In the last lessons we have focused on liquid chromatography (LC). However, from a fundamental perspective, gas chromatography (GC) is much more attractive! Provided that the analytes are volatile, superior plate numbers may be attained. We will therefore shift our focus to these gas-phase separations. This is also a good excuse to revisit plate-height theory. GC typically employs open-tubular columns, yet the Van Deemter equation contains the particle size. We will thus revert to the Golay equation and asks ourselves: Why do we use open-tubular columns in GC and packed-columns in LC?
Learning Goals #
After this first class of the course you will be able to
- understand the unique advantages of gas chromatography (GC).
- explain the structure and functionality of open-tubular GC. columns.
- apply the Golay equation to assess column efficiency.
- understand the fundamentals of open-tubular chromatography columns and why these are dominant in GC, but not in LC.
- interpret and use dimensionless parameters in column comparison for GC.
READ SECTIONS 2.1.1 & 2.1.2
Basic Instrumentation and Operation of GC
1. Introduction to gas chromatography #
Gas chromatography (GC) is the preferred method for separating volatile analytes, for numerous reasons. Gases are ideal mobile phases.
- Compared to the liquid mobile phases they are cheap, readily available in high purity, non-toxic and inert.
- Pure gases provide little or no background, allowing extremely sensitive detection.
- Gasses create no fouling or corrosion, so as to minimize downtime and maintenance.
- The inertness of the mobile phase allows for high operating temperatures and columns tend to be durable.
- Focussing analyte bands, for example after injection, can readily be achieve by cooling (a segment of) the column.
However, the most-important advantages of GC are fundamental. These will be discussed in the next part, where we will be discussing open-tubular GC columns and the reason why these are referred in GC, while LC almost exclusively employs packed columns.
Table 1. SWOT (Strengths, Weaknesses, Opportunities & Threats) analysis of gas chromatography.
1.1. Open-tubular GC columns #
In the vast majority of cases, open-tubular columns are used for gas-chromatography (GC) separations. Packed columns may be used for separating gases at ambient temperature (or higher), thanks to the large surface area available. Packed columns with small particles can also be used for ultra-fast GC, but the relatively high pressures needed are generally prohibitive of such applications.
Figure 1. Schematic depiction of a gas chromatography instrument.
Long open-tubular GC columns are almost always prepared from fused-silica. Again, there is an exception; all-metal columns are used for high-temperature GC applications, such as simulated distillation. A flexible polyimide coating on the outside of fused-silica columns makes them hard to break and heat resistant.
Figure 2. Different stationary-phase formats in GC and properties of fused silica.
1.2. Stationary phases #
Cross-linked polymers are typical stationary phases. Polysiloxanes are by far the most popular, because they show the highest analyte diffusion coefficients. Poly(dimethyl siloxane) (PDMS) is one of the most widely used stationary phases, particularly for non-polar and moderately polar analytes. It is chemically inert, thermally stable, and offers high analyte diffusion coefficients, which contributes to efficient mass transfer and sharp peaks. The polymer is typically crosslinked to ensure it remains immobilized on the column wall and to minimize column bleed, especially at elevated temperatures.
EXERCISE 1 #
Which statements are correct?
This is correct!
This is not correct. Try again! Be sure to check out Sections 2.1.2 and 2.1.4.
Polarity can be tuned by varying the substituents on the polysiloxane backbone, from methyl (non-polar) to phenyl- or cyano-groups and phases in between (e.g. 50% methyl, 50% phenyl). Highly polar “wax” columns are still used for polar analytes.
READ SECTIONS 2.1.4 & 2.1.5
Stationary Phases for GC
Porous-layer open-tubular (PLOT) columns offer a high surface area, which may be beneficial for the analysis of highly volatile compounds at above-ambient temperatures. As such, they combine some of the advantages of packed columns (retentivity) and of open-tubular columns (permeability).
2. Efficiency #
2.1. Reduced plate height parameters #
In Lesson 3 we learned how columns can more objectively be compared by using the reduced plate height (h), also known as the Péclet number (\text{Pe}), and the reduced linear velocity (\nu). We did so for packed-column chromatography. Table 2 shows the similarities and differences for these definitions for open-tubular columns.
In Table 2, H is the plate height, d_{\text{c}} the column internal diameter, d_{\text{p}} the particle diameter of the packed bed, u the average linear velocity of the mobile phase, and D_{\text{m}} the molecular diffusion coefficient of the analyte in the mobile phase. If needed, this would be the suitable moment to revised this concept in Lesson 3, as we will now continue to build on it.
EXERCISE 2 #
Which statements are correct? (* = for example, packed or open tubular)
Correct! Well done.
Unfortunately, this is not correct yet. Only one statement is false.
READ SECTION 1.7.7
Open-tubular columns – The Golay equation
2.2. The Golay equation #
The efficiency (plate count, N=L/H) is commonly described by van Deemter-type plate height equations of the form
Equation 1.30: H=A+\frac{B}{u_{\text{int}}}+C \cdot u_{\text{int}}
The simple geometry of open-tubular columns allow an analytical plate-height equation to be derived, as done by Marcel Golay [1]. The Golay equation reads
Equation 1.125: H=\frac{2D_{\text{m}}}{u}+f(k)\frac{d^2_{\text{c}} \cdot u}{D_{\text{m}}}+g(k) \frac{d^2_{\text{f}} \cdot u}{D_{\text{f}}}
where d_{\text{c}} is the internal diameter of the column, d_{\text{f}} is the thickness of the (stationary-phase) film, D_{\text{m}} is the diffusion coefficient of the analyte in the mobile phase, and D_{\text{f}} is the diffusion coefficient in the stationary-phase film. f(k) and g(k) are functions of the retention factor (k).
Equation 1.126: f(k)=\frac{1+6k+11k^2}{96(1+k)^2}
and
Equation 1.127: g(k)=\frac{2k}{3(1+k)^2}
Both equations are valid for d_{\text{f}} \leq 0.1d_{\text{c}} and in the laminar flow regime. For an extension to the turbulent flow regime see [2].
2.3. Optimal film thickness #
Similar to the van Deemter equation, we can reduce the Golay equation, first by considering the linear velocity, which for open-tubular columns can be reduced by
Equation 1.129: \nu_{\text{opt}}=\frac{u_{\text{opt}} \cdot d_{\text{c}}}{D_{\text{m}}}
For open-tubular columns an additional reduced parameter can be defined, i.e. the reduced film thickness (\delta_{\text{f}})
Equation 1.130: \delta_{\text{f}}=\frac{d_{\text{f}}}{d_{\text{c}}} \sqrt{\frac{D_{\text{m}}}{D_{\text{f}}}}
The resulting expression is the reduced Golay equation
Equation 1.131: h=\frac{2}{\nu}+f(k) \cdot \nu + g(k) \cdot \delta^2_{\text{f}} \cdot \nu
Marcel Golay (1902-1989, Switzerland)
Marcel Golay was born in Switzerland in 1902. He was known as a mathematician, physicist and information scientist (the Savitsky-Golay smoothing algorithm carries his name), but he also made great contributions to separation science. Most importantly, Golay derived an exact equation that described band broadening in open-tubular columns, which we still use today. Golay played a major role in the transition from packed to open-tubular columns for GC. During a large part of his life (from 1955 onward) he was associated with Perkin Elmer (Norwalk, CT, USA).
Similarity Golay and van Deemter #
The effect of the last – g(k) or C_{\text{s}} – term can be kept small relative to that of the f(k) or C_{\text{m}} term if the reduced film thickness does not exceed a value of 0.3 [3], which corresponds to a stationary-phase film thickness that is about one thousands of the column diameter.
Figure 3. The f(k) and g(k) terms of the Golay equation plotted as a function of the retention factor.
Indeed, this is a rule of thumb for open-tubular GC columns. A standard 250-µm i.d. column comes with a film thickness of 0.25 µm. Thicker or thinner films should only be chosen for specific purposes.
EXERCISE 3 #
Let’s see what we understand of this concept of stationary-phase film thickness. Which statements are true?
This is the correct answer! If you have more stationary-phase (i.e. a thicker film) then you’ll be able to facilitate more loadability, because the analytes have more room to interact. Similarly, thicker films may facilitate more interaction. You can also imagine that the stationary-phase film is less fragile if it is thicker. Thicker films are, however, not more efficient. On the contrary, the C_{\text{s}} term increases because the analytes must travel further distances to get out of the stationary phase. Finally, analytes that are not sufficiently volatile cannot be eluted using GC. This is the inherent limit of GC.
Unfortunately, this is not correct. Try again! Two statements are false. Perhaps the following helps: If you have more stationary-phase (i.e. a thicker film) then you’ll be able to facilitate more loadability, because the analytes have more room to interact. Similarly, thicker films may facilitate more interaction. You can also imagine that the stationary-phase film is less fragile if it is thicker.
2.4. Optimal retention factor #
The retention factor has a strong effect on the efficiency in open-tubular chromatography. When the effect of the C_{\text{s}} term is minimized (i.e. \delta_{\text{f}} \leq 0.3) the steepness of the C-branch of the plate-height curve increases strongly with increasing k. This is shown in Figure 4.
Figure 4. Golay equation plotted for different retention factors, all assuming the reduced film thickness to be 0.3.
Figure 4 suggests a trade-off between better efficiency (very low k) and sufficient retention to actually a separation (higher k). In practice this is overcome by the ubiquitous use of temperature programming in open-tubular GC. By increasing the temperature during the run, analytes move mostly through the column once their volatility gets high and their effective retention factor is low.
EXERCISE 4 #
We will now use the Golay equation to evaluate the efficiencies of open-tubular columns in GC. For our calculations we embed the conclusions from the previous two sections and therefore, assume k = 1, and \delta_{\text{f}} = 0.3.
First, calculate the optimum reduced linear velocity (\nu_{\text{opt}} and the minimum reduced plate height (h_{\text{min}}) under standard conditions from the reduced Golay equation. To achieve this, first calculate f(k) and g(k). Then use Equation 1.131 to isolate b = 2, and c = f(k)+g(k)\delta^2_{\text{f}} to then be able to calculate \nu_{\text{opt}} (Equation 1.82; see Lesson 3) and h_{\text{min}} (Equation 1.83).
Round \nu_{opt} to one decimal.
Correct!
This is not correct. Remember, b = 2, and c should be calculated.
Round h_{\text{min}} to one decimal.
This is correct!
This is not correct. Remember, b = 2, and c should be calculated.
The standard conditions apply to columns of any length or diameter, provided that the reduced film thickness is about 0.3. This condition holds for a standard GC column of L = 30 m length and d_{\text{c}} = 250 µm internal diameter.
How many theoretical plates (N) does the above standard column yield at the optimal linear velocity that you just calculated? For this you will need to use the relation H_{\text{min}} = h_{\text{min}} \cdot d_{\text{c}} (Equation 1.128). Equations 1.29 from Lesson 1 can be used to obtain N. Round as integer number (no decimals).
Correct!
Incorrect. Check your calculation. You should have arrived at H_{\text{min}} of 0.000176 m with column length of 30 m.
And what is the dead time (t_0) in seconds? A typical value for the diffusion coefficient of a low-moleular-weight analyte under GC conditions is D_{\text{m}} = 5\cdot10-6 m2\cdots. For this you will need the relation u_{\text{opt}} = (\nu_{\text{opt}} \cdot D_{\text{m}})/d_{\text{c}} (Equation 1.129). Equations 1.11 from Lesson 1 can be used to obtain t_0. Report the dead time in seconds. Round as an integer number (no decimals).
Correct!
Incorrect. Check your calculation. You should have arrived at u_{\text{opt}} of 0.1137 m s-1 with column i.d. of 0.00025 m.
The volumetric flow rate may be estimated – again neglecting the mobile-phase compressibility – by multiplying the average linear velocity u with the cross-sectional area of the column, yielding
which yields a volumetric flow rate of about 0.33 mL\cdotmin-1.
We can conclude from the above exercises that open-tubular GC columns yield very high efficiencies in a reasonable time. This is mostly how GC is operated. High-resolution separations of complex samples are obtained on columns that run up to 60 m in length. Short separations can be obtained on open-tubular columns with a length down to 1 m, for example as second-dimension column in (comprehensive) two-dimensional GC (see Lesson 17). Common internal diameters are 250 or 320 µm. Narrower columns (e.g. 100 µm i.d.) may be used for fast analysis, but the pressure drop across such columns may no longer be negligible. Wide bore columns (e.g. 530 µm i.d.) are slower and less efficient, but they do provide a greater sample loadability.
2.5. Diffusion coefficients #
A general equation for the hold-up time is
This equation shows that if we want to discuss the speed of analysis for the same performance (i.e. same plate count N) under standard conditions (e.g. \nu=\nu_{\text{opt}}=5.7 and h=h_{\text{min}}=0.7) we should focus on the ratio d^2_{\text{c}}/D_{\text{m}}.
3. Open-tubular LC #
We see dazzling plate numbers leading to incredible efficiency in open-tubular chromatography for GC. This raises the question, why don’t we use such column formats in liquid chromatography? Instead of explaining it to you, you can apply your gained knowledge to discover this yourself in the next and final exercise of this lesson.
EXTENSIVE EXERCISE #
The above equations also allow a comparison of open-tubular columns for use in GC and in LC. The main difference between gaseous and liquid mobile phases is the rate of analyte diffusion. Whereas we use a typical value of D_{\text{m}} = 5\cdot10-6 m2\cdots for the diffusion coefficient of a low-molecular-weight analyte in a gas under GC conditions, a typical value in a liquid mobile phase is D_{\text{m}} = 10-9 m2\cdots. Assuming the same (PDMS) stationary phase in both cases, we may assume D_{\text{f}} = 5\cdot10-9 m2\cdots.
Part A
First, calculate (i) the thickness of a PDMS film in a standard GC column (250 µm i.d., \delta_{\text{f}} = 0.3); (ii) the length needed for to obtain 100 000 theoretical plates with such a column; (iii) the average linear velocity.
You can rearrange Equation 1.130 to
For (ii), note that the reduced minimum plate height was earlier in this lesson defined as h_{\text{min}} and the reduced linear velocity as \nu_{\text{opt}}. Finally, remember the following relations:
You can use Equation 1.129 for the reduced linear velocity, and for the volumetric flowrate you can use the equation in Section 2.4 of this lesson.
Part B
Next, we calculate (iv) the hold-up time (t_0) and retention time t_{\text{R}} of an analyte eluting with k=1; (v) the standard deviation of a peak eluting with a retention factor of k=1 in time units; (vi) the standard deviation of a peak eluting with a retention factor of k=1 in volume units.
Use the relation t_0=L/u for (iv). For a retention factor of 1 we obtain t_{\text{R}}=2 \cdot t_0.
It has been some time, but we in earlier lessons worked with the useful definition of the plate number:
Use
(iv) t_0 = 154 s, t_{\text{R}} = 307 s.
(ii) \sigma_{\text{t}} = 0.97 s
(iii) \sigma_{\text{V}} = 5.43 μL
Part C
What do you make of the value of \sigma_{\text{V}}? Is it reasonable? Take into account that, if we assume that the peak follows the normal distribution (i.e. is a Gaussian), then the peak can be considered to be 4 \sigma wide. Consider in your deliberation that the peak must be detectable.
This is correct. A peak volume of about (4\cdot5.43) 20μL, and a peak width in time of about 4 seconds are reasonable values for gas chromatography, which is logical given that the columns of 250 μm internal diameter are among the most common in open-tubular GC.
This is not correct. A peak volume of about (4\cdot5.43) 20μL, and a peak width in time of about 4 seconds are reasonable values for gas chromatography, which is logical given that the columns of 250 μm internal diameter are among the most common in open-tubular GC.
Part D
We now have reference numbers from GC. To see why open-tubular columns are not feasible for LC we’ll repeat the calculation with some different values. Diffusion coefficients in liquids are much smaller. For the liquid mobile phase therefore assume D_{\text{m}} = 10-9 m2\cdots-1. In an attempt to mitigate the effect of a much lower diffusion coefficient on the linear velocity (given that \nu_{\text{opt}}=u_{\text{opt}}d_{\text{c}}/D_{\text{m}} = 5.7 remains the same), we will reduce the column diameter to d_{\text{c}} = 5 μm. Now proceed through the calculations similar to the previous parts to arrive at \sigma_{\text{V}}.
Calculate \sigma_{\text{V}} for the open-tubular LC case. What answer do you find?
This is correct! For a 5 μm i.d. diameter column we obtain a thickness of 0.34 μm, H_{\text{min}} of 3.52 μm, L_{\text{req}} of 0.35 m. This system then runs at 1.14 mm\cdots-1, which translates to flow rate of a meagre 1.34 nL\cdotmin-1. We then obtain t_0 of 309 seconds, and a t_{\text{R}} of 619 seconds. Consequently, we find \sigma_{\text{t}} of 1.96 seconds, and therefore a \sigma_{\text{V}} of… 43.7 pL….. That’s right: pL! There simply is not a detector fast enough to detect pL!
This is not correct. Retry your calculation! Some of the first steps repeating the calculations from the earlier parts yield: for a 5 μm i.d. diameter column we obtain a thickness of 0.34 μm, H_{\text{min}} of 3.52 μm, L_{\text{req}} of 0.35 m. This system then runs at 1.14 mm\cdots-1, which translates to flow rate of a meagre 1.34 nL\cdotmin-1.
Concluding remarks #
We have seen in this lecture that gas chromatography is an absolutely amazing technique that provides tons of advantages at just one disadvantage: analytes must be sufficiently volatile. We used the Golay equation to explain why GC columns are typically open tubular, where the stationary-phase material is a film on the column wall instead of packed particles. Such columns yield powerful efficiencies superior to packed-column chromatography.
Open-tubular chromatography is fantastic, yet not really an option for liquid chromatography due to the very limited peak volumes that are simply not feasible for detectors.
With the fundamentals of GC ready, we can now proceed to learn more about two crucial factors in gas chromatography: injection and detection.
Please find here an Excel sheet that contains the calculates above. In order to facilitate your capability to really learn these calculations it is STRONGLY recommended that you first try them yourself without the sheet.
Download: (SS_09, .XLSX).
References #
[1] M.J.E. Golay, Theory of chromatography in open and coated tubular columns with round and rectangular cross-sections, in Gas Chromatography, ed. D. H. Desty, Academic Press, New York, 1958, 36-55.
[2] F. Gritti, J. Chromatogr. A, 2017, 1492, 129–135, DOI: 10.1016/j.chroma.2017.02.044
[3] P. J. Schoenmakers, J. High Resolut. Chromatogr., 1988, 11, 278–282, DOI: 10.1002/jhrc.1240110311
