BuiltWithNOF


      Alternative Chromatography Theory based on Time

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.

First however:

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 rudolf.kaiser@t-online.de

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

Table 1:
Accurate and precise raw retention time data tms and b05; by tm we got k and bm values The Trennzahl TZ calculation however needs only tms and b05 values in GC and HPLC.

flow 0.9 ml/min

tms [sec]

b05 [sec]

tm [sec]

k=(tms-tm)/tm

bm [sec]  )*

TZ

homol methyl

291.48

9.39

164.7

0.760

6.922  (73 %)

2.67

ethyl

366.00

10.91

 

1.222

(63 %)

3.70

propyl

479.40

13.21

 

1.911

(52 %)

5.08

butyl

661.92

16.80

 

3.019

(41 %)

6.35

pentyl

951.36

22.54

 

4.776

(31 %)

-

flow 0.3

tms [sec]

b05 [sec]

tm [sec]

k=(tms-tm)/tm

bm [sec]

TZ

homol methyl

779.76

22.44

441.42

1.004

16.77(74 %)

3.13

ethyl

978.48

25.70

 

1.515

(62 %)

4.35

propyl

1281.06

30.83

 

2.292

(51 %)

5.96

butyl

1768.26

39.09

 

3.544

(41 %)

7.48

pentyl

2540.76

51.99

 

5.530

(30 %)

-

flow 0.15

tms [sec]

b05 [sec]

tm [sec]

k=(tms-tm)/tm

bm [sec]

TZ

homol methyl

1565.52

41.20

887.16

0.764

29.244 (71 %)

3.48

ethyl

1964.22

47.76

 

1.214

(61 %)

4.73

propyl

2571.24

58.22

 

1.898

(50 %)

6.32

butyl

3549.24

75.38

 

3.000

(39 %)

7.71

pentyl

5097.00

102.17

 

4.745

(29 %)

-

 

 

 

 

 

 

 

graph-hplc-fig1

Figure 1:
Y = b05; X = (tms-tm)/tm; mean diff. Y calc. versus Y given = 0.16%
standard deviation Y calculated = +- 0.17 %; correlation coeff. = 0.999985

b05-bild

Figure 2: b05 over tm time units = k = (tms - tm)/tm showing a, bm, bs

figure 2:

Because of the perfect linearity the drawing left was made showing the simplicity of the found function b05 versus k
k  is proportional to the retention time values on a tm scale. k=4 means 4*tm.

At k= 0 we find the peak width value bm, which has nothing to do with chromatography.


At k= 1 we find a column quality factor a.
bs = a * k, the chromatography peak width.

hplc-b05-k-fig3

Figure 3:
Mathematical details to the perfect linearity of the correlation peak width in halfe height (in seconds) and retention time values based on the tm time unit axis which is
(tms - tm )/tm. This function b05 = a*k + bm has been called “abt” in the past and was critizised as “theoretical impossible”. However the “theory” behind this critical statement “impossible” was based on a bad model: the statistical GAUSS function of peak shapes. Using accurate and precise enough time measurements in the time based chromatography show: it is obvious, that chromatography peaks have nothing to do with a GAUSSIAN structure of data
Test chromatograms with homologues allow to calibrate qualitatively isocratic separations and offer a very precise basis even for structure elucidations which helps when structure detectors like MS, MS/MS, NMR, UV... fail because of too limited sensitivity or because of decompositions of chemically critical traces prior to the spectrum measurement. More and more important today in the drinking water are correct ppb-ppq- trace analysis, which are basically prone to systematic errors. The data for the correlation analysis seen in the figures 1 to 3 are given in table 1 below, see the lines under “flow 0.9 ml/min”

 

Details to Table 1:

To the data example for HPLC
)* = the values in brackets express the quite strong separation efficiency loss by the non chromatographic peak width value bm in % of the  b05 value. The reduction of the Trennzahl by bm is remarkable: The TZ value (methyl-ethyl ester under a quite high flow rate of the mobile phase - 0.9 ml/min - reached 2.67 peaks. Let us check, how large would be the separation power of the used column data if bm could reach (theoretically) the value zero, one realizes, that a practical theoretical concept should take care of the non chromatographic completely external negativ effect of an additional peak width enlargement by a weak instrument design.

TZ reduction by bm = zero at flow 0.9 ml/min:
TZ (poor hardware) = (tms ethyl - tms methyl) / (bs methyl + bs ethyl + 2bm) - 1 = 2.67
TZ (theoretycaly free from bm defects) =  (tms ethyl - tms methyl) / (bs methyl + bs ethyl) - 1 = 10.50

TZ reduction  by bm = zero at flow 0.15 ml/min:
TZ (poor hardware) = (tms pentyl - tms butyl) / (bs pentyll + bs butyl + 2bm) - 1 = 7.71
TZ (theoretycally free from bm defects) =  (tms pentyl - tms butyl) / (bs pentyl + bs butyl) - 1 = 12.00

As a most important result you see an equal Trennzahl from the first to the fifth homologue pair at a low flow of the mobile phase as well as at the higher flow of the mobile phase. The improvement of the separation efficiency expressed as Trennzahl is nearly plus 400 percent. Whilst in a (quite poor) standard HPLC instrument in the substance range comparable to 2,5-dinitro-benzoic-methylester  up to the n-pentyl ester one can baseline separate just ONLY 18 peaks a perfect optimized HPLC instrument could baseline separate 42 peaks at a flow speed of 0.9 ml/min. Nearly an equal improvement could be found at a flow speed of 0.15 ml/min. Are those perfectly made instruments on the market ?

How qualified are these critical statements ? How realistic is the analysis of the peak width in halfe height data ?

We found these facts more or less already in the seventies for capillary gas chromatography. We published  details in CHROMATOGRAPHIA. Whilst some colleagues started to find similar effects a few classical theoretical colleagues stated: this is nonsense. Kaiser puts the peak width in halfe height into correlation with the adjusted retention times ts. The tm-calculation is questionable. There is no lineaer correlation as this is theoretically impossible. All is diffusion controlled and the diffusion based peak width enlargement is NON linear. Forget his “a - b - t “ concept was suddenly a majority statement. This is a standard effect if the discussion basis is called THEORY.
Now the THEORY (THIS THEORY) is in question. Time based quality measurements and analysts own-made software is powerful, because the analyst knows what the software should be able to do and check prior to the success, to find a programmer, who also understands chromatography or at least understands physical measurments and some mathematical statistics. So a team of analysts and a team of top programmers learned chromatography and programming on both sides, crossed that and got it.

We used now accurate and sharp retention time values by fifth degree polynomial interpolation of the peak top using seven top precise signal over time values for a very accurate retention time, a true peak width halfe height value, a very precise and sharp peak width by linear regression of four signal over time at front and four at the back of the peak and checked with polynomial interpolation the data correlation y-axis = b05 values in seconds, x-axis in k = (tms-tm)/tm units . This cuts the x-axis in tm-units. The dead time tm is the residence time in the mobile phase. So we check the correlation of the peak width with the chromatogram time but measured in dead time units.
In this site we show the perfect linearity as a result of accurate and sharo time measurement in isocratic HPLC.

As a result we find , that b05 = a * k + bm
a is a quality value for the goodness of the stationary phase package.
tm is also a measure of the mobile phase flow speed.

The linear regression line cuts the Y-axis at bm, the non chromatographic and against separation working poor part of the peak width growth with time. Important to understand weak techniques or wrong dimensions, remixing dead volumes, poor sampling including too slow transfer of the substances into a solution in the mobile phase.

The quality of polynomial regression is measured by the mean difference of given over calculated time values and the corresponding standard deviation or by the regression coefficient. All values b05 over k given in the table 1 above correlate LINEAR by a correlation factor of 0.99999.
Well: nothing in nature is absolutely linear resulting in a correlation coefficient of 1. A second and a third degree polynom can be calculated and show (may be) a correlation coefficient of 0.999993 at the second degree versus
0.999997 for the first degree, but the practical polynom is of first degree - since we could use time data down to the hundreds or thousands of a second accurate and sharp. In isocratic HPLC !

Lets now check isocratic gas chromatography

The retention time is a time mix:  retention time tms = ts + tm.
The peak width in half height is a time mix: b05 = bs + bm

Neither tm nor bm can directly be measured, both are the result of applied multi substance based regression calculations.
The best condition for these measurements and calculations is isocratic chromatography.
The best substances to find out chromatography quality are homologues.
Besides many others these are especially usable in Gas Chromatography : the n-alkanes from methane till up to n-C40
. The n-alkanes are the reference substances for Kovats retention indices.l

Trennzahl TZ formula :

TZ(I to II) = (tms II - tmsI ) / (b05 II + b05 I) - 1
tms II = raw retention time of the homologue II (may be of n-C12)
tms I = raw retention time of the homologue one CH2 group smaller here = n-C11
the b05 II and b05 I are the corresponding peak width halfe height values. TZ(I to II) would then mean: this is the Trennzahl between alkan n-C11 and n-C12 and tells how many peaks can be baseline seperated in this index range from 1100 to 1200. The total separation power given by a TZ sum is given for the whole working range based on the user decision: how much time do you allow for a total
analysis ? What is outside the working range expressed by tms MUST BE BACKFLUSHED after each analysis, otherwise the column/capillary changes its selectivity. Don’t “heat out”.

See the facts: we don’t need ts, but get it as tms II - tms I = ts II - ts I. TZ values are ‘damaged’ by large bm values.

TZ (I to II) = (ts II - ts I) / (bs II + bs I + 2*bm) - 1

The TZ`s are true chromatography related values (because of ts differenzes) AND depend on the instrument quality, because of 2 * bm .
Trennzahlen are applicable in Gas-Chromatography AND in HPLC. In programmed techniques they show a maximum at the best program rate. Trennzahlen and column length correlate roughly in GC: One m separation path length equals about ONE TZ unit.  TZ and the separation selectivity for a given analytical problem correlate: If the retention indices ‘I’ of two to be separated substances differ by DELTA ‘I’, the necessary Trennzahl for a baseline separation of these two is  TZ (necessary) = 100 / (DELTA ‘I’) + 1

Data example based on the alternative theory for Gas Chromatography, capillary GC

Top linearity is given also in the function [retention index unit] with LN(tms-tm), not only in b05 over k  resp. b05 in tm over (tms-tm)/tm .

The retention index unsits - for instance the Kovats Index in GC - are numbers identifying the structure of analytical substances very sharp and accurate, but qualify also very accurate the nature of a stationary phase. Each functional group in an analytical substance has some retention index units and each CH2 group adds hundred index units to the total. Therefore knowing all Index unit parts for functional gropups and their position in a molecule allows the index calculation (bny now only more or less roughly). The high accuracy of tms values plus the accuracy of the tm value results in very precise retention index measurements. Gas flow and temperature must be under control constant. This offers very qualified qualitative analyses of complex sampleas and prepares the solution of peak overlapping problems. This allows qualified column combinations in multi column instruments, basis  for auto selectivity optimizations. Something not yet available (to the best knowledge of the author) from any instrument company in the chromatography field. In programmed GC we do not use nor need tm data, but tms data correlate with the retention index perfectly under third degree polynom calculation, again very accurate if the program rate is perfectly constant.

Systematically correct retention index data offer identification help where MS fails. If for instance MS data tell the molecule structure is HO-CH2-phenyl-CH2-OH but the retention index found by GC is MUCH larger than 950, forget the MS result. It is falsifyed. The wrong structure resuilt is caused by decomposition in the inlet system. You may have a look into the ‘Retention index’ part of this site, which by now is under translation from D to E and will be added as extra part later.

All further facts discussed under Table 1 with HPLC data exist  the same way in Gas Chromatography but its Trennzahl values are drastically larger especially in capillary GC when compared to HPLC - except in latest high speed long and thin temperature programmed HPLC columns.


Data example based on the alternative theory for GC

Table 2 shows accurate and sharp time data for the homologue series of the n-alkanes C 13 to C 16.
Capillary
L = 25000 mm; i.d. = 0.3 mm;

 

pressur 1.4 bar

tms [sec]

b05 [sec]

tm [sec]

k=(tms-tm)/tm

bm [sec]  )*

TZ

homol n-C 13

88.32

1.28

tm = 36.0437

1.49397

0.725 (40 %)

10.98

n-C 14

124.44

1.73

 

3.1419

 

13.72

n- C 15

185.64

2.42

 

5.17899

 

16.03

n-C 16

289.14

3.65

 

8.62398

 

-

pr. 1.14 bar

tms [sec]

b05 [sec]

tm [sec]

k=(tms-tm)/tm

bm [sec]

TZ

homol. n-C 14

100.44

1.07

tm = 37.5233

1.67673

0.298 ! !

15.91

n-C 14

145.56

1.60

 

2.8792

 

17.74

n-C 15

223.02

2.53

 

4.94351

 

18.87

n-C 16

356.04

4.16

 

8.4885

 

-

pr. 0.63 bar

tms [sec]

b05 [sec]

tm [sec]

k=(tms-tm)/tm

bm [sec]

TZ

homol. n-C 15

167.28

1.76

tm = 68.88

1,42489

0.846 (48 %)

15.89

n-C 14

235.74

2.29

 

2.41728

 

19.93

n-C 15

351.84

3.26

 

4.10027

 

23.03

n-C16

548.82

4.94

 

6.95569

 

-

pr. 0.20 bar

tms [sec]

b05 [sec]

tm [sec]

k=(tms-tm)/tm

bm [sec]

TZ

homol. n-C 16

437.82

4.11

tm = 185.6794

1.3579

1.925 (47 %)

17.16

n-C 14

613.62

5.57

 

2.3047

 

20.86

n-C 15

912.48

8.10

 

3.91428

 

23.65

n-C 16

1419.9

12.49

 

6.6471

 

-

Table 2

To the data example for GC, here for capillary GC, 25000 mm long 0.3 mm i.d., Test substances are only four n-alkanes n-C12 to n-C16 , mobile phase hydrogen
)* = the values in brackets express the quite strong separation efficiency loss by the non chromatographic peak width value bm in % of the b05 value. The reduction of the Trennzahl by bm is remarkable: The largest TZ value (in the homologue range n-alkan C15 - C16) under a quite wide flow rate of the mobile phase caused by the gas pressure from 0.2 to 1.4 bar - reached at 0.2 bar a maximum of 23.65 peaks. Let us check, how large would be the separation power of the used 25 m long capillary if bm could reach (theoretically) the value zero. One realizes, that a practical theoretical concept should take care of the non chromatographic completely external negativ effect of an additional peak width enlargement by a weak instrumental design.

TZ reduction  by bm = zero at apressure of 1.4 bar:
TZ (poor hardware) = (tms n-C14  - tms n-C13) / (bs C13 + bs C14 + 2bm) - 1 = 10.98
TZ (theoretycaly free from bm defects) =  (tms n-C14- tms n-C13 / (bs methyl + bs ethyl) - 1 = 16.17

TZ reduction  by bm = zero at a pressure of 0.2 bar:
TZ (poor hardware) = (tms n-C16  - tms n-C15) / (bs C16 + bs C15 + 2bm) - 1 = 23.65
TZ (theoretycaly free from bm defects) =  (tms n-C16- tms n-C15 / (bs C16+ bs C15) - 1 = 29.32
 

As a most important result you see a strong effect by a mobile phase flow speed at lower gas pressure but you pay by time - in the data example at 0.2 bar by a factor of ten.

How qualified are these statements ? How realistic is the analysis of the peak width data in halfe height ?

We found these facts more or less already in the seventies for capillary gas chromatography. We published details in CHROMATOGRAPHIA. Whilst some colleagues started to find similar effects a few classical theoretical colleagues stated: this is nonsense. Kaiser puts the peak width in halfe height in correlation to adjusted retention times ts using questionable effects in calculating  the dead time tm. There is however no lineaer correlation because this is theoretically impossible. All is diffusion controlled and the diffusion based peak width enlargement is NON linear. Forget his “a - b - t “ concept  [in b05 = a*(tms-tm/tm) + bm]

We used now accurate and sharp retention time values by fifth degree polynomial interpolation of the peak top with seven signal over time values measured by time.  The 7 data points resulted by polynomial interpolation in a very accurate retention time tms, in a true peak width halfe height value b05, a very precise computer iterated dead time tm, a very precise and sharp peak width value bm  by linear regression (see the last figure of this chapter) in HPLC and in GC.

As a result we found , that b05 = a * k + bm is a perfectly linear function with correlation coefficients of 0.9999 up to 0.99999.
a is a quality value for the goodness of the stationary phase package.
bm is the peak width growth with time.  tm is THE measure of the mobile phase flow speed and the “a and o” for athe strict linear regression  dead time corrected retention time values in log format LN(tms-tm) and the molecular weigt, homologue number or retention index.

The linear regression line cuts the Y-axis at bm, which because of k=0 is definitively a non chromatographic and against separation working poor part of the peak width growth with time. The knowledge about bm is important to understand weak techniques or wrong dimensions, remixing dead volumes, poor sampling modes including too slow transfer of the substances into a solution with the mobile phase.

Well: nothing in nature is absolutely linear resulting in a correlation coefficient of 1.00000. But in the minute we use accurate and precise  tms and b05 values, it becomes obvious, that we should use a theoretical concept which has a better model than the GAUSSIAN shaped peak structure and which uses more than a single substance to understand, improve and maximize the separation and its efficiency. We found it should at best be measured by the Trennzahl TZ at least in analytical chromatography.

All definitions, formulas, calculations up to now have been done with data measured under isocratic conditions.

More than 90 % of all practical chromatography runs owever under non isocratic conditions. The commercial instruments try to keep the rate of the mobile phase, temperature or pressure programming konstant in order to get linear functions.
In non of the programmed chromatography modes we can use tm calculations, theoretical plate numbers or HETP values and non chromatographic peak width data. There are no more linear correlations available.
The retention indices correlate only by third polynoms with the retention time. The only value we can use to quantify and optimize the separation power is the Trennzahl TZ.
 
But the beauty with the efficiency control number TZ is: wether isocratic or programmed measured: it is nearly equal in both worlds of chromatography.

Working range, retention and temperature

NOTE: not too seldom a chromatography beginner injects a sample and takes a chair to look at the data system or recorder to see the final substance eluting. If he would know, what means working range and would be able to use the corresponding information he would be able to see, that (under isocratic conditions) “his last peak” may elute only past the end of his activ industry time, or the total time window of his research project. Therefore here about the concept of the working range.

Every part of our technique has a working range:
detectors start showing the first visible signal only above a minimum of substance amount in gram. There is an  upper limit of substance amount which produces a still growing signal with further substance increase. Similar are the borders of substance amounts columns / capillaries handle: socalled dead  trace amounts are non eluteably sorbed, stationary phases have an upper limit of substance load above which peak shapes deform. Temperatures, flow, pressure have their upper limit above which materials change irrepairably.
The working range below however correlates only the retention index - or molekular weight of substansed with the retention time and steps of programming rates or the column/capillary temperatures. Below is an example for capillaty GC.

Figure 4 below shows the strict linearity of LN(tms-tm) for homologues with their retention index data. By definition the retention or KOVATS index is the C number (of CH2 groups) times 100.
So n-hexadecanes index = 16 * 100 = 1600.

ret-index-gc

Even the perfect linearity shown in Figure 4 results in polynom factors, which allow retention index claculations for not better as +- 0.02 units. Thus the shown large decimal numbers is necessary - logarithmic polynomial interpolation is critical - even the log. calculation works only on a very limited level of accuracy.

Figure 5:
 Working range of a capillary given by the  temperature. The retention index of any substance can roughly be calculated by comparison of the molecular data (C number, O-functions, halogen atoms in the structure a.s.o.) One could of course measure the retention index as well but without a knowledge which is shown in Figure 5 you may need quite some time to get Index data as long as the substance in question is not available as clean enough chemical individuum. Having the index - may be I = 1800 +- 5 - you can go from the y-axis at around 1800 into the time scale direction. You want to have the elution of this substance done in 10 minutes. Figure 5 tells you: select a capillary temperature of 150 degree C. The substance will elute near 10 minutes.  All substances with indices larger than 1800 could at best be back flushed without temperature change. Do not “heat out”.  The time axis is exponential of course, the index axis is linear. This data field can be found by homologue test runs at about three to four constant temperatures, isocratically.


The last figure:
 how we got sharp and accurate time data.  Green points: the seven data pairs signal height and corresponding exact time value to get the accurate retention time tms [sec] by using the fifth degree polynom formula. The red points: 4 on the left side and 4 on the right side of the peak are used to get them connected by a first degree polynom line. It delivers the linear line formulas left and right which are crossed by the peak height halfe line parallel to the base line.
The time distance of the two cross points gives the accurate peak width value b05 [sec]
Note the graph error: the seventh green point is missing.
Working ranges look similar for HPLC based on mobile phase composition and / or temperature.

This SITE will be expanded by topics for which you may ask the author under
rudolf.kaiser@t-online.de
 

timepeak

The “Last Graphical statistics figure” shows how to measure and calculate chromatography data SHARP and correct based on quartz frequency precision. This happesn with a special IQ software, which works in real time during the raw data treatment immediately after the base line position is found prior any integration step.

The following tables compare the complete data analysis values between a precise isocratic tms/b05 determination of four to five homologue test substances in capillary GC and after that in HPLC with tghe same type of results which are found in case the time data are systematically FALSE by one percent relative.

That means as example:  (tms) sharp and accurate may be 100 seconds.
(tms) FALSE is then 101 seconds.
(b0.5) sharp and accurate may be 10 seconds.
(b0.5) FALSE is then 10.1 seconds - the ERROR is 100 millii seconds.
These systematic time errors alternate in plus and minus for the four to five homologue peaks in the isocratic test run.
Have a look by comparing the two tables ‘accurate’  versus ‘lfalse’, you may be shocked.
It may make clear: digital data is something we use more and more exclsuively, but comparing the GRAPHICS we may immediately look more critically to digital data. The systematic final error is MUCH larger than the just one percent in the standard values non adjusted retention time (tms) and the peak width in halfe height (b0.5) .

 

bld-GC-OK

correct sharp data cap GC 200 mB H2

Polynom 1.9271 + 1.5862 * k

data quality:

mean error 0.383% measured to calculated;
std. deviation +- 0.246%

linearity determination coeff. 0.99997


tm = 186 sec;  bm = 1.93 sec ( 47% )

bld-GC-false

poor data accuracy, error = one percent
data example for b0.5


correct:   4.11 sec   bad measured: 4.07 sec
               5.57                                  5.51
               8.10                                  8.18
              12.49                                12.62

Polynom  1.7095 + 1.6272*k

mean error 0.805% measured to calculated
std. deviation +- 0.56%

Linearity determination coeff. 0.9998

tm = 183 sec;  bm = 1.71 sec. ( 42% ) 

NOTE: the peak width data error is  only
- 0.04 sec; 0.06 sec; 0.08 sec; 0.13 sec.
The area around the correlation line is the uncertainty range. The wider the less qualified is the linear correlation.

Data Example for HPLC (see the correct tms / b0.5 values
in table 1 above under ‘flow 0.3 ml/min’

Correct
Polynom 16.725 + 7.4204 * k

data quality:

mean error 0.113 % measured to calculated;
std. deviation +- 0.073 %

linearity determination coeff. 0.999993


tm = 441 sec;  bm = 16.73 sec ( 75 % )

Data Example for HPLC (poor measured peak width values, 1 % systematic error

False
Polynom 17.3806+ 8.0866 * k
data quality:

mean error 0.307 % measured to calculated;
std. deviation +- 0.239 %

linearity determination coeff. 0.9999
tm = 480 sec;  bm = 17.38 sec ( 77 % )

NOTE: the tm-error is already rel 10 % !
no wonder, why some colleagues found non linear correlation “time over b0.5”, which in fact is just PERFECTLY linear and allows for the important “bm” external error analysis to improve chromatographs and methods.

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