Lecture 4 Band broadening.

Lecture 4 Band broadening

Lecture 4 Band broadening
Factors leading to band broadening work against the chromatographic efficiency because they increase peaks widths

Why different peak widths?
Factors that lead to peak broadening in a chromatographic column: Detector signal 1 2 3 4 5 6 7 8 9 10 min

Factors that lead to peak broadening in a chromatographic column: A) Column packing / system dead volumes Detector signal 1 2 3 4 5 6 7 8 9 10 min

Factors that lead to peak broadening in a chromatographic column: A) Column packing / system dead volumes B) Longitudinal diffusion Detector signal 1 2 3 4 5 6 7 8 9 10 min

Factors that lead to peak broadening in a chromatographic column: A) Column packing / system dead volumes B) Longitudinal diffusion C) Resistance to mass transfer Detector signal 1 2 3 4 5 6 7 8 9 10 min

Factors that lead to peak broadening in a chromatographic column: A) Column packing / system dead volumes B) Longitudinal diffusion C) Resistance to mass transfer Detector signal Which of the three that is most important depends on many things, but the the velocity of the mobile phase is among the the critical factors 1 2 3 4 5 6 7 8 9 10 min

A) Column packing (multiple paths)

A) Column packing (multiple paths) Liquid chromatography system
Column packing consists of (spherical) particles

Molecule passing through the column Liquid chromatography system
A) Column packing (multiple paths) Molecule passing through the column Liquid chromatography system

A) Column packing (multiple paths) Liquid chromatography system
Molecule passing through the column Liquid chromatography system Molecules follow different paths

Different paths have different lengths
A) Column packing (multiple paths) Different paths have different lengths Molecules follow different paths

Different paths have different lengths
A) Column packing (multiple paths) Different paths have different lengths The “different paths effect” will affect all compounds equally, irrespective of their retention Molecules follow different paths

A) Column packing (multiple paths)
Different paths have different lengths The “different paths effect” will affect all compounds equally, irrespective of their retention The effect is also independent of the mobile phase velocity Molecules follow different paths

Molecules follow different paths
A) Column packing (multiple paths) Molecules follow different paths spread Constant mobile phase velocity

B) Diffusion

B) Diffusion If we have a band of molecules (e.g. a chromatographic peak) in solvent or gas phase it will gradually spead out because of diffusion

B) Diffusion Diffusion takes time
 The faster we can get the analyte through the column, the less will the peak be broadened by the diffusion

inverse relationship to mobile phase velocity
B) Diffusion spread inverse relationship to mobile phase velocity mobile phase velocity

C) Resistance to mass transfer

Mobile phase

Mobile phase Net transfer

Mobile phase distribution in the mobile phase Mobile phase distribution in the stationary phase Stationary phase

Mobile phase Increased mobile phase velocity increases contribution to spread by resistance to mass transfer distribution in the mobile phase Mobile phase distribution in the stationary phase Stationary phase

Mobile phase Simplified: Once the molecules have entered one of the phases it is “difficult” to get out again Increased mobile phase velocity increases contribution to spread by resistance to mass transfer distribution in the mobile phase Mobile phase distribution in the stationary phase Stationary phase

spread mobile phase velocity Increased mobile phase velocity increases contribution to spread by resistance to mass transfer distribution in the mobile phase Mobile phase distribution in the stationary phase Stationary phase

The sum of effects There are three different effects with different dependence of the mobile phase velocity spread mobile phase velocity

The sum of effects There are three different effects with different dependence of the mobile phase velocity spread The multiple paths effect (effect of column packing) is independent of the mobile phase velocity packing ++ mobile phase velocity

The sum of effects There are three different effects with different dependence of the mobile phase velocity spread The diffusion effect is inversely proportional to the mobile phase velocity packing ++ diffusion mobile phase velocity

The mass transfer effect is proportional to the mobile phase velocity
The sum of effects spread The mass transfer effect is proportional to the mobile phase velocity resistance to mass transfer packing ++ diffusion mobile phase velocity

The sum of effects Sum of effects Sum spread
resistance to mass transfer packing ++ diffusion mobile phase velocity

The sum of effects There exists an optimal mobile phase velocity where the spread of the peaks are minimized spread Optimal velocity Sum resistance to mass transfer packing ++ diffusion mobile phase velocity

The van Deemter equation
Optimal velocity A + B/u + C∙u Plate height C∙u A B/u mobile phase velocity, u

Knowing A, B and C may help you decide where to put the effort if you need higher efficiency Optimal velocity A + B/u + C∙u Plate height C∙u A B/u mobile phase velocity, u

The optimal mobile phase velocity is found where A + B/u + C∙u has a local minimum, meaning that the derivative is 0.  uopt = √B/C Optimal velocity A + B/u + C∙u Plate height C∙u A B/u mobile phase velocity, u

- “Multiple paths term” B/u: - “Longitudinal diffusion term” C•u: - “Mass transfer term” / - “Finite equilibration time” - Mass transfer kinetics of the analyte between mobile and stationary phase - C is often split in one factor for the stationary phase and one for the mobile phase: C = CS + CM

C = CS + CM CM is inversely proportional to the diffusion coefficient of the analyte in the mobile phase. For packed columns CM is proportional to the square of the particle diameter of the coumn packing. For open tubular columns CM is proportional to the square of the column diameter.

C = CS + CM CM is inversely proportional to the diffusion coefficient of the analyte in the mobile phase. For packed columns CM is proportional to the square of the particle diameter of the coumn packing. For open tubular columns CM is proportional to the square of the column diameter. CS is proportional to the square of the film thickness. CS is inversely proportional to the diffusion coefficient. In adsorption chromatography CS is proportional to the time required for a compound to be adsorbed or desorbed (first order rate constant).

Solving the equation Knowing A, B and C may help you decide where to put the effort if you need higher efficiency Since the equation has three unknowns you will need at least three experiments to solve it, but in most cases you will need much more experiments to compensate for experimental error (peak width measurements are rarely very accurate).

Solving the equation Knowing A, B and C may help you decide where to put the effort if you need higher efficiency Since the equation has three unknowns you will need at least three experiments to solve it, but in most cases you will need much more experiments to compensate for experimental error (peak width measurements are rarely very accurate). Moody, Journal of chemical Education 59 (1982)

Alternatives The van Deemter equation is the simplest and most general equation that explains band broadening in chromatography. There exist expanded and alternative forms of the van Deemter eqation, as well as other equations explaining the same or similar relationships: The Giddings equation The Huber equation The Knox equation The Horvath equation Remember that all these are models of reality (they are not laws of Nature). Even though they may not fit reality 100% they are accurate enough to be of practical value and to explain the phenomena that occurs in chromatographic columns.

Factors affecting H in packed columns (LC)
mobile phase velocity, u H B/u A C∙u A + B/u + C∙u Optimal velocity Plate height

A = 2λdp  A  dp λ is a constant dependent on particle quality dp = particle diameter u = mobile phase velocity Smaller particles will give more similar paths, limiting peak broadening by the multiple path effect (The A-term)  Small particles are good

Good Bad Smaller particles will give more similar paths, limiting peak broadening by the multiple path effect (The A-term)  Small particles are good Particles should also be uniform in size and shape

Smaller particles leads to more exchange between the phases and therefore reduce the CM-term: f′(k) is a function of k DM = Diffusion coefficient in mobile phase u = mobile phase velocity dp = particle diameter CM = f′(k) dp2 DM  CM  dp2 u  Small particles are good

Smaller particles leads to more exchange between the phases and therefore reduce the CM-term: f′(k) is a function of k DM = Diffusion coefficient in mobile phase u = mobile phase velocity dp = particle diameter Why is the properties of the stationary phase affecting transfer in the mobile phase, CM ? CM = f′(k) dp2 DM  CM  dp2 u

Smaller particles leads to faster equilibrium between the phases and therefore reduce the CM-term: f′(k) is a function of k DM = Diffusion coefficient in mobile phase u = mobile phase velocity dp = particle diameter Why is the properties of the stationary phase affecting transfer in the mobile phase, CM ? CM = f′(k) dp2 DM  CM  dp2 u The void volumes of mobile phase between the particles are smaller with smaller particle diameter

Smaller particles leads to faster equilibrium between the phases and therefore reduce the CM-term: f′(k) is a function of k DM = Diffusion coefficient in mobile phase u = mobile phase velocity dp = particle diameter Why is the properties of the stationary phase affecting transfer in the mobile phase, CM ? CM = f′(k) dp2 DM  CM  dp2 u The void volumes of mobile phase between the particles are smaller with smaller particle diameter Irregularly shaped particles create void volumes that are not efficently flushed by the mobile phase

The effect of particle size is that both A and C are reduced with reduced particle diameter

The effect of particle size is that both A and C are reduced with reduced particle diameter An approximation that is often valid is: N ≈ 3000 L / dp (L is column length in cm, dp is particle diameter in μm)  H ≈ dp / 3000 (H given in cm, dp is particle diameter in μm)

Because diffusion coefficients in liquids are small compared to gas, the B-term is usually not of importance in LC (only when using verly low flow rates)

Some experimental van deemter plots showing the effect of particle diameter

The bad news about small particles: P = pressure f = “shape function” u = mobile phase velocity η = solvent viscosity L = column length r = column radius dp = particle diameter

The bad news about small particles: The pressure required to give a certain flow is inversely proportional to dp2 P = pressure f = “shape function” u = mobile phase velocity η = solvent viscosity L = column length r = column radius dp = particle diameter

The bad news about small particles: The pressure required to give a certain flow is inversely proportional to dp2 5 10 15 20 25 1 2 3 4 Particle diameter (μm) Pressure relative to 5 μm 1 P d 2 p

The bad news about small particles: The pressure required to give a certain flow is inversely proportional to dp2 5 10 15 20 25 1 2 3 4 Particle diameter (μm) Pressure relative to 5 μm 1 P d 2 p Going from 5 to 2.5 μm particles would roughly double the plate number, but pressure would increase 4 times (if other parameters are kept constant) Going to 1 μm would increase the pressure 25 times!

The bad news about small particles: The pressure required to give a certain flow is inversely proportional to dp2 5 10 15 20 25 1 2 3 4 Particle diameter (μm) Pressure relative to 5 μm 1 P d 2 p Conventional (HPLC) systems are typically designed for particles down to ≈5 μm

The bad news about small particles: The pressure required to give a certain flow is inversely proportional to dp2 5 10 15 20 25 1 2 3 4 Particle diameter (μm) Pressure relative to 5 μm 1 P d 2 p State of the art systems today can operate with particle diameters between 1.5 and 2 μm. This is often referred to as ultra high performance LC (UHPLC)

Factors affecting H in open tubular columns (GC)

The van Deemter equation: Eq (13) mobile phase velocity, u H B/u A C∙u A + B/u + C∙u Optimal velocity Plate height

An open tubular capillary column has no packing: Eq (13) mobile phase velocity, u H B/u C∙u B/u + C∙u Optimal velocity Plate height

The Golay equation H = C∙u B u Eq (13b) mobile phase velocity, u H B/u C∙u B/u + C∙u Optimal velocity Plate height

The Golay equation with separate CM and CS terms H = CM∙u + CS∙u B u Eq (13b) mobile phase velocity, u H B/u C∙u B/u + C∙u Optimal velocity Plate height

The Golay equation with separate CM and CS terms B u H = CM∙u + CS∙u Eq (13b) C M = 1 + 6k + 11 k 2 24( + 1) r D C S = 2k 3( k + 1) 2 d f D B = 2·DM k is the retention factor. DM is the diffusion coefficient of the analyte in the mobile phase. DS is the diffusion coefficient of the analyte in the stationary phase. r is the radius of the column. df is the thickness of the stationary phase.

Longitudinal diffusion: mobile phase velocity, u H B/u Optimal velocity Plate height B is a direct function of the diffusion coefficient of the analyte in the mobile phase, DM: B = 2·DM B/u + C∙u C∙u

Diffusion and mass transfer term(s) mobile phase velocity, u H B/u Optimal velocity Plate height Mass transfer in the mobile phase: r is the column radius k is the retention factor DM is the diffusion coefficient in the mobile phase B/u + C∙u 1 + 6k + 11 k 2 r 2 C∙u C = M 24( k + 1) 2 D M Þ C 1/DM M

Diffusion and mass transfer term(s) mobile phase velocity, u H B/u Optimal velocity Plate height Mass transfer in the stationary phase: df is the thickness of stationary phase layer k is the retention factor DS is the diffusion coefficient in the stationary phase B/u + C∙u 2k d 2 C∙u C = f S 3( k + 1) 2 D S Þ C 1/DS S

Diffusion and mass transfer term(s) C S 1/DS C M 1/DM B DM Valid both for capillary and packed columns, and both for GC and LC

Diffusion and mass transfer term(s) C S 1/DS C M 1/DM B DM High dffusion gives more transfers between the phases Diffusion in this direction is good Diffusion in this direction is bad

Diffusion and mass transfer term(s) C S 1/DS C M 1/DM B DM High dffusion gives more transfers between the phases High diffusion also means more longitudinal diffusion Diffusion in this direction is good Diffusion in this direction is bad

Diffusion and mass transfer term(s) Diffusion in one direction is bad, diffusion in the other direction is good – but we cannot control the direction of it. Diffusion in this direction is good Diffusion in this direction is bad

Column dimensions and mass transfer term(s) Diffusion in one direction is bad, diffusion in the other direction is good – but we cannot control the direction of it. Downscaling the diameter and phase thickness has the same effect on the C term as increased diffusion. Diffusion in this direction is good Diffusion in this direction is bad

A molecule with random motion in a hollow tube. Each time it hits the wall there is a possibility for entering the other phase. 163 contacts d

A molecule with random motion in a hollow tube. Each time it hits the wall there is a possibility for entering the other phase. 163 contacts d 350 contacts d/2

A molecule with random motion in a hollow tube. Each time it hits the wall there is a possibility for entering the other phase. 163 contacts d 350 contacts d/2 602 contacts d/4

Mass transfer term(s) mobile phase velocity, u H B/u Optimal velocity Plate height Mass transfer in the mobile phase: r is the column radius k is the retention factor DM is the diffusion coefficient in the mobile phase B/u + C∙u 1 + 6k + 11 k 2 r 2 C∙u C = M 24( k + 1) 2 D M Þ C r 2 M

Mass transfer term(s) mobile phase velocity, u H B/u Optimal velocity Plate height Mass transfer in the stationary phase: df is the thickness of stationary phase layer k is the retention factor DS is the diffusion coefficient in the stationary phase B/u + C∙u 2k d 2 C∙u C = f S 3( k + 1) 2 D S Þ C d 2 S f

Column scaling: CM  df2 CS  r2 Assume that we have achieved plates on 25 m column with internal diameter 0.25 mm and phase thickness of 0.25 μm, and that we are using a high mobile phase velocity  the B term is insignificant What is the effect of scaling down the column to 10 m with internal diameter 0.10 mm and phase thickness of 0.10 μm? H H Optimal velocity Optimal velocity C·u Plate height Plate height B/u B/u mobile phase velocity, u u

Column scaling: CM  df2 CS  r2 Assume that we have achieved plates on 25 m column with internal diameter 0.25 mm and phase thickness of 0.25 μm, and that we are using a high mobile phase velocity  the B term is insignificant What is the effect of scaling down the column to 10 m with internal diameter 0.10 mm and phase thickness of 0.10 μm? The plate height on the 25 m column would be: 2500 cm / = cm-1. H = L / N

Column scaling: CM  df2 CS  r2 Assume that we have achieved plates on 25 m column with internal diameter 0.25 mm and phase thickness of 0.25 μm, and that we are using a high mobile phase velocity  the B term is insignificant What is the effect of scaling down the column to 10 m with internal diameter 0.10 mm and phase thickness of 0.10 μm? The plate height on the 25 m column would be: 2500 cm / = cm-1. The plate height on the 10 m column would be: (10/25)2 · cm-1 = cm-1

Column scaling: CM  df2 CS  r2 Assume that we have achieved plates on 25 m column with internal diameter 0.25 mm and phase thickness of 0.25 μm, and that we are using a high mobile phase velocity  the B term is insignificant What is the effect of scaling down the column to 10 m with internal diameter 0.10 mm and phase thickness of 0.10 μm? The plate height on the 25 m column would be: 2500 cm / = cm-1. The plate height on the 10 m column would be: (10/25)2 · cm-1 = cm-1 The number of plates on the 10 m column would be 1000 cm / cm-1 =

Column scaling: CM  df2 CS  r2 Note that this is pure theory! There will be factors that hinders us to achieve this gain in efficiency: u must be high to make the B-term insignificant, which leads to large effect of C (since H  C·u) There are other effects that will contribute to increased H in any system (extra-column effects) The plate height on the 25 m column would be: 2500 cm / = cm-1. The plate height on the 10 m column would be: (10/25)2 · cm-1 = cm-1 The number of plates on the 10 m column would be 1000 cm / cm-1 =

Column scaling: CM  df2 CS  r2 But it may be possible to achieve increased efficiency by scaling down the system and you save a significant amount of time when columns are shorter (and time = money)  The driving force behind “fast GC” (GC equivalent to UHPLC)

The effect of carrier gas properties:

The effect of carrier gas properties: Mass transfer term: Longitudinal diffusion term: B = 2·DM

The effect of carrier gas properties: Mass transfer term: Longitudinal diffusion term: B = 2·DM There are three gases that are common in gas chromatography and the diffusion coefficients, DM, of any analyte increase in the following order: N2 << He < H2 When DM increases CM decreases and B increases, this shifts the optimum mobile phase velocity to higher flow

The effect of carrier gas properties: Optimal velocities are typically around 15 cm/s for N2 (and air), around 25 cm/s for He and around 35 cm for H2

The effect of carrier gas properties: Optimal velocities are typically around 15 cm/s for N2 (and air), around 25 cm/s for He and around 35 cm for H2 Note that H2 does not give lower H than the other gases when used at optimal velocity, but the “punishment” for using higher than optimal flow is lower for H2 Hydrogen is therefore commonly applied well above optimal velocities in GC.

The effect of carrier gas properties: Note that curves for helium and hydrogen are steeper to the left than to the right of the optimum.  If you are not certain where the optimum is, be certain that you do not select too low mobile phase velocity. (if you are going to be wrong – be wrong in the right way…)

Extra column effects

Extra column effects In any chromatographic system there are effects from the system in addition to the effects in the column that wil contribute to broader peaks and lower N. These are typically referred to as extra-column effects.

Extra column effects In any chromatographic system there are effects from the system in addition to the effects in the column that wil contribute to broader peaks and lower N. These are typically referred to as extra-column effects. Examples are: Effects of the injection volume and poor focusing of the analytes

Extra column effects In any chromatographic system there are effects from the system in addition to the effects in the column that wil contribute to broader peaks and lower N. These are typically referred to as extra-column effects. Examples are: Effects of the injection volume and poor focusing of the analytes Dead volumes in tubing and fittings

Extra column effects In any chromatographic system there are effects from the system in addition to the effects in the column that wil contribute to broader peaks and lower N. These are typically referred to as extra-column effects. Examples are: Effects of the injection volume and poor focusing of the analytes Dead volumes in tubing and fittings Band spreading in the detector

The extra column effects adds to the contributions from the column
A + B/u + C∙u + E C∙u Plate height E (Extra column effect) A B/u mobile phase velocity, u

Extra column effects The extra column effects adds to the contributions from the column H If it looks like this there is little to gain by minimizing A, B and C A + B/u + C∙u + E E (Extra column effect) C∙u Plate height A B/u mobile phase velocity, u

Extra column effects The extra column effects adds to the contributions from the column H If it looks like this there is little to gain by minimizing A, B and C A + B/u + C∙u + E Note that it is extremely important to minimize extra column effects in fast GC and UHPLC which relies on short columns with very low H. Even a moderate extra column effect may be the dominating source of band spreading with these techniques. This is one of the reasons why fast GC and UHPLC does not give the theoretical gain in N. E (Extra column effect) C∙u Plate height A B/u mobile phase velocity, u

Extra column effects The extra column effects adds to the contributions from the column H If it looks like this there is little to gain by minimizing A, B and C A + B/u + C∙u + E The contribution from the extra column effects are not independent of column length, as the contribution from A, B and C are. Band spreading caused by extra column effects are independent of column length, i.e. equal for all column lengths The effect on H will therefore be inversely proportional to the column length since H = L / N  It is strictly not correct to show E in the van Deemter curve as done here  The contribution from E to H is more severe with short columns than with long columns E (Extra column effect) C∙u Plate height A B/u mobile phase velocity, u

Sum of effects The sum of effects leading to peak broadening in chromatography follows common error propagation laws with contributions from the terms in the van Deemter equation and extra column effects:

Sum of effects The sum of effects leading to peak broadening in chromatography follows common error propagation laws with contributions from the terms in the van Deemter equation and extra column effects: Since the terms are squared, the largest effects will be very dominating. It is therefore important to know which effects that are most significant in the system and focus on reducing these

Summary of equations and relationships
The van Deemter equation Eq (13) The Golay equation H = CM∙u + CS∙u B u Eq (13b) A term (multiple paths) Eq (14) B term (longitudinal diffusion) Eq (15) CM term in packed columns Eq (16) CM term in open tubular columns Eq (17) CS term in open tubular columns 2 Eq (18) Sum of broadening effects Eq (19)

Lecture 4 Band broadening.

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Lecture 4 Band broadening.

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