## 1. Introduction

The origin of theoretical plates for electrophoresis in general sense has been obtained from the work of Giddings^{¹}. Giddings has derived the following equation for the number of theoretical plates.

where coefficient Ө=unity when molecular diffusion acts alone and exceed unity when other process contributes.R=gas constant and T=temperature and

For an ideal process, in which Ө=1 and T=298K,this equation reduces

where we have used the Faraday constant F=96500 coulombs/mol.This voltage drops V in the range of 100-10000 v with charge number Z=1 capable of yielding 2000-200000 theoretical plates, a range comparable to that found in chromatographic system. Jorgenson and Lukacs ^{²},also in their land mark paper provided a theory for capillary zone electrophoresis (CZE) in which they proposed two fundamental equations for resolution and migration time. Resolution and number of theoretical plates are the focus of this work. This is how number of theoretical plates are derived for capillary zone electrophoresis and used in resolution equation by Jorgenson and Lukacs ^{²} by starting with Gidding number of theoretical plates.

V=potential across capillary is substituted by EX where E=electric field and X=length of the capillary. Z=charge number for ion is assumed to be 1, hence

Eqn.6 where

By substitution of equation(6) into equation (5) one obtains

There is mistake occurs in Jorgenson and Lukacs work^{²}, t in eqn. (7) is replaced by eqn.(8), this is the case when electroomosis is absent.

By making this mistake substitution eqn (8), what has been obtained by Jorgenson and Lukacs for N, eqn.(2) is

E,X,is the, V, voltage across the capillary and

By using Einstain relation eqns.(11) and (12) we substitute for

Where D is the Diffusion constant of the analyte by this substitution and simplification eqn.13 is obtained

This is the equation for the number of theoretical plates the Jorgenson and Lukacs use to obtain the resolution equation. N is driven by mistake^{²}. The resolution equation is given by them

substituting eqn.(13) for N into eqn.(14), the resolution can be expressed as

The above eqn.(15) is the same equation that was obtained by Jorgenson and Lukacs^{²}.

This equation is incorrect. In the present work equation for resolution which is cosidered correct equation is proposed.

## 2. Theory

According to Gidding’s theory^{¹} the evaluation of the ultimate capabilities of zone electrophoresis is possible. To calculate the number of theoretical plates and separable zone achieveable in ideal zone electrophoresis, the electrostatic force exerted on a mole of charged particles on electric field of strength E is

where z=net charge of a single particle in proton units and F=Faraday constant the negative chemical potential drop across the separation path.

In which X´ is the distance where f, force applied in capillary eletrophoresis. For conventional mode of capillary zone electrophoresis, electroosmotic and electrophoretic velocities are in opposite direction. This conventional mode is similar to tread mill and electric stairs where the moveing object has two movements one walking and the other one movement of the stairs.

There X´ in eqn. (17) is not the length of capillary, it is effective distance where the electric force which is applied is greater than the length of capillary is called

Then by substituting eqn.(18) into eqn.(1), the eqn.(19) for the number of theoretical plates results.

X´=effective length which looks like a treadmill, the solute is like a person which can run for miles on the mill but actually he has stayed stationary, f=FE force is Faraday constant times electric field. By substituting force in eqn. (19) one gets

By definition, the effective distance the analyte travels under the force of electric field, X´ divided by the relation times, t, is equivalent to electrophoretic

Where

By the help of eqn.(21) X´ can be substituted into eqn.(20). Now eqn.(22) can be rewritten to include electrophoretic velocity.

Replacing the electrophoretic velocity variable with the product of electrophoretic mobility and electric field

In order to observe the dependence, efficiency has on capillary length and electroosmosis, a substitution for the time variable in eqn.(21) is made. Net displaccment of the analyte or capillary length, X, is related to the retention time, t as shown

Rearragement of eqn.(24) given eqn.(25)

substituting eqn.(25) into eqn.(20) yields the expression,

By making additional substitution for electrophoretic and electroosmotic velocity produced,

plate height is the ratio of effective length X´ to efficiency N is

Substituting equation into eqn.(28) yields an expression for plate height.

This interesting result shows that the theoretical plate height is independent of electroosmotic flow when it is based on the effective distance the analyte travels rather than the capillary length. Instead, plate height has a simple inverse relation with the electric field strength.

Fig.1 Shows the invers relation between theoretical plate height and electric field strength. Electric field strength is the voltage across two ends of capillary divided by the length of capillary.

Equation for resolution: Based width resolution is the quantitative measure of ability to separate two analytes. For two adjacent peaks with similar elution times, peak showed be nearly identical:

where

.Assuming eqn.(30) is true, resolution for species A and B expressed in terms of their retention times and the peak base width for either species^{²}.

The conventional expression for separation can be written with parameters related to either species A or B, shown here using the retention time and peak base width for species B².

By combining eqns. (31) and (32) an equation for chromatography is produced that expresses resolution in terms of efficiency and retention time^{4}.

The resolution time variable

when electrophoretic velocity,

In order to make a comparison between new resolution eqn.(35) obtained in present work and Jorgenson Lukacs equation for resolution eqn.(15) could be transformed to the following equation by using Einstein relation relating diffustion constant to mobility

Jorgenson and Lukacs equation (eqn.36) and new derived Ghowsi equation (eqn.38) for resolution are given as

This interesting observation that for the absence of electrosmotic flow in Jorgenson and Lukacs equation the resolution eqn.36 is converted to

The other resolution equation obtained in present work Ghowsi’s eqn.(35) with presence of electroosmotic flow

It is interesting that only for this case when electroosmosis is absent the resolution equation of Jorgenson and Lukacs eqn.(15) with the help of Einstein relation is equal to Ghowsi’s derived equation eqn.(38) at present for capillary zone electrophoresis.

## 3. Micellar electrokinetic capillary chromatography (MECC)

By converting figure of Merits in MECC to electrochemical parameters^{5} and pursuing similar procedure we applied to capillary electrophoresis and using effective length³ solute travels rather than length of capillary and thenconverting the resolution equation in terms of chromatography parameters new equation for resolution could be found which is published in another paper^{7}.

## 4. Optimization of micellar electrokinetic capillary chromatography (MECC) as a nano separation technique using three dimensional and two dimensional plottings of characteristic equations

Feyman with the lecture of plenty of room at the bottom at an American Physical Society at Caltech on December 29,1959 cosidered the possibility of direct manipulation of individual atoms as a more powerful forms of synthetic chemistry than those used at the time. In conventional chromatography there are two phases involved one is the stationary phase and one is the mobile phase^{6}. Terabe etal proposed Micellar Electrokinetic Capillary Chromatography^{8}, MECC,which has the smallest pseudo stationary phase within nano range called micelle.

The very high strength of separation comes from these nano sized materials. That is why we call this technique MECC.

## 5. Nano separation technique

There are several work which have been done to final the optimum conditions of this Nano Separation Technique^{5},^{8}-^{10}.

In all optimization characteristic equation is the focus.

What is the characteristic equation?

In column chromatography the resolution equation is given as^{6}

Where k´ is the capacity factor, α is the selectivity and N is the number of theoretical plates.Terabe^{8} etal proposal MECC for the first time and they proposed the resolution equation for MECC:

Where N,α and k´ were already defined, a new term is appearing in equation (41) for resolution is

This characteristic equation has two variables k´ and the ratio^{7}. Based an this model

Variables of this equations were defined in the previous work^{7}. The characteristic equation of

In a comparison between the characteristic equation (42) obtained by Terabe^{8} and the characteristic equation obtained based on the new model^{7},equation(44), the difference is the power of ^{8} as

Two characteristic equations for

## 6. Two dimensional and tree dimensional plots of characteristic equations

In present work by the help of modern technology of computer and three dimensional sofware of Dplot direct access to the plot of characteristic equations (42) and (44) are possible.

Figure 2 shows the three dimensional plot of characteristic equation (42) obtained by Terabe^{8} et al.

In this work

These maximums are local maximums.

In figure 3 the characteristic equation (44) based an proposal thread mill model^{7} is plotted. This plot shows no local maximums but there is a single maximum occurs at

Similar information could be obtained for equation (44) from level curves plotted by Dplot software presented at this work fig.5. This two dimensional plot is the image of three dimensional plot on the surface of the plane

Foley’s characteristic equation

Ghowsi’s characteristic equation (47) has also been plotted three dimensionally figure 8 and two dimensional level curves figure 9. The information obtained from either these two plots is the following : Maximum

## 7. Conclusion

The fundamental eqn.12 for the number of theoretical plates has been used to obtain resolution equation by Jorgenson eqn.14. The number of theoretical plates equation in previous work by Jorgenson is wrong, automatically makes the resolution equation wrong.

With the consideration that length X in eqn.17 is the distance where force is applied in capillary electrophoresis, number of theoretical plates which depends on eqn.17 is discussed.

Another equation is the result obtained in eqn.28 which indicates the new result based on effective length. In comparison with eqn.12 Jorgenson equation different eqn.26 has been used to obtain eqn.34 which is the new resolution equation.Similar procedure could be applied to obtain the number of theoretical plates for effective length for MECC and the new resolution equation could be obtained too.This work for MECC was shown in reference 7.

In present work eight graph are presented as characteristic equations of MECC. Four of them are three dimensional and four of them are two dimensionals. It is very interesting observation that the Terabe’s etal characteristic equation shows local maximums in three dimensional plots but the three dimensional plot of characteristic equation obtained by us based on thead mill model shows no local maximums on the surface of three dimensional plot.