Equations related to resolution Most of these are directly or indirectly related to k

L σ2 tR′ = t R – tM H = = Lecture 5 N L

√ Non-ideal conditions tR′ nAnalyte in the stationary phase N k = = Rs = (γ – 1) tM nAnalyte in the mobile phase 4

t′ k B R(B) (B) H = A + + C·u α = = u tR(A)′ k(A)

cs 2 (tR(B) – tR(A)) ∆t Kc = Rs = Rs = cm wb(A) + wb(B) wb

2 2 √ tR t′R N α–1 k(B) N = 16 Neff = 16 Rs = wb wb 4 α 1–k(B)

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Equations related to resolution Equations related to resolution If k is not constant: k is not constant under the following conditions:

L σ2 tR′ = t R – tM H = = Gradient N L Gradually increasing solvent strength and decreasing k

′ n √N tR Analyte in the stationary phase Sample or analyte overload k = = Rs = (γ – 1) t nAnalyte in the mobile phase 4 M The stationary phase changes properties because it is influenced

t′ k B by the analyte R(B) (B) H = A + + C·u α = = u tR(A)′ k(A) In adsorption chromatography there may be competition for the stationary phase (saturation)

cs 2 (tR(B) – tR(A)) ∆t Kc = Rs = Rs = Multiple retention mechanisms (“active sites”) cm wb(A) + wb(B) wb More than one k per analyte 2 2 t t′ √N α–1 k R R (B) Ad sorption in partition chromatography N = 16 Neff = 16 Rs = wb wb 4 α 1–k(B)

AM0925 - Quality parameters and optimization in Chromatography AM0925 - Quality parameters and optimization in Chromatography

Equations related to resolution Gradient elution Isothermal (in GC) or isocratic (in LC) conditions are conditions where This section covers the following topics temperature and solvent strength is constant throughout the separation leading to constant k Reasons for non-ideal conditions Elution patterns typically follow an exponential How do we measure and describe efficiency under non- function ideal conditions Too low eluent strength = too long time How do we measure and describe selctivity under non- ideal conditions

Reasons for asymmetric peaks Time

How do we measure and describe non-ideal peak shapes “The general elution problem in chromatography” (tailing/fronting)

Too high eluent strength = too low resolution

Time

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1 Gradient elution Gradient elution The general elution problem is solved by applying a gradient of Eluent strength (elution/solvent strength) temperature in GC or of elution strength in LC 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 High solvent strength k = 80/20 = 4 k = 50/50 = 1 Gradient elution: The first compounds eluted with low eluent strength, the

Solvent strength Solvent last compounds eluted with high eluent strength

Time 20 % 50 % 80 % 50 % Gradient elution is suitable for a large range of analyte properties

Amount of analyte in the stationary phase Detector response Retention factor, k = Amount of analyte in the mobile phase

AM0925 - Quality parameters and optimization in Chromatography Time AM0925 - Quality parameters and optimization in Chromatography

Gradient elution Gradient elution Eluent strength (elution/solvent strength) One of the problems with gradient elutions is that plate number and plate height is no longer meaningful concepts For the same compound and the same stationary (solid) phase, a higher eluent 2 strength will lead to higher portion of the analyte in the mobile phase, and lower k. t N = 16 R Low solvent strength High solvent strength wb k = 80/20 = 4 k = 50/50 = 1 Why is plate number a meaningless concept in programmed chromatography?

- An extreme case… 20 % 50 % 80 % 50 %

Amount of analyte in the stationary phase Retention factor, k = Amount of analyte in the mobile phase

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Gradient elution Gradient elution

One of the problems with gradient elutions is that plate number and plate One of the problems with gradient elutions is that plate number and plate height is no longer meaningful concepts height is no longer meaningful concepts

2 2 t t N = 16 R N = 16 R wb wb

Assume we have the k ≈ ∞ following gradient (which has a typical shape). And

Solvent Solvent strength two peaks in the Solvent strength Time chromatogram Time If we prolong the trapping phase the k in the beginning of the retention time would increase, but chromatogram is ≈ infinite peak widths and separation would be (solvent trapping) the same (since analytes are not moving) ⇒ N is icreasing but there is no gain in separation Time Time

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2 Gradient elution Efficiency in gradient elution

One of the problems with gradient elutions is that plate number and plate How do we describe chromatographic efficiency under programmed height is no longer meaningful concepts conditions?

2 t Separation number, SN (or “Trennzahl”, TZ): N = 16 R wb The number of peaks that can be resolved in the space between two members of a homologous series with Rs ≈ 1

It is mathematically possible to calculate N with gradient C13 chromatography, but it is a completely meaningless concept from a C12 C14 chemical point of view SN = 10 ⇒⇒⇒ We need an alternative way of measuring chromatographic Alkanes efficiency separated by GC

However, even though N is a meaningless concept with gradient elution it Detector response is still a useful concept in describing the quaity of chromatographic columns Time A column that has a high N will still be a good column in gradient tR(z+1) – tR(z) chromatography. SN = + 1 (Eq. 20) wh(z) + wh(z+1)

AM0925 - Quality parameters and optimization in Chromatography AM0925 - Quality parameters and optimization in Chromatography

Efficiency in gradient elution Efficiency in gradient elution

How do we describe chromatographic efficiency under programmed How do we describe chromatographic efficiency under programmed conditions? conditions? Separation number, SN (or “Trennzahl”, TZ): Separation number, SN (or “Trennzahl”, TZ): The number of peaks that can be resolved in the space between two The number of peaks that can be resolved in the space between two

members of a homologous series with Rs ≈ 1 members of a homologous series with Rs ≈ 1 The separation number is a measure of efficiency that is valid both in C12 C13 C14 programmed chromatography and in isocratic/isothermal chromatography so it is useful for comparing programmed chromatography with non-programmed SN = 10 chromatography. Alkanes You need a homologous series w separated wh(z) h(z+1) The SN may be different in different regions of the chromatograms, so it is by GC important to give the references (z and z+1) together with the number

Detector response You may have to log-transform the retention scale if applied with isocratic or isothermal chromatography Time tR(z) tR(z+1)

tR(z+1) – tR(z) tR(z+1) – tR(z) SN = + 1 (Eq. 20) SN = + 1 (Eq. 20) wh(z) + wh(z+1) wh(z) + wh(z+1)

AM0925 - Quality parameters and optimization in Chromatography AM0925 - Quality parameters and optimization in Chromatography

Efficiency in gradient elution Efficiency in gradient elution

How do we describe chromatographic efficiency under programmed How do we describe chromatographic efficiency under programmed conditions? conditions? Separation number, SN (or “Trennzahl”, TZ): Separation number, SN (or “Trennzahl”, TZ): The number of peaks that can be resolved in the space between two The number of peaks that can be resolved in the space between two

members of a homologous series with Rs ≈ 1 members of a homologous series with Rs ≈ 1 n-alkanes is frequently applied But any homologous series can be used n-alkanes is frequently applied But any homologous series can be used

n-Hexane CH 3 - Saturated fatty acids n-Hexane CH 3 - Saturated fatty acids CH CH 3 - Polysiloxanes 3 - Polysiloxanes n-Heptane CH 3 CH 3 n-Heptane It CHis3 quite commonCH 3 to find homologous seres in - Alcolhols samples for GC-analyses,- Alcolhols often you will also n-Octane CH 3 n-Octane CH 3 CH 3 - Alkyl phenols have homologousCH 3 series- in Alkyl reversed phenols phase LC. CH CH n-Nonane CH 3 3 n-Nonane CH 3 3 - Alkyl benzenes If there is no homologous- Alkyl series benzenes that is suitable n-Decane CH 3 See G. Castello / J. Chromatogr . A 842 n-Decane CH 3 See G. Castello / J. Chromatogr . A 842 CH → ”peak capacity”CH 3 (1999) 51–64 for alternatives 3 (1999) 51–64 for alternatives n-Undecane CH 3 CH 3 n-Undecane CH 3 CH 3

etc… etc…

tR(z+1) – tR(z) tR(z+1) – tR(z) SN = + 1 (Eq. 20) SN = + 1 (Eq. 20) wh(z) + wh(z+1) wh(z) + wh(z+1)

AM0925 - Quality parameters and optimization in Chromatography AM0925 - Quality parameters and optimization in Chromatography

3 Efficiency in gradient elution Gradient elution

The “peak capacity”, P, is loosely defined as the total number of Relative retention, α, is applied also in chromatography. However, peaks that can be baseline resolved within a chromatographic the equation involving k is not strictly valid, and the transferability separation between systems is poor

It is often difficult to define the start and end of a chromatogram, so it is t′ α = R(B) often useful to give the peak capacity relative to some reference points , tR(A)′ e.g. the number of peaks that can be separated between compound A and compound B

PAB = 26

2.8

Solvent Solvent strength k ≈ ∞ Time

tM = 2.2 min 19.4 22.0 Signal strength Signal 22.0 – 2.2 α = = 1.15 Retention time 19.4 – 2.2 ∆t P = n + 1 (Eq. 21) (1/ n)Σ1wb Time

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Gradient elution Gradient elution

Relative retention, α, is applied also in chromatography. However, Relative retention, α, is applied also in chromatography. However, the equation involving k is not strictly valid, and the transferability the equation involving k is not strictly valid, and the transferability between systems is poor between systems is poor t′ t′ α = R(B) α = R(B) Even though the distance between the peaks (selectivity) has not changed, α will tR(A)′ tR(A)′ change 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 14.5 14.5 not accurate. Solvent Solvent strength Solvent strength

Time Time

tM = 2.2 min 31.1 33.7 tM = 2.2 min 31.1 33.7

33.7 – 2.2 33.7 – 2.2 α = = 1.09 α = = 1.09 31.1 – 2.2 31.1 – 2.2

Time Time

AM0925 - Quality parameters and optimization in Chromatography AM0925 - Quality parameters and optimization in Chromatography

Gradient elution Gradient elution The retention index The retention index Selectivity in gradient elution systems are best described using retention indices. Retention indices are independent of column dimensions and efficiency. A certain compound will Retention indices was originally developed for isothermal GC but it can therefore have similar values on all columns with the also be applied with temperature programmed GC, and there are also same type of stationary phase applications on LC Retention indices are widely used for identification The principle is that elution is described relative to a series of homologs, puroposes and for describing elution patterns in GC traditionally n-alkanes

C12 C13 C14 C15 C16 A series of n-alkanes C12 C13 C14 C15 C16 A series of n-alkanes 1200 1300 1400 1500 1600 1200 1300 1400 1500 1600 Each n-alkane is assigned Each n-alkane is assigned a value 100 times the a value 100 times the 1460 number of carbons in the 1460 number of carbons in the 1225 chain. 1225 chain. 1375 1550 Other compounds are 1375 1550 Other compounds are assigned values relative to assigned values relative to the reference series the reference series

10 20 30 40 min 10 20 30 40 min

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4 Gradient elution Gradient elution The retention index The retention index

Retention indices are independent of column Retention indices are independent of column dimensions and efficiency. A certain compound will dimensions and efficiency. A certain compound will therefore have similar values on all columns with the therefore have similar values on all columns with the same type of stationary phase same type of stationary phase Retention indices are widely used for identification Retention indices are widely used for identification puroposes and for describing elution patterns in GC puroposes and for describing elution patterns in GC

C12 C13 C14 C15 C16 Retention indices are C12 C13 C14 C15 C16 Retention indices are 1200 1300 1400 1500 1600 independent of 1200 1300 1400 1500 1600 independent of efficeincy efficeincy 1460 1460 1225 1225 Retention indices are independent of total 1375 1550 1375 1550 retention

10 20 30 40 min 10 20 30 40 min

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Gradient elution Gradient elution Fatty acids analysed by different temperature and pressure programs on the same GC column The retention index Originally, retention indices were developed for isothermal (a) DE32-D5 (162-1.5-25) 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 log 10 t′R. The equation for isothermal

DE5-F4 (190-4-20) retention indices, I, is therefore:

log tR(i)′′– log tR(z) + z (B is usually 100 or 1) (Eq 22)

Retention time scale I = B 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 log tR(z+1)′′– log tR(z) 18:0 Minutes

18:1 n-9 18:2 n-6 20:0 • B is a constant denoting the number of indices between two homologs 18:3 n-3 20:3 n-3 18:3 n-6 22:0 DE32-D5 (162-1.5-25) 20:1 n-9 20:2 n-6 20:4 n-6 (b) 22:1 n-9 22:2 n-6 26:0 20:3 n-6 20:5 n-3 24:0 • i is the analyte of interest 19:0 22:5 n-3 25:0 22:4 n-6 24:1 n-9 22:6 n-3 C12 23:0 • z is thereference eluting immediately before i 21:0 1200 • z+1 is the reference eluting immediately after i and having one more carbon than z C13 18:0 20:3 n-3 1300 20:0 20:4 n-6 DE5-F4 (190-4-20) t′R(z) t′R(i) t′R(z+1) 18:1 n-9 18:2 n-6 18:3 n-3 22:0 26:0 C14 18:3 n-6 20:1 n-9 20:2 n-6 20:5 n-3 20:3 n-6 24:0 22:1 n-9 22:2 n-6 22:4 n-6 1400 C15 23:0 22:5 n-3 C16 19:0 24:1 n-9 22:6 n-3 21:0 1500 25:0 1600

Retention index scale 10 20 30 40 min

18 19 20 21 22 23 24 25 26 B = 100 Equivalent chain lengths (ECL) AM0925 - Quality parameters and optimization in Chromatography AM0925 - Quality parameters and optimization in Chromatography

Gradient elution Gradient elution The retention index The retention index

• B is a constant denoting the number of indices between two homologs • B is a constant denoting the number of indices between two homologs log t′′– log t • i is the analyte of interest log t′′– log t • i is the analyte of interest I = B R(i) R(z) + z I = B R(i) R(z) + z • z is thereference eluting immediately before i • z is thereference eluting immediately before i log tR(z+1)′′– log tR(z) log tR(z+1)′′– log tR(z) • z+1 is the reference eluting immediately after i and having one more carbon than z • z+1 is the reference eluting immediately after i and having one more carbon than z

log 24.4 – log 18.8 The retention index for the peak at 24.4 min: I = 100 + 15 = 1539 (24.4 min) log 36.2 – log 18.8

C12 C12 1200 1200

C13 18.8 min 24.4 min 36.2 min C13 18.8 min 24.4 min 36.2 min 1300 t′ t′ t′ 1300 t′ t′ t′ C14 R(z) R(i) R(z+1) C14 R(z) R(i) R(z+1) C15 C15 1400 C16 1400 C16 1500 1500 1600 1600

10 20 30 40 min 10 20 30 40 min B = 100 B = 100

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5 Gradient elution Gradient elution The retention index The retention index Originally, retention indices were developed for isothermal gas chromatography • B is a constant denoting the number of indices between two homologs using n-alkanes as reference compounds (Kováts’ retention index) t – t T R(i) R(z) • i is the analyte of interest I = B + z • z is thereference eluting immediately before i tR(z+1) – tR(z) Under temperature programmed conditions one assumes a direct linear • z+1 is the reference eluting immediately after i and having one more carbon than z

relationship between the number of carbons in a homologous series and tR. The equation for temperature programmed retention indices, IT, is therefore:

T 24.4 – 20.2 tR(i) – tR(z) I = 100 + 15 = 1564 IT = B + z (B is usually 100 or 1) (Eq 23) 26.8 – 20.2 tR(z+1) – tR(z)

• B is a constant denoting the number of indices between two homologs • i is the analyte of interest 24.4 min • z is thereference eluting immediately before i • z+1 is the reference eluting immediately after i and having one more carbon than z 20.2 min 26.8 min

t′R(z) t′R(i) t′R(z+1) t′R(z) t′R(i) t′R(z+1) C12 C13 C14 C15 C16 C12 C13 C14 C15 C16 1200 1300 1400 1500 1600 1200 1300 1400 1500 1600

10 20 30 40 min 10 20 30 40 min B = 100 B = 100

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Gradient elution Gradient elution The retention index The van Deemter equation in gradient Retention indices are basically used with GC elution Although isthermal conditions allows you to apply α for describing relative The van Deemter equation is given as: retention, retention indices are more important and usually more precise than α B also in isothermal GC. H = A + + C·u u Kováts indices, using n-alkanes as calibration series are the most important, but there are numerous other systems. See Castello, J. Chromatography A 842 Because H is not valid under programmed conditions (since N is not valid) the (1999) 51-64 for a review. van Deemter equation is not strictly valid. Application of retention indices in LC are rare. But it is possible to use such However, it is possible to replace H with other meaningful values representing systems, particularly in reversed phase LC, if there is a suitable homologous the inverse of the separation efficiency, such as: series that spans the polarity range of your analytes. 1/SN (SN=separation number) There are also retention indexsystems that is not based on homologous series 1/ P (P=peak capacity) (e.g. Lee indices for polycyclic aromatic hydrocarbons) Peak width expressed on a retention index scale (instead of time scale)

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Gradient elution Gradient elution The van Deemter equation in gradient The van Deemter equation in gradient elution elution

The effects underlying A, B and C are the same in programmed chromatography The effects underlying A, B and C are the same in programmed chromatography as in isothermal and isocratic chromatography as in isothermal and isocratic chromatography 1 B 1 B = A + + C·u ⇒ Conditions that are good in= Aisocratoc/isothermal + + C·u chromatography SN u will be good also in programmedSN chromatographyu

∝ ∝ A dp dp is particle diameter A dp dp is particle diameter

B = 2 DM DM is diffusion coefficient B = 2 DM DM is diffusion coefficient

C ∝ d 2 d is particle diameter C ∝ d 2 d is particle diameter Valid relationships also in M p p Valid relationships also in M p p programmed ∝ 2 programmed ∝ 2 CM r r is column radius CM r r is column radius chromatography ∝ chromatography ∝ CM 1/ DM DM is diffusion coefficient CM 1/ DM DM is diffusion coefficient ∝ ∝ CS df df is stationary phase thickness CS df df is stationary phase thickness ∝ ∝ CS 1/ DM DM is diffusion coefficient CS 1/ DM DM is diffusion coefficient

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6 Equations related to resolution Peak asymmetry k is not constant under the following conditions: “Tailing” and “fronting” peaks

Gradient elution

Gradually increasing solvent strength and decreasing k Mobile phase flow 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”) A symmetric peak More than one k per analyte Ad sorption in partition chromatography

Retention time

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Peak asymmetry Peak asymmetry “Tailing” and “fronting” peaks “Tailing” and “fronting” peaks

Mobile phase flow Mobile phase flow

A “tailing” peak A “fronting” peak

Retention time Retention time

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Peak asymmetry Peak asymmetry Why do peaks tail? Why do peaks tail?

Mobile phase flow Mobile phase flow

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

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7 Peak asymmetry Peak asymmetry Why do peaks front? Langmuir isotherms

Ideally, the relationship Mobile phase flow between the concentration in the

stationary phase, cS, and the concentration in the

mobile phase cM should be independent of the Fronting is usually an indication of stationary phase overload in total amounts. partition chromatography Kc = cS / cM High concentration of the analyte will influence the properties of the Effects that increase K stationary phase so it becomes more similar to the analyte c with total amounts will Since “like dissolves like” in chemistry, this will usually lead to lead to fronting peaks. increased retention (higher k). Effects that decrease Kc The first molecules in the peak will not experience this effect since with total amounts will concentration in the beginning of the peak is low (and because of lead to tailing peaks mass transfer effects) Retention will therefore be lower in the beginning that at the end of the peak.

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Peak asymmetry Peak asymmetry Measuring asymmetry Measuring asymmetry

The asymmetry factor, Af, is the right peak width The tailing factor, Tf, is the total peak width divided by the left peak width. Peak widths are divided by two times the left peak width. Peak measured at 10% of the peak height widths are measured at 5% of the peak height

Af < 1 Af = 1 Af > 1 Tf < 1 Tf = 1 Tf > 1

w w wright,10% w5% left right (Eq 24) (Eq 25) A = 5% wleft wright T = 10% f f wleft,10% 2·wleft,5%

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Peak asymmetry Peak asymmetry

Both factors have the value 1 for symmetric peaks Why do we have two different measures for

The asymmetry factor gives values between 1 and ∞ for tailing asymmetry? peaks and between 0 and 1 for fronting peaks. - same reason as why we have numerous measures for length, volume and weight: it seems to be difficult to agree on The tailing factor gives values between 1 and ∞ for tailing peaks and between 0.5 and 1 for fronting peaks. a system… (they are equally good)

wright,10% wright,10% Af = (Eq 24) Af = (Eq 24) wleft,10% wleft,10%

w w w5% w w w5% left right (Eq 25) left right (Eq 25) 5% T = 5% T = 10% f 10% f 2·wleft,5% 2·wleft,5%

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8 Peak asymmetry Summary of equations Measuring plate number with assymetric t – t Separation number SN = R(z+1) R(z) – 1 (20) w + w peaks h(z) h(z+1) ∆t P = – 1 Accurate calculation of the plate number requires symmetric peaks, but Peak capacity n (21) (1/n)Σ 1 wb there is an equation for estimating the plate number with asymmetric

peaks: Isothermal retention log tR(i)′′– log tR(z) Plate number for asymmetric peaks I = B + z (B is usually 100 or 1) (22) index log t′′– log t R(z+1) R(z) 2 Temperature 41.7 (t R / w10% ) tR(i) – tR(z) programmed IT = B + z (B is usually 100 or 1) (23) N ≈ t – t Af + 1.25 retention index R(z+1) R(z)

wright,10% Asymmetry factor A = (24) f w left,10% w Tailing factor T = 5% (25) f 2·w left,5%

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AM0925 - Quality parameters and optimization in Chromatography

9