Download (direct link):
oc = 2enr2dp(l +k')Vn............................. (17)
Now, if the column is to effect a particular separation, where the pair of solutes that are eluted closest together have a separation ratio of (a) and the first of the pair, ( solute A), is eluted at a capacity ratio value of (ê'ä), then the value of (n) is given by the Purnell Equation, which was discussed in the chapter on The Applications of the Plate Theory, viz
4(1 + kA)
, n ............................................ (18)
Substituting for (-Jn), In equation (17) from equation (18), and simplifying,
ßãÿã^ã!,,/ 1+k'* ^2
°C=-77 , ....... ...........................................(19)
The optimum value for (ê'ä), that will provide the maximum resolution and consequently, the minimum elution time, (for an unoptimized column) has been deduced by Grushka and Cook, (14) and Katz et al (15) to be about 2.5. As a result, the composition of the mobile phase is usually adjusted to achieve this value for (k) if it is at all possible.
Thus, taking the accepted value of (e) to be c.a.0.7 and rearranging,
Then, or = 86 I6r2—— ................................................ (20)
Substituting for (oc) in equation (9) from equation (20),
? ^ p
oE = 27.6r —— a-1
Rearranging to obtain an expression for (r).
r = 0.019
Equation (19) shows that the minimum radius will increase as the square root of the extra column dispersion and as the square root of (a-l) but, increase inversely with the square root of the particle diameter. (However,it will be shown later that, that if the column is packed with particles of optimum diameter for the particular separation then the column radius will become linearly related to the function (a-1))
Nevertheless, for unoptimized columns, and for simple separations, the minimum column radius will be relatively large and for difficult separations the minimum column radius will be relatively small. It will be seen later, that it is highly desirable to operate with a column of minimum diameter as this will provide the maximum mass sensitivity from the
w. -----"I------ f - ......
To calculate the minimum diameter for a column of length (1) packed with particles of diameter (dp) which has not been optimized for a particular
separation it is necessary to return to equation (17). From equations (15) and (17) it is seen that,
Substituting for (Vn) from equation (22) in equation (17) and rearranging,
Again, taking the accepted value of (e) to be c.a. 0.7, substituting for (oc) in equation (9) from equation (23) and rearranging, it is seen that,
Equation (24) allows the minimum column radius to be calculated from its length, the particle diameter of its packing and the extra column dispersion of the chromatographic system. Unfortunately, the extra column dispersion is rarely known and very few manufacturers even provide data on the overall dispersion of the detector. When values are given for the detector dispersion, it is often for the sensing cell alone and does not include internal connecting tubes and, as a consequence, can be very misleading.
The total extra column dispersion can be easily measured by removing the column and connecting the tube from the sample valve directly to the tube leading to the detector. A very small sample (0.2jil or less) is then injected into the system and the dispersion calculated as follows,
where, (Q) is the flow rate in ml/min,
(x) is the standard deviation of the eluted peak in cm, and (j) is the chart speed in cm/min.
Employing equation (24), the minimum column diameter was calculated for a column 10 cm long packed with particles of different diameters. The standard deviation of the extra column dispersion was assumed to be 5, 10 and 15 microlitres respectively, values that embrace those that would be
0E = Qx/j
1 < 1
expected in practice from a well designed chromatographic system. The results obtained are shown in figure (8)
Figure 8 Graph of Minimum Column Diameter against Particle Diameter
? S.D. 0.005mi ¦ S.D. 0.010ml ¦ S.D.0,015ml
Particle Diameter (micron)
It is seen that if the length of the column is constant, the minimum column diameter does not change much with particle diameter once it exceeds about 7 microns. However, the minimum column diameter increases rapidly and becomes very large when the particle diameter is reduced to 3 micron or less. It is also seen that the extra column dispersion has a profound effect on the minimum column diameter for particles of all sizes.
As the extra column dispersion becomes large, the column diameter must be increased, to ensure that its variance contribution remains no more than 10% of the that of the eluted peak.
As a finite mass of solute is placed on the column, any increase in peak volume necessary to compensate for high extra column dispersion will dilute the solute concentration as sensed by the detector. Consequently, as the sensitivity, or minimum detectable concentration of the detector, has a limit, increasing the column diameter will result in a reduced mass sensitivity.