GC Theory: Resolution and Carrier Gas Choice

Introduction to Gas Chromatography

Gas chromatography is one of the most common analytical methods for measuring concentrations of organics or other volatile compounds present in a liquid or gas sample.  During GC analysis, analytes are injected into a heated inlet chamber and mixed with carrier gas to vaporize as much material as possible and carry it through to the front of the column.  Most modern GC columns are fused silica capillary columns typically ≤ 0.5 mm in diameter, with a very thin film (~1 µm) of stationary phase.  During the run program, temperature is frequently varied using a series of linear ramps programmed in the method.  Carrier gas flows through the column, pulling along bands of analyte compounds at different rates depending on their affinity for the stationary phase.  A detector on the column outlet generates a response signal that is usually proportional to concentration and the response factor specific to each component that elutes. 

Components in a mixture are separated based on affinities for the gas chromatography column’s stationary phase (Courtesy Agilent Technologies) [1]

GC Theory

The purpose of a gas chromatograph is to separate components in a sample with respect to time so that only one component leaves the column at a time. These eluting components are observed by the detector vs time as roughly Gaussian-shaped peaks centered at retention time t_r with a peak width at baseline W_b or peak width at half height W_1/2.

Illustration of two well-separated peaks and how to measure their retention times and peak widths (Courtesy Agilent Technologies)[1]

Calculating parameters from peaks

Resolution

The resolution R_s describes the ability of a column to separate the peaks of interest. To reliably quantify component concentrations, the resolution must be greater than or equal to 1. It can be calculated from retention times and peak widths measured from a chromatogram using a simple formula:

This formula does not readily illustrate why a specific GC method results in the observed resolution.  We want to know how we can manipulate the run conditions to allow sufficient resolution to meet the analysis requirements while keeping run times down for practical usability.

Retention factor

As a special case, compounds with no affinity to the column’s stationary phase (e.g., CH₄ with a DB-1 column) elute from the column at the same rate as the carrier gas. The retention time of an unretained compound (t₀ or t_M) is called the holdup time. The holdup time is used to calculate the retention factor k’ for a peak with retention time t_r :

The retention factor is also known as the partition ratio or capacity factor. An unretained compound has, by definition, a retention time of t₀, so its retention factor k₀’ = 0.

Selectivity

Once component k’ values are determined, the selectivity α of the stationary phase for a particular pair of analytes can be calculated.  Note that separation factor and separation coefficient are synonymous with selectivity:

Theoretical Plates

The column efficiency, or number of theoretical plates N, is a third factor affecting peak resolution. Efficiency is another measure of column performance and is conceptually related to the number of theoretical stages used in distillation column calculations. N can be obtained from a chromatogram using measured values for either W_1/2 or W_b, and for ideal peak shapes the two expressions are equivalent:

A theoretical plate is a hypothetical stage where two phases of a substance establish an equilibrium.  In gas chromatography, the two phases involved are the mobile (gas) phase flowing through the column and the stationary (liquid or solid) phase on the walls of the column (capillary columns) or on the outside of the particles in a porous bed (packed columns).  Column efficiency increases with increasing column length and decreasing particle size (for packed columns), but other factors can affect N, some of which are outside the scientist’s control.  High values of N result in sharp peaks, while low N leads to broad peaks.

Height equivalent to a theoretical plate (HETP)

Theoretical plate count works just as one would expect when considering columns of different lengths: N is directly proportional to column length L.  The length dependence on plate count can be broken out by dividing by L to obtain the height equivalent to a theoretical plate H, also known as HETP:

HETP is a concept that we will use to evaluate different carrier gases later in this article.

Fundamental Resolution Equation

With expressions in hand for the retention factor k’, selectivity α and number of theoretical plates N, we can now express resolution in terms of adjustable chromatographic parameters with the fundamental resolution equation consisting of three physically meaningful terms:

Each term depends on a single measure of column performance.  From left to right, the term categories are:

1. Efficiency: Describes the separation power of the column as a function of theoretical plate count.

2. Selectivity: Has the most influence on the resolution, with small changes in selectivity leading to large resolution changes.

3. Retention: Has a significant influence at small k’ values, but small effect for large k’.

Note that since R_s increases with the square root of N, and N is directly proportional to L, doubling the column length only increases peak resolution by the square root of 2, which is only about a 40% increase in resolution at a cost of double the analysis time. This shows the usefulness of the fundamental resolution equation: it lends itself to back of the envelope calculations that provide meaningful insights into which adjustments will have the most “bang for the buck,” whether it be in terms of analysis time requirements or monetary cost.

Carrier gas and efficiency

Van Deemter equation

Column dimensions and stationary phase are not the only adjustable parameters that influence the column efficiency N.  The type of carrier gas, flowrate, temperature and pressure profile also affect N, sometimes with drastic effect.  Packed columns have additional considerations like particle size, packing irregularity and tortuosity/labyrinth factor.  All these factors affect mass transport in some way, whether it be via diffusion within a phase (mobile or stationary), mass transfer between phases, or variability of potential pathways that gas can follow when traveling through the column.

The van Deemter equation is the basic interpretive equation of column efficiency, describing how different factors can cause peaks to broaden as they move down the column.[2] Published in 1956, Jan van Deemter developed the equation by applying rate theory to the chromatographic process, the first of its kind.[3]  It expresses theoretical plate height HETP as a function of average linear velocity ū.  For packed columns, the general van Deemter equation is:

where A is a coefficient describing multiple path effects (eddy diffusion), B is the longitudinal diffusion coefficient and C is the coefficient of resistance to mass transfer in the stationary and mobile phases. 

The extended van Deemter equation (for packed columns), which includes ­an extra term for axial diffusion in the mobile phase, can be written in terms of physically meaningful parameters [4]. With H = HETP (synonyms):

where λ is the correction factor for the irregularity of the column packing, γ is labyrinth factor or tortuosity, with 0 < γ < 1, d_p is the particle size of the packing, d_f is the thickness of the stationary phase layer coating the packing’s particles, k is the retention factor, D_m is the analyte diffusion coefficient in the gas phase, D_s is the diffusion coefficient of the analyte in the stationary phase, and f(k), g(k) are functions of the retention factor k.

The van Deemter equation includes parameters that depend on the nature of the carrier gas. Estimates for the A, B and C terms can be calculated a priori if the values for parameters like λ, γ, D_s and D_m are known. They can also be fit to experimental data by measuring HETP at different values of ū to obtain empirical curves for different carrier gases.

Golay Equation

While the van Deemter equation was developed for packed columns, the analogous Golay equation was developed for capillary columns, and they are very similar.[5] Capillary columns do not conrtain a particulate packing, so the A term for eddy diffusion is not included:

where the terms B, C_s and C_m describe longitudinal diffusion, mass transfer to and from the stationary phase, and mass transfer in the mobile phase, respectively. These terms can be written as: [6]

where d_c is the column diameter, and all other parameters the same as those described for the van Deemter equation in the previous section.

Carrier gas selection

Effect of carrier gas type

A van Deemter plot shows how HETP depends on carrier gas velocities for a particular carrier gas.  Recall that lower HETP values translate into higher column efficiencies, since more theoretical trays fit in a given column length.  Overlaying the van Deemter plots for different carrier gases shows different curves for ū, revealing that the type of carrier gas can have a substantial effect on column performance at a given linear velocity.  The three most common carrier gases used for GC are helium, hydrogen and nitrogen, and representative examples of their van Deemter curves are plotted below.

Example van Deemter plots showing how HETP varies with carrier gas velocity for hydrogen, helium and nitrogen. Dashed lines highlight where the optimal velocities are found for each carrier gas.

Some key takeaways about carrier gas effects illustrated by these curves are:

• For a given carrier gas, there is an optimal value of ū where efficiency is maximized (HETP minimized)
• Different carrier gases have different optimal linear velocities. Hydrogen has the highest ū_opt, followed by helium, while nitrogen has the lowest ū_opt of the three
• Efficiency quickly drops (HETP increases) when carrier gas velocity falls below ū_opt
  • Axial diffusion (B / ū term) starts to dominate at low values of ū
• Efficiency with nitrogen also drops quickly above ū_opt, whereas He and H₂ are not as sensitive
• Column efficiency with hydrogen is the least seneitive to velocity variation of the three gases

The efficiencies at ū_opt for the three most common carrier gases are all about the same. In some cases the maximum available efficiency with nitrogen is very slightly higher versus hydrogen and helium, but the difference is almost negligible. Using nitrogen to achieve very slightly higher efficiencies comes at a great time cost, however: since ū_opt for N₂ is less than half of ū_opt for He, runs using nitrogen take more than twice as much time to finish compared to helium runs with the same sample. The maximum possible resolutions are all about the same when running at optimal velocities because k’ and α do not depend on carrier gas properties to a first approximation. Since resolution is proportional to the square root of N, and typically N >> 1, so any small increase in efficiency results in a much smaller relative improvement in resolution.

With hydrogen carrier gas, ū_opt is almost twice as fast as ū_opt for helium. Runs can be even shorter using hydrogen for the same peak resolution. Compressing runs into shorter timeframes has two benefits:

1. Able to run more samples per day on an instrument.
2. Peak widths are narrower, leading to higher peaks since the total area is independent of velocity. Higher peaks mean lower detection limits (greater sensitivity).

Global helium shortage

Graph that shows global helium demand continues to grow while US supply declines. Courtesy globalhelium.com

Helium is the most common carrier gas used for GC because it is inert and analysis times are acceptable.  Unfortunately, helium is a non-renewable resource: it forms underground when alpha particles (which are helium nuclei) are emitted by the decay of certain radioactive elements in the earth’s crust and mantle undergo alpha decay.  It is then co-extracted with natural gas from certain helium-rich wells and cryogenically separated.  The United States is the world’s leading helium producer, followed by Qatar.  Helium demand has increased over the last two decades, with MRIs and semiconductor manufacturing consuming 40% of the supply on their own.  While demand increased, supply became more constrained, and prices skyrocketed. 

Historical prices for bulk helium through 2019. Current prices are higher still, with bulk helium selling for about $650/mcf in 2023, which is below 2022 peaks where spot prices were up to $2000/mcf. Courtesy globalhelium.com

What’s worse, in 2019 the global helium supply became so short that availability became an issue.  GCs require a continuous supply of carrier gas because they must constantly flow carrier gas through the column, the inlet, the septum purge vent and sometimes the detector.  Since a typical GC goes through about two 8200 L cylinders of helium per year, larger labs may need hundreds of cylinder swaps per year.  In 2019, many of those larger labs struggled to reliably find enough helium to keep instruments running because of the global supply crunch.  This led some savvy lab managers to seek out alternative carrier gases like hydrogen to ensure supply continuity while keeping gas costs down.

Why is helium carrier gas so common if hydrogen works so well?

• In contrast to hydrogen, helium is compatible with all common GC detectors, including mass spectrometry (MS), helium ionization detectors (HID), pulsed discharge detectors (PDD), pulsed discharge helium ionization detectors (PDHID) and discharge ionization detectors (DID).

• Helium is a nonreactive noble gas, in contrast with hydrogen that can act as a reductant toward active compounds like unsaturated hydrocarbons and certain pesticides.

• Last but not least, hydrogen is extremely flammable and can form explosive mixtures in air. Additional engineering controls need to be in place with hydrogen carrier gas, such as adequate ventilation and a modern GC with sensors that automatically shut off gases when carrier gas flowrate is too high, e.g. due to a broken column.

Summary

In this article we discussed how to measure peak parameters to quantify chromatogram quality, we defined peak resolution and described the parameters that affect resolution with the fundamental resolution equation. The concept of theoretical plates was introduced as a measure of column efficiency, and used the height equivalent to a theoretical plate (HETP) as a means to decouple column efficiency from column length. The roles of diffusion and mass transfer processes on column efficiency were considered with descriptions of the van Deemter and Golay equations. Representative van Deemter curves for different gases illustrated how the type of carrier gas affects column performance, followed by a brief discussion about pros and cons of different carrier gases to consider when selecting gases for specific applications.

References

1. Agilent Technologies, “Fundamentals of Gas Chromatography: Theory,” 2016, Publication number 5991-5422EN.

2. P.J. Marriott, “GAS CHROMATOGRAPHY: Principles,” in Encyclopedia of Analytical Science, 2nd ed., 2005, pp 7-18.

3. van Deemter, J.J.; Zuiderweg, F.J., Klinkenberg, A., “Longitudinal diffusion and resistance to mass transfer as causes of non ideality in chromatography,” Chem. Eng. Sci., 1956, 5, 271-289

4. Holler, F.J.; Skoog, F.J.; Crouch, S.R. Principles of Instrument Analysis, 2018, (7th ed.), ISBN 978-1-305-57721-3, pp 720-745.

5. Golay, M. “Gas Chromatography Terms and Definitions,” Nature, 1958, 158, 1146-1147

6. Engewald, W.; Dettmer-Wilde, K.; “Theory of Gas Chromatography,” in Practical Gas Chromatography, 2014, pp 21-57, Springer-Verlag, Berlin, DOI 10.1007/978-3-642-54640-2_2