Lecture 2 ChromatographyTheory HVCH V2

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Part II Chromatographic Theory 

Separation process ‐ Elution O

 Partition between mobile phase/stationary phase: phase/stationary  phase: Sm ⇋ Ss

Sm: the solute in the mobile phase Ss: the solute in the stationary phase. stationary phase.

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Theories Two approaches explain the separation process:

O

Plate theory –Martin and Synge (1941), based on an analogy  with  with distillation and countercurrent extraction.  Rate theory –  van Deemter (1956), accounts for the dynamics of a of  a separation. 

 Advantages o  Limitations o

Solute Retention Factor O

equilibrium constant), K partition coefficient ( equilibrium



 

 

 

K is assumed to be independent of concentration.

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Solute Retention Factor O

Capacity ratio, k

  

    



      

t′ R - the adjusted retention t ime

Retention time O

retention time (tR ): time between sample injection and an analyte peak reaching a detector at the end of the column

O

The time taken for the mobile phase to pass through the column is called tM.

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Solute Retention Factor O

Solute fractions

 ,  

1 1

 ,  



If constant mobile phase velocity:

 1

  

   

: retention factor, k, the capacity factor, the capacity ratio, and the partition ratio, and is sometimes given the symbol k ′ .

Example 1 In a chromatographic analysis of low molecular weight acids, butyric acid elutes with a retention time of 7.63 min. The column’s void time is 0.31 min. Calculate the retention factor for butyric acid.

K butyric= (tr − tm) / tm= (7.63 min − 0.31 min) / 0.31 min = 23.6

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Example 2 Relationship between elution time and distance is proportional, we can measure tm, tr,1, and tr,2 using a ruler. The measurements are 7.8 mm, 40.2 mm, and 51.5 mm, respectively. Chromatogram for a two-component mixture. , sample injected time t = 0. : The retention factors for solute A and solute B are: k 1 = (tr − tm) / tm = (40.2 mm − 7.8 mm) / 7.8 mm = 4.15 k 2 =(tr − tm) / tm = (51.5 mm − 7.8 mm) / 7.8 mm = 5.60

Retention time and volume O

, VR – volume of mobile phase required to elute a solute to a maximum from a column.

O

, tR – time required to reach the same maximum at constant flow.       

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Retention time and volume O

 



 



, tM

O

t   1  

Component separation O

             O

2 1 



2  1

tR2 > tR1

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Selectivity factor O

the separation of two species (A and B) on the column;

Example 3 In the chromatographic analysis for low molecular weight acids, the column’s void time is 0.31 min. The retention time for isobutyric acid is 5.98 min. What is the selectivity factor for isobutyric acid and butyric acid (retention time of 7.63 min)?

Calculate the retention factor for isobutyric acid and butyric acid. k iso= (tr− tm) / tm = (5.98 min − 0.31 min) / 0.31 min = 18.3 The selectivity factor, therefore, is α = k but / k iso =23.6 / 18.3 = 1.29

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Example 4 O

Determine the selectivity factor

The selectivity factor is α = k 2 / k 1 = 5.60 / 4.15 = 1.35

Column Efficiency 

  

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Plate theory of Chromatography   A chromatographic column is mathematically divided into theoretical plates (N). O There is an equilibrium partitioning of the solute between the stationary phase and the mobile phase. O

O

The analyte moves down the column by transfer of equilibrated mobile phase from one plate to the next.

Determination of N 

Column efficiency in terms of the number of theoretical plates, N, N = L / H

L - column’s length; H – plate height. Note: more theoretical plates (the better)    Column efficiency improves   chromatographic peaks become narrower. 

Height Equivalent to a Theoretical Plate HETP = L / N  the

smaller the better.

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Determination of N If peak broadening: H is the variance per unit length of the column (.

H = σ2 / L σ - standard deviation, sec or min, Time of elution, T ( σ /solute’s average linear velocity, ν  ). τ = σ / ν  = σtr  / L The solute’s average linear velocity is the distance it travels, L, divided by its retention time, tr.

Determination of N For a Gaussian peak shape, the width at the baseline, w , is four times its standard deviation, τ. w = 4τ Height of a theoretical plate in terms of tr and w : H = Lw 2 / 16tr2 Number of theoretical plates: N = 16(tr 2 / w 2)

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Determination of N Since it is difficult to accurately measure the beginning and end of a peak, it is common to use the width at half height and assume the peak is Gaussian.

Example 5 A chromatographic analysis for the chlorinated pesticide Dieldrin gives a peak with a retention time of 8.68 min and a baseline width of 0.29 min. What is the number of theoretical plates? Given that the column is 2.0 m long, what is the height of a theoretical plate in mm?

The number of theoretical plates: N = 16 tr 2 /w 2 = N = 16(8.68 min) 2 / (0.29 min)2 = 14300 plates The average height of a theoretical plate: H = L / N = (2.0 m / 14300 plates) × (1000 mm / m) = 0.14 mm/plate

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Resolution O

Resolution, R s , how well two neighboring peaks completely separated from each other.

O

The resolution of two peaks A and B:

Resolution O

controlling the capacity factor, k , separations greatly improved: O

by changing the temperature (in GC) or the composition of the mobile phase (in LC).

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Example 6 In a chromatographic analysis of lemon oil a peak for limonene has a retention time of 8.36 min with a baseline width of 0.96 min.  γ-Terpinene elutes at 9.54 min with a baseline width of 0.64 min. What is the resolution between the two peaks? Please give your comments of the results.

The resolution is RAB = 2∆t / r (w B + w A) = 2(9.54 min − 8.36 min) / (1.64 min + 0.96 min) = 1.48

Example 7 O

This Figure shows the separation of a two component mixture. What is the resolution between the two components? Use a ruler to measure ∆tr, w A, and w B in millimeters.

Measurements are 8.5 mm for ∆tr, and 12.0 mm each for w A and w B Using these values, the resolution is RAB = 2∆tr / (w A + w B) = 2(8.5 mm) / (12.0 mm + 12.0 mm) = 0.70

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Peak Capacity  o

o

A measure of the number of solutes that can be separated, nc. Based on isothermal/isocratic conditions. nc = 1 + (√(N)/ 4)ln(V max / V min)

nc - column’s V min  and V max  - the smallest and the largest volumes of mobile phase in which we can elute and detect a solute.

Example 8 O

A column with 10 000 theoretical plates can resolve no more than: nc = 1 + (√(10000) / 4)ln(30 mL / 1 mL) = 86 solutes

assumed V min =1 mL and V max = 30 mL.  consideration a column enough theoretical plates to

separate a complex mixture.

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 Asymmetric Peaks O

Poisson distribution

O

Gaussian distribution

Optimizing Chromatographic Separations Define the effects of  solute retention factor  selectivity,  column efficiency  resolution of two closely eluting peaks. If the two peaks (A and B) have similar retention times, it is reasonable to assume that their peak widths are nearly identical: RAB = (tr,B − tr,A) / (0.5(w B +w A)) ≈ (tr,B − tr,A)/(0.5(2w B)) = (tr,B −tr,A)/w B RAB= (√(N) / 4) × (tr,B − tr,A) / tr,B

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Optimizing Chromatographic Separations Reretention times of solutes A and B.

tr,A = k AtM + tM

tr,B = kBtM + tM

R AB = (√(N) / 4) × ((kB - k A) / (1 + kB)) RAB= (√(N) / 4) × (( α−1) / α) × (k B / (1 + k B)) tr,B= (16RAB2H / u) × (α / (α – 1)) 2 × ((1 + k B)3 / k B2) u - mobile phase’s velocity

Using the Retention factor to Optimize Resolution

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Using Selectivity to Optimize Resolution

Using Column Efficiency to Optimize Resolution

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Rate Theory of Chromatography  O

O

: O O O O O O O O

Phase properties Phase thickness Solute diffusivities Support size Partition coefficients Support porosity Phase velocity Flow rates

Rate Theory of Chromatography  O

O

(inside a column): time for the solute to equilibrate between the stationary and mobile phase. band shape of chromatographic peak is af fected by: O O

O

the rate of elution the different paths available to solute molecules as they travel between particles of stationary phase.

band broadening: O O O O

variations in paths length longitudinal diffusion mass transfer in the stationary phase, and mass transfer in the mobile phase.

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Equation of parameters The height of a theoretical plate is a summation of the contributions: H = Hp + Hd + Hs + Hm An alternative form of this equation is the accounting for the dynamics of the separation process.

 Van Deemter equation

 – factor characteristic of packing dP – particle diameter  – factor for irregularity of interparticle spaces Dg  – diffusion coefficient of compound in gas Dl – diffusion coefficient of compound in liquid u – linear gas velocity k – capacity ratio df  – liquid phase effective film thickness H – height of theoretical plate

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 Van Deemter equation The equation consists of three basic terms. Packing related term

Gas (mobile phase) term

Liquid (stationary phase) term

 Van Deemter equation HETP = A + B / u + C u u - average velocity of the mobile phase. O

O

O

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 Van Deemter plots

Plot showing the relationship between the height of a theoretical plate, H, and the mobile phase’s velocity, u, based on the van Deemter equation.

Other equations 

H = (B / u)+ (C s + C m)u

C  s and C m - mass transfer for the stationary phase and the mobile phase



H = Au1/3 + (B / u) + Cu

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