HPLC Column Cross-Sectional Area and Scaling

Here is a simple formula to use when scaling up or down Internal Column Diameter to maintain retention values (under constant linear velocity). Flow rate must be adjusted to account for any changes made to the column's cross-sectional area. We usually refer to these types of changes as the "Scaling Factor". To determine the scaling factor, we need to know the internal column diameters of the two columns we are scaling from (actually, we need to know the radius, but once we have the diameter, we simply divide the diameter by 2 to obtain the radius). *In this discussion, changes in cross-sectional area are the only parameters we are concerned with as column length does not affect scaling.

• Scaling Factor = (S);
• Column #1 Radius =  (R1);
• Column #2 Radius =  (R2).

     S = R2² / R1²

Example #1: 250 x 4.60 mm column scaled down to a 250 x 2.10 mm column. 

          Answer = 0.208. 

• If the original flow rate was 1.000 mL/min, the the scaled down flow rate would be 0.208 of the original or 0.208 mL/min for the 2.10 mm ID column. *For practical use and application, we often use either 200 ul/min or 210 ul/min to simplify the value.

Example #2: 250 x 4.60 mm column scaled up to a 250 x 10.00 mm ID semi-prep column.

          Answer  = 4.726. 

• If the original flow rate used was 1.000 mL/min with the 4.60 mm ID column, then we would increase the flow rate to 4.726 mL/min on the 10.00 mm ID column to maintain the same relative velocity (and relative retention). *For practical use and application, we often use 5 mL/min to simplify (round off) the value. 

Notes:

1. Flow rate optimization should always be carried out by running a standard at different flow rates and plotting the plate height (N) vs the flow rate. Test flow rates that are slightly below the predicted linear velocity and up to 2 times higher than that rate to find and optimize the flow rate for your sample (it must be determined through experimentation for your specific method). 
    
2. HPLC Columns packed with sub 2 micron supports may have optimum flow rates 2 to 5 times more than the predicted std linear flow rate so actual testing is critical to determining the most efficient flow rate. I recommend optimizing the flow rate used with analysis methods which use any particles which are 2.5 microns or smaller in diameter.
   

HPLC to UHPLC Conversion Notes (Gradient Time Program Adjustment)

In an earlier article we discussed how to adjust the flow rate, injection volume and column dimensions when scaling an HPLC method UP or Down. The formula's needed to do this are fairly simple. If we adjust for changes in the column dimensions or flow rate, what types of changes are needed to adjust the gradient time? The formula to make this adjustment is also very simple. Here is the information you need.

Terms Used in Formula:

Time in minutes, Gradient (Initial): Tg1

Time in minutes, Gradient (New):   Tg2

Flow Rate in mL/min, Column (Initial): Fc1

Flow Rate in mL/min, Column (New): Fc2

Column Diameter, mm (Initial): Dc1

Column Diameter, mm (New): Dc2

Column Length, mm (Initial): Lc1

Column Length, mm (New): Lc2

• Tg2 = Tg1 x (Fc1/Fc2) x ((Dc2²) / (Dc1²)) x (Lc2 x Lc1)

Here is an example problem to solve for. 

If we start with a flow rate of 1.000 mL/min (Fc1) on a 4.6 x 250 mm column with 5 micron support (Dc1 & Lc1) and have an initial Gradient Time of 10 minutes (Tg1), then what would the new gradient time be if we switched to a sub 2 micron support in a 2.1 x 50 mm column (Dc2 & Lc2) at 0.200 mL/min (Fc2)? 

To solve the equation we will plug-in the values for each part of the equation separately, then multiply them to obtain the result.

  (Fc1/Fc2):    1.000/0.200 = 5

  (Dc2²) / (Dc1²)  4.41/21.16 = 0.21

   (Lc2 x Lc1) = 50/250 = 0.20

  Tg2 = 10 x 5 x 0.21 x 0.20 

  Tg2 = 2.10 (or 2.10 minutes)

If a 2.1 x 50 mm column was substituted for the 4.6 x 250 mm AND the flow rate was changed from 1.000 mL/min to 0.200 mL/min, then the initial programmed gradient time of 10 minutes would be changed to 2.1 minutes

Number of theoretical plates (N), Calculation Formulas

Number of theoretical plates (N):

Often used to quantify the efficiency (performance) of a column (HPLC or GC). "Plates" are expressed per meter of column length and should be calculated based on a retained peak with ideal peak shape or symmetry. 

   N = Plates; tr = Retention Time of Peak; w = Peak width;  w_0.5 = Peak width measured at half height.

Two popular formulas are:

Tangent: USP (United States Pharmacopeia / ASTM)

Best for Gaussian peaks. Peak width is often determined at 13.4% of the peak height (w). Inaccurate for peaks which are non-Gaussian, poorly resolved or tail.

   N = 16 (tr / w)²

Half Peak Height: (European Pharmacopeia)

For peaks which are less Gaussian in appearance, using a slightly different formula with the peak width measurement made at the half-height (W_0.5). Less accurate for peaks which are poorly resolved or tail.

   N = 5.54 (tr / w_0.5)²

 

• Other formulas, not included, for calculating Plate numbers include: Half Width, Variance Method, Area / Height & Exponential Modified Gaussian (EMG).
• Caution. HPLC column "Plate" values should not be used for a final determination of efficiency unless you are comparing all results on the same exact HPLC system, which is setup and run under identical conditions each time. Since the result is based on many possible variables, including how your HPLC system is plumbed (dwell volume & tubing ID), the peak's Kprime and symmetry, the detector used, sampling rate, integration quality, flow cell volume, flow rate, actual column used (to name a few), it can easily be manipulated to be very large or small.

HPLC K Prime. Also known as: Retention Factor, Capacity Factor): One of the Single Most Important HPLC Parameters of All

The role of Capacity Factor / Ratio (K prime) in liquid chromatography is to provide a calculation or ratio which defines how much interaction the solute (sample peak) has with the stationary phase material relative to the mobile phase (IOW: the relative time the sample spends interacting with the support vs. the mobile phase). If this interaction is too short (i.e. K prime less than 1.0), then no chromatography has taken place and you have just developed a "flow-injection" method (the same as if no column was used) instead of a chromatography method. The method fails all validation and is NOT fit for purpose. Sample Retention must be long enough to demonstrate that the method developed is specific to the sample and shows good selectivity (retention) for the sample analyzed. This is true for most, but not all modes of liquid chromatography (3). 

• New and ever 'experienced' users lacking training in HPLC often make this error, developing methods where the sample has little to no retention on the column. We routinely review methods where the actual K prime of the key sample is measured to be at or near 0, failing to show any chromatography took place in the method (*The Journals are filled with thousands of examples of invalid HPLC methods of analysis). This mistake is made because the author(s) have no formal training in liquid chromatography and do not understand the basics of the technique. 
• Note: Slowing down the flow rate to "show" a later elution time DOES NOT increase K prime (a very common novice mistake) !!! 
  

Observance of the fundamentals of chromatography are key to developing high quality HPLC methods. For most modes of HPLC, highest on this list of basic fundamentals is that the sample(s) be retained on the HPLC column used and not eluted out at or near the column's void volume (we often refer to this time in minutes as, "T-zero" or "t0"). Many chromatography methods fail this simple test of retention and are invalid as written. Knowing what a compound's retention or capacity factor is allows us to be confident that it has been retained and eluted past this critical point, but to first calculate K prime, we first need to know the HPLC column's void volume. 

Calculation and/or measurement of the Column Void Volume should be one of the very first chromatography method development tasks you learn to perform. Knowing the column void volume allows you to determine the retention time of an unretained sample and the resulting retention factor (K prime) of each sample eluted after it. To do this, you must calculate the column void volume AND inject a sample which will not be retained by the column to determine what time an unretained sample will be eluted off the column. This establishes the 'T' zero time, or T(0). The time it takes an unretained compound to elute off the column is critical to know. If your HPLC method does not retain the sample on the column long enough past this time, then you are not allowing any chromatography to occur. Once you have this T(0) value, you can then determine the retention factor (the "K Prime") of your actual sample(s) using the simple ratio formula below. Your final method should baseline separate all compounds apart and, if properly developed, each sample peak will often have K Prime values between 2.0 and 10.0. K prime values of greater than 10 are acceptable, but often show minimal improvements to resolution. Try and insure that the earliest eluting peak in your sample has a K Prime of  >1.5. Do not develop methods which only result in K Primes of less than 1.5 (an indication of poor quality chromatography). 

Note 1: Many regulatory agencies (e.g. The USA FDA) requires that K prime values for HPLC separations be equal to or greater than 2.0 to meet Specificity acceptance criteria (System Suitability/Method Validation). After all, if it elutes at or near the void volume, then your method is not specific for anything. Besides being unscientific in design, your method will fail System Suitability and fail validation. IOW: It does not meet this basic requirement.

• K Prime, K1 (Capacity Factor or Retention Factor) Formula:

       k1 = [T(R) - T(0)] / T(0)

• (where T(R) equals the retention time of the peak in minutes and T(0) is
  the retention time of an unretained peak). 
• *The 'K Prime' of your sample must be > 1.00. A value greater than 1.5 should be your goal.

Example #1: 
 T(0) found to be 2.90 minutes and the sample elutes at 5.80 minutes. k1 = 5.80 - 2.90 / 2.90. k1 = 1.00.

Example #2:
 T(0) found to be 2.90 minutes and the sample elutes at 9.10 minutes. k1 = 9.10 - 2.90 / 2.90. k1 = 2.13.

Example #3:
 T(0) found to be 1.75 minutes and the sample elutes at 1.74 minutes. k1 = 0. No retention and no chromatography have taken place at all. The method is invalid.


Note 2: I see and read published HPLC methods (including "Validated Methods" !) every week which ignore this fundamental requirement and present data showing little to no retention of the primary sample on the column. Most are RP methods run on popular C18 columns and show the main peak of interest eluting out as a nice sharp peak right at the void volume.  These methods often describe the sample analyzed as "100% pure" and are fully validated (because the person doing the work may not have had any HPLC experience or training)! A mixture will always look like a single peak by HPLC when no 'chromatography' is employed to separate out all of the possible components. The sample must be retained on the column for a period of time before we can conclude anything about its purity by the method employed.

Note 3: In some cases, when other modes of chromatography are utilized (e.g. ion exchange, size exclusion chromatography (SEC / GPC), K prime is not as relevant. The mode of chromatography can affect the interpretation. For example: This is because size exclusion chromatography relies on the sample's interaction with well defined pores inside the support (inclusion/exclusion) to separate based on molar size. A variety of pore sizes can be used to "filter" the sample. So a large K prime value might be normal for a molecule that is low in molecular weight and spends a lot of time working its way through the column. A high molecular weight sample might just "shoot" through the column due to little or no interaction with the pores. You still need to have retention on the column, but now it is determined by how long it takes the sample to find its way out of the column. SEC columns are bracketed by Pore Size (e.g Mw. Excluding all samples that do not "fit"). With size exclusion columns, determine the Retention and Exclusion times, not the K prime). This article is specific to thr more common cases where traditional HPLC NP or RP modes are used. In these cases, low K prime values indicate no retention took place and the method fails all claims of specificity for the sample (selectivity is absent or poor). HPLC methods with little to no selectivity fail scientifically as no chromatography has taken place.

HPLC Column PORE SIZE (or Pore Diameter) and Retention Time

Think of your typical porous bare silica support as a big sponge full of holes. All of those holes (pores) are where the sample will migrate through before emerging out the other side. With conventional chromatography supports, most of the interaction takes place inside the particle, not on the surface. The size and number of these openings relate to retention time. Besides particle size (particle diameter), pore size is one of the most important characteristics of silica based chromatography supports.


The pore size or pore diameter is often expressed in Angstroms (i.e. 80 A = 8 nm). The degree of porosity relates to the hydrodynamic volume of your sample and is inversely related to the surface area of the support. The larger the surface area of the support (smaller pore size), the longer the possible retention of the sample. For small drug molecule samples under 1,000 daltons (an estimate only) we often use high surface area supports with small pore sizes between 60 and 150 Angstroms (~ 200 to 500 square meters per gram). These provide high retention characteristics useful in separating apart many small compounds in one analysis run. For larger molecules (i.e. peptides and proteins), we employ supports with larger pore sizes (~300 Angstroms). Particles with small pores have larger surface areas which can provide more interaction with the sample. Note: Pore size is often determined using the BET Nitrogen adsorption/desorption equation. Due to endcapping of the support (e.g. C8 or C18), the actual value obtained is often 20-30% less than the original value.

When comparing bare silica columns or trying to identify similar conventional columns for use in a method, pore size must be considered. Manufacturer's publish the pore size in Angstroms (*sometimes in nm) for their different supports. Choosing columns with similar pore sizes is just one of many parameters needed to provide similar retention characteristics.