CHAPTER 11: CHROMATOGRAPHY FROM A MOLECULAR VIEWPOINT

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1 CHAPTER 11: CHROMATOGRAPHY FROM A MOLECULAR VIEWPOINT Contrasting approaches 1. bulk transport (e.g., c c = W ) t x + D c x goal: track concentration changes advantage: mathematical rigor (for simple models). molecular transport (e.g., random walk model) goal: track molecular pathway (in statistical sense) advantage: better insights for complex processes MOLECULAR BASIS OF MIGRATION molecular pathway: composed of random steps random steps from: Brownian motion erratic flow random kinetics-sorption-desorption 11.

2 statistical viewpoint many different pathways through column probability of pathway = fraction taking pathway (i.e., probability distribution concentration distribution) zone migration defines position, broadening of zone outward appearance: migration smooth molecular level: motion erratic; stop & go 11.3 Mechanism of Retardation for all molecules, general velocity states velocity = <v> (mobile phase) velocity = 0 (stationary phase) differential : from different time fractions in two states from different probabilities of two states Examples 50% probablity of <v>: 10% probability of <v>: 11.4

3 retention ratio R V R = < v > 1945 LeRosen = probability of occupancy of mobile phase ( 1 molecule) = fraction of molecules in mobile phase (n molecules) 50 % pro R =0.5 10% R is equivalent to the average fraction of time that a molecule of a given kind spends desorbed MOLECULAR BASIS OF ZONE SPREADING Fluctuations from Mean Behavior (statistical theory of zone broadening first by Giddings & Eyring) zone velocity, = mean velocity of molecules zone broadening - due to fluctuations in molecular velocities fluctuations from numerous random steps Gaussian zone (central limit theorem) 11.6

4 Random-Walk Model advantages: simplicity captures essential features of fluctuations review of random walk equations: variance for one walk: variance for > or more independent walks: H for one walk: H for independent walks: σ = l n 11.7 σ from diffusion: σ = Dt capillary GC diffusion in M. phase ---- serious substitution: D Dm, t L / v σ becomes: H becomes: 11.8

5 packed column GC consider numerous obstructions in flow path. prevention of diffusion along a straight line diffusion retardation by obstructive factor (~ 0.6) substitutions: D γdm, t L / v σ becomes: H becomes: note: eq 11.8 valid for capillary GC with = packed column LC consider longitudinal diffusion in st. ph. by contrast to GC: D s ~ D m becomes: σ = σ + m σ s H becomes: H = H m + H s as before: to get use: 11.10

6 time in stat. phase: thus: for all chromatography longitudinal diffusion H D where B = constant (independent of v) STATIONARY PHASE SORPTION-DESORPTION kinetics mean transfer times: t d mobile phase stationary phase consider sorbed molecule step direction: backward (relative to peak center) step length: = zone velocity (V) x td = R v td no. of desorptions: n d = t t s d (1 R) t = t d r 1 R = t d L Rv no. of steps: n = n d 11.1

7 thus: get: H for stat. phase given: H = l n / L adsorption chromaotgraphy: t ds : av. time needed for molecular detachment from the adsorbing surface for uniform surface: --S--M S + M for t ds use: thus: partition chromatography (desorption is diffusion controlled) we use: H becomes: 11.14

8 11.5 FLOW AND DIFFUSION IN THE MOBILE PHASE flow in a packed column - chaotic migration local velocity proportional to size of interstitial channel particle surface obstacles importantly dp chromatographic band broadening source from velocity increment molecule travel faster 1. at the center of narrow flow channel. in flow channel bet. particles than in pores 3. near the column wall than the center 4. in some channels than in others close by S : velocity increment distance n = L/S length of step: distance by which a fast-track molecule surges ahead of the av. molecule in S. v l = S v f l n H = L = plate height becomes: 11.16

9 optimization: minimize S & ω β value of S: governed by mechanisms of removal of molecules from velocity biases By how is a molecule removed from a velocity bias? mechanism a: flow S = S f scaled to d p : H = H f becomes: Eddy Diffusion Term mechanism b: diffusion Figure 11.3.b If diffusion terminates the velocity bias, S is the distance a molecule is carried downstream in the av. time t D S = S D given by s = d vt D t D = time to diffuse distance d' between channels: 11.18

10 by statistics t D = d D ' m since d' scaled to d p as d' = ϖ α d p : then Thus S D becomes: since: we get: Nonequilibrium or Mass Transfer Term Combined effects of H f and H D H = H f + H D? (additive) test additivity at two limits, v and v 0 limit: v Flow causes rapid exchange between velocity extremes. No time for diffusion. (Diffusion plays no role.) Thus: (11.9) If: H = H f + H D = A + C m v Then: H as v 0 Thus eq 11.9 not consistent with additivity 11.0

11 Limit: v 0 all exchange by diffusion Thus: If: Then: H A as v 0 Thus eq not consistent with additivity H additivity wrong. additivity is valid only for random process number of random steps -- only additive thus H becomes n = n f + n D rearrangement gives L S = n f + n D 11.

12 substitute each component H gives overall H c = H i OVERALL PLATE HEIGHT Add the longitudinal diffusion (11.8) the stationary phase term (11.18) the mobile phase term (11.37) H = γ D m v + qr(1 - R) d v Ds 1 + λ i d p + D m -1 ω i d p v longitudinal mass transfer eddy diffusion diffusion shows dependence on: v, d p, d, R, D m, D s problem: constants γ, q, ω, λ not well known i i condensed form: H = B + 1 v 1/ Ai + 1 / C mi C v + s v How to reduce H? optimization 11.4