Lecture 5 Non-ideal conditions.

Lecture 5 Non-ideal conditions

Equations related to resolution
H = L N

Most of these are directly or indirectly related to k H = L N

If k is not constant: H = L N

k is not constant under the following conditions: Gradient elution / Temperature programming Gradually increasing solvent strength and decreasing k Sample or analyte overload The stationary phase changes properties because it is influenced by the analyte In adsorption chromatography there may be competition for the stationary phase (saturation) Multiple retention mechanisms (“active sites”) More than one k per analyte Adsorption in partition (absorption) chromatography

This section covers the following topics Reasons for non-ideal conditions How do we measure and describe efficiency under non-ideal conditions How do we measure and describe selctivity under non-ideal conditions Reasons for asymmetric peaks How do we measure and describe non-ideal peak shapes (tailing/fronting)

Isothermal / isocratic chromatography
Isothermal (in GC) or isocratic (in LC) conditions are conditions where temperature and solvent strength is constant throughout the separation leading to constant k Elution patterns typically follow an exponential function Too low eluent strength = too long time “The general elution problem in chromatography” Too high eluent strength = too low resolution

Programmed chromatography
The general elution problem is solved by applying temperature programming in GC or a gradient of increasing elution strength in LC Solvent strength Gradient elution: The first compounds eluted with low eluent strength, the last compounds eluted with high eluent strength Time Gradient elution is suitable for a large range of analyte properties Detector response Time

20 % 50 % 80 % Programmed chromatography
Eluent strength (elution/solvent strength) For the same compound and the same stationary (solid) phase, a higher eluent strength will lead to higher portion of the analyte in the mobile phase, and lower k. Low solvent strength k = 80/20 = 4 High solvent strength k = 50/50 = 1 20 % 80 % 50 % Amount of analyte in the stationary phase Amount of analyte in the mobile phase Retention factor, k =

One of the problems with temperature or solvent programming is that plate number and plate height are no longer meaningful concepts N = 16 tR wb 2 Why is plate number a meaningless concept in programmed chromatography? - An extreme case…

One of the problems with temperature or solvent programming is that plate number and plate height are no longer meaningful concepts N = 16 tR wb 2 Solvent strength Assume we have the following gradient (which has a typical shape). And two peaks in the chromatogram Time Time

One of the problems with temperature or solvent programming is that plate number and plate height are no longer meaningful concepts N = 16 tR wb 2 Assume we have the following gradient (which has a typical shape). And two peaks in the chromatogram k in the beginning of the chromatogram is ≈ infinite (analyte trapping) Solvent strength k ≈ ∞ Time Time

One of the problems with temperature or solvent programming is that plate number and plate height are no longer meaningful concepts N = 16 tR wb 2 Solvent strength Time If we prolong the trapping phase the retention time would increase, but peak widths and separation would be the same (since analytes are not moving)  N is icreasing but there is no gain in separation Time

One of the problems with temperature or solvent programming is that plate number and plate height are no longer meaningful concepts N = 16 tR wb 2 It is mathematically possible to calculate N with gradient chromatography, but it is a completely meaningless concept from a chemical point of view  We need an alternative way of measuring chromatographic efficiency

One of the problems with temperature or solvent programming is that plate number and plate height are no longer meaningful concepts N = 16 tR wb 2 It is mathematically possible to calculate N with gradient chromatography, but it is a completely meaningless concept from a chemical point of view  We need an alternative way of measuring chromatographic efficiency However, even though N is a meaningless concept with gradient elution it is still a useful concept in describing the quality of chromatographic columns A column that has a high N will still be a good column in gradient or temperature-programmed chromatography.

Efficiency in programmed GC
How do we describe chromatographic efficiency under programmed conditions?

How do we describe chromatographic efficiency under programmed conditions? Separation number, SN (or “Trennzahl”, TZ): The number of peaks that can be resolved in the space between two members of a homologous series with Rs ≈ 1 C12 C13 C14 Alkanes separated by GC Detector response Time

How do we describe chromatographic efficiency under programmed conditions? Separation number, SN (or “Trennzahl”, TZ): The number of peaks that can be resolved in the space between two members of a homologous series with Rs ≈ 1 C12 C13 C14 SN = 10 Alkanes separated by GC Detector response Time

How do we describe chromatographic efficiency under programmed conditions? Separation number, SN (or “Trennzahl”, TZ): The number of peaks that can be resolved in the space between two members of a homologous series with Rs ≈ 1 C12 C13 C14 SN = 10 Alkanes separated by GC Detector response Time (Eq. 20) SN = tR(z+1) – tR(z) wh(z) + wh(z+1) ‒ 1

How do we describe chromatographic efficiency under programmed conditions? Separation number, SN (or “Trennzahl”, TZ): The number of peaks that can be resolved in the space between two members of a homologous series with Rs ≈ 1 C12 C13 C14 SN = 10 Alkanes separated by GC wh(z) wh(z+1) Detector response tR(z) tR(z+1) Time (Eq. 20) SN = tR(z+1) – tR(z) wh(z) + wh(z+1) ‒ 1

How do we describe chromatographic efficiency under programmed conditions? Separation number, SN (or “Trennzahl”, TZ): The number of peaks that can be resolved in the space between two members of a homologous series with Rs ≈ 1 The separation number is a measure of efficiency that is valid both in programmed chromatography and in isocratic/isothermal chromatography so it is useful for comparing programmed chromatography with non-programmed chromatography. You need a homologous series The SN may be different in different regions of the chromatograms, so it is important to give the references (z and z+1) together with the number You may have to log-transform the retention scale if applied with isocratic or isothermal chromatography (Eq. 20) SN = tR(z+1) – tR(z) wh(z) + wh(z+1) ‒ 1

How do we describe chromatographic efficiency under programmed conditions? Separation number, SN (or “Trennzahl”, TZ): The number of peaks that can be resolved in the space between two members of a homologous series with Rs ≈ 1 CH3 n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Undecane etc… n-alkanes is frequently applied But any homologous series can be used - Saturated fatty acids - Polysiloxanes - Alcolhols - Alkyl phenols - Alkyl benzenes See G. Castello / J. Chromatogr. A 842 (1999) 51–64 for alternatives (Eq. 20) SN = tR(z+1) – tR(z) wh(z) + wh(z+1) ‒ 1

How do we describe chromatographic efficiency under programmed conditions? Separation number, SN (or “Trennzahl”, TZ): The number of peaks that can be resolved in the space between two members of a homologous series with Rs ≈ 1 CH3 n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Undecane etc… n-alkanes is frequently applied But any homologous series can be used - Saturated fatty acids - Polysiloxanes - Alcolhols - Alkyl phenols - Alkyl benzenes See G. Castello / J. Chromatogr. A 842 (1999) 51–64 for alternatives It is quite common to find homologous seres in samples for GC-analyses, often you will also have homologous series in reversed phase LC. If there is no homologous series that is suitable → ”peak capacity” (Eq. 20) SN = tR(z+1) – tR(z) wh(z) + wh(z+1) ‒ 1

The “peak capacity”, P, is loosely defined as the total number of peaks that bcan be baseline resolved within a chromatographic separation It is often difficult to define the start and end of a chromatogram, so it is often useful to give the peak capacity relative to some reference points, e.g. the number of peaks that can be separated between compound A and compound B P = 33 Signal strength Retention time (Eq. 21)

The “peak capacity”, P, is loosely defined as the total number of peaks that bcan be baseline resolved within a chromatographic separation It is often difficult to define the start and end of a chromatogram, so it is often useful to give the peak capacity relative to some reference points, e.g. the number of peaks that can be separated between compound A and compound B PAB = 26 Signal strength Retention time (Eq. 21)

Selectivity in programmed chromatography

Relative retention, α and γ, is applied also in programmed chromatography. However, the equation involving k is not strictly valid, and the transferability between systems is poor α = tR(B) tR(A) ′

Relative retention, α and γ, is applied also in programmed chromatography. However, the equation involving k is not strictly valid, and the transferability between systems is poor α = tR(B) tR(A) ′ Solvent strength 2.8 k ≈ ∞ Time tM = 2.2 min 19.4 22.0 22.0 – 2.2 19.4 – 2.2 α = = 1.15 Time

Relative retention, α and γ, is applied also in programmed chromatography. However, the equation involving k is not strictly valid, and the transferability between systems is poor α = tR(B) tR(A) ′ Solvent strength 14.5 Time tM = 2.2 min 31.1 33.7 33.7 – 2.2 31.1 – 2.2 α = = 1.09 Time

Relative retention, α and γ, is applied also in programmed chromatography. However, the equation involving k is not strictly valid, and the transferability between systems is poor α = tR(B) tR(A) ′ Even though the distance between the peaks (selectivity) has not changed, α will chang because it is influenced by the time where the analytes are trapped in the stationary phase or move very slowly. Solvent strength 14.5 Time tM = 2.2 min 31.1 33.7 33.7 – 2.2 31.1 – 2.2 α = = 1.09 Time

Relative retention, α and γ, is applied also in programmed chromatography. However, the equation involving k is not strictly valid, and the transferability between systems is poor α = tR(B) tR(A) ′ Even though the distance between the peaks (selectivity) has not changed, α will chang because it is influenced by the time where the analytes are trapped in the stationary phase or move very slowly.  Using a single reference point to describe selectivity in gradient elution is not accurate. Solvent strength 14.5 Time tM = 2.2 min 31.1 33.7 33.7 – 2.2 31.1 – 2.2 α = = 1.09 Time

The retention index Selectivity in gradient elution systems are best described using retention indices. Retention indices was originally developed for isothermal GC but it can also be applied with temperature programmed GC, and there are also applications on LC

The retention index Selectivity in gradient elution systems are best described using retention indices. Retention indices was originally developed for isothermal GC but it can also be applied with temperature programmed GC, and there are also applications on LC The principle is that elution is described relative to a series of homologs, traditionally n-alkanes C12 C13 C14 C15 C16 A series of n-alkanes 10 20 30 40 min

The retention index Selectivity in gradient elution systems are best described using retention indices. Retention indices was originally developed for isothermal GC but it can also be applied with temperature programmed GC, and there are also applications on LC The principle is that elution is described relative to a series of homologs, traditionally n-alkanes C12 1200 C13 1300 C14 1400 C15 1500 C16 1600 A series of n-alkanes Each n-alkane is assigned a value 100 times the number of carbons in the chain. 10 20 30 40 min

The retention index Selectivity in gradient elution systems are best described using retention indices. Retention indices was originally developed for isothermal GC but it can also be applied with temperature programmed GC, and there are also applications on LC The principle is that elution is described relative to a series of homologs, traditionally n-alkanes C12 1200 C13 1300 C14 1400 C15 1500 C16 1600 A series of n-alkanes Each n-alkane is assigned a value 100 times the number of carbons in the chain. Other compounds are assigned values relative to the reference series 1460 1225 1375 1550 10 20 30 40 min

The retention index Retention indices are independent of column dimensions and efficiency. A certain compound will therefore have similar values on all columns with the same type of stationary phase Retention indices are widely used for identification puroposes and for describing elution patterns in GC C12 1200 C13 1300 C14 1400 C15 1500 C16 1600 A series of n-alkanes Each n-alkane is assigned a value 100 times the number of carbons in the chain. Other compounds are assigned values relative to the reference series 1460 1225 1375 1550 10 20 30 40 min

The retention index Retention indices are independent of column dimensions and efficiency. A certain compound will therefore have similar values on all columns with the same type of stationary phase Retention indices are widely used for identification puroposes and for describing elution patterns in GC C12 1200 C13 1300 C14 1400 C15 1500 C16 1600 Retention indices are independent of efficeincy 1460 1225 1375 1550 10 20 30 40 min

The retention index Retention indices are independent of column dimensions and efficiency. A certain compound will therefore have similar values on all columns with the same type of stationary phase Retention indices are widely used for identification puroposes and for describing elution patterns in GC C12 1200 C13 1300 C14 1400 C15 1500 C16 1600 Retention indices are independent of efficeincy Retention indices are independent of total retention 1460 1225 1375 1550 10 20 30 40 min

Fatty acids analysed by different temperature and pressure programs on the same GC column Retention time scale Low temperature rate High temperature rate Retention index scale

The retention index Originally, retention indices were developed for isothermal gas chromatography using n-alkanes as reference compounds (Kováts’ retention index) Under isothermal conditions there is a linear relationship between the number of carbons in a homologous series and log10 t′R. The equation for isothermal retention indices, I, is therefore: (Eq 22) B is a constant denoting the number of indices between two homologs i is the analyte of interest z is thereference eluting immediately before i z+1 is the reference eluting immediately after i and having one more carbon than z C12 1200 C13 1300 C14 1400 C15 1500 C16 1600 10 20 30 40 min

The retention index Originally, retention indices were developed for isothermal gas chromatography using n-alkanes as reference compounds (Kováts’ retention index) Under isothermal conditions there is a linear relationship between the number of carbons in a homologous series and log10 t′R. The equation for isothermal retention indices, I, is therefore: (Eq 22) B is a constant denoting the number of indices between two homologs i is the analyte of interest z is thereference eluting immediately before i z+1 is the reference eluting immediately after i and having one more carbon than z C12 1200 C13 1300 t′R(z) t′R(i) t′R(z+1) C14 1400 C15 1500 C16 1600 10 20 30 40 min B = 100

The retention index B is a constant denoting the number of indices between two homologs i is the analyte of interest z is thereference eluting immediately before i z+1 is the reference eluting immediately after i and having one more carbon than z The retention index for the peak at 24.4 min: C12 1200 18.8 min 24.4 min 36.2 min C13 1300 t′R(z) t′R(i) t′R(z+1) C14 1400 C15 1500 C16 1600 10 20 30 40 min B = 100

The retention index B is a constant denoting the number of indices between two homologs i is the analyte of interest z is thereference eluting immediately before i z+1 is the reference eluting immediately after i and having one more carbon than z log 24.4 – log 18.8 log 36.2 – log 18.8 I(24.4 min) = 100 + 15 = 1539 C12 1200 18.8 min 24.4 min 36.2 min C13 1300 t′R(z) t′R(i) t′R(z+1) C14 1400 C15 1500 C16 1600 10 20 30 40 min B = 100

The retention index Originally, retention indices were developed for isothermal gas chromatography using n-alkanes as reference compounds (Kováts’ retention index) Under temperature programmed conditions one assumes a direct linear relationship between the number of carbons in a homologous series and tR. The equation for temperature programmed retention indices, IT, is therefore: (Eq 23) B is a constant denoting the number of indices between two homologs i is the analyte of interest z is thereference eluting immediately before i z+1 is the reference eluting immediately after i and having one more carbon than z t′R(z) t′R(i) t′R(z+1) C12 1200 C13 1300 C14 1400 C15 1500 C16 1600 10 20 30 40 min B = 100

The retention index Originally, retention indices were developed for isothermal gas chromatography using n-alkanes as reference compounds (Kováts’ retention index) Under temperature programmed conditions one assumes a direct linear relationship between the number of carbons in a homologous series and tR. The equation for temperature programmed retention indices, IT, is therefore: (Eq 23) B is a constant denoting the number of indices between two homologs i is the analyte of interest z is thereference eluting immediately before i z+1 is the reference eluting immediately after i and having one more carbon than z The equation is the same as for isothermal chromatography except that log tR′ (adjusted retention time) is replaced with tR (unadjusted) t′R(z) t′R(i) t′R(z+1) C12 1200 C13 1300 C14 1400 C15 1500 C16 1600 10 20 30 40 min B = 100

The retention index B is a constant denoting the number of indices between two homologs i is the analyte of interest z is thereference eluting immediately before i z+1 is the reference eluting immediately after i and having one more carbon than z 24.4 – 20.2 26.8 – 20.2 IT = 100 + 15 = 1564 24.4 min 20.2 min 26.8 min t′R(z) t′R(i) t′R(z+1) C12 1200 C13 1300 C14 1400 C15 1500 C16 1600 10 20 30 40 min B = 100

The retention index Retention indices are basically used with GC Although isthermal conditions allows you to apply α for describing relative retention, retention indices are more important and usually more precise than α also in isothermal GC. Kováts indices, using n-alkanes as calibration series are the most important, but there are numerous other systems. See Castello, J. Chromatography A 842 (1999) for a review. Application of retention indices in LC are rare. But it is possible to use such systems, particularly in reversed phase LC, if there is a suitable homologous series that spans the polarity range of your analytes. There are also retention index systems that are not based on homologous series (e.g. Lee indices for polycyclic aromatic hydrocarbons)

The van Deemter equation in gradient elution

The van Deemter equation in programmed chromatography
The van Deemter equation is given as: H = A + B u + C∙u Because H is not valid under programmed conditions (since N is not valid) the van Deemter equation is not strictly valid.

The van Deemter equation is given as: H = A + B u + C∙u Because H is not valid under programmed conditions (since N is not valid) the van Deemter equation is not strictly valid. However, it is possible to replace H with other meaningful values representing the inverse of the separation efficiency, such as: 1/SN (SN=separation number) 1/P (P=peak capacity) Peak width expressed on a retention index scale (instead of time scale)

The effects underlying A, B and C are the same in programmed chromatography as in isothermal and isocratic chromatography = A + B u + C∙u 1 SN

The effects underlying A, B and C are the same in programmed chromatography as in isothermal and isocratic chromatography = A + B u + C∙u 1 SN In temperature-programmed GC peak widths measured in retention index units instead of time can also be used as a measure of the inverse efficiency [Waktola & Mjøs, J. Sep. Science 38 (2015) 3014–3027] B u wb(RI)= A + + C∙u

The effects underlying A, B and C are the same in programmed chromatography as in isothermal and isocratic chromatography = A + B u + C∙u 1 SN Valid relationships also in programmed chromatography 2

The effects underlying A, B and C are the same in programmed chromatography as in isothermal and isocratic chromatography  Conditions that are good in isocratic/isothermal chromatography will be good also in programmed chromatography = A + B u + C∙u 1 SN Valid relationships also in programmed chromatography 2

k is not constant under the following conditions: Gradient elution / Temperature programming Gradually increasing solvent strength and decreasing k Sample or analyte overload The stationary phase changes properties because it is influenced by the analyte In adsorption chromatography there may be competition for the stationary phase (saturation) Multiple retention mechanisms (“active sites”) More than one k per analyte Adsorption in partition chromatography

“Tailing” and “fronting” peaks
Peak asymmetry “Tailing” and “fronting” peaks Mobile phase flow A symmetric peak Retention time

Peak asymmetry “Tailing” and “fronting” peaks Mobile phase flow A “tailing” peak Retention time

Peak asymmetry “Tailing” and “fronting” peaks Mobile phase flow A “fronting” peak Retention time

Peak asymmetry Why do peaks tail? Mobile phase flow
Tailing is caused by adsorption phenomena where there is limited capacity to bind the analyte

Tailing is caused by adsorption phenomena where there is limited capacity to bind the analyte The first molecules in a peak will meat a stationary phase with high density of available sites to bind to Later eluting molecules will meat a stationary phase where much of the available sites are already occupied This will lead to stronger interactions with the stationary phase in the beginning than at the end of the peak

Tailing is caused by adsorption phenomena where there is limited capacity to bind the analyte The first molecules in a peak will meat a stationary phase with high density of available sites to bind to Later eluting molecules will meat a stationary phase where much of the available sites are already occupied This will lead to stronger interactions with the stationary phase in the beginning than at the end of the peak High k Low k

Tailing is caused by adsorption phenomena where there is limited capacity to bind the analyte The first molecules in a peak will meat a stationary phase with high density of available sites to bind to Later eluting molecules will meat a stationary phase where much of the available sites are already occupied This will lead to stronger interactions with the stationary phase in the beginning than at the end of the peak Tailing in partition (absorption) chromatography is an indication that there are significant adsorption phenomena taking place, so called ”active sites” (which we should usually not have)

Peak asymmetry Why do peaks front? Mobile phase flow
Fronting is usually an indication of stationary phase overload in partition chromatography

Fronting is usually an indication of stationary phase overload in partition chromatography High concentration of the analyte will influence the properties of the stationary phase so it becomes more similar to the analyte Since “like dissolves like” in chemistry, this will usually lead to increased retention (higher k). The first molecules in the peak will not experience this effect since concentration in the beginning of the peak is low. Retention will therefore be lower in the beginning that at the end of the peak.

Fronting is usually an indication of stationary phase overload in partition chromatography High concentration of the analyte will influence the properties of the stationary phase so it becomes more similar to the analyte Since “like dissolves like” in chemistry, this will usually lead to increased retention (higher k). The first molecules in the peak will not experience this effect since concentration in the beginning of the peak is low. Retention will therefore be lower in the beginning that at the end of the peak. low k high k

Peak asymmetry Langmuir isotherms
Ideally, the relationship between the concentration in the stationary phase, cS, and the concentration in the mobile phase cM should be independent of the total amounts. Kc = cS / cM Effects that increase Kc with total concentration will lead to fronting peaks. Effects that decrease Kc with total concentration will lead to tailing peaks

Peak asymmetry Measuring asymmetry

Peak asymmetry Measuring asymmetry wright,10% wleft,10% Af = Af < 1
The asymmetry factor, Af, is the right peak width divided by the left peak width. Peak widths are measured at 10% of the peak height Af < 1 Af = 1 Af > 1 Af = wright,10% wleft,10% (Eq 24) wleft wright 10%

Peak asymmetry Measuring asymmetry w5% 2∙wleft,5% Tf = Tf < 1
The tailing factor, Tf, is the total peak width divided by two times the left peak width. Peak widths are measured at 5% of the peak height Tf < 1 Tf = 1 Tf > 1 Tf = w5% 2∙wleft,5% (Eq 25) wleft wright 5%

Peak asymmetry Measuring asymmetry w5% 2∙wleft,5% Tf = Tf < 1
The tailing factor, Tf, is the total peak width divided by two times the left peak width. Peak widths are measured at 5% of the peak height In pharmacy the tailing factor may be referred to as the “symmetry factor” (Do not confuse with asymmetry factor) Tf < 1 Tf = 1 Tf > 1 Tf = w5% 2∙wleft,5% (Eq 25) wleft wright 5%

Peak asymmetry wright,10% wleft,10% Af = w5% 2∙wleft,5% Tf = wleft
Both factors have the value 1 for symmetric peaks The asymmetry factor gives values between 1 and ∞ for tailing peaks and between 0 and 1 for fronting peaks. The tailing factor gives values between 1 and ∞ for tailing peaks and between 0.5 and 1 for fronting peaks. Af = wright,10% wleft,10% (Eq 24) Tf = w5% 2∙wleft,5% (Eq 25) wleft wright 5% 10%

Peak asymmetry Why do we have two different measures for asymmetry?
- same reason as why we have numerous measures for length, volume and weight: it seems to be difficult to agree on a system… (they are equally good) Af = wright,10% wleft,10% (Eq 24) Tf = w5% 2∙wleft,5% (Eq 25) wleft wright 5% 10%

Measuring plate number with assymetric peaks
Peak asymmetry Measuring plate number with assymetric peaks Accurate calculation of the plate number requires symmetric peaks, but there is an equation for estimating the plate number with asymmetric peaks: Plate number for asymmetric peaks N ≈ 41.7 (tR / w10%)2 Af

Summary of equations

Lecture 5 Non-ideal conditions.

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