bandwidthCM: mobile phase–mass transferCM = [(dP)2]/DM packed columnsCM = [(dC)2]/DM open columns dP: particle diameter dC: column diameter
13. 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 bandwidthThe analyte takes a certain amount of time to equilibrate between thestationary and mobile phase. If the velocity of the mobile phase is high,and the analyte has a strong affinity for the stationary phase, then theanalyte in the mobile phase will move ahead of the analyte in thestationary phase. The band of analyte is broadened. The higher thevelocity of mobile phase, the worse the broadening becomes.
14. 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. 15. Van Deemter model Figure 1 H = A + B/u + u [CM +CS]
16. 16. Decreasing particle size has been observed to limitthe effect of flow rate on peak efficiency—smallerparticles have shorter diffusion path lengths,allowing a solute to travel in and out of the particlefaster. Therefore the analyte spends less timeinside the particle where peak diffusion can occur.Figure 2 illustrates the Van Deemter plots forvarious particle sizes. It is clear that as the particlesize decreases, the curve becomes flatter, or lessaffected by higher column flow rates. Smallerparticle sizes yield better overall efficiencies, orless peak dispersion, across a much wider rangeof usable flow rates.
17. 17. Smaller particle sizes yield higher overall peakefficiencies and a much wider range of usable flow rates (Figure 2)
18. 18. Resolution• Ideal chromatogram exhibits a distinctseparate peak for each solute in reality: chromatographic peaks often overlap• We call the degree of separation of twopeaks: resolution which is given as resolution = peak separation/average peak width
19. 19. Resolution•Resolution =∆ tr / wavg•let’s take a closer look at the significance of the problem:
20. 20. Resolution•So, separation of mixtures depends on:–width of solute peaks (want narrow)efficiency–spacing between peaks (want large spacing)selectivity
21. 21. Example•What is the resolution of two Gaussianpeaks of identical width (3.27 s) and heighteluting at 67.3 s and 74.9 s, respectively?•ANS: Resolution = 2.32