We continue our excursion through the fundamental concepts of chromatography. In this module we explore the thermodynamic and physical principles that govern retention and migration in chromatography. We discuss the structural properties of stationary phases, such as porosity and particle morphology, and their impact on efficiency and selectivity. Finally, we analyze flow behavior and column permeability, laying the groundwork for understanding chromatographic performance and limitations in modern high-pressure systems.
Learning Goals #
After this lesson you should
Understand the thermodynamic basis of retention in chromatography
Be able to describe the influence of temperature and pressure on chromatographic behaviour
Know relevant structural properties of stationary phases and their role in chromatography
Understand flow profiles and their effect on efficiency
Be able to quantify column permeability and predict back pressures
READ SECTION 1.6.1
Thermodynamics of Retention
1. Thermodynamics of Chromatography #
We saw in the previous lesson that chromatography concerns the distribution of analytes between two phases, governed by the distribution coefficient K_{\text{d}}. Although often treated as a constant, we will see in this section that K_{\text{d}} is in fact a thermodynamic property.
To understand retention in chromatography at a fundamental level, we must therefore turn to thermodynamics. The thermodynamics of the retention of an analyte are governed by its chemical potential in the two phases.
1.1. Chemical Potential #
The chemical potential (\mu) of an analyte in a given phase is a thermodynamic quantity that reflects the escaping tendency of the analyte from that phase. It can be expressed as:
Equation 1.44: \mu_{\text{f},i}=\mu^{0}_{\text{f},i}+R \cdot T \ln{a_{\text{f},i}}
where \mu^{0}_{\text{f},i} is the standard-state chemical potential, R the universal gas constant, T the absolute temperature, and the a_{\text{f},i} activity of the analyte I in phase f.
At equilibrium, the chemical potential of the analyte in the stationary phase equals that in the mobile phase, and from this equality, the distribution coefficient K_{\text{d}} can be defined in terms of activities. However, in practice, we often relate this to mole fractions and activity coefficients (\gamma), making it more accessible experimentally and computationally.
In the book it is demonstrated that the activity coefficients (\gamma) themselves encapsulate molecular interactions: favourable interactions between an analyte and a phase lower the activity coefficient, whereas unfavourable interactions increase it.
This principle underlies the familiar behaviour in reversed-phase liquid chromatography, where nonpolar analytes interact more strongly with the nonpolar stationary phase than with the polar mobile phase, resulting in retention.
EXERCISE 1
Which of these statements is correct?
Correct! The equilibrium between the stationary and mobile phases is defined by the equality of chemical potentials for the solute in both phases. Solutes spontaneously move from regions of higher to lower chemical potential until equilibrium is reached. The solute migrates toward equalizing the chemical potential, not simply to the phase with the highest potential. Chemical potential is affected by variables such as temperature, pressure, and concentration. Retention time depends on the distribution (related to chemical potential), but not directly on which solute has the highest chemical potential.
Incorrect. Remember that chemical potential reflects the thermodynamic tendency of a substance to move or redistribute. In chromatography, solute molecules move to equalize their chemical potentials between phases, not necessarily to the phase with the highest value
1.2. Effect of Temperature #
A particularly useful result arises when considering how temperature influences retention. By examining the temperature dependence of the distribution coefficient (see book for derivation), one arrives at the van ’t Hoff equation:
Equation 1.51: \ln{k_i}=\ln{K^{x,\infty}_{\text{d},i}}+\ln{\frac{n_{\text{s},i}}{n_{\text{m},i}}}=\frac{-\Delta H}{R \cdot T}+\frac{\Delta S_i}{R}+\ln{\frac{n_{\text{s},i}}{n_{\text{m},i}}}
where k is the retention factor, and \Delta H and \Delta S are the enthalpy and entropy changes, respectively, associated with transferring the analyte from the mobile to the stationary phase. This equation enables a linearization: plotting \ln{k} versus 1/T in a so-called van ‘t Hoff plot yields a straight line, the slope and intercept of which can provide thermodynamic insights in the separation mechanism.
Figure 1. Example of a Van ‘t Hoff plot.
Plots such as the one in Figure 1 are frequently used in retention-mechanism studies, but their interpretation requires care. They assume constant enthalpy and entropy across the temperature range and a homogeneous phase system. In addition, scientists have criticized the concept, because retention in, for example, reversed-phase liquid chromatography, typically is affected by several mechanisms.
Moreover, in gas chromatography (GC), and LC at high pressures, one must account for the compressibility of the mobile phase, meaning that modified forms of the van ’t Hoff equation that incorporate pressure and density terms are needed.
1.3. Selectivity #
Selectivity, the ability to distinguish between two compounds, is also rooted in thermodynamics. We saw in the previous lesson that selectivity can be expressed as a ratio of retention factors. Ultimately, it can also be expressed as a difference in Gibbs free energy between analytes:
Equation 1.57: \ln{\alpha_{j,i}}=\frac{\Delta \Delta G_{i,j}}{R \cdot T}
This deceptively simple expression emphasizes the extraordinary resolving power of chromatographic systems. For example, a modern LC column with N = 20 000 theoretical plates can resolve two analytes that differ in ΔG by as little as 50 J mol-1, which is about 400 times weaker than hydrogen bonds in water. This sensitivity also explains why retention and selectivity are so challenging to predict from first principles. Even minor inaccuracies in estimating intermolecular interactions can ruin predictions of chromatographic selectivity.
1.4. Effect of Pressure #
Finally, pressure also plays a role in retention, especially in high-pressure LC systems. In the book it is demonstrated how we can derive a modified van ’t Hoff expression that includes the partial molar volume change \Delta V upon analyte transfer:
Equation 1.59: \ln{k}=- \frac{\Delta U}{R \cdot T}-\frac{P \Delta V}{R \cdot T}+\frac{\Delta S}{R}+\ln{\frac{n_s}{n_m}}
Here, \Delta V reflects the volume difference between the analyte in the stationary and mobile phases. In most systems, analytes occupy less volume in the stationary phase, so \Delta V<0, meaning that an increase in pressure leads to increased retention.
This effect, while long considered negligible, has gained renewed attention with the advent of ultra-high-pressure LC (UHPLC) and the analysis of large biomolecules. For example, peptides and small proteins can exhibit dramatic increases in retention with rising pressure, as their molar volumes change more significantly during partitioning. These effects are visualized in Figure 1.18 in the book, which shows the nonlinear increase in retention as pressure increases, particularly for systems with large \Delta V.
EXERCISE 2
Different parameters affect retention to a different relative degree. Below are four parameters. Rank each from weakest (top; little to no effect) to strongest (bottom; large relative effect) in liquid chromatography in terms of their effect on retention factors (k). Assume that all other parameters stay the same.
This is correct! While pressure has an affect on retention, its effect is arguably relatively less than that of temperature. Column length has no effect at all on retention factors (assuming that all other factors stay the same).
Unfortunately, this is not correct. Try again! While pressure has an affect on retention, its effect is arguably relatively less than that of temperature. One parameter has no effect at all on retention factors. Consider that we are looking at the effect on retention factor, not retention time!
READ SECTION 1.6.2
Phase Ratio
2. Stationary Materials #
In column chromatography, the stationary phase is usually either a bed of packed spherical particles in a cylinder, or a thin film on the wall of a capillary. While both formats can theoretically be used for gas (GC), liquid (LC) and supercritical fluid chromatography (SFC), we will learn later that spherical particles are more favourable for LC and SFC, whereas thin films are favourable for GC. To introduce you to some key concepts of stationary phases, we for now will stick with packed spherical particles.
2.1. Morphology #
In LC, the spherical particles usually comprise of silica, the surface of which can be chemically modified to realize the targeted retention mechanism.
We saw in the previous lesson that such stationary phases can also be porous. leading to large pore volumes and large surfaces areas. One could compare the benefit of a larger surface area with a larger parking lot. The more spaces are available, the more analytes will be able to interact with the stationary phase.
Figure 2. Development of particle size over time. Data is not exact, but serves to illustrate the trend.
Another relevant parameter is the particle size. In Lesson 3 we will see that smaller particles are more efficient than larger particles: smaller particles reduce band broadening (and increasing the number of theoretical plates) by shortening diffusion paths and enabling faster mass transfer. The particle-size distribution also matters, because narrow distributions promote uniform flow and consistent packing. In modern high-efficiency chromatography, tightly controlled particle sizes are essential for high-resolution separations.
2.2. Porosity #
The pore-size distribution of the stationary phase critically affects mass transfer and analyte accessibility, influencing both retention and band broadening. A well-optimized distribution ensures efficient separation by allowing analytes of various sizes to interact appropriately with the stationary phase. In size-exclusion chromatography elution volumes are exclusively determined by the pore-size distribution in relation to the size of analyte molecules.
Figure 3. Different representations of stationary phase, with a column, a schematic 2D representation of stationary phase, a SEM photograph of a particle, and a SEM photograph of stationary phase. Photo of particle from Schure et al. Chem. Eng. Sci. 174, 2017, 445-458 [1], photo of stationary phase from Xia et al. J. Chromatogr. A, 2016, 1471, 138-144 [2].
Porosity in chromatography determines the fraction of the column volume available to the mobile phase, directly affecting analyte retention, flow resistance, and sample loading capacity. It influences to what extent analytes can penetrate in the stationary phase particles and impacts both retention time and mass-transfer kinetics (see Lesson 3). We saw in Lesson 1 that typical values for the porosity are between 0.56 and 0.7.
2.3. Phase Ratio #
Lesson 1 showed us that, ultimately, the partitioning of analytes between the two phases is governed by the phase ratio. In adsorption chromatography, the phase ratio is proportional to the surface area (and thus related to the porosity) of the stationary phase, which we have seen is very large for porous particles, and much larger than a film on a capillary wall. For absorption chromatography, the phase ratio is proportional to the stationary phase thickness and the surface area. In all cases also the column internal diameter is of relevance.
EXERCISE 3
In liquid chromatography, increasing the phase ratio (i.e., increasing the relative amount of stationary phase compared to mobile phase) typically leads to higher retention factors (k), while higher column porosity generally corresponds to lower phase ratios and lower retention.
This is correct! The phase ratio influences how much time an analyte spends in the stationary phase. A higher amount of stationary phase (lower phase ratio) increases analyte interaction and typically increases retention. In contrast, column porosity reflects the fraction of the column volume that is accessible to the mobile phase. A high porosity means more mobile phase and less stationary phase per unit volume, resulting in a lower phase ratio and typically lower retention.
This is incorrect! The phase ratio influences how much time an analyte spends in the stationary phase. A higher amount of stationary phase (lower phase ratio) increases analyte interaction and typically increases retention. In contrast, column porosity reflects the fraction of the column volume that is accessible to the mobile phase. A high porosity means more mobile phase and less stationary phase per unit volume, resulting in a lower phase ratio and typically lower retention.
READ SECTION 1.6.3
Flow Profiles
3. Flow Profiles #
Now lets turn our attention to the mobile phase. We noted in Lesson 1 that a mobile-phase delivery system is required to transport the bulk mobile phase through the column. The resulting flow profile can have a significant effect on the efficiency of the chromatographic system, so it is useful to investigate.
3.1. Fluid dynamics #
When the fluid is pushed through the column by the pump, it will flow through the channel in velocity layers, sometimes called “streamlines“. Near the column wall there will be more friction and thus the flow there will be much lower, generating a flow profile as shown in Figure 4A. We can explain this as two forces being at play:
- Inertial forces: the momentum of the fluid; and the
- Viscous forces: the friction of the fluids between different streamlines
Figure 4. Schematic representation of laminar flow (A) and turbulent flow (B).
As long as the viscous forces are dominant, we obtain a laminar flow (Figure 4A), and the flow profile will be parabolic as described by the Hagen-Poisseulle equation:
Equation 1.64: F=\frac{\Delta P \cdot \pi \cdot d^4_{\text{c}}}{128 \eta \cdot L}
where \Delta P is the pressure drop across the tube, and d_{\text{c}} its diameter.
The theoretical background provided in the book shows that when the inertial forces become too strong (i.e. the flow rate becomes too high), the radial pressure differences increase to the point that streamlines start to mix by random fluctuations in the form of eddies, giving rise to a mixing phenomenon known as eddy diffusion, and ultimately turbulent flow (Figure 4B).
At this stage it is useful to note that other mechanisms than pressure are also used to facilitate transport of mobile phases and solute molecules through a separation channel. For example, a high voltage generates an electro-osmotic flow (with a very flat profile, Figure 1.19C in the book) through the column in capillary electrophoresis. This is discussed in Chapter 5 and in later lessons.
EXERCISE 4
In liquid chromatography, flow through the column is typically laminar rather than turbulent. What is the main reason for this behavior?
Correct! The Reynolds number in chromatography is typically low due to small column diameters, high fluid viscosity, and dense packing of particles. This results in viscous forces dominating, maintaining laminar flow.
Oops! Not correct! High velocities increase inertial forces, potentially leading to turbulence, but LC operates well below that threshold. The mobile phase in LC is typically a liquid, which is incompressible, so this does not prevent turbulence.
READ SECTION 1.6.4
Column Permeability
4. Permeability #
For pressure-driven separations, the transport of mobile phase through a column requires a certain pressure drop across the column (cf. Equation 1.64). This is known as the column pressure or backpressure. The permeability describes how easily a fluid can flow through a chromatographic column. It depends on factors such as particle size, porosity, and the type of stationary-phase packing. A lower permeability means a higher back pressure.
The backpressure is one of the most-important limiting parameters for LC separations and very important to consider during method development. It can be estimated using the Darcy equation. See Section 1.6.4 for more details.
Equation 1.69: \Delta P=\frac{\Psi \cdot \eta \cdot L \cdot u_0}{d^2}
Here, \Psi is the column flow-resistance parameter, \eta the viscosity of the liquid, and d the diameter of the column, d_{\text{c}} (for open-tubular chromatography; GC), or the diameter of the packed particles, d_{\text{p}} (for packed-column chromatography; LC).
According to Equation 1.69, the backpressure will increase significantly for smaller particle sizes. We will discuss this in Lesson 3.
All systems have maximum operating pressures. Typically, high pressures give good efficiency. Higher pressure limits usually cost more resources, although the most advanced systems do not tolerate infinite pressures.
Table 1. Overview of the relevant size parameters and typical values for the column resistance factors for different column geometries. The domain size is defined as the diameter of an (average) flow channel plus that of a structural element (pilar or “skeleton” of a monolith). For monolithic columns it is easier to specify the permeability coefficient (see Section 1.6.4).
| Type | Size Parameter | \Psi |
|---|---|---|
|
Cylindrical tube |
Column diameter (d_{\text{c}}) |
32 |
|
Flat channel |
Channel height |
12 |
|
Packed bed of solid spheres |
Particle diameter (d_{\text{p}}) |
500 |
|
Packed bed of fully-porous spheres |
Particle diameter (d_{\text{p}}) |
1000 |
|
Pillar array |
Domain size |
50 – 200 |
|
Monolith |
Domain size |
20 – 100 |
The flow-resistance parameter is shown for different column formats in Table 1. Readers interested in this can see Section 1.6.4 where this parameter is placed into context of the permeability coefficient and the Kozeny-Carman equation.
Figure 5. Different SEM photographs of stationary-phase formats relating to the values for flow-resistance parameter in Table 1. Panels B and C adapted from Refs. [3] and [4], respectively, with permission from the Royal Society of Chemistry. Panel D adapted from Ref. [2], Copyright (2016), with permission from Elsevier. Panel E reproduced from Ref. [5] with permission from Springer Nature, Copyright 2020.
One other important parameter that highly affects backpressure is the viscosity of the mobile-phase solvents. Some solvents yield much higher viscosities and therefore also pressures, which renders them less attractive.
EXERCISE 5
In the previous lesson we calculated various separation characteristics for the separation of two compounds in a 100 mm long column (L) with an internal diameter (d_{\text{c}}) of 2.1 mm. The column is packed with fully porous particles of (d_{\text{p}}) 2.7 μm. Now calculate the backpressure (\Delta P) in bar (105 Pa = 1 bar = 14.5 psi) when the mobile phase composition is 40% methanol in water. For this exercise, you will need to estimate the viscosity from Figure 6 (Figure 3.22 in the book; round to 1 decimal), and obtain the flow resistance parameter from Table 1 (Table 1.1 in the book). The linear velocity (u_{\text{0}}) is 0.0044 m s-1.
The backpressure in bar is (round to an integer; no decimals):
This answer is correct!
Unfortunately, this answer is not correct. Try a viscosity of 1.6 mPa s (which is 0.0016 Pa s). Pay good attention to units and convert everything to meters! The column flow resistance parameter should be 1000.
Concluding remarks #
Flow profiles, thermodynamics in chromatography and phase ratio are concepts that seem far away from daily practice in the lab. In this lesson we learned about some of the thermodynamics in chromatography which allowed us to describe the effect of pressure and temperature in chromatography. We have also investigated important properties of stationary-phase structures and how they affect column permeability. The latter is a direct link to backpressure which for LC separations is a limiting factor.
We also saw how profiles impact the efficiency of a chromatographic system. In the next lesson, we will stick with the efficiency and kinetics by considering the plate height theory in detail.
EXTENSIVE EXERCISE
In an HPLC system a 250×4.6 mm column is used packed with 5.0 μm porous particles. The mobile phase is a 60:40 mixture of aqueous buffer and methanol. The flow rate is set to 1 mL·min-1. Estimate the dead time of the column at this flow rate (t_0) in seconds. Your answer should be an integer (no decimals). Assume the porosity to be 0.65.
This is correct!
This is not correct. When assuming a porosity of 0.65, your dead volume should be 2701 mm3.
Calculate the corresponding linear velocity (u_0) in mm s-1. Round to four decimals.
Correct!
This is not correct. Did you confuse the units?
Estimate the backpressure of the column in bar. Look up the flow resistance parameter in Table 1 (Table 1.1 in the book), and the viscosity in Figure 6 (Figure 3.22 in the book). Round your answer of the backpressure to an integer (no decimals).
This is correct!
This is not correct. Did you use a column flow resistance parameter of 1000 (fully porous particles), and a viscosity of 0.0016 Pa s?
Your colleague is considering to use even more efficient particles of a size of d_p 1.8 μm. Based on your earlier calculations and regarding Equation 1.69, do you think it is feasible if the pressure limit of the system is 1000 bar?
Correct. The backpressure is already close to the limit. Going for 1.8 μm would really be well above the limit.
Incorrect. Consider the fact that the particle size is in the denominator of Equation 1.69. In other words, decreasing the particle size will dramatically increase the backpressure which is already close to the limit.
Another colleague proposes to increase the temperature and to switch the organic solvent from methanol to acetonitrile. Which of the statements is true?
Indeed, all statements are correct!
Unfortunately, this is not correct! For the effect of temperature, consider what you know about thermodynamics. How does the temperature affect the pressure?
Please find here an Excel sheet that contains the calculations 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_02, .XLSX).
References #
[1] R. Schure, R. S. Maier, T. J. Shields, C. M. Wunder and B. M. Wagner, Chem. Eng. Sci., 2017, 174, 445–458, DOI: 10.1016/j.ces.2017.08.024
[2] H. Xia, G. Wan, J. Zhao, J. Liu and Q. Bai, J. Chromatogr. A, 2016, 1471, 138–144, DOI: 10.1016/j.chroma.2016.10.025
[3] T. Sun, X. Jiang, Q. Song, X. Shuai, Y. Chen, X. Zhao, Z. Cai, K. Li, X. Qiao and S. Hu, RSC Adv., 2019, 9, 28783–28792, DOI: 10.1039/C9RA05085J
[4] M. Callewaert, J. O. De Beeck, K. Maeno, S. Sukas, H. Thienpont, H. Ottevaere, H. Gardeniers, G. Desmet and W. De Malsche, Analyst, 2014, 139, 618–625, DOI: 10.1039/C3AN02023A
[5] X. Feng, J. Cai, H. Zhao and X. Chen, Chromatographia, 2020, 83, 749–755, DOI: 10.1007/s10337-020-03893-0
