1 gas chromatography: theory and deﬁnitions, retention …gas chromatography: theory and...

1Gas Chromatography: Theory and Definitions, Retentionand Thermodynamics, and SelectivityGlenn E. Spangler

Due to the diversity of the published literature of the day, the writing of this chap-ter on gas chromatography was complicated. An additional exasperation is thenumber of processes dynamically involved during the separation process. Toaddress the issue, a disciplined theoretical approach was taken while writing thechapter with liberal cross-referencing to the open scientific literature. This is unlikeprior treatments where mere sequential mathematical demonstrations are pro-vided. The advantage of the approach is that it ties nomenclature to theory andallows a critical assessment of the published scientific literature. Because not allreaders appreciate such detail, Section 1.1 provides an overview of the historicaldevelopments in gas chromatography followed by a brief discussion of the princi-ples of operation. Important key words are italicized. This is continued in Sec-tions 1.2.1 and 1.2.2 where the simple concepts of the early theories of gaschromatography are provided. The development of a comprehensive theory for gaschromatography then begins in Section 1.2.3 and continues through the remainderof Section 1.2. Section 1.3 discusses issues related to sensitivity, resolution, andtemperature programming that are of interest to the user. Finally, the nomencla-ture section, Section 1.4, is not just a rehash of parameters used in the chapter,but an additional source of relationships. When properly digested, the chapter con-tains a lot of information for all readers, regardless of interest and experience.

1.1Introduction

Gas chromatography is an analytical technique used in many research and indus-trial laboratories to determine the composition, assess the quality, and/orimprove the purity of a sample compound. Instrumentally, the basic componentsof a gas chromatograph are as shown in Figure 1.1. The sample is analyzed byinjecting it into a mobile phase (carrier gas), vaporizing it in a heated injector,reducing the injected quantity through a splitter [1,2], separating the split samplethrough an internally coated column, and detecting the eluting constituent com-ponents (or analytes) for electronic recording. Like other chromatographic

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Analytical Separation Science, First Edition. Edited by Jared L. Anderson, Alain Berthod,Verónica Pino Estévez, and Apryll M. Stalcup. 2015 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2015 by Wiley-VCH Verlag GmbH & Co. KGaA.

techniques, the component vapors distribute or partition themselves betweenthe mobile phase and the stationary phase (internal column coating) as theypass through the column. James and Martin introduced gas–liquid chromatogra-phy (GLC) to the world when they first filled a 4–11 foot long, 4-mm internaldiameter glass tube with kieselguhr (size-graded Celite 545) and mixed it withsilicone gum during their Nobel Prize winning work of 1952 [3]. It replacedTswet’s [4,5] gas–solid chromatography (GSC) (or absorption chromatography)of 1906. James and Martin’s packed column chromatography technology wasmarketed as packed stainless steel column technology during the 1960s and1970s when packings were often nonuniform (i.e., contained voids). After estab-lishing theoretical feasibility, Golay introduced the wall-coated open tubular(WCOT) column in 1956 [6]. The WCOT column was a very long (90–180m)narrow-bore glass tube whose internal wall was coated with stationary phase.With time, the WCOT column technology evolved into capillary column tech-nology; but before that could happen, the fragility of the then favored fused silicacapillaries had to be addressed. Stability against breakage was accomplished byexternally coating the fused silica tubing with polyimide (low-to-intermediatelyhigh-temperature operation) or internally impregnating a stainless steel tubewith silica using a proprietary Silcosteel treatment (high-temperature opera-tion). These developments allowed capillary–column chromatography to gainwide acceptance during the late 1970s to early 1980s. Support-coated open tubu-lar (SCOT) columns were also developed to provide access to liquid phases notpossible with fused silica. The early column technology, however, was plagued bynonlinear retention due to the sample adsorbing to active sites within theunevenly coated columns. Acid washing and silylation of the columns before/after installation and/or between chromatographic runs helped to mitigate theproblem somewhat, but the nuisance remained until Grob introduced the immo-bilized phase in 1981 [7]. The liquid phase then became a cross-linked polymerwith the OH-terminated and Carbowax phases chemically bonded to the glasssurface. The new immobilized columns provided thicker coatings, washability ofthe coatings, reduced bleeding, and a wider practical range of operating temper-atures. As the bonded or immobilized phases became solidly entrenched inapplication, the higher capacity (more liquid phase) of packed columns contin-ued to be indispensable for preparing (preparative chromatography) and purify-ing specialty chemicals of interest to the chemical, pharmaceutical, andbiotechnology industries. This led to the development of highly repeatablepacked and micropacked columns with bonded and cross-linked stationaryphases. These later columns had lifetimes longer than their earlier predecessors.

Figure 1.1 Basic components of a gas chromatograph.

776 1 Gas Chromatography: Theory and Definitions, Retention and Thermodynamics, and Selectivity

When the sample components elute from the end of a gas chromatographiccolumn, they produce a series of time-sequenced peaks or a gas chromatogram.Such a gas chromatogram is illustrated in Figure 1.2, where the detectorresponse is shown vertically and the time after injection is shown horizontally.The first peak is an unretained air peak that corresponds to the dead volume ofthe instrument. Later, the sample component peaks elute with characteristicretention times (tR). While idealized symmetrical Gaussian peak shapes are

shown in Figure 1.2 with a full-width-at-half-maximum of wh � 2σffiffiffiffiffiffiffiffiffi2ln2

p, a

width between inflection points of wi � 2σ at 0:607h and a baseline width ofwb � 4σ, where σ is the standard deviation, non-Gaussian peak shapes are alsopossible. The task of modern vendors of gas chromatographic equipment is todesign instruments that avoid elution of non-Gaussian peaks. Because peakwidths increase with retention time, peak heights are not indicative of relativesample composition. That information is contained in the area under the peaks(shown brown in Figure 1.2). Advanced data-processing algorithms can be usedto help resolve overlapping peaks.

1.2Fundamental Theory of Operation

Because additional details on instrumentation and application are discussed else-where in this volume, only the fundamental theory of operation will beaddressed here. After discussing some simple concepts associated with the earlytheories of gas chromatography in Sections 1.2.1 and 1.2.2, the advanced theorywill be systematically developed in the remainder of Section 1.2, until the reten-tion of methanol on a Rtx-5 (5% diphenyl/95% dimethyl polysiloxane) column

Figure 1.2 Typical gas chromatogram.

1.2 Fundamental Theory of Operation 777

coupled to an injector and detector can be described. Because of the incompletenature for current gas chromatography theory, areas needing additional researchwill be noted.

1.2.1

Retention Volume Theory

Retention volume theory is the oldest and simplest theory for gas chromatogra-phy [8]. It is based on the idea that a volume of mobile phase (carrier gas) isneeded to transport an entrained sample component through the column. Twocontributions to retention volume V R are known: (1) the holdup volume VM

corresponding to the volume required to elute an unretained peak (e.g., air) and(2) the retained volume given by the distribution coefficient for the sample com-ponent in the stationary phase times the volume or area of the stationary phase.When added together, the retention volume becomes

V R � VM � KCV S � 1 � k� �VM � VM

RM(1.1)

for gas-liquid chromatography and

V R � VM � KS AS � 1 � k� �VM � VM

RM(1.2)

for gas–solid chromatography. If equilibrium conditions exist everywhere withinthe column, the distribution coefficients are the Henry’s law constants, KC �cS=cM and KS � cA=cM, respectively.Instead of retention volumes, retention times are more practical to measure. Expe-

rience has shown that the significant retention time is not just the retention volumedivided by carrier gas flow, but other adjustments are necessary [9,10]. While reten-tion time/volume theory is not developed further here, the reader may consult thereferenced literature for additional information.

1.2.2

Plate Theory

To describe the shape of a gas chromatographic peak, Martin and Synge devel-oped plate theory that was later refined by van Deemter, Zuiderweg, and Klin-kenberg [11,12]. The theory imagines that the column consists of a largenumber of bins (theoretical plates) through which the sample must pass beforeexiting the column. As the migration proceeds, the migrating zone spreads ran-domly to neighboring plates with a probability distribution that gradually evolvesinto a Gaussian distribution before exiting the column. While a number of gooddescriptions exist for plate theory [13,14], it has at least five limitations: (1) itassumes the existence of discrete/discontinuous plates, (2) it assumes plate-wideequilibrium, (3) no accounting is made for longitudinal diffusion, (4) it is unableto describe mass transfer, and (5) it sidesteps kinetic processes that might alsocontribute to the separation process [15]. For these reasons, plate theory will not


developed further here, but the reader may consult the referenced literature foradditional information.

1.2.3Fluid Dynamics

When viewed as a fluid dynamic flow tube, a pressure gradient is needed to pushthe mobile phase (carrier gas) through the column. Depending on the magnitudeof the pressure gradient, the flow may be laminar or turbulent. Gas chromatog-raphy is typically operated under Hagen–Poiseuille laminar flow (low Reynoldsnumber) conditions [16] with the average linear velocity for the carrier gas satis-fying Darcy’s law [17,18]:

vk � �Kmean

η

dPdz� Fc

εMAc: (1.3)

Here, Kmean � ∫K vdAc=Ac is r2c=8 for an open tubular column [19]

andD2

413� 64Dπ5W

X1n�1

tanh�πW 2n � 1� �=2D�2n � 1� �5

( )��→

W>>D

13

D2W 2

D2 �W 2 for a rect-

angular column [20–22]. While Darcy’s law applies to both open tubular andpacked columns, Guiochon discusses additional considerations that should begiven when applying Darcy’s law to packed columns [23].Boyle’s law, PoFc�o� � PFc, requires that the pressure gradient satisfy

dPdz� � ηPoFc�o�

KmeanεMAc

� �1P: (1.4)

After integrating using the boundary conditions P � Pi at z � 0 and P � Po atz � L, Equation 1.4 becomes [13]

zL� P2

i � P2

P2i � P2

o

: (1.5)

The first and second derivatives, after rearrangement, then become

dPdz� P2

o � P2i

2L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP2i � z

LP2i � P2

o

� �r and@2P@z2� � P2

o � P2i

� �24L2 P2

i � zL

P2i � P2

o

� �� 3=2 :(1.6)

The derivatives are plotted in Figure 1.3 (normalized to 1.0 at z = L), where rPis nearly constant and r2P is near zero for high Pi=Po ratios. Thus, the averagelinear velocity for the mobile phase through the column is nearly constant (seeDarcy’s law of Equation 1.3). While this operating condition overlooks convec-tion arising from the pressure drop at the end of the column z � L� �, the condi-tion can be encouraged experimentally by slightly restricting the flow of carriergas exiting the column. This is done, for example, in supercritical fluid chroma-tography, a form of high-pressure chromatography [24].


1.2.4

Continuum Theory

Shortly after the work of James and Martin, a move was made to develop a “con-tinuum theory” for gas chromatography. Lapidus and Amundson [25] realizedthat due to the large number density of molecules involved in gas chromatogra-phy, the separation process is better described by classical diffusional mass trans-port theory [19,26,27]. Kinetics might also be included to address boundaryconditions, but this is secondary to the overall scheme. Compared to the reten-tion volume and plate theories, continuum theory was very comprehensive anddescribed not only the flow profile for the carrier gas through the column, butalso addressed transport and dispersion of the sample through the column,retention of the sample in the stationary liquid phase, and loss of sample byreversible and irreversible adsorption on the stationary phase. That is, a veryfundamental understanding of gas chromatography was needed to formulate thetheory. Until recently, continuum theory has been avoided by most practicingchromatographers because of its complexity.

1.2.4.1 Sample Transport in the Mobile PhaseAccording to the continuum theory, the transport of sample through a gas chro-matographic column is described by the continuity (hence the name) equation

dcMdt� @cM

@t� v

)� r)

cM � r

)� D

$

M � r

)

cM

� cM r

)� v)

; (1.7)

Figure 1.3 rP (solid lines) and r2P (dashed lines) versus relative position z=L along thecolumn and for different inlet/outlet pressure ratios Pi=Po.


where dcM=dt is a substantive derivative in a reference frame moving with themobile phase, v

)

is the stream velocity (not a particle velocity) for the mobilephase (carrier gas) [19,28], D

$

M is a three-dimensional diffusion tensor, andcM r

)� v)

is a convection term [29]. For packed column chromatography, theeffects of eddy circulation (the splitting and recombination of flows around thepacking material) can be included by adding an extra term to the diffusioncoefficient, DM�eff� � DM � εD [12,30,31].Typically, Equation 1.7 is averaged over the cross-sectional area Ac of the column

@cM z� �@t

� r)� D

$

M � r

)

cM z� �

� 1Ac ∫

Ac

0v

)�r)

cMdAc � 1Ac ∫

Ac

0cM r

)� v)

dAc;

(1.8)

where the average concentration cM z� � � ∫Ac

0 cMdAc=Ac remains a function z.cM z� � will differ from cM z� � only if the mobile phase flows faster than a lag time,τL M� � � r2c=6DM ?� �, the time required to establish equilibrium perpendicular tothe longitudinal flow through the column [32]. Using

1Ac∫

Ac

0v

)� r)

cM z� �dAc � 1Ac ∫

Ac

0vk@cM z� �@z

dAc � 1Ac ∫

Ac

0v?

@cM z� �@r

dAc

' vk@cM z� �@z

;

(1.9)

∫Ac

0cM z� � r)� v)

dAc � ∫Ac

0cM z� � @vk

@zdAc � ∫

Ac

0cM z� � @v?

@rdAc

' cM z� � @vk@z

;

(1.10)

where the transverse velocity v? is assumed zero, Equation 1.8 can be rewritten as

@cM z� �@t

� r)� D

$

M � r

)

cM z� �

� @ vkcM� �@z

: (1.11)

Since τL M� � is typically 2ms for DM ?� � � 0:059 cm2=s across a rc � 0:54 mm mac-robore column, equilibrium is normally established within 1mm of the columnentrance.

1.2.4.2 Sample Transport in the Stationary PhaseSimilar to the mobile phase, the concentration of solute molecules cS in the sta-tionary phase is given by

dcSdt� @cS

@t� r

� D

$

S � r

cS

: (1.12)

Note that the terms that contain the linear stream velocity in Equation 1.7 arezero in Equation 1.12. Because the sample component molecules are now free to


interact with the polymeric backbone of the stationary liquid phase, the diffusioncoefficient DS is thermally activated and increases with temperature [32,33]

DS � DS 0� � exp � ED

RDT

: (1.13)

For DS? �5:3�10�7 cm2=s through a df �1�10�6m thick silicone film [34], the lagtime for establishing equilibrium, τL S� � �d2

f =6DS?; is typically on the order of 3ms.

1.2.4.3 Sample Transport through the ColumnFor the reason that the sample zone must interact with both the mobile and thestationary phases while passing through the column, Equations 1.11 and 1.12 canbe added together to describe total transport. To assist, the volume of a differentiallength (slice) of the column, Δz; is shown in Figure 1.4. The total volume for theslice is ΔV c � ΔVM � ΔV S � AcΔz: Because the number of sample componentmolecules contained in the slice is Equation 1.11 multiplied by ΔVM plus Equa-tion 1.12 multiplied by ΔV S, the total concentration of the migrating sample satis-fies

@cT z� �@t

� 1ΔV c

ΔVM@

@zDM k� � @cM z� �

@z

� ΔV S

@

@zDS k� � @cS z� �

@z

� ΔVM

@ vkcM z� �� @z

� �:

(1.14)

Since DS k� � << DM k� �, the second term on the right-hand side can be neglected(unlike liquid chromatography). Defining RM � cM z� �ΔVM=cT z� �ΔV c �cM z� �ΔVM=cT z� �ΔV c as the fractional amount of sample in the mobile phase,Equation 1.14 becomes

@cT z� �@t

� RM@

@zDM k� � @cT z� �

@z

� @ vkcT z� ��

@z

� �: (1.15)

Equation 1.15 is known as the general rate model for gas chromatography.The above derivation assumed no intracolumn adsorption. In the presence of

intracolumn adsorption, a third component equation can be written to accountfor the amount, cA, of sample attached to the adsorption sites. This third equa-tion is then combined with Equations 1.11 and 1.12 to develop an expandedgeneral rate model. Additional terms will also appear in each of the equations

Figure 1.4 A sample zone (light gray) migrating through a gas chromatographic column witha slice (dark gray) of width Δz selected for analysis.


due to kinetic coupling dictated by assumed adsorption isotherms (or changingadsorption isotherms, in the case of temperature programming) [25,35].Unfortunately for a nonuniformly coated column, RA will in general be a func-tion of location within the column to create either an indescribable chromato-gram or make the theoretical description intractable [36]. Similarly exposedcouplers and fittings will aid in unevenly distributing the sample vapors as theypass through the system. It is clear that a uniformly coated or thoroughly deacti-vated sample train is necessary to avoid complications associated withunnecessary sample holdup during a gas chromatographic run.

1.2.4.4 Solution to the General Rate ModelBecause vk and DM k� � are functions of pressure within the column, Equation 1.15is not easily solved without careful accounting. The pressure dependence of vk isgiven by Equation 1.3 and a similar relationship exists for DM k� � [15,37–39]:

DM k� � �0:00186T3=2 1

Mw 1� �� 1Mw 2� �

1=2

P atm� �σ212Ω 1;1� �* T*� � : (1.16)

Like vk, DM k� � is proportional to pressure gradient through Equation 1.4. Com-bining Equations 1.3, 1.4, and 1.16 with 1.15, the transport equation becomes

@cT z� �@t

� RM

� 0:00186T3=2 �1=Mw 1� �� 1=Mw 2� �� 1=2

σ212Ω1;1� �* T*� �

KmeanεMAc

ηPoFc�o�

@2P@z2

@cT z� �@z

� @P@z

@2cT z� �@z2

1

Mw 1� �� 1Mw 2� �

�Kmean

η

@2P@z2

cT z� � � @P@z

@cT z� �@z

:

8>>>>>>>>>>><>>>>>>>>>>>: (1.17)

Since viscosity η is independent of pressure [38], it can be moved outside thedifferential operator along with the other constants! Equation 1.17, however, isno easier to solve than Equation 1.15. But if r2P � 0 as discussed in connectionwith Figure 1.3, Equation 1.17 can be written as

@cT z� �@t

� RMDM�k�@2cT z� �@z2

� RMvk@cT z� �@z

: (1.18)

Einstein showed that a solution to Equation 1.18 is [40,41]

cT z; t� � � Affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4πRMDM k� �t

q exp � z � RMvkt� �24RMDM k� �t

!; (1.19)

where A is an unspecified constant.Sternberg explained that Equation 1.19 can be used in combination with (or

convoluted with) a source function to describe total system performance [42].


This is accomplished using Green’s theorem:

cT z; t� � � ∫G z � z´; t � t´� � S z´; t´� �dt´dz´; (1.20)

where S z´; t´� � is the source function and G z � z´; t � t´� � is a normalized Green’sfunction [43]. For plug injection [44], the appropriate source function S z´; t´� � is

S z´; t´� � � cT 0� � H z´ � RMvkti=2� � �H z´ � RMvkti=2

� � �δ t´� �; (1.21)

which causes the integration to be performed over z´, rather than t´. This is nec-essary to bypass the time-dependence of Einstein’s standard deviation,

σ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2RMDM k� �t

q. The appropriate Green’s function is Equation 1.19 with A = 1.

Substituting Equations 1.19 and 1.21 into Equation 1.20, we get

cT z; t� � � cT 0� �2

erfz � RMvk t � ti=2� �2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRMDM k� �t

q0B@

1CA � erf

z � RMvk t � ti=2� �2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRMDM k� �t

q0B@

1CA

0B@

1CA;

(1.22)

where the definition of the error function is needed to obtain the final result [45].Equation 1.22 explicitly specifies the time t and location z of the migrating sam-ple zone after injection. It is plotted in Figure 1.5 for two injection times, ti, andtwo column lengths, L. RM is set equal to 0.67 to describe the chromatogramproduced by methanol after injection onto a Rtx-5 (5% diphenyl/95% dimethylpolysiloxane) column with a helium carrier gas flow rate of 25 cm/s at35 °C [46]. For the longer injection time (1.5 s), the error functions produce aflat-topped peak for a short (6m) column; and for the shorter injection time(0.5 s), the error functions approximate a Gaussian peak shape. This indicatesthat a fast injector is always needed when designing a fast- or short-column gaschromatograph [47]. Equation 1.22 corresponds to Sternberg’s equation 204when ti is negligibly small [42].

Figure 1.5 Simulated gas chromatograms for 1.5 s (solid) and 0.5 s (broken) methanol pluginjections into 6m (a) and 30m (b) columns.


1.2.4.5 Relationship to Retention VolumeIn Equation 1.22, z � RMvkt � L � RMvktR is equal to 0 at the column exit.Because tR � tM 1 � k� � and k � K cV S=VM;

L � RMvktR→L � RMvktM 1 � k� �→L � RMvktM 1 � K cV S

VM

� 0 (1.23)

Multiplying by the volumetric flow rate Fc,

LFc � RMvktMFc 1 � K cV S

VM

! V R � VM 1 � K c

V S

VM

� 0 ; (1.24)

which corresponds to

V R � VM � KCV S: (1.25)

Equation 1.25 is equivalent to Equation 1.1!

1.2.4.6 VarianceVariance can be discussed from two points of view [42]: (1) equivalence to thecentralized second moment [48–50],

μ2 �∫all ξ

ξ �M1� �2cT ξ� �dξ

∫all ξ

cT ξ� �dξ; (1.26)

or (2) additivity of second moments for “independent Gaussian partial contributions”

σ2 �Xnk�1

σ2k : (1.27)

By convoluting a Gaussian with a Gaussian, Sternberg showed that Equation 1.27follows from Equation 1.26 and total variance is a geometric mean of “statisti-cally significant and independent variances.” Since the direct application ofEquation 1.26 to find variance is not yet available, Equation 1.27 has been usedto less rigorously estimate variance. For example, Spangler and Collins usedEquation 1.27 when writing the variance for ion mobility spectrometry (usingcurrent chromatographic terminology) as [51]

σ2 � 2DM k� �tRRMv2k

� t2i4: (1.28)

Equation 1.28 has been extensively applied to optimize the performance andconstruction of ion mobility spectrometer sensor cells [52–59].Another approach to computing variance is to take the Laplace transform of

Equation 1.15 and use [49]

∫1

0ξkCT ξ� �dξ � �1� �k lim

s!0

dk ~C s� �dsk

(1.29)


to compute the kth moment. The second centralized moment corresponds tok= 2. For plug injection, Wolff, Radeke, and Gelbin showed that this Laplacetransform method yields (using current chromatographic terminology) [60]

σ2 � 2zDM k� �R2Mv

3k� t2i12

(1.30)

for RMzvk >> DM k� � in the absence of intracolumn adsorption [61]. More work isneeded to compare a direct moment computation using Equation 1.26 in Equa-tions 1.28 and 1.30.

1.2.4.7 Nonequilibrium TransportIn an effort to provide a bridge between plate theory and continuum theory [12],Golay and Giddings argued that the sample component concentration must departslightly from equilibrium εi as it leaves one region (plate) of the column containinga high concentration of sample to enter another region (plate) of the column con-taining a low concentration of sample, and vice versa. That is [15,62–65],

ci � c*i 1 � εi� �; (1.31)

Then by performing a variational analysis on Equation 1.7, they showed that themigration of sample through an open tubular (or capillary) column can bedescribed by an effective diffusion coefficient

DM eff� � �DM k� �1 � k� � �

1 � 6k � 11k2

48 1 � k� �3v2k r2cDM k� � : (1.32)

The first term arises from Einstein diffusion and the second term arises fromresistance to mass transfer in the mobile phase. For an open rectangular column,the result is similar

DM eff� � �DM k� �1 � k� � � f �k� v

2kD2

DM k� � ; (1.33)

where f k� � is a function of the partition coefficient k only [66]. Aware of vanDeemter’s work [12], Golay also tried to compute the resistance to mass transferin the stationary phase, but equation 76 in reference [61], reproduced in refer-ence [67], was incorrect. It took Giddings to show [68] that

DM eff� � �DM k� �1 � k� � �

1 � 6k � 11k2

192 1 � k� �3v2k d

2c

DM k� � �2kv2k d

2f

6 1 � k� �3DS: (1.34)

Much later, Spangler confirmed Equation 1.34 using Golay’s more complicatedtheoretical approach [69]. Using Equations 1.32 to 1.34, Golay further arguedthat the variance for a gas chromatographic peak can be expressed as

σ2 � 2DM eff� � 1 � k� �Lvk

� HL; (1.35)

where H � B=vk � CM � CS� �vk is van Deemter’s height equivalent to a theoreti-cal plate (HETP) [12].


Grushka derived equations similar to Equations 1.34 and 1.35 using the Laplacetransform method mentioned above [70,71]. However because he felt that thefinite rate of mass transfer across the mobile stationary phases is normally assumedinfinite in gas chromatography, he came up short. His result is sufficiently com-plete to suggest that nonequilibrium phenomena do indeed play at least a partialrole when describing the variance of a gas chromatographic separation.

1.2.5

Extracolumn Effects

Besides the column, other elements of a gas chromatograph contribute to peakbroadening [72]. The two most important are the dead volumes for the (1) injec-tor and (2) detector. If instantaneous concentrations of sample cI and cD areintroduced into the internal volumes of these elements, the concentrations expo-nentially decay in accordance with [73]

dcIdt� � F I

V I�eff�cI or

dcD�t�dt

� � FD

VD�eff�cD�t�: (1.36)

The exponential decay is caused by the flow of mobile phase (carrier gas) pro-gressively diluting the initial injected concentration. The effective volumes,V I�eff� and VD eff� � are not only geometric volumes, but also include equivalentvolumes of sample vapors adsorbed on the internal walls [73]

V I�eff� � VM I� � � cAcM

AS I� �

� VM I� � � KSAS I� �or

VD�eff� � VM D� � � cAcM

AS D� �

� VM D� � � KSAS D� �:(1.37)

The solutions to Equation 1.36 are

cI � ARMvkτI

expz � RMvktRMvkτI

and cD t� � � A

vk od� �τDexp

z � vk od� �tvk od� �τD

!;

(1.38)

when expressed in terms of a reference frame moving with the linear velocity ofthe mobile phase. Equation 1.38 is essential to modeling extracolumn contribu-tions due to peak broadening in gas chromatography where τI and τD are timeconstants.

1.2.5.1 Injector (Plug Injection with Exponential Dilution)Applying Green’s theorem to the injector and using the first relationship ofEquation 1.38, the source function can be written as

S z´; t´� � � cI 0� �Fc

Fc � SFexp

z´ � RMvkt´RMvkτI

H z´ � RMvkti=2� � �H z´ � RMvkti=2

� � �δ t´� �;(1.39)


after taking sample dilution by the splitter into account; SF is the splitter ventflow. Introducing Equation 1.39 and Equation 1.19, A � 1, into Equation 1.20

cT z; t� � � cI 0� �Fc

2 Fc � SF� �

expz � RMvktRMvkτI

� DM k� �tRM vkτI� �2

! erf

z � RMvk�t � ti=2� �2DM k� �tvkτIffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4RMDM k� �tq

0BBB@

1CCCA�

erf

z � RMvk�t � ti=2� �2DM k� �tvkτIffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4RMDM k� �tq

0BBB@

1CCCA

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCA

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

;

(1.40)

where formula 7.4.32 of Abramowitz and Stegun is used to complete the integra-tion [45]. Equation 1.40 is a generalized statement of Sternberg’s equation 20,which in the limit of small ti, produces an “exponentially modified Gaussian.”An exponentially modified Gaussian is often used to describe this type ofinjection [42,74,75].Equation 1.40 is plotted in Figure 1.6. The flat-topped peak of Figure 1.5 is

now an exponentially decaying peak with the slope extending over the durationof the injection pulse. Also, the arguments of the error functions displace thepeak maximum by 2DM k� �t=vkτI.1.2.5.2 DetectorSince the detector receives effluent from the column, the correct source functionis Equation 1.40 evaluated at z � L and t � tR

Figure 1.6 Simulated gas chromatograms for methanol eluting from 6m (a) and 30m (b) longcolumns using the same operating conditions as in Figure 1.5 but with a 1 s injector timeconstant.


S z´; t´� � � cI 0� �Fc o� �Fc

2FD Fc � SF� � expz´ � RMvkt´RMvkτI

�DM k� �t´

RM vkτI� �2

0BBB@

1CCCA

erf

z´ � RMvk�t´ � ti=2� �2DM k� �t´vkτIffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4RMDM k� �t´q

0BBB@

1CCCA�

erf

z´ � RMvk�t´ � ti=2� �2DM k� �t´vkτIffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4RMDM k� �t´q

0BBB@

1CCCA

266666666666664

377777777777775δ t´ � tR� � 1 �H�z´ � L�� :

(1.41)

Here, Fc o� �=FD adjusts for dilution by gas expansion and/or addition of makeupgas at the column exit.If the Green’s function is assumed to be the second relationship of Equa-

tion 1.38, A � 1, the final solution to Equation 1.20 is

cT z; t� � � cI 0� �Fc o� �FcRMvkτI2FD Fc � SF� � RMvkτI � vk�od�τD

� � exp � tRτI� tRτD� DM k� �tRRM vkτI� �2

!

�

expRMvkτI � vk�od�τD� �

L

RMvkvk�od�τIτD

0@

1A

erfL � RMvk�tR � ti=2� �

2DM k� �tRvkτIffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4RMDM k� �tRq

0BBB@

1CCCA�

erf

L � RMvk�tR � ti=2� �2DM k� �tR

vkτIffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4RMDM k� �tR

q0BBB@

1CCCA

266666666666664

377777777777775�

expRMvkτI � vk�od�τD� �2

RMDM k� �tRRMvkvk�od�τIτD� �2

0B@

1CA

erf



q �

RMvkτI � vk�od�τD� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

RMDM k� �tRq


0BBBBBBBBB@

1CCCCCCCCCA�

erf



q �

RMvkτI � vk�od�τD� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

RMDM k� �tRq


0BBBBBBBBB@

1CCCCCCCCCA

266666666666666666666666664

377777777777777777777777775

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

;

(1.42)


where again integration formula 7.4.32 of Abramowitz and Stegun is used tocomplete the integration [45]. Because Sternberg viewed the detector as a mirrorimage of the injector, he did not derive an equation similar to Equation 1.42 [42].Equation 1.42 is plotted in Figure 1.7, where the time constants τI and τD

enter as τ�1I � τ�1D . Interestingly, when Figure 1.7 is compared with Figures 1.5and 1.6, the shapes of the eluting peaks appear to be much more Gaussian! Inthe open literature, Schmauch has shown that the response time, sensitivity,and resolution of a flow-through thermal conductivity detector (TCD) improvesover a diffusion-controlled TCD by incorporating a flow-through design [76].Similarly, Jennings describes a configuration for the flame ionization detector(FID) where the sample is introduced into the throat of the detector aftermakeup gas is added (see insert in Figure 1.7) [77]. Both of these approachesshorten the effective time constant for exponential decay in accordance withEquation 1.42, causing the detected peak shape to be more Gaussian! This resultis independent of whether the detector is a concentration or mass flow rate sen-sitive detector [78,79].

1.2.5.3 VarianceUsing the Laplace transform method and noting the difficulty that might beencountered when taking the inverse transform, Sternberg showed that the vari-ance for an exponential convoluted with a Gaussian is μ2 � σ2 � τ2I at a center ofgravity displaced from the peak maximum [42,80]. Because of his finding,Gasper, Annino, Vidal-Madjar, and Guiochon have postulated that the variancefor a total gas chromatographic system is σ2t � σ2c � σ2ec where σ2c and σ2ec are thepartial column and extracolumn contributions, respectively [72]. Delley has fur-ther suggested that the total variance associated with an injector and detectorcoupled to a column is σ2t � σ2 � λIτ2I � λDτ2D [81]. This is unlike Equation 1.42where τ�1I of Equation 1.40 is replaced by τ�1I � τ�1D . More work, theoretical andexperimental, is needed to resolve this issue.

Figure 1.7 Simulated gas chromatograms for methanol eluting from 6m (a) and 30m (b) longcolumns using the same operating conditions as in Figure 1.5 but with a 1 s injector time con-stant and a 0.35 s detector time constant, FD � 4:3 � Fc o� �.


1.3Other Operational Considerations

Other operational considerations for gas chromatography are its sensitivity, reso-lution, and the improved performance accompanying temperature programming.

1.3.1

Sensitivity

A good starting point for discussing sensitivity is the detector. It is the mostsensitive element of a gas chromatographic system and can also provide specific-ity of detection. As shown in Table 1.1, modern detectors are more sensitivethan older versions. The reasons are not completely clear, but the recent trendsin detector development show the use of smaller internal volumes with polished/deactivated interior walls and to more tightly couple the chromatographic col-umn to the detector using makeup gas. Variations on this theme depend on thedetector type. The FID is sensitive to the carbon content of the sample [82–87],but the exact relationship with carbon number depends on compound class [88].If the nitrogen–phosphorous detector (NFD) is viewed as a hydrogen enrichedchemically reactive boundary layer near a hot ceramic bead, it ionizes nitrogenand phosphorous heteroatoms in a sample by electron extraction from theceramic bead after decomposition [97]. The pulse-flame photometric detector is

Table 1.1 Minimum detection limits for gas chromatographic detectors.

Olderspecifications [88–94]

Newerspecifications [95,96]

MDL Dynamicrange

MDL Dynamicrange

Thermal conductivity detector 1–10 pg/s(0.01–1.0 μg)

103–104 400 pg >105

Flame ionization detector 20 pgC/s 106–107 <1.5 pgC/s >107

Microelectron capture detector 10 fg/s 102–103 <5.5 fg/s >53104

Nitrogen–phosphorous detector <0.2 pgN/s0.01 pgP/s

<0.1 pgN/s<0.003 pgP/s

>105 N>105 P

Flame photometric detector 0.01 pgP/s <60 fgP/s<3.6 pgS/s

104 P103 S

Sulfur chemiluminescent detector <0.5 pgS/s >104

Nitrogen chemiluminescentdetector

<3 pgN/s >104

Barrier discharge ionizationdetector

1 pgC/s 105

Photoionization detector 1 pg 104

1.3 Other Operational Considerations 791

highly selective for sensitive detection of phosphorous and sulfur com-pounds [98]. The electron capture detector (ECD), originally a DC detector [99],is now a pulsed constant-current (frequency-modulated) detector that greatlyincreases its dynamic range [100–102]. Its ability to ionize electronegative com-pounds is typically specified relative to chlorinated compounds [103,104]. Thephotoionization detector (PID) ionizes compounds with unsaturated bondsusing high energy photons emitted by a UV lamp: 8.4 eV (Xe-Sapphire), 9.6 eV(Xe-MgF2), 10.0 eV (Kr-CaF2), 10.2 ev (D2-MgF2), 10.6 eV (Kr-MgF2), and11.7 eV (Ar-LiF). The ionization potential of the compound, rather than the rela-tive number of π-electrons, is more important to explaining the sensitivity of thedetector, which is typically specified relative to benzene [105,106]. While thethermal conductivity detector is the least sensitive of all the detectors, it is par-ticularly suited for preparative chromatography because of its nondestructivedetection capabilities.The addition of a chromatographic column to a detector reduces sensitive by at

least an order of magnitude [107]. Some of this loss can be regained by usinggood sample collection and preparation techniques [108–113]. In recent years,this has been done using solid-phase microextraction (SPME) [114–116] devel-oped by Pawliszyn’s group at the University of Waterloo [117–119]. SPMEinvolves exposing a coated fiber to sample vapors and inserting it into the heatedinjector of the gas chromatograph to thermally desorb the enriched vapors.Because no solvent is used, inlet volumes can be reduced by replacing the glassinsert with one having a reduced internal diameter. Limits of detection achievablewith SPME and related techniques are summarized in Table 1.2. For SPME, theminimum detectable levels range 1–20 pg/ml for aqueous to 62–364 pg for airsamples. A moderately large standard deviation accompanies the use of SPME.

1.3.2Resolution

The resolution of a gas chromatograph is defined as the ratio of the difference inretention times to the average base width wb� � between two peaks

Rs � tR2 � tR1wb1 � wb2� �=2 : (1.43)

If the two peaks are Gaussian in shape, then wb1 � 4σ1 and wb2 � 4σ2 with theσi’s being the standard deviations for the two peaks. Satisfactory resolutionoccurs when Rs � 1, and baseline resolution occurs when Rs � 1:5 [146,147]. Fornon-Gaussian peaks, the relationship of Rs to variance is not established.A related measure of resolution is resolving power [55,57]

Rm � tRwh� tR

2σffiffiffiffiffiffiffiffiffiffiffi2 ln 2

p : (1.44)


Table

1.2

Minim

umde

tectionlevels(M

DLs)for

gaschromatog

raph

yusingvario

ussamplecollectionan

dinjectiontechniqu

es.

Autho

rsSa

mple

Sample,collection,

injection

Detector

MDL

Nacsonet

al.[120]

Exp

losive

vapo

rsin

air,residu

esPreconcentratevapo

rsSw

abbedresidu

eECD

5–20

ppt(highV.P.)

<1–

5ng

(low

V.P.)

Batile

etal.[121]

Nitroarom

aticexplosivevapo

rsSP

E,S

FE,H

ypercarb

PGCtrap

N–P

62–36

4pg

Hableet

al.[122]

Exp

losivesvapo

rsin

air

XAD-2

samplecartridg

e,isoamylacetate

extraction

ECD

0.4–

2.0μg

WelschandBlock

[123]

Exp

losivesin

water

Liquid–

liquidextraction

ECD

1–24

0ng

/l

Hableet

al.[12

4]Exp

losivesin

water

Toluene

andisoamylacetateextractwith

autosampling

ECD

0.00

3–6μg/l

Walsh

andRanney[125–127]

Exp

losivesin

water

SPEdisks/cartridg

es,d

eactivated

injector

ECD

0.04–0.4μg/l(lab)

1–40

μg/l(field)

BarshickandGriest[128

]Exp

losivesin

seaw

ater

Various

SPME

DVD

porous

polymer

ITMSNCI

5pp

t(2ADNT)

10pp

t(4ADNT,T

NT)

325pp

t(RDX)

Dou

se[129]

Exp

losivesin

ethylacetate

Syring

einjection

ECD

1–10

0pg

Hew

ittandJenk

ins[130

]Nitroarom

atic,n

itramine

explosives

insoil/water

Acetone

supernatant

SPE

N–P

ECD

0.38–0.52

mg/kg

1.63–1.97

μg/l

Walsh

[127]

Exp

losivesin

soil

Metho

d8330

extraction

ECD

0.73–26

μg/kg

Erickson

etal.[131]

Aqu

eous

tributylph

osph

ate

PDMSSP

ME

IMS

9.8–

196μg/l(w

/oSP

ME)

0.49

μg/l(w

SPME)

Luoet

al.[132]

Organop

hospho

rous

pesticides

Graph

eneSP

EECD

0.83–11

.5ng

/l

Berijani

etal.[133]

Organop

hospho

rous

pesticides

Dispersiveliq

uid–

liquidextraction

FPD

3–20

ng/l

Anjos

andAnd

rade

[134]

Pesticidesin

coconu

twater

Sing

ledrop

microextraction

EI-MS

0.10–0.88

μg/l (c

ontinu

ed)


Table

1.2

(Con

tinued)

Autho

rsSa

mple

Sample,collection,

injection

Detector

MDL

Lakso

andNg[135]

Aqu

eous

chem

ical

warfare

agents

PDMSandCBDVXSP

ME

PDMSD

VBSP

ME

FID

0.05

ppb(G

A,G

B,G

D)

0.5–

1.0pp

b(V

X)

Palitet

al.[13

6]Aqu

eous

chem

ical

warfare

agentsandrelated

compo

unds

SPME(w

salt)

EI-QMS

10–75

μg/l

Denget

al.[13

7]Fo

urhydrocarbo

ns0.4μl

syring

e14

:1split

injection

Miniature

FID

0.43–0.51

ng

Tsujin

oandKuw

ata[138

]Atm

osph

erichydrocarbo

nsTedlarbag

2cryogenictraps

FID

0.4–

31pp

b

PotterandPaw

liszyn[139]

Aqu

eous

BTEX

PDMSSP

ME

ITMS

1–15

pg/m

l

PotterandPaw

liszyn[140]

Aqu

eous

PAH,P

CB

PDMSSP

ME,

Carbo

pack

BSP

ME

ITMS

1–20

pg/m

l

Fanet

al.[14

1]Aqu

eous

PAH,n

-alkanes

Graph

eneSP

E(w

salt)

FID

9.8–

50ng

/l(PAH)

Kialeng

ilaet

al.[14

2]Aqu

eous

VOCs

Fullevaporationheadspace

FID

MSSIM

14–86

ng/vial

AlvaradoandRose[143]

VOCsin

soil

HP7694

headspacesampler

ECD

0.9–

10pp

t

Baim

andHill

[144]

Dod

ecanein

gasolin

eSyring

eIM

S10

0pg

YangandPe

ppard[145

]Flavor

analysis

PDMSSP

ME

aqueou

s(w

salt)andheadspace

GC

(1mm

IDlin

er)

GC/M

S

0.01–10

ppb


1.3.3

Temperature Programming

Originally, gas chromatography was performed by applying a constant tempera-ture to the column, but this isothermal approach led to what is known as the“general elution problem”

1) A given analysis is limited to a narrow range of boiling points or volatility.2) Early eluting peaks are either too close or coeluting.3) More strongly retained peaks are broad and reduced in amplitude.

To improve separations, modern gas chromatographs are equipped with tem-perature programmable ovens. Early in a run when the column temperature islow, the migration of the more volatile components is slowed down; later in arun when the column temperature is high, the migration of the lesser volatilecomponents is speeded up.Temperature programming is effective because the partition coefficients of

Equations 1.1 and 1.2 are functions of temperature. For gas–liquid chromatogra-phy, KC satisfies [148–150]

KC � exp �Δgmix=RDT� � � exp � Δhmix � TΔsmix

� �=RDT

� �; (1.45)

where the changes in molar free energy Δgmix and molar enthalpy Δhmix accom-pany dissolution of the sample component vapors into the stationary phase. Fora dilute mixture of nonpolar compounds, the molar free energy and molarenthalpy of mixing satisfy [151,152]

Δgmix � RDTXM ln γMXM� � � RDTXS ln γSXS� �' RDTXM ln XM � RDTXS ln XS; (1.46)

Δhs �@

Δgmix

T

@1T

� XMV S M� � � XSV S S� ��

δM � δS� �2ϕMϕS: (1.47)

For polar compounds, the polarity of the stationary phase can be adjusted toaccommodate “like dissolves like” [153].Because of the heterogeneity of active sites in gas–solid chromatogra-

phy [154,155], it is best performed on gases and otherwise more volatilecompounds at high temperature using isothermal conditions. The high tem-perature causes the active adsorption sites with low heats of adsorption tocontribute less to the separation process than the active adsorption sites withhigh heats of adsorption. This, of course, places demands on column tech-nology since high-temperature operations require special fabrication consid-erations. Porous layer open tubular (PLOT) columns are an example of afairly recent development in gas–solid chromatography where a solid porouslayer is applied to the inner surface of a capillary column. A good PLOT


column will perform gas–solid chromatographic separations otherwise notpossible with gas–liquid chromatography.

1.4Nomenclature

The International Union of Pure and Applied Chemistry (IUPAC) has made rec-ommendations for the nomenclature to be used when describing gas chromatog-raphy [156]. Those recommendations have been followed as closely as possiblewhile writing this chapter. Where no guidance was given by IUPAC, nomencla-ture consistent with the original scientific literature was followed. The followinglist is provided to consolidate terminology:

A - Unspecified constant.

Ac - Cross-sectional area of the column. Ac � πr2c � πd2c=4 for an open tubular

column and Ac �WD for a rectangular column.

AS - Surface area available for sorption on solid phase. ΔAS is a differentialelement (slice). AS I� � and AS D� � are the sorbing surface areas in the injec-tor and detector, respectively.

cD - Concentration (mass/unit volume) of sample component in the detector.cD 0� � is the initial concentration.

cI - Concentration (mass/unit volume) of sample component in the injector.cI 0� � is the initial concentration.

ci - Concentration (mass/unit volume) of sample component in ith phase: cMin the mobile phase, cS in the stationary phase, and cA in the sorption

phase. cM z� � � ∫Ac

0 cMdAc=Ac:

CM - Resistance to mass transfer in the mobile phase. CM � 1 � 6k � 11k2

96 1 � k� �2d2c

DM k� �for an open tubular capillary column.

CS - Resistance to mass transfer in the liquid stationary phase. CS �2k

3 1 � k� �2d2f

DSfor an open tubular column.

cT - Total concentration (mass/unit volume) of sample component in column.cTV c � cMVM � cSV S � cAAS, and cT 0� � is the initial concentration.

dc - Internal diameter for an open tubular or capillary column. dc � 2rc:

df - Average film thickness for stationary phase.

D - Depth of a rectangular column.

D

$

M - Diffusion tensor describing molecular diffusion in the mobile (carrier gas)phase. DM jj� � and DM ?� � are the parallel and perpendicular components.


D

$

S - Diffusion tensor describing molecular diffusion in the stationary phase.DS k� � and DS ?� � are the parallel and perpendicular components.

ED - Activation energy for diffusion in the stationary phase.DS � DS 0� �exp �ED=RT� �:

Fc - Volumetric flow rate for the mobile (carrier gas) phase through the col-umn at column temperature. Fc o� � at the column outlet.

FD - Volumetric flow rate for mobile (carrier gas) phase through the detectorat column temperature. FD is Fc o� � plus an additional contribution fromadded makeup gas.

F I - Volumetric flow of mobile (carrier gas) phase through the injector at col-umn temperature. F I � Fc � SF:

f k� � - Functional k dependence for the resistance to mass transfer in themobile phase.

G z � z´; t � t´� � - Normalized Green’s function.

H - Height equivalent to a theoretical plate.

H z´� � - Heaviside step function. H z´� � � 0 for z´ < 0, H z� � � 1 for z´ � 0:

k - Retention or capacity factor – time spent by the sample in the stationaryphase relative to mobile phase. k � RS=RM � KCV S=VM for RA � 0:

KC - Distribution coefficient for gas–liquid chromatography:

KC � mass of sample=unit volume of stationary phasemass of sample=unit volume of mobile phase

� W S=V S� �WM=VM� � :

KS - Distribution coefficient for gas–solid chromatography:

KS � mass of sample=unit surface area of sorbent phasemass of sample=unit volume of mobile phase

� WA=AS� �WM=VM� � :

K v x; y� � - Darcy’s Law permeability. Kmean � ∫K vdAc=Ac:

L - Column length.

Mn - nth moment for an eluting peak, Mn � ∫all t tncT t� �dt= ∫all t cT t� �dt.

Mw - Molecular weight. Mw 1� � and Mw 2� � for two different molecules.

n - Number of moles.

P - Pressure of the mobile gas phase. Pi at column inlet and Po at column outlet.

r - Radial coordinate. r is a unit vector pointing perpendicular to the axis ofrotation.

rc - Internal radius of curvature for an open tubular or capillary column.rc � dc=2:

RD - Rydberg gas constant. 8.32 joules/(mole)(°K) = 1.98 cal/(mole)(°K)

Ri - Fraction of total sample component in the ith phase.Xi

Ri � RM � RS � RA � cMVM

cTV c� cSV S

cTV c� cAAS

cTV c� 1:

Rm - Resolving power. Rm � tR=wh:

1.4 Nomenclature 797

Rs - Resolution or time separation between two peaks relative to average peak

width at the base. Rs � tR2 � tR1wb1 � wb2� �=2 :

SF - Split vent flow rate. Split ratio � SF=Fc; fraction sample transferred =Fc= Fc � SF� � .

S z´; t´� � - source term.

T - Column temperature.

ti - Injection time or time to inject sample into a flowing volume of mobile(carrier gas) phase.

tR - Retention time or time for the migrating sample zone to travel the length ofthe column. ΔtR � tR2 � tR1 is the retention time difference between twopeaks.

V c - Internal geometric volume for the column. ΔV c is a differential element(slice). V c � VM � V S and ΔV c � ΔVM � ΔV S:

VD eff� � - Effective volume of detector = geometric volume for the detector plusanother volume of sample adsorbed on the internal walls of thedetector.

V I eff� � - Effective volume of injector = geometric volume for the injector plusanother volume of sample adsorbed on the internal walls of theinjector.

VM - Volume or holdup volume of mobile gas phase in column. ΔVM is adifferential element (slice). VM I� � and VM D� � are the geometric volumesfor the injector and detector, respectively.

V R - Retention volume for a retained peak V R � VM � KCV S � �1 � k�VM � VM=RM:

V S - Volume of stationary phase in column. ΔV S is a differential element(slice).

v

)

r� � or v

)

x; y� � - Linear velocity vector for the flow of mobile phase throughthe column. vk and v? � 0 are the parallel and perpendicularcomponents, respectively.

vk - Average linear velocity for the mobile phase, vk � ∫ v

)�ndAc=Ac. vk o� � at thecolumn exit, vk od� � at the column exit after gas expansion and addition ofmakeup gas.

W - Width of a rectangular column.

w - Width of a gas chromatographic peak. See Figure 1.2. wb1 � wb2 is thebaseline sum for two peaks.

xi - Mole fractional composition of mobile phase.Xi

xi � xM � xS � 1:

xM � moles of sample in mobile phasemoles of sample � carrier gas in mobile phase

:


xS � moles of carrier gas in mobile phasemoles of sample � carrier gas in mobile phase

:

X i - Mole fraction composition of the stationary liquid phase.Xi

X i � XM � XS � 1:

XM � moles of sample in stationary phasemoles of sample � liquid phase in stationary phase

:

XS � moles of liquid phase in stationary phasemoles of sample � liquid phase in stationary phase

:

x - Direction perpendicular to mobile gas flow in a rectangular column.

y - Direction perpendicular to mobile gas flow in a rectangular column.

z - Direction of mobile gas flow through a column. z is a unit vector pointingin the direction of the flow, Δz is a differential element (slice).

β - Phase ratio or volume of the mobile phase to volume of the stationaryphase β � ΔVM=ΔV S � VM=V S.

δM - Solubility parameter for sample component vapors.

δS - Solubility parameter for stationary liquid phase (w/o dissolved sample).

δ t´� � - Delta function. δ t´� � � 1 when t � 0, δ t´� � � 0 when t ≠ 0.

εD - Correction term for eddy diffusion. DM eff� � � DM � εD:

εi - Fractional volume of the column occupied by the ith phase:εM � ΔVM=ΔV c�VM=V c, εS � ΔV S=ΔV c�V S=V c, εA � ΔVA=ΔV c �VA=V c; and

Xi

εi � εM � εS � εA � 1:

εi - Nonequilibrium departure term for the ith phase.

ϕi - Volume fraction composition of stationary phase.

ϕM � volume of samplevolume of sample � stationary phase

' 0:

ϕS � volume of stationary phasevolume of sample � stationary phase

' 1:

γi - Activity coefficient for sample in mobile phase (γM) and in stationary phase(γS).

η - Dynamic viscosity of mobile (carrier gas) phase.

λi - Weighting factors for extracolumn time constants, λI for injector and λDfor detector.

μn - nth central moment for the eluting peak μn � ∫all t

t �M1� �ncT t� �dt=∫

all tcT t� �dt n > 1:

σ - Standard deviation. σ2 or σ2i is variance.

σ12 - Collision diameter

1.4 Nomenclature 799

τD - Response time constant for the detector τD � VD eff� �=FD:

τL - Lag time associated with establishing equilibrium by diffusion. τL M� � �r2c=6DM ?� � for mobile phase and τL S� � � d2

f =6DS ?� � for stationary phase.τI - Response time constant for the injector τI � V I eff� �=F I:

Ω* T *� � - Reduced collision integral. Ω 1;1� �* T*� � applies to diffusion.

ξ - Dummy variable to support change of variable in a computation.

@P=@z - Pressure gradient along the column length.

Δgmix - Mixing free energy change per mole of sample component with sta-tionary phase. Δgmix � Δhmix � TΔsmix:

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