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Now, from the Plate Theory, oc = Vn(vo +Kvs) and (vo +Kvs)= Vr/n
where, (n) is the efficiency of the column,
(vo) is the volume of mobile phase per plate,
(v§) is the volume of stationary phase per plate, and (K)is the distribution coeff lcient of the solute between the two
oe = 0 32oc
oe = 0.32Vr/Vn
Vr = Vo + KV's = Vq( 1 +k')
where, (Vo) is the total volume of mobile phase in the column,
(Vs) is the total volume of stationary phase in the column
oe = 0 32V0( 1 +k')/Vn ..................................(23)
Vo * cnr2lopt ....................... (24)
where, (r), is the optimum column radius,
(l0pt) is the optimum column length, and (c) is the fraction of the column occupied by the mobile phase.
and for an optimized column.
lopt - nHrnin
Thus, substituting for (lopO from equation (25) in (24)
Vo - atr2nHmin ................................. (26)
Substituting for (Vo) from equation (26) in equation (23),
oe = 0.32E7ir2VnHrfiin( I +Þ
Substituting for (Vn) from equation (1) and for (Hmin) from equation (12) in equation (26),
oe = 0.32cnr
2A + (U6k' + I Ik'2) .0.5^
Y î 3(1 +ê')
Substituting for (dp) in equation (27) from equation (18),
0? = 10.24c nr*
k'2 (a -A2
(l+6k'+i Ik'2) .05' f „ \ 0.5 j
21+ 2ß. 3Y(l+kf
Y î ifl+k'r V ' ' pp (l+6k’+l Ik'2) 4 +Y
Rearranging and solving for ropt,
()+6k‘+1 Ik'2 2U ó--~
Graph of Optimum Column Radius against Separation Ratio
D 2000 p.s.i ¦ 4000 p.s.i. ¦ 6000 p s.i.
It is seen from equation (29) that (ropt) is directly proportional to (a-l) and inversely proportional to the fourth root of the pressure. Thus, difficult separations (where a=l.01) would be carried out on long, r.3rrow diameter columns packed with relatively large particles. In contrast, simple separations (where a=1.12) would be achieved on short, wide diameter columns packed with very small particles Employing equation (29), the values of (ropt) were calculated for different values of (a) and the results plotted as curves relating (r0pt) to (a) in figure (4). It is seen that the
linear relationship between (r(0pt)) and the function (a-1) and the small dependance on the inlet pressure is confirmed.
The Optimum Flow Rate
The optimum flow rate is obviously the product of the fraction of the cross-sectional area occupied by the mobile phase and the optimum mobile phase velocity, i.e.
Qopt = OTopt^Uopt
Substituting for (ropt) and (uopt) from equations (29) and (9) respectively, and simplifying,
Çó Î + ê'Ã
(l+6k'+l Ik'2) Ë * i
Substituting fordpopt from equation (18) and simplifying,
It is seen from equation (30) that the optimum flow-rate is also proportional to the extra column dispersion and, as a consequence, the total volume of mobile phase employed in an analysis will also depend on the extra column dispersion. It follows that the economy of the analysis lies in the hands of the designer of the chromatograph, a responsibility for which, many instrument makers are not aware. Steps taken in the design of the chromatographic system that would reduce the extra column dispersion by a factor of two would also halve the volume and cost of solvent used in the
analysis host instrument makers, at this time, do not publish a value for the extra column dispersion of their instruments and indeed, many may not even know their true value. The progressive concern over waste disposal of solvents and their potential toxic hazard may well change this attitude in the future. It may well be that, in future, the analyst may also be more concerned in employing optimized chromatographic systems for the very same reason.
The optimum flow rate is also proportional to the applied pressure, the fourth power of (a-1) and the inverse of the viscosity, it follows, somewhat surprisingly that by selecting a solvent system of low viscosity would also permit a higher flow rate. Whether this will also increase the solvent consumption will be seen in due course. Employing equation (30) curves were constructed relating optimum column flow-rate to the separation ratio of the critical pair and these are shown in figure (5).
Graph of Log. Optimum Flow-Rate against Separation Ratio
? 2000 p.s.i.
¦ 4000 p.s.i.
* 6000 p.s.i
It is seen that the optimum flow-rate increases rapidly as the separation becomes less difficult and, in fact, there is a flow-rate change that extends over three orders of magnitude. Again the trend in column design becomes more apparent, simple separations are earned out on short, wide columns, packed with relatively large particle and operated at at high flow-rates.