Time can be measured sharp and accurate. Not so flow or volume data.
Probably because of the very limited precision and accuracy of volume and flow based peak height - peak width- peak shape measurements and mainly because of the majority of classical theoretical concepts based on a “single substances chromatography” the peak shape has been misinterpreted as a GAUSSIAN shaped one.
We used time as the quantitation technique to analyze hundreds of peak shapes under many differing chromatography conditions like isocratic or programmed runs in short or long packed columns, by HPLC or open tubular columns in GC. Non of the peaks measured had a GAUSSIAN shape. The discrepancy for instance in HPLC reached 40 % if expressed as the variance, which is the square peak width in 60.7 % of the peak height of a truly GAUSSIAN shaped one.
The chromatographic position of substance peaks on the time axis is again time. All details of the peak shape can also be given in time - so the peak width in half height or in 60.7 % of the peak height can exactly be measured in time just by using statistical graphics.
There is no any problem to get these time values by on-line real time based computer software - see later.
An intensive discussion with M.J.E. GOLAY about this serious systematic error in the classical chromatography theory resulted in the following statement, for which M.J.E. GOLAY and colleagues used quite large digital and necessarily even special analog computers: If into a nearly unlimited long column or capillary we inject by amount nearly nothing of a single substance its peak shape will be GAUSSIAN if eluted in the future. Thus no real peak in practical chromatography can be treated the way given by the classical theory - which for instance defined the theoretical plate number N and the height of a theoretical plate HETP shown here:
Theoretical plate number N:
N = [tms / s]2 and the theoretical plate height HETP
HETP = L / N = L * [s / tms ]2
tms = non adjusted (raw) retention time of a peak eluted exclusively under isocratic conditions:
s = peak width in 60.7 % of the peak height
L = length of the separation system (packed column or capillary)
From now on we would like to use data which correspond to CHROMATOGRAPHY ONLY. As we need tools to do chromatography which are mechanical instruments we may falsify FUNDAMENTAL VALUES CORRESPONDING WITH true chromatography data BY instrument related methods - construction based condition values or others and lets look on TIME values only, as only those can be measured precisely and accurately. Technical instruments consist of sampe inlet systems, connection mechanics, columns or capillaries, outlet tubes, inlet tubes, detectors which need mechanical inlet systems, signal measuring systems, electronic signal controllers / amplifiers / recording systems. All of them produce or depend on time constants and mechanical structures. These “add-on” data alltogether have nothing to do with chromatography.
Thus one problem is: theoretical chromatograpohy related data you find with the instrument made by company HP, or PE, or I, or produced in Germany, Russia, China, may produce chromatography data falsified or overlapped by instrument - company - country based facts resulting in a mix of real chromatography correlated values and those which are practicalloy measured but who have nothing exactly to do with chromatography, at least and always by part. For instance a dead time value, during which nothing happens in a chromatographic separations is NOT chromatography. A signal peak width width value, which depends on the technical construction is no clean chromatography value. Accoprding to all which has been publised since 1952 and 2015 contains systematic errors of many prcentages from 90 to at least some percentages resulting in a bad - poor or false chromatography model. The limit “2015” may grow -. at least by now the authors cannot see fundamental hardware and software corrections by any of the harware and software partners of the chromatographer.
And lets from now on stick with TIME data. Thease can be measured incomparably sharper than volume values, which still since the early fiftieth of the last century are “regulated” by “theoreticiants” as basis of theoretical consideratiosn. At least non of them has realized, that TIME is it, not volume, flow, pressure.....
We will quickly realize, that some fundamental values cannot be measured directly but are only the result of statistically assured extrapolations or interpolations and that graphical statistics is a good concept to understand correlations just by looking on graphical presentations. And that LINEARITY is well understood by a majority, allthough some experts still believe that the early reasults of this author are “wrong” - refusing what qualified coworkers of the one or the other theoretician only could find out with no trouble when using computer power of even simpe statistics.
Data example based on the classical theory :
classical theory, N, HETP for the hydrocarbon n-C13 by isocratical Gas Chromatography:
tms of n-C13 = 100.44 sec
L of the capillary = 25000 mm
u = 666.67 mm/sec
s (peak width in 60.7% of the peak height in the GAUSSIAN shape considered peak
= s (for n-C13) = 0.4551 [sec]
therefore N = (tms / s)2 = 48706 as classical theoretical plates
HETP = 25000/48706 = 0.51 [mm] for n-C13 as height equivalent of one theoretical plate
at a mobile phase speed of 667 [mm/sec].
NOTE: this is the mobile phase speed measured at the column outlet. Gas is compressible.
These data offer no any information about the separation power or separation capacity of a given column or capillary as at least two substances with differing tms retention are the minimum.
The separation capacity could be defined as the number of peaks for a mix of substances base line separated per second or minute. Only based on such values we would be able to quantitize and optimize chromatographic separation which we need for analytical information about a sample.
It is no question that a mix of homologue TEST substances would be best to take as they can be separated for sure at least by the sum of their peak width values found in half of their peak heights.
Thus retention times and peak width at half height, both given in time, would be a better basis to understand the separation power of a column or capillary. As mentioned above, time data in chromatography can be measured very accurate and sharp. This is the way we did it automatically and at the time of the chromatogram data storage.
How could data like the retention time and the peak width (in any height) be measured sharp and accurate - by the second or if needed by milli seconds ?
The detector signals are digitized in practice by an a-to-d-converter over time. The digital data are signal height/time data pairs taken at very precise quartz controlled frequencies.
Let us separate a series of forthcoming homologues like n-hexan to n-dodecan in GC or dinitrobenzoic-esters from methyl to n-pentyl in HPLC. All of these homologues would show single peaks on a straight base line, there is no overlapping and no base line drift.
By summarizing after digital smoothing into portions at an apropriate very accurate frequency we get per each peak at least twenty signal-over-time data pairs. An elegant base line analyzing computer program finds the peak start and end over a base line value. Inside this data portion we find one signal-to-time pair as the largest one. Most of the chromatography data handling packages take this maximum as the retention time of the peak. But if we select three data pairs in front of the maximum and three behind, we can let calculate the fifth polynom through this 7 points. Its formula can be used to calculate the maximum as sharp as wanted. Its corresponding time value will differ from the time position of the highest signal-to-time data pair of these about twenty portions taken along the peak basis. The found peak maximum-time value is the real retention time easily given in hundreds to thousands of a second.
Because of the utmost precision of quartz driven a-to-d-converters this tms-value is even quite qualified by the signal to noise reduction based on seven data pairs. The same is true for the peak maximum, now very probably differing from the bevore highest signal-to-time value.
This interpolated maximum height minus the base line level offers a correct half height level, which easily provides two height-to-time data pairs below it and two above it on both sides of the peak. Through four data points a linear polynom program can easily find a line and its formula YL = aL + bL on the left peak side and YR = aR + bR on the right side. As we know the half height value the software sets YR = YL = halve height which defines a parallel line crossing the two tangent lines. Probably even chemists understand, that the two cross points provide a peak width halve height value of b05, which is based on the statistically save 8 data values, this way providing a very accurate and sharp peak width value b05 for each of the homologues. Connected software packages provide these on most simple mathematics based reduced chromatography data automatically nearly in real time seconds after the chromatogram integration - see the last figure of this SITE part on “Alterntive Chromatography Theory based on Time”.
The retention time of any substance in the chromatography instrument is the sum of two time values: the residence time tm in the moving mobile phase and the retardation time ts in the stationary phase. tm is equal for all substances under isocratic conditions. Only ts differences mean chromatography.
We can measure only the sum of both tms = tm + ts
If we have a detector, which for instance can measure Helium in a HPLC equipment we may get the residence time tm in the mobile phase as Helium has no retardation time ts in the stationary phase.
Chemical homologues which differ only by the number of CH2-groups are perfect test substances for HPLC, GC and even PLC (see µPLC ) and allow the tm calculation.
We have in gas chromatography as well as in HPLC excellent linear correlations between the
LN(ts) values for homologues and the retention index I (a next new value, see later). A homologue pair covers by time 100 retention index units in GC as well as in HPLC. If we take a computer iteration procedure, which checks how much retention time must be subtracted from each tms value until the LN(reduced retention) correlates strictly linear with the corresponding index value of the homologue. The found time value to be subtracted equals exactly the dead time tm.
Why dead time ? well, during tm seconds there is no chromatography seen at the column/capillary outlet. Inside we have already separation as in PLC but we only can see it in planar chromatography.
Only a retention time larger than tm shows retardation in the stationary phase.
Below the strict linearity of LN(tms-tm) over the homologue-number * 100 (which is the retention index) is shown in a figure and the data quality of the linear regression line is given for a HPLC and a GC example. Four consecutive homologue pair data are sufficiant for such tm-calculation.
It may sound that too much effort is needed to get so many qualified test data as discussed here, but a correct tm-value is for all isocratic chromatography very helpfull.
Having a correct tm value we could correctly reduce the raw retention time tms to the truly chromatographic residence time ts in the stationary phase. This way we realized that the peak width in halfe height b05 is the sum of a width part bs caused by chromatography and a second part bm wwhich has nothing to do with chromatography but with the technical weaknesses of the used instrument. We see that b05 = bm + bs.
Just one example to chromatography instrument weaknesses:
if connection tubes from the sample injection part into the column or capillary and later into the detector show stepwise drastic tube diameter changes this results in back mixing dead volume
parts. It not only makes peaks broader but give them an e-function like tailing.
There are more chromatography laws which become available only in case we use accurate and sharp measured time data for the retention time tms and the peak width in half height b05 under isocratic conditions and very stable constant temperature, pressure, phase flow conditions.
By the way: poor or no linear correlation data are indicators for instrument trouble. Qualified flow measurements are technically not so easy. Temperature measurements depend from the position and type of the temperature sensor. Pressure measurements and pressure constancy regulation is not so qualified when instrument makers cut investments or users insist in cheapies.
At ICI laboratories in the UK one of my top colleagues responsible for hundreds of equipments in GC first removed the pressure regulators completely from incoming new instruments and replaced them by systems, which could really keep the mobile phase pressure constant or could truly program it linearly. Basically the author repaired brand new instruments prior to the first real use always.
Only this way we found an unexpected perfect linear correlation between the corrected retention time ts = (tms - tm) and the peak width b05 = bs + bm.
Unfortunately in both cases, the retention time sum of the non chromatographic tm-part plus the residence time ts in the stationary phase and the peak width sum of the chromatography based part bs and the non chromatographic part bm we cannot directly measure tm and bm.
Both, tm and bm are the result of regression calculations.
In order to find the dead time value tm we reduce the LN(tms) values of 4 to 5 test substances stepwise by time portions through computer iteration until we find linearity between the now reduced LN(tms-tm) value versus the row of 100, 200, 300, 400 500 until we get a linear correlation with the retention index values = numbers of 100, 200, 300.... There is even no need to enter retention index values, just the input of the forthcoming tms values does it. This improvement of the “tm - regression - mathematics” procedure we found by chance.
Now we use the found tm value and change this way tms data into k-values: (tms - tm)/tm = k. These k values we check versus the corresponding b05 data. It shows perfect linearity. This way we find a correct non chromatographic bm value as the non chromatographic part of b05 by the strictly linear function b05 = a * k + bm.
Strictly means: time data for tms and b05 better than +- 0.1 % relativ.
The best condition for these measurements and calculations is isocratic chromatography.
The best substances to find this way of instrument improvement for highest separation efficiency are homologues as test substances.
The methyl - ...n-dodecyl esters of 2,5-dinitro benzoic acid are good applicable homologues in HPLC and n-alkanes are perfect for Gas Chromatography, see below.
Homologue test runs calibrate qualitatively columns or capillaries - see under “Working range”. Some special detectors need other homologues than the mentioned ones, but there are many. It is no need to use very clean test substances. Impurities can easily be detected, as an isocratic run produces a strictly linear [ LN(ts) over the test number correlation ] - for the methyl to ... n-butyl to n-dodecy-ester. The corresponding retention index numbers are 100, 200, 300, a.s.o. for the esters in HPLC as for the n-alkans in GC or any other test homologue family members. The retention index data rule is: CH2-number times 100 is the retention index for the homologues on any stationary phase, in any mode of chromatography. As a next example: n-hexadecane has the index number 1600 always. Who did not check the retention index (or KOVATS index) technique cannot believe in the analytical power of this concept. Its only weak aspect was critical in the last century, as chemists had problems to work with logarithm based calculations, but since log(x) as well as LN(x) is in each clever handy - or even better - since chemists know how to get software which does all by a few mouse clicks - the retention index should return back to chromatography. Clever on-line software prints out automatically next to tms and b05 values also retention index data. Nothing is hand calculated.
In the following we let calculate by computer iteration the dead time tm in order to get true chromatography retardation time values ts. Having tm we can analyze the peak width parts bm and bs of b05 in order to find stepwise the separation power decreasing instrument parts. Mechanical corrections - homologue test run again -next improvement a.s.o. It pays back, as a better tool plus better separations by more than a few relative percentages remains from now on.
NOTE: for the logarithmic data LN(tms-tm) calculations we (or the software) must use tms and b05 data with at least 6 decimals. Otherwise the linear correlation control value cannot reach a correlation coefficient of 0.9999 or better - see figure 3.
The used software mathematics must be highly qualified. Even in populare languages we find often only a medium good level. The author uses own programs written in PURE-BASIC, a quite young near to machine oriented language normally used for high speed graphical computer games.
Check with www.purebasic.com or consult the author at firstname.lastname@example.org
tm calculation by iteration:
As an example the following values given in table 1 below for HPLC at 0.15 ml/min are used - see table 1 below:
To homologue 1: tms = 1565.52 sec; To 2: tms = 1964.22 sec; To 3: 2571.24 sec;
To 4: 3549.24 sec; To 5: 5097.0 sec.
After (only) 30 iteration steps the result is: tm = 887.1645 sec.
Linearity of the LN(tms-tm) data over the homologue retention index values 100 to 500 is seen in the correlation coeff. = 0.99998.
See also figure 1 below.
bm and bs calculation to split the peak width value mix b05 :
Now we have a correct dead time value tm and can continue to check the instrument quality with the analysis of the peak width value MIX: b05 contains the chromatography related bs value plus the non chromatographic peak width part bm. To get these data we use the retention time values given just under the tm-iteration part above and correlate with the corresponding b05 values:
b05= 41.20 sec; b05= 47.76 sec; b05= 58.22 sec; b05= 75.38 sec; b05= 102.17 sec.
The above used tms-values are changed into k which is (tms-tm)/tm;
the chromatographic retention value k correlates proportional with the partition coefficient K, which is valid for the statinary phase and the test homologue used. However we need this static value K in no case, as chromatography is a dynamic process.
We use a polynomial interpolation program with a wide number range (16 digits), again written in
PURE-BASIC. The entered data result in a perfectly linear function :
b05 = a * (tms-tm)/tm + bm ; bs = b05 - bm see figure 2 and figure 3.
Let us use here already a new test value “Trennzahl” abbr. TZ - and see later for the definition.
Just one single result in more details: the peak width b05 = 41.20 sec for the methyl homologue tms = 1565.52 sec. contains a non chromatographic part of bm = 29.24 sec. This quite large part reduces the separation efficiency which is Trennzahl TZ = 3.48. If we would find the weak instrumental parts and could (theoretically) reduce or even avoid the non chromatographic peak width part bm = 29.24 sec, the Trennzahl would reach TZ = 5.68. This would improve the most impartent job of chromatography by 160 %.
Well: HPLC is not too strong with its separation efficiency. Capillary gas chromatography could reach a Trennzahl total (this is the sum of all possible homologue ranged TZ values) to the world record of TZ total = 800 (J.Roeraade in the seventieth) see table 1.
Trennzahlen are a perfect measure of the separation power a column/capillary has. It is valid for isocratic and programmed chromatography and needs only directly measurable data, see the formulas below.
Data example based on the alternative theory for HPLC