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where P is the inlet pressure to the column, i] is the viscosity of the mobile phase, Ô is the d'Arcy's Constant.
Now, 1 = nHmin thus, substituting for 1, and rearranging,
nHmin uopt —
Ï ' 4.02Dmn
factor of two. it is also seen that the higher efficiencies will be obtained with mobile phases of low viscosity and for solutes of low diffusivity Solvent viscosity and solute diffusivity tend to be inversely proportional to each other and so the sensitivity of the the maximum obtainable efficiency to either solvent viscosity or solute diffusivity will generally not be large. The approximate length of a column that will provide the maximum column efficiency when operated at optimum velocity is given by, 1 = nHmin
Thus, from equations (12) and (16),
It is seen that the column length varies inversely as the product of the solute diffusivity in the mobile phase and the mobile phase viscosity in much the same way as the column efficiency does when operating at the optimum velocity. As would be expected the column length is directly proportional to the inlet pressure but, less obviously is also proportional to the cube of the particle diameter.
The analysis time for a solute mixture in which the last peak is eluted at a capacity ratio of k'p is given by,
t = (l + k'F)t0 = (l + k'F)— uopt
Substituting for (u0pt) and (I) from equations (8) and (17),
1 -62 Dm
It is seen from equation (18) that the analysis time is proportional to the fourth power of the particle diameter and inversely proportional to the square of the diffusivity in the mobile phase. In a similar manner to column length, the analysis time is also directly proportional to the applied inlet pressure and inversely proportional to the mobile phase viscosity.
Summarizing, if a column is operated at its optimum velocity and the solute concerned is eluted at a relatively high k' value, and assuming the film thickness of the stationary phase is small compared with the particle diameter (a condition that is met in almost all LC separations) then,
Hmin- ~ 2.48dp
. (8) (12)
As already stated, these equations are insufficiently precise to be used for accurate column design but, they can give a general indication of the more important performance specifications that could be expected from an approximately optimized chromatographic system.
it would be of interest to use the equations to calculate the maximum efficiency, column length and analysis time that would be obtained from columns packed with particles of different diameter under conditions that commonly occur in LC analyses.
The diffusivity of solutes having molecular weights ranging from about 60 to 600 vary from about 1X10"5 to 4X10~5 cm2/sec.(6) Thus, an average
value of 2X10“5 cm2/sec will be taken in the following calculations. In a similar way a typical viscosity value for the mobile phase will be assumed as that for acetonitrile (7), i.e. 0.375X10'2 poises. The inlet pressure taken will be 3,000lb/p s i. or about 200 atmospheres. Many LC pumps available today will operate at 6,000 p.s.i. or even 10,000 p.s.i., however, it is not the pump that determines the average operating pressure of the chromatograph, but the sample valve. Sample valves can, for a limited time, operate very satisfactorily at very high pressures but their lifetime at maximum pressure is significantly reduced. Most valves, designed for high pressure operation will have extensive lifetimes when operated at 3,000 p.s.i. and so this is the pressure that will be employed in the using the above equations. The value assumed for the d'Arcy constant,(ô) was 35 when Ô) is measured In p.s.i. which had been determined experimentally (8). The capacity factor of the last eluted peak was taken as 5 for calculating the elution time.
Graph of Maximum Efficiency against Particle Diameter
Particle Diameter (micron)
Employing the above values, equation (16) was used to calculate maximum efficiency attainable for columns packed with particles of different diameters. The results obtained are shown in figure 2.
It is seen from figure 2 that changing the particle diameter from l to 20 micron results in an efficiency change from about 3500 theoretical plates to nearly 1.5 million theoretical plates and furthermore, this very high efficiency is achieved at an inlet pressure of only 3000 p.s.i.. It is also seen that the maximum available efficiency increases as the particle diameter increases. This is because, as already discussed, if the pressure is limited, in order to increase the column length to accommodate more theoretical plates the permeability of the column must be increased to allow the optimum mobile phase velocity to be realized. It is possible to increase the inlet pressure to some extent, but ultimately the pressure will be limited and the effect of particle diameter will be the same but at higher efficiency levels.