Chromatographic Theory

7/28/2019 Chromatographic Theory

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CHROMATOGRAPHIC THEORY

Prof. Derick CarbooChemistry DepartmentUniversity of Ghana

LegonE-mail: [email protected]

2/14/2010 1Prof Derick Carboo


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Individual solutes interact withstationary phase to differentdegrees.The interactions may be adsorption,relative solubility, charge, or dipole-dipole interaction, van der Waals etc.

So they are retarded by the st.phase differently.For example: A component which is quitesoluble in the st. phase will take longer to travelthrough the column than one which is lesssoluble in the st. phase but more soluble in themobile phase.



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Chromatographic theory:As a result of these differences in mobilities,

sample components will become separatedfrom each other as they travel through the

column.The sample is transported through the

column by continuous addition ofmobile phase (Elution).

The average rateat which an analyte movesthrough the column is determined by thetime it spends in the mobile phase. (It isassumed that solute does not move in the

st. phase)So if it has a higher affinity for the st. phaseit would be retarded even more.



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Chromatographic theory:

The Retention time, tr for each component is the time

needed after injection of the mixture onto the

column until it reaches the detector.

The Void time, tm (or dead time) is the time taken by the

(un-retained) mobile phase to travel through the

columnThe adjusted retention time for a solute is the

additional time required for the solute to travel the

length of the column beyond the time required by

the unretained mobile phase (solvent).

Adjusted retention time : tr = trtm



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Relative retention (also known as selectivity factor) :For any two components 1 and 2 the Relative retention

is given by the ratio of the adjusted retention times:

= tr2/ tr1 (where tr2> tr1 , so > 1 ) is a measure of the separation between thetwo components. The greater , the greater

the separation between the two solutes (on thechromatogram).

Relative retention is fairly independent of flowrate and can therefore be used to help identifypeaks when flow rate changes. can also be related to the partition coefficients of thesolutes:

= Ka/Kb (the ratio must be greater than unity) If Ka >Kb , then solute A is more retained on the stationaryphase than solute B.



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2. Capacity factor (also known as retention factor, capacity ratioor partition ratio)

Is a measure of solute velocity through a columncompared to mobile phase

In a chromatographic column, the solute is distributed between thestationary phase and the mobile phase.

The fraction in the mobile phase moves at the same velocity as the

mobile phase

The fraction in the stationary phase is considered as having zero velocity.

For solute A, the capacity factor is given by :

kA =time solute spends in stationary phasetime solute spends in mobile phase

Or kA = trtm/tm



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The longer a solute is retained by the columnthe greater is the capacity factor.

Very short capacity factor (20) means elution

time too long (leading to band broadening) Ideal capacity factor 1 5. Capacity factors can be manipulated in GC by

changes in Temp and column packing

(packed column) , and In LC by changes in mobile phase

composition and stationary phase.



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Efficiency is how well a chromatographicsystem can separate compounds. Thisdepends on two factors:

1. Differences in the retention times of thesolutes: the further apart the peaks the

better the separation2. How broad the peaks are: the broader the

peaks the poorer the separation.So the Efficiency of the column also refers to

the extent of band broadening that occurswhen solute traverses the column.



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As a band of solute moves through the chromato-

graphic column it tends to spread. This manifestsitself in a broadening of the chromatographic peak.

Typical injected volume is 5-20 L; Typical collectedvolume is 1000L.

Band Broadening is caused by:

1) the non-even flows around and insidethe porous particles,

2) slow adsorption kinetics,3) longitudinal diffusion, and other factors.



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The longer the component is retained, the more

broad its zone.

Band broadening is, in general, dependent on the

a) adsorbent particle size,b) adsorbent porosity,

c) adsorbent pore size,

d) column size, shape,

e) and packing performance.



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In the ideal case, the chromatographic peak can be

represented by a Gaussian curve with the standarddeviation .

The ratio of standard deviation to the peak retentiontime /tr is called the relative standard deviation, which

is independent on the flow rate.

The width of the curve is measured by :

(1) Peak width at half height, w1/2 = 2.35

(2) Peak width at base, w = 4

(3) Peak width at 0.67 height w0.67 = 2



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In chromatography resolution of two peaks is afunction of the separation between the peaks. Its

defined quantitatively by:Resolution = tr / wav = Vr / Wav

tr or Vr = separation between the peaks ; Wav = average width of the two peaksat base

Example: Given trA = 407s, width at base = 13s;

trB = 424s, width at base = 16s

Find the resolution.

Resolution = tr / wav = (424 404) / 0.5(13 + 16)= 1.17



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time(s)

Prof Derick Carboo


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Diffusion is the spontaneous movement of solutemolecules from a region of high concentration toa region of low concentration

Fick's first law of diffusion describes the number ofmoles crossing each square meter per second:

J(mol/m2s) =D dc/dx

where J is the flux ; D is diffusion coefficient;dc/dx is concentration gradient

The () sign shows the decrease with distance.In general: Diffusion in liquid is 10,000 times slower than ingases. Macromolecules diffuse 10 to 100 times slower

than small molecules.



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If the solute was injected as infinitely sharplayer with m moles per unit cross section area

and spreads by diffusive mechanism as ittravels, then the Gaussian profile of the bandis described by :

c = m.e-x2/4Dt

4Dt

where t is time, and x is distance along thecolumn from the current center of thebandand c is the concentration mol/m3.

The standard deviation of the band is:= 2Dt



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Two approaches can be taken to explain the separationprocess:

The PLATE THEORY proposed by in 1941 byMartin and Synge. The theory is based on ananalogy with distillation and counter currentextraction.

RATE THEORY proposed by van Deemter in1956 accounts for the dynamics of a

separation.Each has its own advantages and limitations.



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The Plate theory: Distillation is a technique used to separate

liquids over their volatilities or boiling points. Amixture of the liquids is heated and the vapoursare in equilibrium with the liquid. The morevolatile or the one with the lower boiling pointrises and is collected by cooling it in acondenser.

We can identify simple distillation where theboiling points of the components are verydifferent ( 30oC )so separation is achievedwithout a fractionating column.

In case where the boiling points are closerfractional distillation is used. Here a column isplaced in between the vessel and condenser toallow a longer path for the components toseparate.



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In fractional distillation a mixture of two or more liquidshaving slightly different boiling points can be separated

using a fractionating column. A typical fractionatingcolumn is the bubble cap column. This contains a numberof shallow trays or plates capable of holding a thin layer ofliquid. Each plate has an overflow, which allows excess

liquid to flow to the plate below, and several bubble capsthrough which vapour rising upward can escape only afterbubbling through the liquid. The vapour is condensed atthe top of the column and part, called reflux, is allowed toflow back down the column.

In the bubble cap column actual plates exist where vapourpasses through the liquid phase causing the phases to mix;

The height of a plate can often be directly measured as theplate height



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In packed distillation column the plates cant beobserved but can be calculated; so they arecalled theoretical plates, According to the PLATE THEORY the chromatographic

column, is likened to a distillation column whichcontains a large number of separate layers, calledtheoretical plates.

Separate equilibrations of the sample between the

stationary and mobile phase occur in these "plates".

Amob Ast

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Prof Derick Carboo


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The analyte moves down the column by transfer of equilibrated

mobile phase from one plate to the next.

The more equilibrium points there are in the column the

narrower the plate height.

The smaller or narrower the plate height, the narrower the

band width; hence the better separated the bands will be.

1. An efficient column has a small plate height

2. An efficient column has more theoretical

plates.



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The band has a gaussian profile with a standard deviation of

If a solute has travelled a distance of x at a linear flow rate of ux

(m/s),then the time it has been on the column is:

t = x/ux

; therefore 2 = 2Dx/ux

Replacing 2D/ux with H : 2 = (2D/ux)x = H x

H = 2 /x

H is called the plate height and is the width ofthe band after the solute had travelled a certain

distance. H is proportional to the variance of the band

H is the height equivalent of a theoretical plate (HETP).

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Dt2

Prof Derick Carboo


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HETP is a quantity relating the width of a band to

the distance travelled through the column.The smaller the plate height, the narrower theband width. So

The ability of a column to separate the

components of a mixture is improved bydecreasingplate height. Therefore anefficient column has more theoretical plates.

Different solutespassing through the samecolumnwill have different plate heightbecause each diffuses differently in thecolumn. ( D is different for different solutes)



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N can be calculated from the chromatogram:

N = L/H = 16L2/w2N = 16tr2/w2 = 5.55tr2/w21/2

Generally plate height for GC are between 0.1 1.0mm

HPLC plate height 10m Capillary electrophoresis plate height


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The relationship between the number of

plates N and resolution R is given by:R = N (1) k2

4 () (1+ kav ) = selectivity factork2 = capacity factor for the more retained component

kav = average capacity for both compounds

It follows that : R N L

Because N is proportional to the length of column doublingthe column length increases resolution by 2



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Band broadening outside the column. The solute cannot be injected as infinitely thin zone.

The band has some finite width even before entering

the column.

If the band is applied as a plug of width t, the

contribution to the final variance is:

2injection= (t)

2/ 12

The broadening in the detector holds the

same relationship, because some finite timeis required for the sample to pass trough.

2detector= (t)

2/ 12



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Given :Elution rate = 1.35 ml/min;

w1/2 for the collected band is 16.3 s.Volume of the sample applied is 0.30 ml.Detector volume is 0.20 ml.

Find :1. the variances introduced by injection anddetection.

2. The width at half-height, which is caused bycolumn only.



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SOLUTION.

From w1/2

= 2.35 the observed total variance is:

2obs = (w1/2/2.35)

2 = (16.3/2.35)2 = 48.11s2

The time of injection is:

tinjection=(0.30 ml)/(1.35 ml/min) = 0.222 min =13.3s.2injection = 14.78s

2

Similarly,

tdetector= (0.20 ml)/(1.35 ml/min) = 8.89 s, and 2detector= 6.58 s2.

from 2obs = 2

column + 2

detector + 2

injector

column = 5.17 s.

The width due to column broadening alone is:w1/2= 2.35column = 2.35 x 5.17 s = 12.1 s.



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The Plate Theory assumes that diffusion is the onlysource of the band broadening.

The rate theory realizes that band broadening is a

kinetic effect occasioned by the finite rate at which

mass transfer occurs during migration of solute downthe column and that this effect also depends on the

length of possible passages between the mobile phase

and stationary phase and is therefore proportional to

the flow rate of the eluent. The theory therefore attempts to investigate the

dependence of the plate height on the linear flow rate.



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The Van Deemter Equation.

J.J. Van Deemter proposed in 1956 theequation, which summarizes the on-columneffects that contribute to the plate height.

The equation takes into account three

components:

1. multiple path of an analyte through the

column packing;2. molecular diffusion;3. effect of mass transfer between phases.



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H A + B/ux + C.uxH = plate height

A = multiple paths term (or eddy diffusion)B = longitudinal diffusion term

C = equilibration time (or mass transfer) term

Ux = linear flow rate



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The most significant result is that we can find an optimumeluent flow rate where the column efficiency will be best.

Prof Derick Carboo


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The velocity of mobile phase in the column may vary significantly acrossthe column diameter, depending on the particle shape, porosity, andthe whole bed structure.

For packed columns:A, B, C 0For open-tubular columns: A = 0

For capillary electrophoresis: A = C = 0

Prof Derick Carboo


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Band broadening due to differing flowvelocities can be written in form:A = Hpath = 2 dp

A is theoretical plate height (HETP) arising fromthe variation in the zone flow velocity;dp is average particle diameter;

is the constant (very close to 1), describing

the particle size distribution; The narrower the distribution, the smaller is

.



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The term A may be reduced (efficiencyincreased) by (1) reducing the particlediameter (which will lead to the

increasing of the column back pressure)and (2) by narrowing the size distribution.

Prof Derick Carboo


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The B term arises as a result of dispersion or

mixing of the molecules due to diffusion.The longitudinal diffusion (along the columnlong axis) leads to the band broadening ofthe chromatographic band.

As it was shown before, the variance resultingfrom diffusion is:

= 2Dt

2

= 2Dt = 2D x/ux= (2D/ux)x = HxPlate height due to diffusion:

Hdiffusion = 2/x = 2D/ux B/ux



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The plate height due to diffusion is inverselyproportional to flow rate.

The faster the linear flow rate , the less time is spent in

the column and the less diffusion occurs .

Therefore : The higher the eluent velocity, the lower

the effect on the band broadening.

Longitudinal diffusion is a common source of band

broadening in GC but is of little significance in LCbecause

molecular diffusion in the liquid phase is about five

orders of magnitude lower than that in the gas phase.



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The term Cux arises from the mass transfer term whichrefers to the finite time required for solute toequilibrate between the mobile and the stationaryphases.

Plate height due to finite equilibration time of themass transfer is:

Hmass transfer= Cux = (Cs + Cm)uxwhere Cs describe the rate of mass transfer through stationary phase, and Cm describes mass

transfer through mobile phase.

Two mass transfer coefficients Cs and Cm are needed becausethe equilibrium between the mobile phase and the stationaryphase is established so slowly that the chromatographiccolumn always operate under non-equilibrium conditions.



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Therefore :

analyte molecules at the front of a band are swept aheadbefore they have time to equilibrate with stationary phase and

be retained.

Equilibrium is not reached at the trailing edge of a band, and

molecules are left behind in the stationary phase by the fast-moving mobile phase

The slower the linear flow rate the more complete the

equilibration (or mass transfer) and less zone broadening

occurs

The mass transfer equations are different for LCand GC.



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where

k' is the capacity factor,

d is the thickness of

stationary phase, r is column radius. Dsand Dmare the

diffusion coefficients instationary and mobile

phases.


sDd

k

kCs

2

2

1'3

'2


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Therefore:1. Decreasing the stationary phase thickness d

reduces plate height and increasesefficiency, because solute diffuse fasteracross the stationary phase.

2. Decreasing the column radius r reducesplate height and increases efficiency byreducing the distance trough which thesolute must diffuse to reach the stationaryphase.



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Mass transfer for the modern types of packingmaterials combines two effects:

adsorption kinetics; mass transfer (mainly due to diffusion) inside

the particles. Modern packing materials for HPLC are theforespherical, totally porous, rigid particles with

average diameter ~5 m and pore diameter~100. Ratio of the particle to the pore diameteris 500/1.

There is no pressure propelled flow inside theparticle, and molecules can move there only bydiffusion.



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Adsorption kinetics is almost negligible

compare to the diffusion inside the particles,and band spreading of the peak may bewritten in form:



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The Van Deemter equation can be further expanded to:

H = 2dp + 2GDm/ + (dp or dc)2/Dm + Rd2

f/DsWhere:

H is plate height

is particle shape (with regard to the packing)dp is particle diameterG, , and R are constants

Dm is the diffusion coefficient of the mobile phasedc is the capillary diameterdf is the film thickness

Ds is the diffusion coefficient of the stationary phase.

Chromatographic Theory

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