Learning Goals #
- Use plate-height theory to determine the optimal linear velocity (and flow rate).
- Decompose plate height into the different factors that contribute to band broadening: eddy diffusion, longitudinal diffusion and mass transfer resistance.
- Compare plate-height curves of different columns by reducing parameters.
- Evaluate separation systems for their suitable for high-throughput screening.
READ SECTION 1.4.4
The Van Deemter Equation
1. Plate Height #
In the previous lesson, we introduced the concept of the number of theoretical plates (N) as a measure of column efficiency. However, for practical and comparative purposes, it is often more informative to express efficiency in terms of the plate height (H).
1.1. Van Deemter Equation #
In chromatography, the plate height varies as a function of the linear velocity (u_0) of the mobile phase. This relationship is commonly represented by an H versus u curve and mathematically described using a plate-height equation. The first such formulation was introduced in a landmark paper by van Deemter et al. back in 1956 [1]. The simplified form of this equation, still widely known as the van Deemter equation, is given as:
Equation 1.30: H=A+\frac{B}{u_{\text{int}}}+C \cdot u_{\text{int}}
Equation 1.30 shows a number of contributions to the plate height, depicted as the A, B and C terms. These are also illustrated in Figure 1.
The A-term is ascribed to eddy diffusion and can be imagined as path-length differences for the different molecules moving through the column (Figure 1A). In a particle-packed column, the particles are not homogenously packed. Each analyte molecule may take a slightly different path through the labyrinth of particles. The different molecules pertaining to the same analyte will thus arrive at slightly different times at the detector, translating into a broadened peak.
Figure 1. Graphical illustration of the van Deemter equation, describing the plate height as a function of the mobile-phase linear velocity and schematic illustrations of the three different contributions. Adapted from Ref. [2].
Van Deemter assumed A to be independent of u_{\text{int}}, but we will see that this turned out not to be accurate.
The B-term arises from longitudinal molecular diffusion, as discussed in the previous lesson. Due to concentration gradients within the column, analyte molecules naturally diffuse from regions of higher concentration, such as the apex of a chromatographic peak, toward regions of lower concentration. The lower the linear velocity, the more time there is for this to take place.
Jan J. van Deemter (1918-2004, The Netherlands)
Jan Jozef van Deemter obtained a PhD from the University of Amsterdam (The Netherlands) in 1950, while he was already (since 1947), employed at the Royal Dutch Shell Laboratory in Amsterdam (KSLA). Together with colleagues F.J. Zuiderweg and A. Klinkenberg, he developed a detailed model for dispersion in chromatographic columns that was published back in 1956. We still use the concepts from his work and – most importantly – his name lives on in the numerous times chromatographers talk about the van Deemter equation.
The C-term in the van Deemter equation encompasses several processes with mass transfer as the common denominator.
The C_{\text{s}}-term arises from delayed equilibration between the mobile and stationary phases (Figure 1Cs). At any point the system moves towards an equilibrium between the analyte molecules in the mobile and stationary phases. However, the mobile phase is moving and thus the equilibrium is disturbed. Consequently, analyte concentrations can locally deviate from equilibrium at both the leading and trailing edges of a peak (Figure 1).
The C_{\text{m}} term reflects the impact of non-uniform flow paths within the mobile phase, particularly due to parabolic flow profiles (Figure 1Cm).
1.2. Optimal flow rate #
The most practical application of the Van Deemter curve is to identify the optimum linear velocity (i.e. flow rate). This can be done using the following distinct steps:
- Select one or several compounds
The band broadening for unretained analytes will be different from that of retained analytes. Depending on the application, a recommendation may be to consider one unretained analyte and one that exhibits some retention. - Conduct a series of experiments at different linear velocities
With the column in place, inject your analytes at different flow rates. - Measure the plate height for each experiment
The most convenient method is to use the peak width at half height, and convert it to N, and then H (see Lesson 1). - Plot the plate height against the linear velocity for each compound
This will yield a van Deemter curve for each analyte such as shown in Figure 1. You can fit the Van Deemter equation through these data points using Excel, or more sophisticated tools (see Chapter 9). The optimal linear velocity can be converted into an optimal flow rate.
EXERCISE 1
Select all statements that are correct.
1.3. Interstitial linear velocity #
If you have seen the Van Deemter equation before, you may have noticed that we use u_{\text{int}} instead of u_0. This is on purpose! In the original work of Van Deemter, the intersitial linear velocity (u_{\text{int}}) is used. It differs from the average mobile-phase velocity (Equation 1.16) u_0.
Unretained molecules spend time in the interstitial volume (i.e. the mobile phase outside the packing particles), where they move with average linear velocity u_{\text{int}}, and in the pores of the packing material, where their average lateral movement is zero. Using u_0 will not yield a different optimum, but will yield different B and C parameters, which can be significant for fundamental studies. The open-tubular columns used in GC do not feature particles and thus u_0 can be used. See Section 1.4.4 for more information on how to measure u_{\text{int}}.
1.4. Operational flow regimes #
The van Deemter equation serves as a tool for analyzing column performance and band broadening and Equation 1.30 allows us to measure the van Deemter curve to determine the optimal linear velocity and thus the optimal flow rate.
From this, we may identify three regions in the van-Deemter curve.
- The B-branch, where u_{\text{int}} < u_{\text{int,opt}}
- Below the full potential of the system (N< N_{\text{max}}) and, to make things worse, the retention times also increase.
- Lose-lose situation: this is a place to avoid.
- The optimum, where u_{\text{int}} \approx u_{\text{int,opt}}.
- System performs best (N \approx N_{\text{max}})
- Relatively long retention times and, therefore, place to be when top performance is needed.
- The C-branch, where u_{\text{int}} > u_{\text{int,opt}}
- Sacrifice som performance (N< N_{\text{max}}) to gain time.
- Region for compromise and place to be for high-throughput,
READ SECTIONS 1.7.1-1.7.2
Effect of Diffusion Coefficient and Particle Size
2. Extended van Deemter equation #
From a fundamental perspective, plate-height equations, such as the van Deemter equation, are also eminently useful to understand the physicochemical processes that contribute to band broadening.
2.1. Effect of diffusion coefficient #
Van Deemter originally introduced his plate height equation for packed-column chromatography. Including the particle size (d_{\text{p}}) and the diffusion coefficient of the analyte (D_{\text{m}}), its extended reads
Equation 1.76: H=a \cdot d_{\text{p}}+\frac{b \cdot D_{\text{m}}}{u_{\text{int}}}+ \frac{c \cdot d^2_{\text{p}}}{D_{\text{m}}} \cdot u_{\text{int}}
We can see from Equation 1.76 that a higher value for D_{\text{m}} increases the B-term, but decreases the C-term, causing the optimum to shift to higher values of u_{\text{int}}, while H_{\text{min}} is unaffected. This is reflected by
Equation 1.77: u_{\text{int,opt}}=\sqrt{\frac{B}{C}}=\frac{D_{\text{m}}}{d_{\text{p}}} \sqrt{\frac{b}{c}}
and
Equation 1.78: H_{\text{min}}=a \cdot d_{\text{p}}+2\cdot d_{\text{p}} \sqrt{b \cdot c}
Equations 1.77 and 1.78 underline that at higher D_{\text{m}}-values, separations may be performed faster.
2.2. Effect of particle size #
The other important parameter is the particle size d_{\text{p}}. Equations 1.77 and 1.78 show that H_{\text{min}} increases in proportion with the particle size. Moreover, in the C-branch of the Van Deemter curve we see a sharp increase in the plate height for larger particles. This means that it is much more efficient to work with smaller particles.
Figure 2. Left: Effect of particle size on the plate height as a function of flow rate. Right) Reduced plate heights or the curves shown to the left. Adapted from Ref. [3], Copyright (2017), with permission from Elsevier.
From this we learn that decreasing the particle size allows (i) the column length to be decreased whilst maintaining performance (N), (ii) the velocity to be increased, favouring higher throughput and thus significantly decreasing the analysis time, and, unfortunately (iii) dramatically facing more back pressure, following the Darcy equation from the previous lesson.
Equations 1.77 and 1.78 underline that at higher D_{\text{m}}-values, separations may be performed faster.
READ SECTION 1.7.3
Reduced Plate-Height Curves
3. Comparing Plate-Height Curves #
To fairly compare columns characterized with different mobile phases, or with different particle sizes it is important to use reduced parameters. Reduced parameters are dimensionless values that essentially normalize the parameters in the plate-height equation. For the reduced plate height, h, we must take out the effect of the particle size through
Equation 1.79: h=\frac{H}{d_{\text{p}}}
For the reduced linear velocity we normalize for both the particle size and diffusion coefficient through:
Equation 1.80: \nu_{\text{int}}=\frac{u_{\text{int}} \cdot d_{\text{p}}}{D_{\text{m}}}
3.1. Reduced Van Deemter Equation #
Using Equations 1.79 and 1.80 in Equation 1.76 yields
Equation 1.81: h=a+\frac{b}{\nu_{\text{int}}}+c \cdot \nu_{\text{int}}
Here, the coefficients a, b and c are now dimensionless. Allowing Equation 1.77 to be transformed into
Equation 1.82: \nu_{\text{int,opt}}=\sqrt{b/c}
and Equation 1.78 into
Equation 1.83: h_{\text{min}}=a+2 \sqrt{b \cdot c}
The coefficients are now independent of the diffusion coefficient and particle size. The result is shown in Figure 2 at the right, where we can now see that the curves overlap neatly, allowing them to be compared.
The practical value of the reduced Van Deemter equation is more important than you may expect.
Using the theoretical values for a, b, and c (see above), a h_{\text{min}} of roughly 2 for fully-porous particles. This value will be lower for superficially-porous particles. For non-porous particles a value as low as 1.4 has been reported [6,7].
EXERCISE 2
In a pharmaceutical company a reversed-phase HPLC system is used for quality control of a product. Because of the high number of samples to be analyzed the analytical department considers buying a UHPLC system. In a UHPLC system short columns with very small particles are used.
You are tasked to investigate whether such an upgrade would be useful by theoretically comparing the two systems with respect to the required selectivity of baseline separation, and analysis time at the optimal flow velocity. You consider two systems:
- HPLC: 100 mm long column with 3.5 μm particles; max. pressure 400 bar.
- UHPLC: 50 mm long columns with 1.7 μm particles; max pressure 1200 bar.
First, calculate the optimal linear flow velocity u_0 in m\cdots-1 using Equation 1.80. Assume a = 2; b = 1; c = 0.05, and a diffusion coefficient of D_{\text{m}} = 7\cdot10-10 m2 s-1. For this, you will first need to calculate the optimal reduced linear velocity \nu_0 using Equation 1.83. Round your answer to five decimals (e.g. “0.00072”). Enter your value for the HPLC system in the field and you will obtain the value for the UHPLC system in the answer feedback.
This is correct! For the UHPLC system, this is 0.00180.
Unfortunately, this is not correct. \nu_{\text{opt}} should be 4.5.
Now calculate the required selectivity \alpha_{\text{req}} for baseline separation for the HPLC and UHPLC systems. Assume that k = 2. You will need to first calculate the reduced plate height (h_{\text{min}}; Equation 1.83), then convert it to the plate height (Equation 1.79; H), then calculate the efficiency (N; Equation 1.29), and, finally, calculate \alpha_{\text{req}} using Equation 1.40 (Equation 1.39 would be too complex).
Write in the field \alpha_{\text{req}} (round to 3 decimals) for the HPLC system (the UHPLC value is provided in the answer feedback).
This is correct! For the UHPLC system \alpha_{\text{req}} = 1.082.
Unfortunately, this is not correct. For the HPLC system, h_{\text{min}} = 2.447, H = 8.56\cdot10-6 m and N = 11675. You can then rearrange Equation 1.39 to obtained \alpha_{\text{req}}.
Now calculate the analysis time in seconds. You will first need to estimate the dead time (t_0) for this. Report an integer number (no decimals) for the HPLC system. You will obtain the answer for the UHPLC system in the answer feedback.
Correct! For the UHPLC system this is 81 seconds.
Unfortunately, this is not correct. The dead time should be 112 seconds.
What is your conclusion based on your theoretical assessment about the UHPLC system upgrade?
This is correct! The analysis time will be decreases significantly, thus increasing throughput, while the selectivity is barely changing, meaning that a similar separation chemistry can be used.
This is incorrect. Try again! Consider that a higher throughput means that analysis time is reduced.
From this we can compute for our specific situation in the what the expected plate height should be.
Concluding remarks #
Plate height theory is used extensively for fundamental studies that advance column technology. Examples include the development of superficially-porous particles, or pillar-array columns (see Lesson 2) for which better reduced plate heights may be obtained. Reducing the van Deemter parameters allow different column packings to be compared.
We also have seen how the same theoretical concept can be used to predict the feasibility of improving e.g. throughput for separation systems when considering upgrades.
In the next lesson we will study the band broadening processes in some further depth.
EXTENSIVE EXERCISE
Repeat Exercise 2, but now at the maximum flow velocity. You will need the Darcy equation (Equation 1.69; Lesson 2) for this and rearrange it for u. Assume that the viscosity (\eta) is 0.001 Pa\cdots. The pressure limits of both systems are shown in Exercise 2.
First, calculate the linear velocity at the maximum pressure using the rearranged Darcy equation in m s-1. Round your answer to four decimals (e.g. “0.0034”). Enter the number for the HPLC system. The answer for the UHPLC system will be shown in the answer feedback.
This is correct! For the UHPLC system it is 0.0070.
This is not correct. Ensure that you rearranged your Darcy equation so that u_0=\frac{\DeltaP \cdot d^2_{\text{p}}}{\Psi \cdot L \cdot \eta}
Now, calculate the required selectivity \alpha_{\text{req}}, for both systems. Round to three decimals. Enter the value for the HPLC system in the field. The value for the UHPLC system is provided in the answer feedback.
Correct! For the UHPLC system it is 1.089.
This is not correct. Your plate number for the HPLC system should be around 8749. Make sure the units are correct!
Finally, calculate the analysis time (through t_{\text{R}}) in seconds. Report an integer number (no decimals). Enter the value for the HPLC system in the field. The value for the UHPLC system is provided in the answer feedback.
Correct! For the UHPLC system it is, however, much lower, 22 seconds!! What can we take from this? Well, the UHPLC system at maximum flow velocity does not really sacrifice selectivity, and – despite the lower plate number – manages to seriously decrease the analysis time. In contrast, the HPLC system loses too much efficiency, although it is still faster than the UHPLC system at optimal flow velocity.
Incorrect. Did you use a retention factor of 2, a dead time of 20 seconds and the alpha of 1.096 (previous answer)?
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_03, .XLSX).
References #
[1] J.J. Van Deemter, F.J. Zuiderweg and A. Klinkenberg, Chem. Eng. Sci., 1956, 5, 271–289, DOI: 10.1016/0009-2509(56)80003-1
[2] B.W.J. Pirok, Making Analytical Incompatible Approaches Compatible, University of Amsterdam, Amsterdam, 2019 [LINK].
[3] M.M. Dittmann and X. Wang, in Handbook of Advanced Chromatography /mass Spectrometry Techniques, eds. M. Holçapek and W. C. Byrdwell, Elsevier, 2017, pp. 179–225.
[4] P.A. Bristow & J.H. Knox, Standardization of test conditions for high performance liquid chromatography columns, Chromatographia, 1977, 10, 279–289, DOI: 10.1007/BF02263001
[5] H. Poppe, Some reflections on speed and efficiency of modern chromatographic methods, J. Chromatogr. A, 1997, 778, 3-21, DOI: 10.1016/S0021-9673(97)00376-2
[6] G. Stegeman, A.C. van Asten, J.C. Kraak, H. Poppe, R. Tijssen, Comparison of Resolving Power and Separation Time in Thermal Field-Flow Fractionation, Hydrodynamic Chromatography, and Size-Exclusion Chromatography, Anal. Chem. 1994, 66, 7, 1147–1160, DOI: 10.1021/ac00079a033
[7] A.M. Striegel & A.K. Brewer, Hydrodynamic Chromatography, Ann. Rev. Anal. Chem., 2012, 5, 15-34, DOI: 10.1146/annurev-anchem-062011-143107
