1LC•GC Europe - February 2001 GC connections

Analysts often need to make calculationsrelated to their chromatographic systems.Yet, many do not know what calculationsare appropriate, or they use incorrectformulas that can cause serious errors. Agood understanding of how to measureand calculate various working parametersfrom an instrument and a chromatogramdoes not require strong theoreticalknowledge. Some high-school algebra andthe willingness to make a little extra effortare the only prerequisites for improving theaccuracy and usefulness of commonchromatographic measurements.Calculating and applying thesemeasurements correctly facilitatesinstrument and method maintenance.Effective method transfer from instrumentto instrument requires a good workingknowledge of chromatographiccomputations as well.

In this “GC Connections” column, I’llpresent a number of useful measurementsand calculations, based on a testchromatogram, that can help gaschromatography (GC) users to betterunderstand the analytical process andthereby improve control over the quality oftheir results.

The ChromatogramThe classic elution chromatogram is a two-dimensional representation of the amountof solute exiting the column as a functionof elapsed time since the separationexperiment began. In GC, this plotrepresents detector response during theperiod after solute injection. Many of theparameters that chromatographers must

understand derive from measurements ofretention time and peak width made fromthe chromatogram. These measured andcalculated values and their trends over timeare excellent diagnostic tools that indicatehow well a GC system functions and pointto developing problems before theybecome serious.Unretained peak measurements: Figure 1illustrates a simple chromatogram withthree peaks. The first peak is the detectorresponse to substances that are not retainedin the stationary phase. These componentstraverse the column entirely in the mobilephase, as expressed in the designation oftheir retention time (tM, in which thesubscripted M indicates its non-retainedcharacteristic). If one of the non-retainedsolutes evokes a detector response, then thechromatogram will include a peak at thistime. A thermal conductivity detector will

respond to air in a sample, but a flameionization detector will not. To measure theunretained peak time, analysts purposelyinject an unretained substance to which thedetector responds. Among these substancesare methane or butane for flame ionizationdetectors, air for thermal conductivitydetectors and methylene chloride vapour forelectron-capture detectors.

The unretained peak time can changeduring programmed temperature orprogrammed pressure GC runs —including constant flow control of packedor capillary columns — depending on thepneumatic system and its mode ofoperation. Record the unretained peak atthe initial column temperature andpressure to verify the instrument set-upand then maintain a complete descriptionof the method temperature and pneumaticconditions from beginning to end.

Tracking a column’s performance over its lifetime is not complicated. Gas chromatography (GC) users can learn a greatdeal about their columns by making several measurements and calculations from simple test chromatograms acquiredweekly or more often. This month, John Hinshaw reviews some basic GC measurements and calculations and applies themto questions of column quality.

John V. Hinshaw, ChromSource, Franklin, Tennessee, USA.

Basic GC Measurements and Calculations

Figure 1: Basic measurements from an isothermal chromatogram. mmax is the peak height atits apex and wh is the peak width at half height.

0 tM tR1 tR2

Time

mmaxwh

LC•GC Europe - February 20012 GC connections

columns. For any column at a specific temperature with a knowncarrier gas, the pressure drop across the column dictates theaverage carrier-gas velocity and the column flow-rate according torelationships too complex to discuss in this “GC Connections”column. Chromatographers can use these calculations if they haveaccess to a computer or to a gas chromatograph with electronicpneumatic controls. Electronic pneumatic control systems forcapillary columns incorporate the column pressure–velocity–flowrelationships into the gas chromatograph, and GC users can takeadvantage of this built-in capability. For those without electronicsystems, carrier-gas calculators are available on the Internet: the flowcalculator at http://www.chem.agilent.com/cag/servsup/ssmain.htmlis a good example.

To use a program such as those or the built-in capabilities ofelectronic pneumatic systems, enter the column dimensions,carrier-gas identity, column outlet pressure conditions and — foran external calculator — enter the column temperature. The resultsfrom these calculations are only as good as the information thatusers input. With an electronic pneumatic system, errors in theinput column dimensions will cause the actual operating conditionsto depart considerably from the set points.

Next enter the desired average carrier-gas linear velocity in theinstrument system or flow calculator. The resulting column pressure isthe pressure increment above atmospheric pressure required toproduce the specified linear velocity through a column with the givendimensions at the current oven temperature. An electronic system willadjust the pressure automatically. With an external calculator, set theindicated pressure on the capillary inlet system by adjusting thepressure regulator accordingly.

Then inject an unretained substance and record thechromatogram. Calculate the measured average carrier-gas linearvelocity from the unretained peak time, according to Equation 1,and compare it with the nominal velocity entered into theelectronic system or calculator. The two velocities should be withinapproximately 10% of each other. A larger discrepancy indicates apotential problem with a carrier-gas leak, an error in the columndimensions, the wrong type of carrier gas or the wrong columnoutlet pressure setting.

The column outlet pressure indicates to the pneumatic calculatorwhether the column vents to the atmosphere or to a vacuumsystem. Larger inner-diameter columns or short columns mayrequire inlet pressure less than atmospheric pressure to deliver theset point velocity to a vacuum system; checking the requiredpressure drop before operating the column helps prevent thismistake.

Wide-bore, 0.53 mm inner diameter columns present a problemwith pressure drop and linear velocity correlation because thepressure drops they require for normal operation are very low,ranging from 1.5 psig helium for a 15 m column at 30 cm/s linearvelocity to 6.25 psig for a 60 m column at the same velocity.Mechanical pressure gauges will not indicate these low pressuresaccurately, and even electronic systems will have trouble at the lowend of this range. Use a direct flow-controlled inlet system, not asplit inlet, with columns shorter than 25 m in length. Calibrate theflow controller, set the desired flow-rate, and check the flow at thecolumn exit. A column flow calculator program will compute thecorresponding linear velocity, so it is possible to cross-check velocityand flow for these columns.Column dimensions: Most chromatographers take the columndimensions right off the column box and assume that they arecorrect. In fact, this assumption is a frequent source of problems.The original box or specification sheet indicates the columndimensions correctly, but after the column is removed from the box

GC operators should determine the unretained peak time andexamine its shape after column installation, a carrier-gas supplychange or a GC method change. This step verifies that carrier gas isflowing through the column at the desired rate. The unretained peakshould have a symmetrical shape; tailing or other distortions indicatea problem with the column or its installation that operators shouldcorrect before proceeding. Record the unretained peak at fasterchart speeds — or expand its area in the chromatogram on thecomputer screen — to see its shape better.

For single packed-column analyses, the unretained peak shapehelps diagnose installation or column problems, but GC methodsmost often specify the column flow-rate and not the unretained peaktime. The unretained peak time is difficult to relate to packed columnflow-rate and pressure drop, and often it is too short for accuratemeasurement, so chromatographers should measure the flow at thecolumn exit and the pressure drop across the column. Therelationship between flow and pressure provides a convenientdiagnostic tool. After installing a new column, record the pressuredrop and measure the flow with a flowmeter at the initial and finalcolumn temperatures. The pressure drop will be higher at elevatedoven temperatures — make sure that it does not exceed the upperpressure limit of the pneumatics. Then check the pressure and flowvalues again at regular intervals. A significant change in thepressure drop at either the lower or the upper temperatureindicates changes in the column packing or a developing leak thatwill require attention. Any peak-shape aberrations indicate anongoing problem with the inlet or the column packing.

For capillary columns, the unretained peak time is an essentialparameter that allows chromatographers to calculate the carrier-gas velocity through the column and cross-check it with themeasured pressure drop. The GC method may specify the carrier-gas velocity, the unretained peak time or both in addition to thepressure drop. The velocity in the column is lower near the beginningand higher at the exit, because of expansion of the compressiblecarrier gas as it moves down the column. For simplicity, mostchromatographers concern themselves only with the average carrier-gas velocity along the entire column. The following formula gives theaverage carrier-gas linear velocity (u�) in centimetres per second interms of the unretained peak time (tM) in seconds and the columnlength (L) in centimetres.

u� � L/tM [1]

The carrier-gas velocity affects the column’s efficiency inresolving solute peaks. If the velocity is too high, solutes don’tspend enough time in the column for an efficient separation; if thevelocity is too low, solutes broaden excessively by diffusion throughthe mobile phase. A normal range of operation exists betweenthese two extremes that is approximately 20–40 cm/s for heliumand 30–60 cm/s for hydrogen carrier gases. The exact optimumaverage velocity depends on the column specifications,temperature profile and solute, in addition to the carrier gas. It isimpossible to choose one velocity that yields the best efficiency forall solutes in a diverse multicomponent mixture. Furthermore, largecarrier-gas velocity changes in a temperature-programmed analysisaffect relative peak separation as well as peak widths and maycause adjacent peaks to be coeluted or even reverse their elutionorder. For these reasons, be sure to determine the average carrier-gas linear velocity after column installation, changing carrier-gastanks or making any changes to the inlet or pneumatic systems.

The exact relationships between the carrier-gas pressure dropand the linear velocity for capillary columns provide a diagnostictool analogous to the relationship of pressure and flow for packed

3LC•GC Europe - February 2001 GC connections

there is no certainty that an operator willreturn it to the correct box. Subsequentusers may be surprised to find that thecolumn in the box is not what the boxindicates. Most columns come with aserialized identity tag attached to thecolumn cage for this reason. Even then,the length of a capillary column becomesshorter with each installation — sometimesby as much as 1 m — which quicklyinvalidates the length shown on the tag.Accurate column length is essential forseveral computations, including the carrier-gas linear velocity. With electronicpneumatic control systems, the columnlength and other column dimensions play acrucial role in the instrument system’sdetermination of column flow and splitratio, which have a direct impact onquantitative results.

To avoid these problems, establish aninventory list for both packed and capillarycolumns that correlates column serialnumbers with their original dimensions andpacking or coating specifications. Keep alog of column activities and dates ofinstallation and removal from service. Forcapillary columns, measure and record theapparent column length before installationby counting turns and computing thelength from the following formula:

L � n�d [2]

where n is the number of column coilsand d is the average coil diameter on thecolumn cage. Subtract the total length ofcolumn removed from both ends duringinstallation. The result is accurate enoughfor most applications: an error of one coilon a 15 cm (6 in.) cage is equal toapproximately 0.5 m, or less than 2% of a30 m column length. Do not, under anycircumstances, remove the column from itscage, stretch it out along the laboratoryfloor, and count floor tiles to determine thelength. Not only will this damage thecolumn by attracting dirt on the outercoating, but it may damage otherlaboratory personnel as they trip over thenearly invisible tubing.

GC users can assume that the columninner diameter is very close to its nominalvalue, because of the precision drawingmachines that produce fused-silica capillarycolumn tubing. In general, the tubing’sinner diameter lies within �0.4–0.5% ofits nominal value. With the correct columnlength and carrier-gas identity, the linearvelocity test described above will quicklyidentify a mislabelled column. If the columninner diameter is unknown, a flowcalculator programme will provide an

indication to within roughly 10% of theactual column diameter. Simply adjust theinput column diameter until the calculatedpressure or linear velocity correlates with themeasured value. Then round the apparentdiameter up or down to the closestdiameter available from the specific columnmanufacturer and record that value.

The unretained peak time provides acheck not only of the column linearvelocity that is important for obtaining thebest possible column performance but alsofor checking the column dimensions andthe electronic pneumatic system set-up.Retained peak measurements: The nextitems to examine in Figure 1 are theelution times of the retained solute peaks,designated by the subscripted R. All solutesthat pass through a column spend thesame time in the mobile phase, equal tothe unretained peak time. Any additionaltime spent in the stationary phase is equalto the difference between the peaks’retention times (tR) and the unretainedpeak times. Chromatographers call thisparameter the adjusted retention time (t�R),which is measured in seconds and can becalculated from the following formula:

t�R � tR � tM [3]

The adjusted retention time is not verysignificant by itself, but it plays animportant role in another retentionmeasurement, the retention factor (k). Theretention factor is the ratio of the adjustedretention time to the unretained peaktime. In other words, it expresses peakretention in terms of multiples of theunretained peak time.

The formula for the retention factor is

Interestingly, the retention factor doesn’tdepend on the average carrier-gas linearvelocity. Increasing or decreasing thevelocity changes both the unretained andretained peak times in proportion. Thisbehaviour makes the retention factor veryuseful for column characterization with atest mixture under isothermal conditions.GC operators can detect potentialproblems in their separations byperiodically monitoring the retentionfactors of test substances.

A retention time shift could be causedby changes in the column pressure drop orby changes in the column itself, such asloss of the stationary phase. Pressure-dropchanges do not affect the retention factor;however, stationary-phase changes do. By

k � t'R

tM tM

tR � tM� [4]

regularly determining the retention factorsof test solutes, chromatographers canmonitor the condition of their columns andcan detect small stationary-phase losses orshifts in apparent column polarity beforethey significantly affect peak separationand resolution.

Extended column use at temperaturesclose to a column’s maximum ratedoperating temperature can significantlyreduce the amount of stationary phase inthe column. Other catastrophic events,such as oxygen leakage or chemical stresscan cause a sudden loss of stationary filmor a change in the phase polarity.Chromatographers can track the loss ofstationary phase by monitoring retentionfactors of test-mixture peaks as part ofroutine column qualification. Stationary-phase losses cause peaks to be eluted atlower, earlier retention factors under thesame temperature conditions. A peak shifttowards lower retention of more than 20%will indicate a continuing problem withstationary-phase loss that could affect peakseparation and resolution.Two retained peaks: The relationship oftwo peaks’ retentions also plays a majorrole in chromatographic separations. Twopeaks that are close together require moreresolving power from a column than peaksthat are far apart. Analysts measure therelative separation of two peaks in twosimilar ways: the separation factor (�) andthe relative retention (r). These twomeasurements take the same values, buttheir fields of application are different. Theseparation factor is commonly used incharacterizing the separating power of aparticular column under specific analyticalconditions, and the use of relativeretention appears most often in data-handling systems. The ratio of tworetention factors defines both parameters:

� � k2/k1 for adjacent peaksr � ki/kst for any two peaks [5]

In the first equation k1 and k2 are theretention factors of the first and secondadjacent peaks, respectively, whendetermining the separation factor. For relativeretention, kst defines a reference peakretention factor and ki defines the targetanalyte retention factor, which could begreater or less than the reference peakretention factor. The two peaks for relativeretention calculation don’t have to beadjacent.

The separation factor is always greaterthan or equal to 1, and it always applies totwo adjacent peaks. It indicates the degreeof selectivity obtained with a particular

LC•GC Europe - February 20014 GC connections

column and set of analytical conditions.Drifting separation factors in test mixturechromatograms obtained over time canindicate a continuing problem with columnpolarity shifts caused by stationary-phasedegradation or column contamination.

The relative retention can apply to anytwo peaks in a chromatogram. It is moreindependent of long-term changes incolumn length or film thickness thanabsolute retention times or retentionfactors, and it is often used in data-handling system peak identification.

Significant changes in the separationfactor or relative retention indicate shiftingcolumn polarity in isothermal analyses. Inthe instance of temperature-programmedanalyses, drifting linear velocity may alsohave this effect.Peak width: Each peak has a characteristicwidth, which chromatographers usuallymeasure at a halfway point on the peak.The peak width at half-height (wh), given inseconds, characterizes the aggregateamount of broadening that a solute hasundergone in its passage through the inlet,column and detector. In most situations,the inlet and detector contributions topeak width are negligible and the columnnormally produces most of the broadening.

Figure 1 shows two methods for peak-width determination. The first uses directmeasurement of the peak width at half-height from the chromatogram, which onlyrequires a ruler. The second method usesthe peak area and peak height toapproximate the width at half-height. Atriangle drawn from the peak apex to thebaseline through the data points at half-height has an area approximately equal to94% of the entire peak area (A). Thefollowing equation relates the peak areaand height to the width at half-height:

wh � 0.94A/mmax [6]

where mmax is the peak height at its apex.This formula is convenient when thechromatogram is stored in a computer,because it only requires the peak area andheight. Many data-handling systems includethe capability to report peak widths, whichoperators should choose over theapproximation shown in Equation 6. With apeak width at hand, a GC operator canascertain the apparent column performancelevel in relation to the highest performancethe column could deliver or to laboratorystandards for column performance.

Chromatographers measure columnperformance in terms of the number oftheoretical plates (N) that the columndelivers or in terms of the distance or

height that one theoretical plate occupiesin the column. High performance columnsproduce more theoretical plates andcorrespondingly smaller plate heights.Narrower peak widths translate to higherperformance; the relationship between thepeak width at half-height and themeasured number of theoretical plates(Nmeas) is

Narrower peak widths at longer retentiontimes have higher plate numbers. Forexample, a peak that is eluted at 10 minwith a width at half-height of 4 s has125000 measured theoretical plates.

Dividing the column length by the platenumber yields the height equivalent to asingle theoretical plate (H), in centimetres:

H � L/N [8]

For a 30 m column that yields the peakmentioned above, the measured plateheight is 0.024 cm or 240 µm.

To gauge column performance,chromatographers compute the smallestpossible plate height that a column coulddeliver. The minimum theoretical plateheight (Hmin) is related to the column innerdiameter (dc) and the retention factor bythe following formula:

The value of the square-root function ofk in Equation 9 ranges from 0.6 at a kvalue of 1.0 to 0.95 at a k value of 100. Inthis example, if the average carrier-gaslinear velocity in the 30 m column were 30 cm/s, then the unretained peak timefrom Equation 1 would be 100 s and theretention factor from Equation 4 would be5.0. At a k value of 5.0, the square-rootfunction in Equation 9 is equal to 0.84. Ifthe column has an inner diameter of 0.025 cm (250 µm), then the minimumpossible plate height is 0.021 cm or 210 µm (0.84 3 0.0250 5 0.021 cm).

How does this theoretical value for theplate height compare with the measuredvalue? Chromatographers can calculate theper cent utilization of theoretical efficiency(UTE%) to make this comparison:

In this example, UTE% is 100 � 0.021 0.024, which equals 87.5%. This result

[10]Hmin

Hmeas

UTE% � 100 ( (

[9]Hmin � dc1 6k 11k 2

12 (1 k 2)

[7]tR

wh

Nmeas � 5.545 ( (2

means that the column delivers nearly allthe possible efficiency that it could for thispeak.

The above calculations do not include anadjustment for the effects of thestationary-phase film on the measuredplate height; they assume that no film ispresent. This is untrue, of course, becausethe column wouldn’t retain any peaks if nophase was present, and the peaks allwould be eluted at the same unretainedpeak time. The stationary-phase filmbroadens peaks as the analytes traverse theinterface between it and the mobile gasphase and as they diffuse slowly throughthe film while inside it. The extent of thisstationary-phase band broadening dependsprimarily upon the film thickness. Thin-filmcolumns with coatings of 0.25 µm or lesssuffer only negligible losses, but thick-filmcolumns can lose as much as 50% or moreof the theoretical performance.

Rather than attempt to calculate thedegree of efficiency loss caused by thestationary phase, GC users should measurethe plate height or number of theoreticalplates when a column is new. Make sure thenew column delivers close to 100% or moreof the performance, as recorded on themanufacturer’s test chromatogram under thesame conditions. This level of performancemay be significantly less than the theoreticalmaximum, but it reflects the effect of thestationary film on the plate height. If a newcolumn fails to yield performance similar to itsoriginal test chromatogram, then carefullycheck the column installation and the linearvelocity. For an old column, chromatographersshould expect some performance degradationover time. Replacement is mandated whenthe column no longer delivers a satisfactorychromatogram for the test mixture oranalytical standard, fails to deliver a minimumplate number, or tests to a larger plate heightthan tolerated by the test procedure. Any ofthese column qualification criteria are equallyvalid; operators should choose one and applyit regularly.

ConclusionGC users can perform many basicchromatographic calculations based on atest chromatogram, the pressure dropacross the column, its temperature and thecolumn dimensions. Making these basicmeasurements and calculations canprovide chromatographers with enhancedquality assurance for their analytical results.Testing following installation and at regularintervals will provide a record of columnperformance over time that analysts canuse to gauge when to replace a column, aswell as to identify problems before they

5LC•GC Europe - February 2001 GC connections

become serious and significantly affectanalytical results.

“GC Connections” editor John V. Hinshaw ispresident and principal scientist ofChromSource, Franklin, Tennesse, USA, and amember of the Editorial Advisory Board ofLC•GC Europe.

Direct correspondence about this columnto “GC Connections,” LC•GC Europe,Advanstar House, Park West, Sealand Road,Chester CH1 4RN, UK, e-mail: dhills@advanstar.com

For an ongoing discussion of GC issues withJohn Hinshaw and other chromatographers,visit the Chromatography Forum discussiongroup at www.chromforum.com